A Fuller Explanation
The Synergetic Geometry of R. Buckminster Fuller

Design Science Collection
Series Editor

Arthur L. Loeb

Department of Visual and Environmental Studies
Harvard University

Amy C. Edmondson

A Fuller Explanation: The Synergetic
Geometry of R. Buckminster Fuller, 1987

In Preparation

Marjorie Senechal
and George Fleck(Eds.)
Arthur L. Loeb

Shaping Space: A Polyhedral Approach

Concepts and Images

Amy C. Edmondson

A Fuller Explanation
The Synergetic Geometry of
R. Buckminster Fuller

A Pro Scientia Viva Title

Birkhauser
Boston . Basel . Stuttgart

Amy C. Edmondson
A Fuller Explanation
The Synergetic Geometry of R. Buckminster Fuller

Coden: DSCOEDFirst Printing, 1987

Library of Congress Cataloging in Publication Data
Edmondson, Amy C.
A Fuller explanation. The synergetic geometry of R. Buckminster Fuller.
(Design science collection)"A Pro scientia viva title."
Bibliography: p.
Includes index.
1. System theory. 2. Thought and thinking.
3. Mathematics-Philosophy. 4. Geometry-Philosophy.
5. Fuller, R. Buckminster (Richard Buckminster).
1895- . I. Title. II. Series.
003
86-14791
Q295.E33 1986
CIP- Kurztitelaufnahme der Deutschen Bibliothek
Edmondson, Amy c.:
A Fuller explanation: the synerget. geometry of
R. Buckminster Fuller / Amy C. Edmondson-I.
print. -Boston; Basel; Stuttgart: Birkhauser,
1986.
(Design science collection) (A pro sci entia
viva title)

ISBN 978-0-8176-3338-7
ISBN 978-1-4684-7485-5 (eBook)
001 10.1007/978-1-4684-7485-5

Frontispiece photograph courtesy of Phil Haggerty.
© Birkhauser Boston, Inc., 1987
Softcover reprint of the hardcover 1st edition 1987

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying, recorded
or otherwise, without prior written permission of the copyright owner, Birkhauser Boston, Inc.,
380 Green Street, P.O.B. 2007, Cambridge, MA 02139, U.S.A.

ISBN 978-0-8176-3338-7
Typeset by Science Typographers, Inc., Medford, New York.

To my parents
Mary Dillon Edmondson and Robert Joseph Edmondson

Contents

Series Editor's Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . .. xv
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XIX
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxi
Note to Readers ................................. XXlll
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxv

1.

2.

3.

Return to Modelability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

From Geometry to Geodesics: A Personal Perspective. . . . . .
Operational Mathematics ...........................
Experimental Evidence ...........................
Nature's Coordinate System. . . . . . . . . . . . . . . . . . . . . . . .
Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Return to Mode1ability ...........................

3
6
7
9
11
12
14

The Irrationality of Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15"Nature Isn't Using Pi" . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Accounting System .........................
Which Way Is "Up"? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Visual Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Peaceful Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16
18
19
22
23

Systems and Synergy ..............................

25

Conceptual and Real Systems . . . . . . . . . . . . . . . . . . . . . . .
Limits of Resolution as Part of the Whole-Systems
Approach ...................................
Synergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28
30
33

x

Contents

4.

5.

6.

7.

8.

Tools of the Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Plato's Discovery ................................
Triangles .....................................
Squares ......................................
Pentagons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Limited Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Euler's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Truncation and Stellation ...........................
An Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
"Intertransformability" . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Symmetry ......................................

37
38
40
41
41
43
45
46
48
51
52

Structure and "Pattern Integrity" .....................

54

Pattern Integrity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56
60

Angular Topology ................................

65

Frequency and Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time and Repetition:
Frequency versus Continuum .....................
Topology and Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vector Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Planes of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Applications of Dimension . . . . . . . . . . . . . . . . . . . .
Angular Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular Takeout: An Example . . . . . . . . . . . . . . . . . . . . . .
Angle Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66
66

Vector Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82"Nature's Own Geometry" . . . . . . . . . . . . . . . . . . . . . . . . .
Spatial Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Planar Equilibrium ................... . . . . . . . . . . .
Cuboctahedron as Vector Equilibrium . . . . . . . . . . . . . . . . .
VE: Results ...................................
Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84
85
87
90
91
93

Tales Told by the Spheres: Closest Packing ..............

100

Equilibrium: Equalization of Distances . . . . . . . . . . . . . . ..
Symmetry versus Specificity of Form. . . . . . . . . . . . . . . . ..

100
101

67
68
68
70
72
73
74
75
78
79

Contents

9.

10.

11.

xi

Organization of Identical Units. . . . . . . . . . . . . . . . . . . . ..
New Level of Focus. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Background: Closepacking ..........................
Planes of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Fuller Observations ...............................
Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Vector Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Frequency ....................................
Icosahedron ........................ . . . . . . . . . ..
Further Discoveries: Nests. . . . . . . . . . . . . . . . . . . . . . . . . ..
"Interprecessing" ...............................
A Final Philosophical Note ........................

101
102
102
106
109
110
114
114
117
120
121
124

Isotropic Vector Matrix ............................

127

A Quick Comparison: "Synergetics Accounting" .........
Cells: "Inherent Complementarity" .................. ,A Complete Picture ...............................
Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Locating New Polyhedral Systems. . . . . . . . . . . . . . . . . . ..
Duality and the IVM ..............................
Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Framework of Possibility. . . . . . . . . . . . . . . . . . . . . . . . . . ..
Invention: Octet Truss. . . . . . . . . . . . . . . . . . . . . . . . . . ..

130
131
133
134
135
136
136
137
138
140
141

Multiplication by Division: In Search of Cosmic Hierarchy. ..

143

Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Results: Volume Ratios. . . . . . . . . . . . . . . . . . . . . . . . . ..
Shape Comparisons: Qualities of Space. . . . . . . . . . . . . . ..
Volume: Direct Comparison. . . . . . . . . . . . . . . . . . . . . . ..
Multiplication by Division. . . . . . . . . . . . . . . . . . . . . . . . . ..
Tetrahedron as Starting Point. . . . . . . . . . . . . . . . . . . . . ..
Cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Vector Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Rhombic Dodecahedron ..........................
Multiplication by Division . . . . . . . . . . . . . . . . . . . . . . . ..
Cosmic Hierarchy (of Nuclear Event Patternings) ..........
Volume Reconsidered ............................

143
144
146
147
149
149
150
152
153
154
157
157

Jitterbug.......................................

159

Folding a Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Volume and Phase Changes ....... . . . . . . . . . . . . . . . ..

160
163

Contents

xii

12.

13.

14.

Icosahedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Single Layer versus IVM ..........................
"Trans-Universe" versus "Locally Operative" ...........
Fives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
"S-Modules" ..................................
Icosahedron and Rhombic Dodecahedron .. . . . . . . . . . . ..
Pentagonal Dodecahedron. . . . . . . . . . . . . . . . . . . . . . . ..
Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Complex of Jitterbugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Other Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Topology and Phase .............................

163
164
165
165
167
167
168
169
170
172
172

All-Space" Filling: New Types of Packing Crates .........

175

Plane Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Filling Space ....................................
Complementarity ...............................
Other Space Fillers ..............................
The Search Continues ............................
The Dual Perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Duality and Domain in Sphere Packing. . . . . . . . . . . . . . ..
Truncated Octahedron. . . . . . . . . . . . . . . . . . . . . . . . . . ..
Two to One: A Review ...........................

176
177
178
179
180
181
183
184
185

The Heart of the Matter: A- and B-Quanta Modules. . . . . . ..

189

A-Quanta Modules ........ . . . . . . . . . . . . . . . . . . . . ..
B-Quanta Modules ..............................
Energy Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . ..
Mite ..........................................
Mirrors ......................................
Cubes into Mites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Rhombic Dodecahedra ...........................
Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Volume and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Review: All-Space Fillers. . . . . . . . . . . . . . . . . . . . . . . . . . ..

190
190
193
195
197
198
198
199
201
203

Cosmic Railroad Tracks: Great Circles . . . . . . . . . . . . . . . . ..

206

Why Are We Talking About Spheres? .................
New Classification System . . . . . . . . . . . . . . . . . . . . . . . ..
Great-Circle Patterns ..............................
Least Common Denominator . . . . . . . . . . . . . . . . . . . . . ..
LCD: "Intertransformability" ......................
LCD of 31 Great Circles ..........................
VE's 25 Great Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . ..

207
208
209
213
215
216
217

Ii

Contents

xiii

Operational Mathematics ...........................
Conservation of Angle. . . . . . . . . . . . . . . . . . . . . . . . . . ..
Foldable Systems ...............................
Energy Paths ....................................
Gas Molecules .................................
Great-Circle Railroad Tracks of Energy. . . . . . . . . . . . . . ..
Icosahedron as Local Shunting Circuit. . . . . . . . . . . . . . . ..
Inventory: Seven Unique Cosmic Axes of Symmetry ........
Excess of One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

219
221
223
226
226
228
229
230
230

From Geodesic to Tensegrity: The Invisible Made Visible. . ..

232

Theory Behind Geodesic Structures: Summary. . . . . . . . . ..
Geodesic Design in Nature. . . . . . . . . . . . . . . . . . . . . . . ..
Geodesic Domes: Design Variables. . . . . . . . . . . . . . . . . ..
Tensegrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Nature's Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
New Concept of Construction. . . . . . . . . . . . . . . . . . . . . ..
Modeling the Invisible . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Tensegrity Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Pneumatics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Case in Point: Donald Ingber. . . . . . . . . . . . . . . . . . . . . ..

235
237
240
243
245
249
250
251
255
257

"Design Science" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
"Comprehensive Anticipatory Design Science" ..........
"Comprehensive ... " . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
" ... Anticipatory ... " ............................
Dymaxion Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Suspended Storage Systems ........................
More with Less. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

258
258
259
261
263
266
268

Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

271

A. Trigonometric Calculations . . . . . . . . . . . . . . . . . . . . . . . . ..
B. Volume Calculations for Three Prime
Structural Systems ..............................
C. Sources of Additional Information. . . . . . . . . . . . . . . . . . . ..
D. Special Properties of the Tetrahedron. . . . . . . . . . . . . . . . . ..
E. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

271

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

287
289
295

15.

16.

273
277
279
281

Series Editor's Foreword

In a broad sense Design Science is the grammar of a language ofimages rather than of words. Modern communication techniques
enable us to transmit and reconstitute images without the need ofknowing a specific verbal sequential language such as the Morse
code or Hungarian. International traffic signs use international image
symbols which are not specific to any particular verbal language. An
image language differs from a verbal one in that the latter uses a
linear string of symbols, whereas the former is multidimensional.
Architectural renderings commonly show projections onto threemutually perpendicular planes, or consist of cross sections at different altitudes representing a stack of floor plans. Such renderings
make it difficult to imagine buildings containing ramps and otherfeatures which disguise the separation between floors; consequently,
they limit the creativity of the architect. Analogously, we tend toanalyze natural structures as if nature had used similar stacked
renderings, rather than, for instance, a system of packed spheres,with the result that we fail to perceive the system of organization
determining the form of such structures.
Perception is a complex process. Our senses record; they are
analogous to audio or video devices. We cannot claim, however, thatsuch devices perceive. Perception involves more than meets the eye:
it involves processing and organization of recorded data. When weclassify an object, we actually name an abstract concept: such words
as octahedron, collage, tessellation, dome; each designates a wide
variety of objects sharing certain characteristics. When we devise
ways of transforming an octahedron, or determine whether a given
shape will tessellate the plane, we make use of these characteristics,
which constitute the grammar of structure.
The Design Science Collection concerns itself with various aspectsof this grammar. The basic parameters of structure, such as symme-

xvi

Series Editor's Foreword

try, connectivity, stability, shape, color, size, recur throughout thesevolumes. Their interactions are complex; together they generate suchconcepts as Fuller's and Snelson's tensegrity, Lois Swirnoff's modulation of surface through color, self-reference in the work of M. C.
Escher, or the synergetic stability of ganged unstable polyhedra. Allof these occupy some of the professionals concerned with the complexity of the space in which we live, and which we shape. The
Design Science Collection is intended to inform a reasonably welleducated but not highly specialized audience of these professional
activities, and particularly to illustrate and to stimulate the interaction between the various disciplines involved in the exploration of
our own three-dimensional, and in some instances more-dimensional, spaces.
When R. Buckminster Fuller recalled his days as a schoolboy in
Milton, Massachusetts, he related how his mathematics teacher
would introduce two-dimensional surfaces by placing lines of zerothickness side by side; young Buckminster used to wonder how one
could create a finite surface out of nothing. Similarly, he could not
accept the stacking of planes of zero thickness to create volumes.
Intuitively, he sensed that areas and volumes are as different fromeach other as are forces and velocities: one cannot mix quantities ofdifferent dimensionality. Accordingly, Fuller learned to compare
three-dimensional objects with each other, and hence to add, subtract, and transform them from and into each other rather thancreating them out of objects of lower dimensionality. In doing so hecame to discard the conventional orthogonal system which has
blinded architects as well as solid-state scientists, and followed
natural structure in designing his stable light-weight structures.
Two days before Harvard Commencement in 1983, Amy Edmondson called me from Buckminster Fuller's office in Philadelphia,
saying that Fuller had decided at the last moment to attend theCommencement exercises, and wondered whether I might still befree to have dinner with them the following evening. Amy had
graduated from Harvard with combined honors in Applied Scienceand in Visual and Environmental Studies, and had been working for
Fuller since then. At dinner we planned a working session in August
at Fuller's island off the Maine coast. Unfortunately that Commencement turned out to be Buckminster's last, and when I saw Amy
again it was at the combined service in memory of Buckminster andAnne Fuller. We decided right then and there that the best tributewould be a volume aiming at translating Buckminster Fuller's ideas

Series Editor's Foreword

xvii

and idiom into a language more accessible to the lay audience andmore acceptable to the scientist.
Amy Edmondson has succeeded admirably in conveying to us notjust the idiom but also the atmosphere of Fuller's "office." There
were no professional draftsmen, for the staff was minimal. We
believe that the sense of a direct link to the Fuller office would beenhanced by reproducing Edmondson's own illustrations directly,
just as she would have produced them there.
With A Fuller Explanation we initiate the Design Science Collection, an exploration of three-dimensional space from the varied
perspectives of the designer, artist, and scientist. Through this series
we hope to extend the repertoire of the former to professions byusing natural structure as an example, and to demonstrate the role ofesthetic sensibility and an intuitive approach in the solution ofscientific problems.
ARTHUR L. LOEB

Cambridge, MassachusettsPreface

Buckminster Fuller has been alternately hailed as the most innovative thinker of our time and dismissed as an incomprehensible
maverick, but there is a consistent thread running through all thewildly disparate reactions. One point about which there is little
disagreement is the difficulty of understanding Bucky. "It was great!
What did he say?" is the oft-repeated joke, describing the reaction ofa typical enraptured listener after one of Fuller's lectures.
Not surprisingly then, Fuller's mathematical writing has not attracted a mass audience. Rather, synergetics has become a sort of
Fuller proving ground, into which only a few scientific-minded types
dare to venture. "Oh, I'll never be able to understand it then" hasbeen the response of countless people upon learning that the subjectof my book is synergetics. This reaction would have saddened Bucky
immensely: he was so sure his geometry was appropriate for fiveyear-olds! However, such shyness is understandable; deciphering
Fuller's two volumes, Synergetics and Synergetics 2, requires asizable commitment of time and patience from even the most dedicated reader. Study groups have gone a long way toward helping
individuals unravel the idiosyncratic, hyphenated prose of these twoworks, but the task, still arduous, is not for everyone. However, as
those who dared it will have discovered, the major concepts presented in Fuller's intimidating books are not inherently difficult, andmuch of synergetics can be explained in simple, familiar terms. Thatis the purpose of A Fuller Explanation.
Synergetics, in the broadest terms, is the study of spatial complexity, and as such is an inherently comprehensive discipline. Designers,
architects, and scientists can easily find applications of this study intheir work; however, the larger significance of Fuller's geometry may
be less visible. Experience with synergetics encourages a new way of
approaching and solving problems. Its emphasis on visual andxx

Prefacespatial phenomena combined with Fuller's holistic approach fosters
the kind of lateral thinking which so often leads to creative
breakthroughs.
A Fuller Explanation is geared to readers with no mathematicalbackground, but of course it can be read at many levels. Even if one
is familiar with some of the concepts, Fuller's unique interpretationand development of them will be enlightening. This book should
appeal to anyone interested in patterns and design and how things
work.
Synergetics is also fascinating as a reflection of Fuller himself; hiswide-eyed appreciation of nature and human invention alike exudes
from his expression of these geometric concepts. The primary purpose of this volume is thus to present the nuts and bolts of synergetics, the tools with which to continue exploring this discipline; but
almost as importantly, 1 hope to convey the spirit of Fuller's inquiry
into the organizing principles of nature.
When Bucky Fuller looked around, he saw, not trees and roads
and butterflies, but a miraculous web of interacting patterns. As he
describes these patterns, using his peculiar blend of antiquatedphrases and electronic-age jargon, one cannot help concluding thatno child was ever as startled as the young Bucky to discover that theworld is not what it appears, that the apparently solid and lifelessrock is a bundle of energetic atomic activity. He never lost that awe.
Synergetics is his attempt to give some of it away.
1 have included (along with explanations and definitions) many of
Fuller's own descriptions and invented terms. Although 1 set out tointerpret and explain Bucky in ordinary language-a task 1 foundincreasingly tricky as 1 became more deeply involved in., the project- I began to sense a deep appropriateness to Bucky's peculiar
phraseology. Many passages that seem convoluted at first reading
later seem to express his meaning more precisely than could any
substitutes. As 1 quote Fuller often in this book, readers will be able
to judge for themselves.

Acknowledgments

Arthur Loeb first introduced me to the intricate structure of ordered
space through his wonderful course at Harvard University. As editorof this book, he generously resumes his instructive role-seven years
later, long after his responsibility to a former student has expired.
For this I am deeply grateful. Dr. Loeb has perfected the art ofgentle criticism, and it was an extraordinary privilege to have had
this exposure to his kindness and wisdom.
I am greatly indebted to Bucky's family, especially Allegra Fuller
Snyder and Jaime Snyder, for their enthusiastic endorsement of mydecision to write this book; their commitment to making available
information about Bucky's work and life has enabled A Fuller
Explanation to exist. I also want to thank John Ferry at the Buckminster Fuller Institute for his assistance in locating photographs.
I am grateful to many people for their encouragement and inspiration, but above all to my parents, Bob and Mary Edmondson, whose
support of this project was indispensable.
I want to thank Martha Lerski for her eager and skillful editing;her help was invaluable. Also, I profited enormously from theinsights of many who have attended synergetics workshops, but tonone am I more indebted than to Darrell Mullis, Tony Perry,
Martha Stires Perry, and Carrie Fisher.
My husband, Mark Carpenter, provided the constant sustenancethat nurtured both this book and its author throughout the writing
process. He contributed many of the drawings, hours of editing, andmost importantly his inexhaustible intellectual curiosity.
Finally, Bucky was and remains an endless source of inspiration.
He was the best teacher and friend any 21-year-old just entering the"real world" could have had. It would be redundant to try to express
my gratitude for this experience here, for that is what I hope thisbook will do.
AMY C. EDMONDSONNew York. New York

Note to Readers

Quotations from Fuller's Synergetics: Explorations in the Geometry ofThinking, (which is organized into numbered sections and subsections in such a way that each paragraph is given its own reference
number) are followed by the numerical section reference in parentheses.
Passages from Synergetics 2: Further Explorations in the Geometryof Thinking will be referenced likewise, with the addition of the letter"b" after the section number.
Quotations are reprinted with permission of Macmillan Publishing
Company, New York, from Synergetics: Explorations in the Geometry of Thinking by R. Buckminster Fuller, in collaboration with E.J.
Applewhite. Copyright ©1975 by R. Buckrninster Fuller and from
Synergetics 2: Further Explorations in the Geometry of Thinking by
R. Buckminster Fuller, in collaboration with E.J. Applewhite.
Copyright ©1979 by R. Buckminster Fuller.

Photograph courtesy of ARGO Construction, Inc., Montreal, Quebec, Canada.

Introduction

The scene is Montreal, 1967: travelers from around the world emerge
from a subway station at the Expo site, and catch their first glimpseof an enormous transparent bubble. Looking and exclaiming, theygravitate toward this strange monument, which is the United States
Pavilion, and few notice the stocky white-haired old man, straining
his slightly deaf ears to glean their reactions. Buckminster Fuller,
playing the disinterested bystander, along with Anne Hewlett Fuller,
his wife of exactly fifty years, is a triumphant eavesdropper; thecandid observers have enthusiastically approved his design. Fifteen
years later, he recalls that summer morning with a playful grin,
clearly enjoying the memory of his short stint as detective, and I canalmost see him there, standing next to Anne, silently sharing theknowledge that the years of perserverence-ignoring skepticism andoften decidedly harsh disapproval of his mathematical work-werevindicated.
Bucky is such a gifted story teller that I also imagine I can see thehuge geodesic sphere reflecting the intense summer sun, and it looks
more like one of nature's creations than architecture. But it is steel,
Plexiglas and human ingenuity that have created this glittering
membrane, which was, in 1967, the world's largest dome, spanning
250 feet and reaching an altitude equal to that of a twenty-story
building, without any interior support.
More than just the millions who visited Expo '67 have admired
this architectural feat, and humanity has found countless other uses
for the geodesic dome, as evidenced by the 100,000 such structuresof various materials and sizes that are sprinkled around the globe.
However, the "synergetic geometry," which lies behind Fuller's
remarkable design, has remained almost completely obscure.
The goal of this book is to catalyze a process which I hope willcontinue and expand on its own: to rescue Fuller's fascinating
material from its unfortunate obscurity.
A. C. E.

Experience has shown repeatedly that a mathematical theory with arich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one
could have envisioned before the theory was developed.
William P. Thurston and Jeffrey R. Weeks
The Mathematics of Three-Dimensional Manifolds
Scientific American, July 1984.

1
Return to Modelability1

Synergetic geometry is the product of a mind as comfortable withmathematical precision as with the intuitive leaps associated withvisual and spatial conceptualizing. Buckminster Fuller was guided
predominantly by intuition throughout his 87 years; nonetheless, he
was entirely at ease with the painstaking exactitude of numericalcalculation-such as that required in the development of the geodesic dome in the early 1940s. Years before the pocket calculator, heproduced volumes of intricate trigonometric solutions, manipulatingeight-digit numbers with the patience and precision of a monk.
However, the peculiar language of Fuller's mathematical writings
quickly betrays the intuitive influence and all but conceals that of thehard-nosed engineer. Buckminster Fuller was both the pragmatic
Yankee mechanic and the enigmatic mystic, and synergetics is the
product of that combination.
Above all, he was driven by curiosity-and found nature afar more compelling teacher than the textbooks in his Milton,
Massachusetts, schoolhouse. Frustrated by the apparent lack of aconnection between conventional mathematics and reality, young"Bucky" Fuller adopted his own approach. The resulLing self-directed
exploration into pattern and structure became the most powerful
influence in his remarkable career as inventor, architect, engineer,and philosopher, and produced a geometrical system that provides
useful background for problem-solving of any kind.
Synergetics is the discipline hiding behind Fuller's fantastic visionsof a sustainable future. These reliable patterns were the source of his
unshakable confidence in his design-science philosophy, which-in
short-upholds that innovative application of the principles governing nature's behavior can insure ample life support for all humanity.
While many people around the world have been exposed to Fuller's
ideas and inventions, few have understood or even been aware of thelSee Notes, pp. 289ff.

2

A Fuller Explanation

mathematical principles underlying the elegant efficiency of structures such as the Octet Truss and geodesic dome. Happily, theseprinciples are easily accessible once you get into the spirit of Fuller's
approach: synergetics is a "hands-on" branch of mathematics.
However, listening to one of Fuller's all-encompassing lectures,
you might wonder when the "hands-on" part begins. Tangibility is
not a prominent feature in his spell-binding discourse, the subject ofwhich is no less than "humans in universe." He challenges, in thecourse of a few hours, age-old assumptions about our lives andinstitutions, asking us to reconsider the most commonplace aspectsof experience. Some of his observations are stated so simply, you
may find yourself wondering, "Why haven't I thought about that
before"? For example:
How many of you have said to your children, "darling, look at the beautiful sun
going down"? [A show of many hands.] "Well, we've known for five-hundred years
that the sun isn't going down, and yet we consider it practical to keep on lying toour children!

Or:
When I was born in 1895, reality was everything you could see, smell, touch andhear. The world was thought to be absolutely self-evident. When I was three years
old, the electron was discovered. That was the first invisible. It didn't get in any of
the newspapers; (nobody thought that would be important!) Today 99.99% of
everything that affects our lives cannot be detected by the human senses. We live ina world of invisibles.

And later, he takes his keys out of his pocket and carelessly tosses
them in the air; gravity takes care of the landing.
Nature doesn't have to have department meetings to decide what to do withthose keys, or how to grow a turnip. She knows just what to do. It must be thatnature has only one department, one coordinating system. 2

These simple truths each relate to different aspects of synergetic
geometry. But for all his lighter anecdotes, Fuller's underlying message could not have been more serious:
The fact that 99 percent of humanity does not understand nature is the prime
reason for humanity's failure to exercise its option to attain universally sustainablephysical success on this planet. The prime barrier to humanity's discovery andcomprehension of nature is the obscurity of the mathematical language of science.
Fortunately, however, nature is not using the strictly imaginary, awkward, andunrealistic coordinate system adopted by and taught by present-day academic
science. (OOO.125b)

Nature is instead using the principles embodied in synergetics, which

l. Return to Modelability

3thus provides the way to eradicate this lethal ignorance. Claiming tohave discovered no less than the mathematical system that describes
the coordination of physical and metaphysical phenomena
alike-that is, of both energy and thought-Fuller was urgent in his
insistence that we study these principles:
I am confident that humanity's survival depends on all of our willingness tocomprehend feelingly the way nature works. 2

From Geometry to Geodesics: A Personal Perspective

What college student (or human being for that matter) would not be
overjoyed to receive an invitation to work on "ever-more relevant
affairs" from the person she most admires? That is precisely what I
found in my mailbox in 1980 when Buckminster Fuller actuallyanswered my letter, the timid plea of an undergraduate: "What canpeople do toward furthering your vision of making this planet workfor everyone? And where can I apply the experience of having
studied synergetics"? I was later to understand that Fuller's responding to an undergraduate's letter was not a miracle but instead
revealed his profound trust in the integrity and capability of humanbeings-and especially of youth. His action was typical of his life
and work, which relied heavily on intuition, with a powerful faith in
the willingness of others to apply their minds as diligently andjoyfully as he applied his. We have to take Fuller at his word when
he claims to be not a genius but an "average healthy human being"
who exercised his option to think. He embraced that potential in allof us.
I was introduced to the intricate discipline of geometry in a
Harvard course "Synergetics: the structure of ordered space" taught
by the editor of this series, Arthur L. Loeb and I had been fascinatedby this material for a couple of years. Reconciled to its obscurity, I
was enchanted by the perfection and complexity of this body ofgeometric knowledge, which was all but completely hidden from
popular awareness. In those days Loeb's course was similarly hidden,a bizarre option within the two-inch-thick course catalog, taught in asequestered attic in Sever Hall where one would never wander
accidentally. My peers had no doubt that there was a reason for that.
In fact, my academic pursuits were perceived by most as an irreverent cross between kindergarten games and mathematical torture. My
roommates, forever tripping over cardboard tetrahedra and unsuccessful tensegrity wheelbarrows while gingerly avoiding small de-4

A Fuller Explanation

posits of Elmer's glue, were tolerantly confused. I can't blame myclassmates for their bemused head-shaking; I had trouble taking
myself seriously. Mathematical elegance aside, I felt deep down that
I had chosen an unbelievably fascinating road to nowhere, a choicethat would no doubt ultimately bar me from all chance of meaningful participation in human affairs. But still I was trapped-like an
addict immune from better judgment-in my polyhedral playpen.
Then one February evening, I heard Fuller speak at theMassachusetts Institute of Technology. It's easy to understand how
pivotal the experience could be: in love with geometry but distressedby its nonapplicability, I heard Bucky Fuller that night spin out-in
an omnidirectional web of ideas, predictions, and obscure but brilliantly juxtaposed facts-an unfamiliar version of world history inwhich synergetic geometry (and other aspects of comprehensive
thinking) somehow played a crucial role in rescuing humanity fromits current crisis of squandering vast resources in an unwinnablearms race. We are suddenly at a turning point in history at which it
is possible to provide adequate life support for everyone, declared
Bucky. Malthus is obsolete. (He didn't know about alloys.3) There isno such thing as a straight line, the sun does not go down, and it istime we updated our language.
A funnier, more serious, more mesmerizing discourse I have neverheard. I walked-no, skipped-back home down Massachusetts
Avenue: not that I could have told you exactly how it worked, thisplanetary success, but I was sold. My geometry had relevance! I
worked harder than ever.
I still can't explain exactly how synergetics is going to turn theworld around, but I have found at least that I can explain synergetics. My hope is that if enough other people become aware of theseprinciples, the missing pieces will ultimately come together. So far,this has been a valid working hypothesis. In giving lectures andworkshops to clarify Fuller's material, I have met people who foundsignificant applications in their own work. Synergetics has provided
both useful models to elucidate scientific phenomena and methods ofsolving structural problems. Examples of both aspects will be citedthroughout the book; see especially the end of Chapter 15, "Case In
Point: Donald Ingber" which can easily be read independently of therest of the book.
After Fuller's lecture, the next step was clear: read Synergetics. If
I expected easy answers, I was in for a surprise. Fuller's ambiguous
writing called for considerable interpretation. With the patient guidance of Arthur Loeb, I struggled through Fuller's massive text and1. Return to Modelability

5learned that truth was far more elusive than he had made it soundthat night at MIT. But the geometry was no less seductive, andultimately I decided to risk a thirteen-cent stamp.
And then his unexpected letter arrived-in response to my earnest
but decidedly indirect questions. Even a photocopied list of organizations would have been received with joy. How it was that my letter
filtered through the procedural maze that lay between Buckminster
Fuller and the formidable stack of mail that was opened and sorted
by various trusted assistants every day, I'll never know.
The signature was real:
Dear Amy Edmondson:... I would like to take advantage of your offer to come and work with me .... I
am busier and busier with ever more relevant affairs.
Warmly, Faithfully,
Buckminster FullerEver more relevant affairs! A college student's dream and-she isconvinced after three years of first-hand acquaintance-an accurate
description of Fuller's experience. Even with a healthy dose ofskepticism about some aspects of his philosophy, one could not help
being stunned by his tireless enthusiasm for work. At four times myage, he was awake and working before I arrived and long after I had
crept home to bed exhausted. The secret of this energy was his
conviction that humanity had a viable option of designing an
unprecedentedly successful environment aboard "Spaceship Earth"4,
and that his work just might play an important part in that. I foundthat the more deeply involved in the actual work I became (in mycase, calculations and drawings for Fuller's engineering projects,
including the progressive refinement of geodesic designs), the more
impressed I was by the scope of his vision. Gradually, as is generally
the case, naIve youthful worship disappeared. But as is not generally
the case, its place was taken by a new deep respect for the mind of
my friend, Bucky. (It would be unimaginable for anyone who everspent more than an hour with Bucky to call him anything else, withthe sole exception of his friend and colleague, Arthur Loeb, whocalls him "Buckminster.")
Bucky was extraordinarily generous with his time-perhaps due toan uncontrollable urge to teach-and treated every listener as anintellectual equal. This might be called a brilliant teaching strategy,
except that it was utterly spontaneous. One of the most importantlessons of my three-year experience was the difference between
Bucky on the other side of his desk-spontaneously lapsing into

6

A Fuller Explanation

simple clear explanations as a result of the catalyst of a pair of
expectant human eyes which would cloud into a worried frown whenlost-and Buckminster Fuller's dense polysyllabic prose in the 800
pages of Synergetics. I became accustomed to translating the Fullerese
into lay English for various befuddled readers who went so far as tocall the office for help.
This is not a book about those three years; it is about synergetic
geometry. Here I have only tried to give you some of the backgroundthat has led me to attempt to explain what is in many ways
unexplainable, for no one can speak for Bucky Fuller but himself.
The goal of this volume is to help readers get through the barriers
imposed by Fuller's idiosyncratic use of language, and to introduce
the major concepts of synergetics in an accessible format. The next
steps are up to all of us.
Operational Mathematics

We can imagine the young Bucky, an enthusiastic misfit sensing that
he is alone in his skepticism about the fundamental premises ofgeometry. ("Does no one see what I see? Does no one else sense theterrible problems that lie ahead if we follow these absurd premises totheir logical ends?") While his grade-school classmates were apparently content to go right along with the teacher's strange games
without complaint, Bucky was astonished by the implausible newconcepts.
Bucky would tell us that he tried, constantly, to accept therules-be a good student, make his family proud, submit to andeven excel at the illogical activities-but somehow his efforts at
model behavior were always thwarted. He just couldn't help pointing
out that the teacher's "straight line" was not at all straight, butrather slightly curved and definitely fragmented. Perplexed by herlack of accuracy, Bucky saw a trail of powdery chalk dust left on theblackboard, a trace of the motion of her hand, and it seemed quite
unlike her words.
Bucky was virtually blind until he got his first eyeglasses at the ageof five, so he had truly experienced life without this primary sense.
Now he was insatiably curious about the visual patterns around him.
An "infinite straight line"? He would turn toward the window,
thoughtfully pondering where that "infinite line" stopped. "Out thewindow and over the hill and on and on it goes"; it didn't seem rightsomehow. Bucky, childishly earnest even in his eighties, would tell

1. Return to Modelability

7

this story, explaining that he didn't mean to be fresh, he just couldn't
help wondering if that teacher really knew what she was talking
about.
Much later, fascinated by Eddington's5 definition of science as thesystematic attempt to set in order the facts of experience, Fuller had
a plan. It must be possible to develop a mathematical system
consistent with experience. He concluded that humanity had been on
the wrong track all these years.
Experimental Evidence

It seemed to Fuller that mathematicians arbitrarily invent impossible
concepts, decide rules for their interaction, and then memorize thewhole game. But what did he propose as an alternative?
Starting from scratch. Mathematical principles must be derived
from experience. Start with real things, observe, record, and thendeduce. Working with demonstrable (as opposed to impossible)
concepts, the resulting generalizations would reflect and apply to theworld in which we live. It seemed highly likely that such an experimental approach would lead to a comprehensive and entirely rational set of principles that represented actual phenomena. Furthermore, Bucky suspected that such an inventory would relate tometaphysical as well as physical structure.
Fuller decided that to begin this process of rethinking mathematics he had to ask some very basic questions. What does exist?
What are the characteristics of existence? He proposed that science's
understanding of reality should be incorporated into new models to
replace the no longer appropriate cubes and other" solids" that had
kept mathematicians deliberately divorced from reality since thedays of ancient Greece.
To begin with, there are no "solids"; matter consists exclusively ofenergy. "Things" are actually events-transient arrangements offrenetically vibrating atomic motion. It's almost unthinkable, but
perhaps if we get our vocabulary and models to be more consistent
with an energy-event reality, humanity can be retuned to become
comfortable with the invisible discoveries of science. In short, simplifies Fuller, we have the option to tell children the truth about
nature in the first place.
What if we do go along with the rule that mathematics cannot
define and depend upon a concept that cannot be demonstrated;
where does that leave us? Fuller saw inconsistencies even in thenotion of an "imaginary straight line," for imagination relies on8

A Fuller Explanation

experience ("image-ination" he would say) to construct its images.
Therefore an "operational mathematics" must rely on concepts that
correspond to reality.
Bucky pulls us back into the turn-of-the-century schoolhouse of
his childhood. The teacher stood at the blackboard, made a little dot,
and said, "This is a point; it doesn't exist." ("So she wiped thatout.") Then she drew a whole string of them and called it a "line."
Having no thickness, it couldn't exist either. Next she made a raftout of these lines and came up with a "plane." "I'm sorry to say it
didn't exist either," sighs Bucky. She then stacked them together andgot a "cube," and suddenly that existed. Telling the story, Bucky
scratches his head as if still puzzled seventy years later: "I couldn't
believe it; how did she get existence out of nonexistence to thefourth power? So I asked, 'How old is it'? She said, 'don't benaughty.' ... It was an absolute ghost cube.,,6
Instead of a dimensionless "point," Fuller proposes the widely
applicable "energy event." Every identifiable experience is an energy
event, he summarizes, and many are small enough to be considered"points," such as a small deposit of chalk dust. An aggregate ofevents too distant to be differentiated from one another can also betreated as a "point." Consider for example a plastic bag of oranges
carried by a pedestrian and viewed from the top of the Empire State
Building, or a star-consisting of immense numbers of speeding
particles-appearing as a tiny dot of negligible size despite having
an actual diameter far greater than that of the earth. 7
The mathematician's "straight line," defined as having length but
no width, simply cannot be demonstrated. All physical "lines" uponcloser inspection are actually wavelike or fragmented trajectories:
even a "line of sight" is a wave phenomenon, insists Fuller; "physics
has found no straight lines." 2 But forces exist, and they pull or pushin a line, which can be modeled by a vector. We shall explore vectors
and their role in synergetics in great detail in later chapters. Finally,the "continuous plane" with no thickness must be replaced by amesh of energy events interrelated by fine networks of tiny vectors.
To Fuller, these adjustments were crucial, for mathematicians' gameswith continuous planes and sizeless points were ultimately irrelevant
diversions; however enticing the intellectual pursuit might be, they
do not lead to a better understanding of how nature works.
The essential nature of the above revolution is semantic and caneasily seem trivial. The difficulty in evaluating the impact of suchchanges lies in the subtlety of the effect of words and the images they

1. Return to Modelability

9

produce. Only through experimenting with Fuller's substitutions forsome period of time can we judge the merits of mathematicalterminology that reflects science's new understanding of reality.
Back to the starting point! Nothing can be accepted as self-evident; a new mathematics must be derived though "operational"
procedure. Fuller decided that through sufficient observation of bothnaturally occurring and experimentally derived phenomena without
reference to a specific framework, nature's own coordinate system
might emerge. He sought a body of generalizations describing theway patterns are organized and able to cohere over time. We shallsee how these principles can be discerned both in deliberate experiments with various materials and by recording existing natural
patterns.
Bucky's grade-school skepticism was thus the beginning of alifelong search for "nature's coordinate system." After rejecting
traditional academia through his dramatic departure from Harvard's
freshman class in 1914, 8 he began an independent exploration ofmathematics which he was to pursue for the rest of his life. A sort ofphilosophical geometry gradually began to take shape, consisting ofa rich body of facts and principles, some new and others newly
considered. Synergetic geometry is tied together as one cohesive
system by the unmistakable presence of Bucky Fuller in conventional and bizarre observations alike. Including expositions called"Tetrahedron Discovers Itself and Universe," "Life," "Cosmic
Hierarchy," "Complex of Jitterbugs," and the more conventional"Closest Packing of Spheres," Synergetics does not fall neatly into
any preconceived category. This volume will attempt to clarify these
multifaceted (or polyhedral) observations, which together constitutesynergetic geometry.
Nature's Coordinate System

What accounts for the shape similarities among unrelated phenomena, radically different in both scale and material? Or, more fundamentally, what accounts for nature's magnificent orderliness itself?
Whether honeycomb or conch shell or virus, time after time individual structures turn out true to form. The fundamental hypothesis
behind synergetics-and the work of many other pioneers exploring
the science of form-is that nature's structuring occurs according to
the requirements of minimum energy, itself a function of the interplay between physical forces and spatial constraints.

10

A Fuller Explanation

Wait. The role of physical forces (gravity, magnetism, electricaland chemical attractions) is clearly important, but what are "spatialconstraints"?
We are so used to thinking of "space" as empty nothingness thatthe idea of its having specific properties seems absurd. However, as
will become increasingly clear from the examples throughout thisbook, space has shape. The idea is concisely expressed by Arthur
Loeb in his introduction to Space Structures: "Space is not a passive
vacuum, but has properties that impose powerful constraints on any
structure that inhabits it. These constraints are independent ofspecific interactive forces, hence geometrical in nature." 9 A simple
example is the fact that to enclose space with only four polygons,these polygons must all be triangles~ Nothing else will work, no
matter how hard you try. The limitation is a function of neithermaterial nor size but rather of the nature of space. Fuller alludes tothis active role when he says "natural is what nature permits."
When Bucky points out that nature doesn't have to stop everything she's doing and gather the physics, chemistry, biology, andmathematics departments to decide how to grow a turnip (or build avirus), he is calling our attention to the self-organization of natural
phenomena. Structuring in nature occurs automatically. "Nature has
only one department," declares Bucky, "one comprehensive coordinating system." How does this self-structuring occur? In the mostgene:al terms, according to the path of least resistance, or, as statedabove, according to the "requirements of minimum energy." Inshort, systems automatically find comfortable arrangements, which
are necessarily a result of the balance between specific forces andinherent spatial properties. When Fuller set out to inventory possible
configurations and thereby formulate generalizations, his exploration
was destined to be "geometrical in nature" because of the nature ofsystems, as we shall learn in Chapter 3. "Nature's coordinate system"
is thus a geometry of most economical relationships which govern all
structuring. In Fuller's words, "a geometry composed of a system ofinterrelated vectors may be discovered that represents the complete
family of potential forces, proclivities, and proportional morphosis ... " (215.02).
We shall see how "operational procedure" produced a geometry ofvectors and look at the specific shape of this diagram of potentials in
Chapter 7, "Vector Equilibrium," and Chapter 9, "Isotropic Vector
Matrix." The most notable characteristic of these models is theabsence of perpendicularity. We thus shall explore Fuller's statementthat nature is never operating in perpendicular and parallel direc-

1. Return to Mode1ability

11

tions but rather convergently and divergently in radial growth patterns.
Another important aspect of "nature's coordinate system" is theexistence of rules governing the coherence of structures. What holds
its shape? If nothing is self-evident, we can no longer take" solids,"
or reliable structures, for granted. Chapter 5 examines Fuller's
investigation into structure and the resulting principles governing thestability of systems.
All of this takes a while to sink in. The idea that space has shape is
profoundly reorienting; we are so used to conceiving of space aspassive emptiness on which we impose desired configurations that an
entirely new perception cannot be adopted overnight. Nonetheless,upon further study, this premise begins to feel quite comfortable andnecessary-an all-embracing something ness influencing structural
phenomena. As more specifics are uncovered, this conception, which
is at once so elusive and so ordinary, begins to seem more and more
the latter.
Universe

The ultimate manifestation of nature's coordinate system is "Universe." Fuller deliberately omits the article, for" the universe" implies the possible existence of more than one-just as we do not say"the God" but rather simply "God." Fuller capitalizes "Universe"
for the same reason: Universe is everything; it's all there is. (Or,more poetically, "Universe is all that isn't me and me."6) But Fuller
would never leave it at that; he is indefatigably thorough.
Einstein revolutionized our understanding of Universe, explains
Fuller; prior to his relativity theory, we could think in terms of asingle-frame (simultaneously complete) picture, unimaginably vast,
but still simultaneous at any given moment. This understanding
must now be replaced by a "scenario" concept:
301.10 Universe is the aggregate of all humanity's consciously apprehended andcommunicated nonsimultaneous and only partially overlapping experiences.

He willingly disects his own long-winded definition. To Fuller,aggregate implies a complex that cannot be comprehended in totality at only one moment (302.00): "Consciousness means awareness ofotherness." (That one's easy.) To be apprehended, information mustfirst be within the range of human perception and then actually be
noticed. "Communicated means informing self or others. Nonsimultaneous means not occurring at the same time" (302.00); events of12

A Fuller Explanation

Universe are instead "partially overlapping" -like generations. My
lifetime overlaps my grandmother's and hers overlaps the life of her
grandmother, but I was born long after the death of both my greatand great-great-grandmothers. Such are the events of Universe;
every experience overlaps some but not all other experiences.
Another facet of the "scenario" concept centers on the misconception of the environment as a static whole. For instance, looking out
at a distant star it is all too easy to think we are" seeing" it just as it
is at that moment, while in reality that particular star is so far awaythat its light takes 100 years to reach us. What we are actuallylooking at is a "live show" taking place 100 years ago. We are seeing
an event that occurred before we were born. Universe is the integral(or sum total) of all experience. It cannot be unitarily conceived, but
as thus defined it is all inclusive. "You cannot get out of Universe"(321.02).
Fuller's definition avoids imparting a sense of thingness-part ofhis effort to encourage us to think in terms of "pure principle."
Universe is energy and thought all knotted together by incredibly
complicated webs of relationships. It is ultimately impossible toseparate the physical and metaphysical; both are "experience." Thescientific principles that govern the interactions of energy events-as
timeless statements of truth-are themselves metaphysical. The line
therefore becomes ever more difficult to draw. This is why Fuller's
definition depends upon consciousness. Our awareness of energy
events defines their existence; we cannot go beyond the limits set by
our understanding.
Finally, Fuller assures us that the definition is complete:
People say to me, "I think you have left something out of your definition of
Universe." That statement becomes part of my experience. But never will anyone
disprove my working hypothesis because it will take experimental proof to satisfy
me, and the experiment will always be part of the experience of my definition, ergo
included. (306.01)

He elaborates in a 1975 videotaped lecture: "Someone might ask'what about dreams? I think you left that out,' and I reply, 'no, forthat is part of your experience.''' He seems to have all the angles
covered.
Generalized Principles
The principle of leverage is a scientific generalization. It makes no difference ofwhat material either the fulcrum or the lever consists .... Nor do the special-casesizes of the lever and fulcrum ... in any way alter either the principle or themathematical regularity of the ratios of physical work advantage ....

1. Return to Modelability

13

Mind is the ... uniquely human faculty that surveys the ever larger inventory of
special-case experiences stored in the brain bank and ... from time to time discovers
one of the rare scientifically generalizable principles running consistently through all
the relevant experience set. (Synergetics, p. xxvi)

Fuller spoke frequently and ponderously of the "generalized
principles": those statements-be they verbal or in the shorthand ofmathematical equations-that have been proven to always hold true.
In other words, generalized principles are rules with no exceptions.
From the very simple (the mechanical advantage allowed by leverage) to the highly profound (E = me 2 , equating matter with energy
and quantifying the rate of exchange), these principles, taken all
together, describe our Universe. Applying a generalized principle ina novel way is called invention. In the broadest sense, synergetics isthe search for generalized principles. (" Design science" is the application, as will be discussed in Chapter 16.)
Fuller placed enormous stock in these principles, and saw humanity's role in Universe as discoverer and utilizer of the progressively
uncovered truths. Endowed with "minds" (in contradistinction to"brains," which are merely able to coordinate sensory input), humans are uniquely able to survey successive experiences and detect
reliable patterns, thereby discovering over time such subtle workingsof Universe as gravity. Utterly invisible and unpredicted by theinvestigation of separate objects, gravitational force represents a
profound discovery and is certainly without exception. More remarkable still, human mind was able to express the magnitude ofthis force in precise terms: F = GMmjr2.1O A fantastic leap beyond
sensory-based information, this discovery places tiny humans incontact with great Universal motions.
That the human mind is able to detect eternal truths amidst"special-case" experiences, explains Fuller, is the wealth of humanityand our hope for the future. Many of these Universal laws are widely
known and applied, as for example the principle of leverage, butother equally reliable truths are all but completely unfamiliar, suchas the geometric discoveries described in this book. Fuller, convincedthat an inventory of yet to be explored applications had the potential
to solve humanity's problems, was the ardent champion of unfamiliar principles.
Universe is thus the total web spun by all generalized principles
and their interaction. As rules without exception, they are necessarily"interaccommodative." Fuller's terminology takes some getting used
to: "eternally regenerative Universe" is an ongoing event governedby "the omni-interaccommodative complex of unique and eternal
generalized principles." The principle of synergy, described in14

A Fuller Explanation

Chapter 3, accounts for the incalculable complexity of the whole web
despite thorough comprehension of many of the separate parts. And
finally, "God is the unknowable totality of generalized principles."Return to Modelability

Synergetics is a product of Fuller's passionate concern with models.
Concerned that society's ignorance of science is seriously destructive,
he devoted years of thought to ways of alleviating the resulting crisis.
In the twentieth century, we suddenly find ourselves confronted with
an "invisible" atomic reality in which the average person understands very little about how things work. Although confronted daily
with "incredible technology," which to Fuller includes the natural
phenomena of Universe as well as the ever-expanding inventory ofhuman invention, the vast majority assume such phenomena to beout of their reach. Fuller attributes this widespread discomfort toboth the" invisibility" of science and the devastatingly complicatedmathematics without which, scientists claim, their findings cannot be
described. The dangerous chasm between scientists and lay people,
with the truth guarded by an elite few and the rest resigned toignorance, thus seems inevitable.
The origin of this troubled state of affairs? An incorrect mathematical system! Long ago human beings surveyed their environment
and, seeing a never-ending flat earth, decided upon cubes andorthogonal planes as the appropriate measuring system. Today, says
Fuller, we're still stuck with that uninformed early guess, and as aresult, nature's behavior has seemed irrational, perverse, and difficult
to explain because we're using the wrong kind of yardstick. Withaccurate models, he claims, this ignorance can be eradicated. The
purpose of synergetics is to make the invisible events and transformations of Universe visible, through tangible models that elucidate
the principles behind our energy-event Universe. Human beings will
thereby be able to "coordinate their senses" with a new understanding of reality.
Synergetics is full of tantalizing models; the difficulty comes inassigning them to aspects of physical reality. However, a number ofnotable examples, in which a newly discovered scientific phenomenon is described by one of Fuller's previously developed models,suggest that there may be many more such successes to come. The
immediate goal therefore is to unravel and study the geometricsystem itself.

2
The Irrationality of Pi

Young Bucky Fuller seems to have been haunted by 'IT. Discoveringthat the ratio of a sphere's circumference to its diameter is an
irrational number commonly referred to by the Greek letter 'IT was
perhaps the most disturbing of all geometry's strange lessons. Just as
he could not imagine where its accomplice in deception, the infinite
straight line, stopped, Bucky could not let go of a vision of thousands upon thousands of digits trailing after 3.14159 ... , spilling out
the classroom window and stretching absurdly to the next town andthen farther, never allowed to stop.
He has told the story countless times, each rendition sounding likethe first, with an air of revelation. If he happens to be standing by
the ocean, so much the better; he's bound to talk about the foaming
waves.
"Look at them all"! he marvels, "beautiful, beautiful bubbles,
every one of them!" Bucky recalls that growing up near the ocean
gave him plenty of time to think about the structuring of these
bubbles. Looking back at the wake trailing behind his boat, or
standing knee-deep in breaking waves at the shore, he saw the watercontinuously being laced with white foam. Its whiteness was createdby vast numbers of tiny air bubbles, each one suddenly formed andemerging at the water's surface. "How many bubbles am I lookingat"? Bucky would ask himself, "fantastic numbers, of course." He
could not help wondering: if each and every one of those spheres
involves 'IT, to how many digits does nature carry out the irrational 'IT
in making one of those bubbles, before discovering that it can't becompleted? At what point does nature stop and make a "fakebubble"? And how would the decisions be made? In meetings of thechemistry and mathematics departments? No, the young Bucky
concluded, I don't think nature's making any fake bubbles; I don't
think nature is using 'IT.
It is a decidedly amusing image: nature gathering department
heads, nature's consternation over fudging the numbers, getting

16

A Fuller Explanation

away with imperfect bubbles-and It IS through such deliberate
personification that the ideas become memorable. The story of ayoung man looking out from his ship carries a deeper message.
Fuller uses this particular event-even assigning a time and place,
1917 on a U.S. Navy ship-as a moment of revelation, the thresholdof his conscious search for nature's coordinate system. The scene isbait, to draw us in, make us as curious about nature's structuring asthat young sailor was. It works. If he had started on a headydiscourse about mathematical concepts versus natural structures, hisaudience might have walked away.
It's important to realize the nature of his rebellion: not to challenge the theoretical numerical ratio between the circumference anddiameter of the ideal sphere, but rather to challenge that sphere tomaterialize. Irrational numbers don't belong in tangible experiences.
It is a question of sorting out the demonstrable from the impossible
and then developing models based on the former. Essentially, Bucky
is choosing not to play with 'IT, posing the question "why shouldn't
mathematics deal with experience"?
"Nature Isn't Using Pi"

Nature can have no perfect spheres because she has no continuous
surfaces. The mathematician's sphere calls for all points on its
surface to be exactly equidistant from the center. This "sphere,"
explains Fuller, has no holes. It is an absolutely impermeable container sealing off a section of Universe, a perpetual energy-conserving machine defying all laws of nature. The illusion of a physicalcontinuum in any spherical system is due to the limitations of thehuman senses.
On some level of resolution, all physical "solids" and surfaces
break down into discrete particles. A magnifying glass uncovers thetiny dots of different colors that make up the "pure" blue sky in a
magazine photograph, and a microscope-if it could be set up in thesea-would reveal that nature's bubbles are likewise fragmented,
consisting of untold numbers of discrete molecules located approximately equidistant from an approximate center. If such aphenomenon could be precisely measured, we would find that theratio of a bubble's circumference and diameter is some number very
close to the elusive 'IT, but the point, says Bucky, is that the bubble
differs from the Greek ideal. A physical entity is necessarily demonstrable and finite, while an irrational number such as 'IT is just the
opposite. A real sphere consists of a large but finite number of2. The Irrationality of Pi

17interconnected individual energy events. Moreover, the middle ofeach of the implied chordal (straight-line) connections between eventsis slightly closer to the sphere's center than the ends, thereby
violating the mathematical definition, and insuring some small departure from the ratio 'IT. Any cross-section of this sphere will be, nota circle, but a many-many-sided polygon. Likewise, the mostperfect-looking circle, carefully drawn with a sharpened pencil andaccurate compass, will appear fuzzy and fragmented through a goodmagnifying glass. The most precise ball bearing is imperfect: a good
approximation of 7f, but always imperfect, with bizarre mountains
and valleys dramatically visible after 5000 times magnification. In
Chapter 15 we shall examine the specifics of one alternative modelfor the mathematician's impossible sphere.
"Fake bubbles" led to further contemplation about 'IT. Nature isalways associating in simple whole rational numbers, thought Bucky:
H 20, never H"O. Irrational numbers do not show up in chemical
combinations of atoms and molecules. Clearly, nature employs "onecoordinate, omnirational, mensuration system." (410.011)
None of this is new information; anyone of us could have thought
about it for a while and understood that of course 'IT does not playa
role in the making of bubbles. What happens is that turbulence
introduces air into the water and the air is so light that it floats
toward the surface to escape. It's an easy problem of getting themost air in each pocket with the least surface area of water pushingin to collapse the bubble. A spherical space allows the most volume
per unit of surface area (as we shall learn in later chapters), providing the most efficient enclosure for that air. (Nature is exquisitely
efficient.) But, even if we did contemplate this mystery at theseashore, we probably didn't bother to take our children aside andexplain it to them before they became perplexed and frustrated by
the never-ending 'IT in mathematics class.
If nature's lack of employment of 'IT isn't news, why does Bucky
persist? Why use his own considerable-but still finite-energy tolecture tirelessly on the subject for half a century? The answer lies in
the fact that our education and popular awareness fall short of themark; Fuller felt that society did not sufficiently emphasize thisdiscrepancy between theoretical games and real structures. We introduce "solids" to school children long before they learn the real story
about energy-event reality. If humanity is to feel comfortable with
science, argues Fuller, education must present accurate models in thefirst place. The message behind the bubble's story is: why not tell thetruth from the beginning? He calls for widespread recognition of

18

A Fuller Explanation

discrete energy events, a reality that cannot be perceived by humansenses.
Unfortunately, the issue is not that easily resolved. We shall notever completely escape the brain-teasing presence of irrational numbers. For example, the construction of the simplest polygons, withcarefully measured unit-length edges, frequently produces irrational
diagonals. The distance between nonadjacent vertices in a regularpentagon is the irrational number known as the "golden section,"
which we shall see again in Chapter 11, and the diagonal of a squareis incontrovertibly fi. However, in every actual construction, afinite approximation can be determined according to the specificlimitations of available measuring instruments. That numerical fractions must terminate is a consequence of investigating verifiable
experience rather than theoretical relationships. Just as architects do
not work with irrational-length two-by-fours, measurements obtained through "operational procedure" are necessarily real. Finally,
Bucky's dismissal of these troublesome values calls attention to the
granular constitution of physical reality, as described in the following section.
Finite Accounting System"Y ou cannot have a fraction of an energy event," philosophizes
Bucky. Science's progressive subdivision ultimately reaches indivisible particles, which means that reality consists of whole numbers l ofenergy events. Therefore, we need models that will demonstrate theconcept of structures consisting of discrete, or countable, units. In
synergetics, area and volume are presented as quantities that can becounted, as opposed to measured and described as a continuum.
Area is associated with some whole number of events on the surface,
while volume is tallied as the number of events throughout a system.
Chapter 8 will examine how sphere-packing models illustrate theassociation of volume and surface area with discrete units; Chapter
10 will discuss the concept of volume in general and its role insynergetics.
We are not used to thinking of reality as submitting to a "finite
accounting system," but in fact it does. Consider the phenomenon oflight. Few things seem more continuous than the light that fills yourliving room when you turn on the switch, or the light that seems tofill the whole world on a sunny day at the beach. But we now knowthat even light consists of individual packages of energy, called
photons. Although they could hardly be smaller or lighter, photons

2. The Irrationality of Pi

19

are nonetheless discrete events, countable in theory, even if there arealways too many, jumping around too fast, to make that enumeration possible in practice. With its flawless illusion of continuity
contradicting science's detection of constituent photons, light is aperfect phenomenon to symbolize the validity of a "finite accounting
system." We could hypothesize a system in which volume was tallied
in terms of the number of photons in a given space, and area, thenumber of photons found at the surface; however, it would be a
fantastically impractical system. A more appropriate unit is called
for. For now, however, we simply consider the qualitative implications of the discrete-events concept. A punctuated reality is hard to
get used to.
Which Way Is "Up"?

Fuller proposed a revolution in modes of thinking and problem-solving, which above all else required a comprehensive approach, as will
be discussed in Chapter 16. To Fuller, "comprehensive" means notleaving out anything-least of all humanity's important tool oflanguage. A dictionary contains an inventory of 250,000 agreements,
he explains, specific sounds developed as symbols for 250,000 nuancesof experience. He saw this gradual accomplishment as one of themost remarkable developments in the history of humanity, with its
implied cooperative effort.
One aspect of his revolution thus involves an effort to employ
words accurately. Fuller's discourses on the subject tend to be quite
humorous, almost (but not quite) concealing how deeply serious he
was about the matter. Much of our language is absolutely stuck in"dark ages" thinking, he would lecture. Up and down, for instance.
These two words are remnants of humanity'S early perception of aflat earth; "there is no up and down in Universe!" exclaims Fuller.
When we say" look down at the ground" or "I'm going downstairs"
we reinforce an underlying sensory perception of a platform world.
N either the ground nor Australia can accurately be referred to as
down; three hours after a man in California says that the astronauts
are up in the sky, the shuttle is located in the direction of his feet.
Up and down are simply not very precise on a spherical planet. The
replacements? In and out. The radially organized systems of Universe have two basic directions: in toward the center and radially out
in a plurality of directions. Airplanes go out to leave and back in toland on the earth's surface. We go in toward the center of the earth
when we walk downstairs. The substitutions seem somewhat trivial

20

A Fuller Explanation

at first, but again it is difficult to judge without trying them out.
Experimenting with "in" and "out" can be truly reorienting; unexpectedly one does feel more like a part of a finite sphericalsystem-an astronaut on "Spaceship Earth." (It's hard to keep at itfor long however; up/down reflexes are powerful.)
The sun does not go down, insists Bucky; how long are we goingto keep lying to our children? First of all, we now know there is noup and down in the solar system, and secondly, the sun is not
actively touring around the earth. Rather, we are the travelers, andour language should reflect that knowledge. Oddly, the phraseology
which gives the sun an active role ("darling, look at the beautiful sun
going down") does seem to reinforce the erroneous conception of ayellow circle traveling across the sky. While we know this isn't thecase, it often feels like the way things happen. This effect is noteasily measured and probably varies from individual to individual.
What does all this have to do with geometry? Remember that oneof the goals of synergetics is to help coordinate our senses withreality, that is, to put us in touch with Universe, which to Fuller
involves eradication of the erroneous vocabulary which keeps us
locked into "dark-ages" thinking on a sensorial level. In short, weneed to align our reflexes with our intellect. Instead of sunset andsunrise, reinforcing the sun's active role, Fuller suggests sunclipse
and sunsight, which imply instead that our view of the sun has been
obscured and that an obstacle has been removed, respectively. Who
knows if people could adjust to such substitutions? It might be worth
a try.
The cube is another remnant of flat-earth days, and Fuller has awealth of reasons to prefer another mathematical starting point,
which will be discussed throughout this volume. 2 Our age-old dependence on squares and cubes is honored by an unfortunate verbal
shorthand for "x to the second power" and "x to the third power."
The expressions "x squared" and "x cubed" are so commonly usedthat most people assume these multiplication functions to have atrue and exclusive relationship to squares and cubes. The shorthand"squared" is derived of course from the fact that a square can be
subdivided by parallel lines, with "x" subdivisions along each edge,
into "x to the second power" smaller squares. For example, a squarewith 2 "modular subdivisions" per edge contains 4 small squares,
and similarly 3 subdivisions yield 9 squares, 4 yield 16, 5 yield 25,
and so on (Fig. 2-1). Everyone is familiar with these diagrams, butwhat most people do not realize is that this result is not unique tosquares. Triangles exhibit the same property, as also shown in Figure2-1. Furthermore, explains Fuller, triangles take up only half the2. The Irrationality of Pi

D

11

1

4

21

D

EE

9

Fig. 2-t. "Triangling" versus "squaring."

space, because every square divides into two triangles. Triangles ingeneral, therefore, provide a more efficient diagram for the mathematical function of multiplying a number by itself. Nature is always
most economical; therefore nature is not "squaring"; she is "triangling."
Happily, the same is true in three dimensions: the increasing
volumes of subdivided tetrahedra (if this is an unfamiliar shape, wait
until the next chapter!) supply the third-power values just as accurately as do subdivided cubes: 23 = 8, 33 = 27, 4 3 = 64, etc. 3 Tetrahedra of course take up less room than cubes, and to Bucky, thechoice is clear. We do not, at this point in the text, have thenecessary experience to fully understand the third-power model
shown in Figure 2-2, but the relevant principles will eventually be
discussed. The point to be made now is that squares and cubes
cannot boast a special inherent significance for multiplicative

22

A Fuller Explanation

W

GtE

1

(0

8

4ft

27

64

Fig. 2-2. "Cubing" versus" tetrahedroning."

accounting. Triangles and tetrahedra are equally reliable (and insome ways more reliable, as we shall see). Fuller argues that our
arbitrary habitual references to squares and cubes keep us locked
into a right-angled viewpoint, which obscures our vision of the truth.
From now on, says Bucky, we have to say "triangling" not "squaring" if we want to play the game the way nature plays it.
Visual Literacy

Fuller's concern with fine-tuning communication, developing andusing words that are consistent with scientific reality, is one facet ofthe role of language with respect to synergetics. Another deals withthe difficulty of describing visual and structural patterns. Anyone
who has tried to describe an object over the telephone is well awareof the problems involved; there seems to be a shortage of functional

2. The Irrationality of Pi

23

words. The temptation to use your hands is irresistible, despite thefutility. This scarcity of linguistic aids is especially severe for noncubical structures-which characterize most of nature. That a language of pattern and structure is not widely accessible indicates thathumanity's understanding of such phenomena is similarly underdeveloped. We thus join forces with Fuller in an investigation of thisneglected field, and in so doing we become more and more aware of
the rich complexity of the order inherent in space. Along the way, we
are introduced to some new terminology that includes the lesserknown language of geometry as well as some words invented by
Fuller. More information-useful only, it is clear, if the terminology
is both precise and consistent-leads to better comprehension, which
in turn leads to the ability to experiment knowledgeably. Experiment fosters both greater understanding and invention-in short,
progress.
In conclusion, Fuller's insistence on employing accurate vocabulary is part of an important aspect of human communication. We join
him as pioneers in the science of spatial complexity, the terminologyof which is for the most part unfamiliar. The systematic study ofstructural phenomena is an important and badly neglected aspect ofhuman experience.
Peaceful Coexistence

There is a strong temptation to ignore synergetics on the grounds
that we feel perfectly able to handle mathematical concepts thatcannot be seen. Academic" sophistication" leaves us with a certain
intellectual pride that makes Fuller's observations with their childlike (but-the-emperor-isn't-wearing-any-clothes) ring to them seem
unimportant. Every child is boggled by infinity and surfaces of no
thickness, but these are necessary concepts, natural extensions ofphilosophical "what-ifs." The human mind is not bounded by theconstraints of demonstrability.
True enough. However, it is also possible to define a system ofthought and exploration that is confined to the" facts of experience,"
and moreover such a system is able to reveal additional insights
about physical and metaphysical phenomena that would not necessarily be discovered following the traditional route. Such is the casewith Fuller's synergetics; as we shall see, his hands-on approach led
to a number of impressive geometrical discoveries. Synergetics is adifferent kind of mathematical pursuit, not a replacement for calculus but rather a complementary body of thought. Reading further
will not face you with an ultimate demand for a decision of funda-

24

A Fuller Explanation

mental allegiance: synergetics or the mathematics you learned inschool. Rather, you have the option of being additionally enrichedby a fascinating exploration of structure and pattern that cannot
help but change the way you see the visual environment. When
Bucky reminisces, "There is nothing in my life that equals the senseof ecstasy I have felt in discovering nature's beautiful agreement," heoffers us an enticing invitation. He has taken an alternate route andit has not disappointed him.
Operational mathematics cannot claim exclusive rights to thename of mathematics, any more than other branches of mathematics
can; it is simply a new approach, stemming from the characteristically human drive to experiment. Its claim instead is that exposure tothese concepts fosters an understanding of nature's structuring andtherefore provides an advantageous base of experience. In short,synergetics is an internally consistent system which has produced
significant models with respect to certain physical phenomena andled directly to practical inventions (with "life-support advantage," touse Fuller's terminology). It is likely that thus far we have only
skimmed the surface of the applications of synergetics; very few
people have been exposed to its principles, and so their full significance is as yet untested.
Finally, Bucky's approach is compellingly playful, and we should
read his material with the inclination to enjoy the adventure. "Senseof ecstasy"? Why not? Fuller's unorthodox way of looking at
mathematics-and indeed Universe-can provide a way to circumvent some of our more rigidly held assumptions. Here is aninvitation to start over; with a temporary suspension of disbelief, we
can embrace a new understanding of the exquisitely designed"scenario Universe." One of the challenges of synergetics lies in
opening rusty mental gates that block discovery, for we are asked to
be explorers and "comprehensive thinkers" -job titles not usuallyassigned in our specialized world. The essence of synergetics is"modelability"; anyone can play with these models, and likewise,
claims Fuller, anyone can understand science once they get theirhands on nature's coordinate system. Its accessibility and emphasison experimental involvement makes Fuller's thinking extremely important. He offers us an approach to learning and thinking-an
open-minded, experimental curiosity-which itself is applicable toevery discipline and aspect of life. No question is too simple or too
complex to be asked. Fuller's first words in Synergetics are" Dare to
be naive," reminding us that we have the option to see the world
through new eyes.

3Systems and Synergy

Fuller's mathematical explorations seem to fly out in many directions at once, but they share a common starting point in the conceptof systems. Derived from the Greek for "putting together," the word
system means any group of interrelated elements involved in acollective entity. If that sounds vague, it's meant to. The theme is
widely encompassing.
Long ago, secluded in his room experimenting with toothpicks or
ping-pong balls or whatever available material seemed likely toreveal nature's secrets, Fuller began to see a persistent message ofinterdependence. He was later to discover the precisely descriptive
word" synergy," but even without that lexical advantage a sense ofinteracting parts increasingly dominated his vision. More like thepoets and artists of his generation than the scientists, he was drawn
to relationships rather than objects.
By stating that Fuller looked at systems, we learn very little,
especially in view of the word's current popularity. We have transportation systems and systems analysts, stereo systems and evenskin-care systems, all conspiring to diminish the precision and usefulness of the word. But let us enter into the spirit of Bucky's
half-century search and abandon our twentieth-century sophistication in order to rediscover the obvious en route to the surprising andcomplex. Much can be gained, for alongside our era's growing
consciousness of systems and interdependence is also its ever-increasing specialization. Individuals are encouraged to narrow theirfocus, precluding a comprehensive vision and inhibiting curiosity.
How-things-work questions are reserved for children; as adults weare afraid to step outside our expertise. Furthermore, we are quite
likely to have decided that we have no use for mathematics at all by
the time we reach high school. Both factors-specialization andavoidance of mathematics-cause some aspects of Fuller's synergetics to seem dense while others seem oddly simple. However, thenovelty of his approach serves to give us new insights, so bear with

26

A Fuller Explanation

Bucky as he "discovers the world by himself"; 1 his enthusiasm may
be contagious.
Fuller was unafraid to appear naIve. He announced his observations with equal fervor for the simplest ("only the triangle holds its
shape") and the very complex (the surface angles of the planarrhombic triacontahedron correspond exactly to the central angles of
the icosahedron's fifteen great circles), but on every level he wasconscious of systems.
A system, says Bucky, is a "conceivable entity" dividing Universe
into two parts: the inside and the outside of the system. That's it(except, of course, for the part of Universe doing the dividing; hedemands precision). A system is anything that has "insideness andoutsideness." Is this notion too simple to deserve our further attention? In fact, as is typical of Fuller's experimental procedure, this iswhere the fun starts. We begin with a statement almost absurdly
general, and ask what must necessarily follow. At this point inFuller's lectures the mathematics sneaks in, but in his books the
subject is apt to make a less subtle entrance! (Half-page sentences
sprinkled with polysyllabic words of his own invention have discouraged many a reader.) The math does not have to be intimidating; it's
simply a more precise analysis of our definition of system.
So far a system must have an inside and an outside. That sounds
easy; he means something we can point to. But is that trivial after
all? Let's look at the mathematical words: what are the basic
elements necessary for insideness and outsideness, i.e., the minimum
requirements for existence?
Assuming we can imagine an element that doesn't itself have any
substance (the Greeks' dimensionless "point"), let's begin with twoof them. There now exists a region between the two points-albeit
quite an unmanageable region as it lacks any other boundaries. The
same is true for three points, creating a triangular" betweenness," nomatter how the three are arranged (so long as they are not in astraight line). In mathematics, any three noncollinear points define aplane; they also define a unique circle.
Suddenly with the introduction of a fourth point, we have anentirely new situation. We can put that fourth point anywhere wechoose, except in the same plane as the first three, and we invariably
divide space into two sections: that which is inside the four-point
system and that which is outside. Unwittingly, we have created theminimum system. (Similarly, mathematics requires exactly four noncoplanar points to define a sphere.) Any material can demonstrate3. Systems and Synergy

27

a

d

_ _ _ '~7bcFig. 3-1. Six connections between four events, defining a tetrahedral system.

this procedure-small marshmallows and toothpicks will do thetrick, or pipecleaner segments inserted into plastic straws. The
mathematical statement is unaltered by our choice: a minimum offour corners is required for existence.
What else must be true? Let's look at the connections between thefour corners. Between two points there is only one link; add a third
for a total of three links, inevitably forming a triangle (see if you canmake something else!). Now, bring in a fourth point and count the
number of interconnections. By joining a to b, b to c, c to d, d toa, a to c, and finally b to d (Fig. 3-1), we exhaust all the possibilities
with six connections, or edges in geometrical terminology. Edges joinvertices, and together they generate windows called faces.
This minimum system was given the name tetrahedron (four sides)by the Greeks, after the four triangular faces created by the set offour vertices and their six edges (Fig. 3-1). Fuller deplored the Greek
nomenclature, which refers exclusively to the number of faces- thevery elements that don't exist. ("There are no solids, no continuous
surfaces. . . only energy event complexes [and] relationships." 2)
However, he did not fully develop a satisfactory alternative, so we28

A Fuller Explanation

shall have to work with the time-honored convention. What we losein accurate description of physical reality, we gain in clarity andconsistency.
The tetrahedron shows up frequently in this exploration. This andother recurring patterns seem coincidental or magical at first, but
soon come to be anticipated-endless demonstrations of the orderinherent in space. The process is typical of synergetics: we stumble
into the tetrahedron by asking the most elementary question-whatis the simplest way to enclose space?-and later, everywhere we
look, there it is again, an inescapable consequence of a spectrum ofgeometric procedures.
The straightforward logic of our first encounter with the tetrahedron drives us to wonder if it displays any other unique properties.
It turns out to be a reliable sort of minimum module or "quantum,"as Fuller points out in myriad ways. Not the least impressive
involves counting the edges of all regular, semiregular, and triangulated geodesic polyhedra (from the simple cube to the more complex
rhombic dodecahedron to the vast array of geodesics.) The resulting
numbers are all multiples of the tetrahedron's six. We can therefore
take apart any polyhedral system in these categories and reassemble
its edges into some number of complete tetrahedra. Even though weare not yet familiar with these other polyhedra, the observation
stands as a representative example of the surprising whole-number
relationships which make our investigation increasingly alluring.
Conceptual and Real Systems

Notice that these geometrical systems are purely conceptual: so far
they exist only in our mind, as sets of relationships. They can be lent
substance by any number of materials, as for example the toothpicks
and marshmallows mentioned above. However, the essence of asystem is independent of the choice of materials: six sticks will
create a tetrahedron whether we use wood or metal. Similarly, four
Ping Pong balls or four people constitute a tetrahedral system. The
tetrahedron, being a conceptual entity, is "sizeless and timeless."
Thus Fuller writes in Synergetics, "Size is always a special-caseexperience." (515.14) Size belongs to a different category of parameters than vertices, edges and faces- those which only relate to actual
constructs, such as color, temperature, and duration.
Does he take this concern with terminology too far? His justification is twofold, encompassing first a deep conviction that words
influence the shape of our thinking, and secondly faith in the power3. Systems and Synergy

29of accurate models in problem-solving. For humanity to solve its
complex problems, he was convinced that vocabulary and othermodels had to be absolutely precise. So Fuller's concern that we
recognize conceptual systems as sizeless sets of relationships capableof being physically embodied is an essential part of his geometry of
thinking. In either form, the emphasis is on the relationships.
We begin to see a basis for the phenomenon of vastly different
properties exhibited by systems with identical constituents. One
notable example is the soft grey graphite of pencils in contrast to
sparkling impenetrable diamonds, both consisting exclusively ofcarbon atoms. Geometry alone accounts for their differences. We
shall see how later, but as always our attention will be on shape andvalency (numbers of connections) rather than substance.
Let's get back to our starting point. Any subdivision of Universe
constitutes a system. We have found the simplest example andlearned the mathematical terms; our next step is to look at more
complex systems. It does not stretch our definition to discuss somevery elaborate forms, such as a school, or even a crocodile (bothhave the requisite boundary). For that matter, our entire planet is asystem-unimaginably intricate but still finite. This line of thought,
together with our· geometry lesson, suggests an approach toproblem-solving: a "whole systems" view that demands consideration of the influence of every move on its entire system. Such an
approach, which prohibits short-term or piecemeal solutions tolong-term problems, may sound simplistic or vague at first; however,
the method is based on the assumption of rigorous analytical procedures.
In Synergetics, Fuller introduces the concept with a deliberately
simple example, which provides an analogy for more complex situations. A fictitious child draws on the ground with a stick, announcingthat he has made a triangle. Then Bucky himself intervenes to pointout to the child that he has created four triangles- not justone-because "operational mathematics" requires that a triangle
must be inscribed on something in order to exist. Whether on a pieceof paper or on the surface of the earth, that something is always a. system, with an inside and an outside. Unwittingly, the child has
divided the earth's surface into two areas. Both regions are boundedby three arcs, and therefore both qualify as spherical triangles, 3
despite the fact that one is small and tangible and the other covers
most of the earth's surface. We are not used to thinking in theseterms, philosophizes Fuller, but we must begin to really think about
what we're doing.

30

A Fuller Explanation

Hold on, says the child, that's only two triangles! Why did you say
there were four? Well, Bucky continues, concave and convex 3 are not
the same; when you delineated two concave triangles on the outside
surface, you also created two convex spherical triangles-one verysmall and the other very large-on the inside. But I didn't mean tomake four triangles, protests the bewildered child. That doesn't
matter, his teacher replies; you are still responsible for them.
His story can be considered a parable; its purpose is as much toencourage a sort of holistic morality as to make a mathematicalstatement. The message: tunnel vision is obsolete. As human beings,
we cannot afford to ignore the effect of our actions on the rest of asystem while working on an isolated part. Rather we must become
responsible for whole systems. We didn't mean to make four triangles-or indeed, to make "the big mess of pollution" (814.01).
As playful as this example seems, it calls our attention to what
Fuller perceives as a dangerous "bias on one side of the line"
inherent in traditional mathematics. He points out that our gradeschool geometry lessons involve concepts defined as bounded bycertain lines-a triangle is an area bounded by three lines, forexample-thus excusing us from paying attention to its environment. Once a figure is delineated, we no longer have to consider the
rest of the system. This narrow approach, Fuller argues, instills in us
at an impressionable age a deep bias toward our side of the line; we
see and feel an unshakable correctness about our side's way of"carrying on." On the other hand, "Operational geometry invalidates all bias" (811.04). It forces us to remain aware of all sides. In
Fuller's opinion, being taught in the first place that all four triangles
are "equally valid" would significantly influence our later thinking
and planning.
One consequence of this approach is Fuller's realization that
"unity is inherently plural" (400.08). "Oneness" is impossible, he
explains, for any identifiable system divides Universe into two parts,
and requires a minimum of six relationships to do so. Furthermore,
as illustrated in the above parable, all "operations" produce aplurality of experiences, and awareness itself-without which there
can be no life-implies the existence of "otherness." Ergo, "Unity isplural and at minimum twO.,,4
Limits of Resolution as Part of the Whole-Systems Approach

Another important aspect of Fuller's systems concept is tune-in-ability, which deals with limits of resolution and is best explained byanalogy. Fuller's ready example, as implied by the term, would be to3. Systems and Synergy

31

remind us of the radio waves of all different amplitude and frequency,
filling the room wherever you happen to be reading this page. Thesewaves are as much a part of physical reality as the chair you are
sitting in, but the specific energy pattern is such that you cannot tune
in to the programs without help from a radio. Information andenergy are scattered chaotically throughout your room, mostly undetected, except for the small fraction (chairs, visible light, and so on)that can be directly perceived by human senses. You can turn on theradio and thereby tune in to one program (one system), temporarily
ignoring the rest.
Boundaries change all the time as new elements are incorporated
into a system, or as the focus zooms in to investigate a component ingreater detail. New levels of complexity reveal distinct new systems.
For example, we might look at the system called your living room,and then want to consider its function in the bigger system, yourhouse, or conversely, zoom in to investigate the red overstuffed chair,and the details of its carpentry and upholstery; or further still, one
nail might be of interest as a system. We can also go back out-to
your town, your state, etc. The concept of tune-in-ability allows us totreat a set of events or items as a system despite the involvement ofmany concurrent factors on other levels.
What kinds of things constitute systems? Tetrahedron, crocodile,room, chair, you, thought, .... Wait.
What about thoughts? We recall Fuller's lifelong effort never touse mankind's precious tool of language carelessly: "I discipline
myself to define every word I use; else I must give it up." In a 1975
videotaped lecture, he explains that he would not allow himself theuse of any word for which he did not have "a clear experientiallyreferenced definition."4 Such an effort requires enormous discipline
to avoid automatic associations and thereby enable an objective
analysis of each word. It extends to the most basic words and actions-even" thinking." Fuller formulates his definition analytically, asking, "What is it I am conscious of doing, when I say I am thinking"?
We may not be able to say what it is, but we should be able tospecify the procedure.
Thinking, he explains, starts with "spontaneous preoccupation";the process is never deliberate initially. We then choose to "accommodate the trend," through conscious dismissal of "irrelevancies"
which are temporarily held off to the side, as they do not seem tobelong in the current thought. Fuller places "irrelevancies" in two
categories: experiences too large or too infrequent to influence thetuned-in thought, and those too small and too frequent to play apart. The process he describes is similar to tuning a radio, with its

32

A Fuller Explanation

progressive dismissal of irrelevant (other-frequency) events, ultimately leaving only the few experiences which are "lucidly relevant,"
and thus interconnected by their relationships.
Thinking isolates events; "understanding" then interconnects
them. "Understanding is structure," Fuller declares, for it means
establishing the relationships between events.
A "thought" is then a "relevant set," or a "considerable set":
experiences related to each other in some way. All the rest ofexperience is outside the set-not tuned in. A thought therefore
defines an insideness and an outsideness; it is a "conceptual subdivision of Universe." "I'll call it a system," declares Bucky; "I now
have a geometric description of a thought."
This is the conclusion that initially led Fuller to wonder how many"events" were necessary to create insideness and outsideness. Realizing that a thought required at least enough "somethings" to define
an isolated system, it seemed vitally important to know the minimum number-the terminal condition. He thereby arrived at the
tetrahedron. "This gave me great power of definition," he recalls,
both in terms of understanding more about "thinking" and by
isolating the theoretical minimum case, with its four events and six
relationships.
One example of the development of a thought-by no means aminimal thought-could be found in what to cook for dinner.
Walking to the grocery store, you notice that the leaves of the maple
trees are turning autumn-red, but you consciously push that observation off to the side to be considered later, as it does not relate to thepressing issue of dinner. You begin to pull in the various relevant
items: the food that you already have at home that could become apart of this meal, what you had for dinner last night, special items
that might be featured by the grocery store, favorite foods, how theylook, ideas about nutrition, certain foods that go well together, andso on. Out of this jumble of related events, a structure starts to takeform. After a while, dinner is planned, and your mind is free toattend to some of the other thoughts waiting quietly in the sidechambers.
This kind of digression is typical of Fuller's discourse, both
written and oral. Such juxtapositions of geometry and philosophy
are quite deliberate, for synergetics strives to identify structuralsimilarities among phenomena-both physical and metaphysical.
Fuller encourages us to seek these patterns, which we often missbecause of the narrow focus of our attention.
To conclude: Geometry is the science of systems-which arethemselves defined by relationships. (Geometry is therefore the study3. Systems and Synergy

33of relationships; this makes it sound relevant to quite a lot!) A
system is necessarily polyhedral; as a finite aggregate of interrelated
events, it has all the qualifications. Relationships can be polyhedrally
diagrammed in an effort to understand the behavior of a given wholesystem. Along these same lines, Fuller has described synergetics as
the "exploratory strategy of starting with the whole and the known
behavior of some of its parts and the progressive discovery of theintegral unknowns along with the progressive comprehension of thehierarchy of generalized principles" (152.00). This mouthful canreadily be identified as Fuller's elaboration of Eddington's definitionof science as "the systematic attempt to set in order the facts ofexperience." 5Thinking in terms of systems is a crucial part of Fuller's mathematics. The isolation of systems enables the description of local
processes and relationships without reference to an absolute
origin-an indispensable tool in a scenario universe. And finally, we
pay particular attention to how Fuller's geometry emerges-its
principles developing from the basis of the process of thinking.
Hence the title of Fuller's opus: Synergetics: The Geometry of Thinking.
Synergy

Implicit in the above discussion of systems is a property describedaccurately by only one word. "Synergy" has come into fairly
widespread use recently, perhaps due to Bucky's many years ofchampioning its cause, or perhaps just because we finally needed it
badly enough. Formerly unknown except to biologists and chemists,
this word describes the extraordinarily important property that" thewhole is more than the sum of its parts." In Fuller's words, "Synergymeans the behavior of whole systems unpredicted by the behavior of
their parts taken separately."
Consider the phenomenon of gravity. The most thorough examination of any object (from pebbles to planets) by itself will notpredict the surprising behavior of the attractive force between two
objects, in direct proportion to the product of their masses andchanging inversely with the square of the distance between them.
Another dramatic example is the combination of an explosive metal
and a poisonous gas to produce a harmless white powder that wesprinkle on our food-sodium chloride, or table salt.
Bucky's favorite illustration was the behavior of alloys: "synergy
alone explains metals increasing their strength" (109.01). He en-

34

A Fuller Explanation

thusiastically describes the properties of chrome-nickel steel, whose
extraordinary strength at high temperatures enabled the development of the jet engine. Its primary constituents-iron, chromium,and nickel-have tensile strengths of 60,000, 70,000, and 80,000
pounds per square inch respectively, and combine to create an alloy
with 350,000 psi tensile strength. Not only does the chain far exceed
the strength of its weakest link, but counter-intuitively even outperforms the sum of its components' tensile capabilities. Thus the chain
analogy falls through, calling for a new methodology which will
incorporate interaugmentation.
A flood of examples of "synergy" is so readily available that one
might wonder how we got along without the word. Bucky wondered
and concluded that humanity must be out of touch with its environment. Synergy is certainly how nature works; though we pay little
direct attention to the phenomenon, we are still familiar with it. Few
are surprised by complex systems arising out of the interaction of
simpler parts.
Fuller took it a step further. He saw the age-old forms of geometry
as models of synergy, comprehensible only in terms of relationships.
His eye drawn to their vector edges, he simply did not perceive the"solid" polyhedra of Plato. Self-exiled from the formal mathematicalcommunity which would have told him otherwise, Fuller saw thestatic constructs of geometry as ready and waiting to elucidate thedynamic events of physical Universe.
Determined to model the new "invisible" energetic reality, Fuller
began to refer to his accumulated findings as "energetic geometry."
As the search for "nature's coordinate system" progressed and therecurring theme of synergy became more and more prevalent, theterm evolved to "synergetic-energetic geometry" and finally to "synergetics." Fuller's vocabulary tended to develop organically in response to his changing needs for emphasis. He felt a great responsibility to get it just right. In the thirties, enchanted by certain
properties of the cub octahedron, Fuller replaced the Greek namewith his own trademark word, "Dymaxion," less from egotism thanfrom frustration in being unable to invent exactly the rightname-one with enough impact. 6 Later, he found the perfect term toexpress its unique property, and eagerly renamed this indispensible
shape "vector equilibrium." The name remained unaltered; when
Fuller found his truth, he never wavered.
The term "synergetics," then, was a response to the single mostimportant characteristic of energetic reality. As discussed in Chapter
1, Fuller's overriding goal was to collect the "generalized principles."3. Systems and Synergy

35

The law of synergy, although too all-encompassing to seem a valid
starting point for such an inventory, dictates a basic strategy ofstarting with a whole system and then investigating its parts. The
most painstaking study of its separate components will never reveal
the behavior of a system. All other generalized principles therefore
must be subsets of this fundamental truth: the whole is not equal to
the sum of its isolated parts.
N ow we take some time out to look at aspects of conventional
geometry that will illuminate Fuller's work, despite the fact that it is
not directly included in Synergetics.

4
Tools of the Trade

Whether or not the thought of high-school geometry class stirs
unpleasant memories, chances are that most of the actual material islong forgotten. Moreover, few of geometry's more pleasing properties are taught, leaving volumes of elegant transformations in therealm of esoteric knowledge. Lack of exposure to these age-old
discoveries is the primary barrier to understanding synergetics.
By now familiar with Fuller's underlying assumptions, we shalltake time out to introduce some background material. The origins ofhumanity's fascination with geometry can be traced back four thousand years, to the Babylonian and Egyptian civilizations; two millennia later, geometry flourished in ancient Greece, and its development
continues today. Yet most of us know almost nothing about theaccumulated findings of this long search. Familiarity with some ofthese geometric shapes and transformations will ease the rest of thejourney into the intricacies of synergetics.
A little experimentation with basic geometric forms and procedures reveals the important role of space itself. The work of otherthinkers reinforces the fundamental premise of synergetics: space has
shape-all structures are formed according to spatial symmetries andconstraints. It turns out that the number of symmetrical arrangements allowed by space is surprisingly limited; perpetual (synergetic)
interaction of relatively few patterns accounts for the seemingly
endless variety of form.
Even for readers whose background in geometry is already strong,reviewing it can be enjoyable and perhaps even illuminating. There
are so many significant connections among polyhedral shapes andoperations that even experienced geometers continually discover new
ones.
Bucky's delight in a new-found truth never lost its intensity. He
promptly adopted each discovery into his growing synergetics inventory. If, in his enthusiasm, he appeared to be taking credit for4. Tools of the Trade37

age-old discoveries, let us-rather than judging-try to enter into
the spirit of his search. And if egotism seemed to have gone hand inhand with enthusiasm, it is because both grew out of his constantwillingness to see everything as if for the first time. Both were part of
the whole system Bucky.
Our most rewarding course is to immerse ourselves in the geometry. We shall first set the stage with some of the Greek basics, andthen move on to the work of Arthur L. Loeb of Harvard University- whose "Contribution to Synergetics" 1 provides an analyticalcounterpart to the 800 pages of Fuller's more intUltlve
approach-before moving on to the thought-provoking twists ofFuller's mathematical thinking.
As this thinking is encompassing and holistic, it would be counterproductive to scrutinize isolated parts of Fuller's geometry. Synergetics must be critically examined as a whole system before judging its
contribution to mathematics. The emphasis (and certainly the excitement) in Fuller's ongoing research was in the inherent "omni-interaccommodation" -that is, uncovered principles are never contradictory but rather augment each others' significance. Fuller's developing
inventory was thus continually strengthened, becoming more andmore integral to his thinking. He was especially gratified, in his later
years, to receive numerous letters from noted scientists, reinforcing
his discoveries with their observations from research in other fields.
Bucky's guiding purpose in developing synergetics was to reacquaint us with Universe. Such eagerness is compelling. ("Beautiful, beautiful bubbles ... ," "eternally regenerative Universe,"" ... ecstasy in discovering nature's beautiful agreement.") We do
him and ourselves a disservice to expect his approach to fit comfortably into the traditional scholarly framework. Unacknowledged byacademic institutions for most of his life, he felt unencumbered bytheir conventions. (But we are not, and this volume will attempt tolocate sources wherever possible.)
Onward! Let's take another look at geometry; the journey itself isfascinating, as well as full of useful tools for understanding Fuller's
work.
Plato's Discovery

The Platonic polyhedra, as their name suggests, have been aroundfor almost two millenia. However, despite the fact that the five
shapes (also called regular polyhedra) are well known, few people are

38

A Fuller Explanation

aware of what exactly defines this group-and fewer still of theimplications about space itself.
The requirements seem lenient at first. They are two: the faces of apolyhedron must be identical, and the same number of them mustmeet at each vertex. The tetrahedron fits, with its four triangles andfour equivalent (three-valent) vertices. 2 (Notice the roots of the wordequivalent.) From these two criteria alone, one might suspect theexistence of many more regular polyhedra.
Triangles

Let's study the possibilities step by step, beginning with the simplest
polygon (fewest sides) and the smallest number of edges meeting at
each vertex. In keeping with Fuller's use of vectors as edges, thisstudy will be confined to "straight" edges. It is quickly apparent thata minimum of three edges must meet at each vertex of a polyhedron,for if vertices join only two straight edges, the resulting array isnecessarily planar. This lower limit can also be expressed in terms offaces, for we can readily visualize that a corner needs at least threepolygons in order to hold water-which is another way of saying theinside is separated from the outside, as specified by Fuller's definition of system.
The minimal polygon is a triangle. Three triangles around one
vertex form a pyramid, the base of which automatically creates afourth triangular face. As all corners and all faces are identical, thefirst regular polyhedron-a tetrahedron-is completed after one step
(Fig. 4-1a).
N ext, a second regular polyhedron can be started by surrounding
one vertex with four triangles, resulting in the traditional square-

(a)

(e)
(b)

Fig. 4-1

4. Tools of the Trade

39

(a)

(b)

/\ - -- -7',
I

I

\

I

\

/\--~-~/\\

I\'---- -

(c)Fig. 4-2

based pyramid (Fig. 4-1b). But, as our specifications for regularity
indicate that all vertices must connect four edges, an additional edge
is required at each of the four vertices around the pyramid's base,bringing the total to twelve edges. By connecting the four danglingedges (Fig. 4-1c), we introduce a sixth vertex, which is also surrounded by four triangles. The result is an octahedron, with eight
triangular faces and six four-valent vertices. Both criteria for regularpolyhedra are satisfied, and so the octahedron is added to our list.
As the procedure is thus far simple and successful, analogy suggests the next step. Five triangles around one vertex form a shallow
pyramid (Fig. 4-2a). Paying attention as before to nothing but thetwo rules, we continue to employ triangular faces while making surethat five of them surround each corner. The structure essentiallybuilds itself in that there is only one possible outcome-and we
don't even have to know what it is to be able to finish the task. The
icosahedron, with twenty (in Greek, "icosa") triangles and twelve
five-valent corners, is indeed regular (Fig. 4-2b).
Once more then. Bring six triangles together at a corner. But wait!
Six 60-degree angles add up to 360 degrees, or the whole plane (Fig.

40

A Fuller Explanation
Fig. 4-34-2c). An unprecedented result, this indicates that we could surroundvertices with six equilateral triangles indefinitely and never force thecollection to curve around to close itself off. A space-enclosing
system is therefore unattainable with exclusively six-valent vertices.
Is it possible we have exhausted the possibilities for regular polyhedra out of triangles? We simply cannot have fewer than three ormore than five triangles around all vertices and create a closed finite
system. We thus encounter a first upper limit.
SquaresLet's back up and start the procedure over, with another kind ofpolygon. Continuing the step-by-step approach, we change fromtriangles to squares by increasing the number of sides by one. The
method is reliable if plodding. What happens if two squares cometogether? Again, that's just a hinge (Fig. 4-3). So start with three. If
there are three squares around one corner, the same must be true forall corners, and as before, the structure is self-determining. Continue
to join three squares at available vertices until the system closes itself
off. With six squares and eight corners, this is the most familiar
shape in our developing family of regular polyhedra-the ubiquitous
cube.
Next, we gather four squares at a vertex and immediately hit
ground. Four times 90 degrees is 360 degrees, the whole plane again.
And while such groups of four will generate graph paper indefinitely,
they can never close off as a system. So that's it for squares.

4. Tools of the Trade41

Fig. 4-4

Pentagons

Triangles, squares, ... , now we come to pentagons. Three of themaround each corner works. A system exists as soon as there are
twelve pentagons and twenty three-valent corners. Called the pentagonal dodecahedron ("dodeca" is Greek for twelve), it adheres tothe definition of regular polyhedra (Fig. 4-4).
Try four around a corner. Another new situation. The interior
angle of a regular pentagon measures 108 degrees (see Appendix A);
four of them together add up to 432 degrees, which is more than theplanar 360. This indicates that four pentagons simply will not fit
around one point. Not only have we reached a stopping point for
regular polyhedra out of pentagons, but this example also shows that
not all regular polygons can be made to fill a page (or tile a floor).
Specific spatial constraints apply in two dimensions as well as inthree. We shall investigate these patterns in more detail in Chapter
12.
Having exhausted the pentagonal possibilities, we go on to hexagons. Three of their 120-degree angles total 360 degrees, and are
therefore planar right away-creating the hexagonal pattern seen inhoneycomb and frequently used for bathroom floor tiles (Fig. 4-5).
Three heptagons with angles totalling 385.71 degrees just won't fit
together. Neither will octagons, nor any polygons with more vertices.
A limited Family

A quick review reveals the surprisingly limited inventory of fiveregular polyhedra: tetrahedron, octahedron, icosahedron, cube, andpentagonal dodecahedron (Fig. 4-6). Like it or not, we have reached
the end. A child in kindergarten, with the two rules carefully
explained, will discover the same five shapes. Space takes over,
imposing that upper limit. This idea runs counter to the bias of ourmathematical background that space is passive emptiness and we42

A Fuller Explanation
Fig. 4-5

impose desired configurations. Invisible, unyielding constraints soundmore like mysticism than science.
If we dwell on this subject, that is because it is crucial tounderstanding the framework of Bucky's investigation. Space has
specific characteristics, and we want not only to list and understandthem, but also to begin to really feel their embracing qualities-a
sense of structured space permeating all experience.
The regular polyhedra provide a good starting point from which tobranr;h out in all directions. An eighteenth-century mathematician,Leonhard Euler (1707-1783), greatly simplified our task with his~

Fig. 4-6. Fi" "gulac polyhoo".

4. Tools of the Trade43

realization that all patterns can be broken down into three elements:
crossings, lines, and open areas. He thereby introduced the basic
elements of structure (vertices, edges, and faces) which underlie allgeometrical analysis. Bucky saw this contribution as a breakthroughof equal importance to the law for which Euler is known, for this
precise identification of terms enabled Euler's other, more famous
observation.

Euler's law

Euler's law states that the number of vertices plus the number offaces in every system (remember Fuller's definition) will always
equal the number of edges plus two. It may not sound like much at
first, until you reflect on the variety of structures-from the tetrahedron to the aforementioned crocodile-that all obey this simple
statement. Every system shares this fundamental relationship. The
number of vertices can be precisely determined by knowing thenumber of faces and edges, and so on.
Denote the numbers of vertices, faces, and edges by V, F, and E
respectively. Then we have V + F = E + 2. What about that constant 2? The other numbers might be extremely large, or as small as
four, yet by Euler's equation the difference between the number ofedges and the sum of the number of vertices and faces will always be
exactly two. It seems unlikely at first.
To gain confidence in this principle, let's try it out on the regularpolyhedra. Remember the four vertices and four faces of the tetrahedron; four plus four is eight, exactly two more than its six edges.
Not bad so far. Similarly, we can count to check the other four
regular polyhedra. The results are displayed in Table I.
The persistent 2 has led to some controversy. Fuller long ago
assigned his own meaning to the recurring number: 2 occurs in the
Table I
Tetrahedron
Octahedron
Cube
Icosahedron
Pentagonal
dodecahedron

E

2

Total

2
2
2
2

8
14
14
32

20 + 12 = 30 + 2

32

V

F

4+ 4= 6+
6 + 8 = 12 +8+ 6 = 12 +12 + 20 = 30 +

44

A Fuller Explanation

equation to represent the "poles of spinnability." That requires someclarification.
All systems, Fuller explains, can be spun about a central axis. An
axis has two poles (e.g., north and south), and thus at any given time,
two vertices must be poles. Subtract the two poles from the total
number of vertices to get a number exactly equal to the combined
number of edges and faces. Simply stated, in Fuller's view, thepermanent 2 represents two vertices which, as "poles of spinnability,"should be subtracted from the total number of vertices to equalize
the equation. It's a puzzling explanation.
There is another way to view that constant element, as you may
have guessed. What are the characteristics of the variables includedin Euler's Law? Zero-dimensional points, one-dimensional lines, andtwo-dimensional areas-each a level higher than the last. The law
compares all aspects of structure-almost. Something's missing.
Three-dimensional space is the next and only absent parameter; its
geometrical units (corresponding to vertices, edges, and faces) arecells. Why are cells left out of this fundamental relationship? The
other view, as expounded by Loeb, says they're not. 3
Recall Bucky's definition of a system: a subdivision of spacecreating an inside and an outside, both equally important. Two cells!
Could the constant 2 in the equation be incorporating the otherwise
only missing dimension? Indeed it is. Euler's law is actually a special
case of Schlaefli's formula for any number of cells. In other words, if
the number of cells, C, is substituted for 2, the equation holds true
for multicellular structures, that is, arrangements with more than two
cells: V + F = E + C, even when C is greater than 2.4
Evaluation of significance is a tricky business, but we cannot avoid
indulging in it altogether, as Fuller's Synergetics overflows with such
speculation. It is ultimately puzzling that Fuller, with his emphatic
observation that every polyhedron is a system dividing Universe into
two parts (inside and outside), would not connect Euler's constant 2
with the implied two cells. His insistence that both parts of a system
must be considered equally important provides a truly new orientation in geometry.
At some point in any discussion about Euler and the" polar two,"
Fuller would speak of a structural system's inherent "constantrelative abundance." The meaning of the term eluded many. Fuller
observes in Synergetics that the number of faces in triangulated
systems is always two times the number of vertices minus two (thesubtraction, he says, again taking account of the two poles).5 He

4. Tools of the Trade45

further states that the number of edges is three times the number ofvertices less two.
Turning these two statements into simple equations, we have
F = 2(V - 2)

andE = 3(V - 2).

Simplifying,F = 2V - 4,
E = 3V - 6,
F + 4 = 2V,
E + 6 = 3V.

Combining the two equations by subtraction, we haveE + 6 = 3V-(F + 4 = 2V)
E - F + 2 = V,or
E+2=V+F.
So his observations directly substantiate Euler's law. Constant relative abundance refers to the everpresent two faces and three edges
for each vertex in triangulated polyhedra, (excepting of course thetwo" poles," which are not included, according to Fuller's rationale).

Duality
Table I reveals a curious pattern. Notice the relationship between
cube and octahedron, along with the similar pairing of pentagonal
dodecahedron and icosahedron. Polyhedra thus related, each withthe same number of vertices as the other has faces, are called eachother's dual. 6 We are thereby introduced to duality as a numerical
relationship; the vertex and face tallies are simply switched. Reassuringly, the significance extends: Loeb observes the geometric manifestation of duality in precise matching of vertex to face. Two dual
polyhedra line up with every corner of each meeting the center of awindow of the other, as the correspondence implies. (Figure 4-7ashows the dual relationship of the cube and octahedron.)
We have conspicuously ignored one member of Plato's polyhedralfamily. What is the tetrahedron's dual? Following Loeb's example,
we count elements to predict the answer, and find the same numberof vertices as faces. Therefore, by interchanging the two elements tofind the tetrahedron's dual, we generate another tetrahedron (Fig.

46

A Fuller Explanation
Fig. 4-7. (a), (b) Dual polyhedra. (c) Two
tetrahedra intersect to form the eight
vertices of a cube: "star tetrahedron."

(a)

(b)

(c)

4-7b). Once again, the minimum system stands out: only the tetrahedron is its own dual. Put two of them together to check: the four
windows and corners line up, and curiously, the combined eight
vertices of two same-size tetrahedra outline the corners of a cube, as
Fuller never tired of explaining. ["Two equal tetrahedra (positive
and negative) joined at their common centers define the cube"(462.00, figure; Fig.4-6c).]

Truncation and Stellation

The introduction of two operations explored by Arthur Loeb willfurther elucidate the relationships between polyhedra. By alteringand combining the five regular polyhedra, we can generate newshapes-one of the geometric phenomena that inspired Fuller's
coinage "intertransformabilities." Loeb's observations will help toclarify the meaning of this polysyllabic Fullerism.
"Truncation" involves chopping off corners so that they are
replaced by surfaces (Fig. 4-8a). 7 Truncation of a three-valent vertex
will generate a triangle; four-valent vertices become squares, and soon. The number of edges determines the number of sides of the newpolygon. Notice that the definition does not specify how much of thecorner is sliced away in truncation. Loeb's work reinforces our4. Tools of the Trade

(a)

(b)

(c)

47

(d)

Fig. 4-8. Degenerate truncation of a tetrahedron.

emphasis thus far on topology (studying numbers of elements, or
valency) rather than size. The location of slicing is therefore unimportant, until the truncation planes move inward far enough to toucheach other. At that point the edges between the new planes disappear, and so the topology changes. Figure 4-8 shows various possibilities, including that final chop at the mid-edge point. This special
limit case, called by Loeb degenerate, yields some interesting results,
as we shall see. (For example, the "degenerate truncation" of atetrahedron unexpectedly turns out to be another member of the
Platonic Family, the octahedron, as revealed by Fig. 4-8d.)
Readers interested in learning more about these concepts and themathematical analyses involved should read Space Structures by
Arthur Loeb. Although we introduce only "vertex truncation,"
Loeb's studies extend to "edge truncation" as well. For our purposeof becoming familiar with the interconnectedness of basic polyhedra,
we explore just one of Loeb's discoveries in the following pages.
Then when we see these shapes again in the context of synergetics,
some of their important relationships can be anticipated.
Whereas truncation cuts off corners, "stellation" is accomplishedby the addition of a corner-imposing a shallow pyramid on aformerly flat face. How many sides the pyramid will have is de-

Fig. 4-9. Stellated cube and octahedron.

48

A Fuller Explanation

termined by the number of edges of the face to be stella ted. Figure
4-9 compares a stellated cube and a stellated octahedron. Notice that-as required by their different faces-the cube's superimposed
pyramids have four sides, while the octahedron's have three. Degenerate stellation occurs when the altitude (height) of the added pyramids is such that adjacent faces of neighboring pyramids become
coplanar, as will be illustrated below.
An Experi ment

What happens if we truncate both members of a pair of dual
polyhedra? The octahedron and cube provide a representative pairfor Loeb's elegant experiment. Chop off the corners of the cube,
creating eight new triangle faces, while changing the six squares into
octagons (Fig. 4-10a). The truncated octahedron (also called tetra-

~

(b)

(a)

(d)
(e)

Fig. 4-10. (a), (b) Truncation of cube and octahedron. (c), (d) Degenerate truncationof cube produces the same polyhedron as degenerate truncation of octahedron:cub octahedron.

4. Tools of the Trade

49

kaidecahedron for its fourteen faces) (Fig. 4-10b) is a space-fillingshape-a subject that we shall explore more fully in Chapter 12.
Now, notice what happens if we allow all truncation planes toexpand. The truncated cube's triangles and octahedron's squares
independently spread to the "degenerate case," with truncation
planes meeting at mid-edge points (Fig. 4-10c, d). Suddenly octagons
and hexagons are phased out, becoming squares and triangles respectively. The two different systems turn into the same polyhedron.
Conventionally called the cuboctahedron for reasons now apparent(and renamed" vector equilibrium" by Fuller, for a reason exploredin Chapter 7), this polyhedron plays a crucial role in Synergetics. In
the present context it merely provides an exemplary illustration of
the interactions of duality and truncation.
Loeb's further investigation reveals that degenerate truncation ofany dual pair leads to the same polyhedron. For instance, degenerate
truncation of both the pentagonal dodecahedron and the icosahedroncreates the icosadodecahedron, named for its twenty triangles andtwelve pentagons. The pentagons, created by slicing the twelve
five-valent vertices, alternate with the twenty icosahedral triangles.
Looked at another way, twenty triangles have resulted from chopping off all the three-valent vertices of the pentagonal dodecahedron(Fig. 4-11). With either outlook, we begin to see the effect of duality.
(Take special note of the system's twelve fivefold elements; theirpresence will soon be interpreted as a fundamental law under certain
conditions.)
We now need a new category as we uncover new polyhedra thatare not regular, but certainly far from random or irregular. The
cub octahedron and the icosadodecahedron are alike in having only
one kind of vertex but two different kinds of faces. Such polyhedra
are called semiregular. It will not come as a surprise that their duals
have one kind of face and two kinds of vertices. You can begin to

Fig. 4-11. Icosadodecahedron.

50

A Fuller Explanation

Fig. 4-12. Degenerate stellation of cube produces the same polyhedron as degeneratestellation of octahedron: rhombic dodecahedron.

imagine the potential for generating new systems, as reflected in theterm "intertransformabilities."
The elegant results of degenerately truncating dual polyhedra
inspire further questions. Does it follow that degenerate stellation of
a dual pair will create the same polyhedron? And if so, how do we
define degenerate stellation? Experimentation answers both questions.
To check the hypothesis with our reliable octahedron-cube pair,
we stellate both systems independently. Try to imagine the transition: six shallow square pyramids are superimposed on the cube,
while eight triangular pyramids are added to the octahedron, as seenin Figure 4-9. We then increase the altitude of all pyramids, until
triangles of adjacent pyramids just become coplanar. In both cases,
twenty-four individual triangular facets suddenly merge into twelve
rhombic (or diamond) shapes, thereby creating the rhombic dodecahedron, named for its twelve rhombic faces. The original edges of the
cube form the short diagonals of the twelve faces, while the octahedral edges turn into the twelve long diagonals (Fig. 4-12). We
thereby have illustrated degenerate stellation.
Notice that both members of the original dual pair have the same
number of edges. This turns out to be a necessary condition ofduality, which follows logically from the nature of the geometriccorrespondence. In order for each vertex to line up with a face, theedges of two dual polyhedra must cross each other, as can be seen in4. Tools of the Trade51

Figure 4-6. We further observe that whereas degenerate truncation
produced quadrivalent vertices (cuboctahedron), degenerate stellation produced quadrilateral faces (rhombic dodecahedron). A generalized principle emerges.
"I ntertransformabil ity"

As Fuller so often observed, nature consists exclusively of endlesslytransforming energy. Atoms gather in "high-frequency" clusters,
disassociate periodically, and regroup elsewhere-new patterns, different substances. We interpret these transient events as solids andliquids because of the limitations of our five senses. To Fuller,"exquisitely transformable" polyhedra were highly logical models
with which to elucidate nature's behavior. "Intertransformability"
thus applies to both nature and her models.
We now consider a final experiment from Loeb's research, to bring
the subject to a close. Start by truncating a representative polyhedron such as the cube. Continue to the degenerate polyhedron (inthis case the cub octahedron, as illustrated in Fig. 4-10c), and put it
aside for a moment. Now take that same cube, the degeneratestellation of which yields the rhombic dodecahedron (as seen in Fig.
4-12) with its twelve rhombic faces.
Twelve identical faces? Have we accidentally overlooked a regularpolyhedron? No, for the other requirement, of identical vertices, isnot satisfied. The diamonds' obtuse angles 8 come together at eight
three-valent vertices, and their acute angles 8 at six four-valentvertices. (Refer to Fig. 4-12.)
Those numbers are familiar. Twelve identical faces, along with six
four-valent and eight three-valent vertices, correspond to the twelve
identical vertices, six four-valent faces (squares), and eight three-valent faces (triangles) of the cub octahedron (Fig. 4-13). Loeb thus

Fig. 4-13. Cuboctahedron andcahedron are dual polyhedra.

rhombic dode-

52

A Fuller Explanationshows that duality extends to semiregular polyhedra. This fact would
have enabled us to predict the rhombic dodecahedron's existence, byspecifying twelve similar quadrilateral faces to correspond to twelve
four-valent vertices, and so on. Now we can examine the results ofthe experiment. A pair of dual polyhedra was created by applying
the two inverse operations to the same initial shape.
Had we started with an octahedron, the cube's dual, the results
would have been the same. Degenerate stellation creates the rhombic
dodecahedron (Fig. 4-8), and degenerate truncation, the cuboctahedron, as described above (Fig. 4-9). Thus, the two operations
generate dual polyhedra from one starting point. Loeb concludes
that degenerate truncation and stellation are dual operations.
To generalize, Loeb discovers that if both members of a pair ofdual polyhedra are truncated (or both stellated) to the degeneratecase, they will lead to the same result. Conversely, separate degenerate truncation and stellation of the same shape create a new dual
pair. Finally, regular polyhedra usually generate semiregular ones.
Symmetry

The foundation is almost in place. A final tool to pick up is anunderstanding of symmetry: "exact correspondence of form or constituent configuration on opposite sides of a dividing line or plane or

,

M

(a)

R
(b)

Fig. 4-14

4. Tools of the Trade53

about a center or axis." This somewhat abstruse definition from The
American Heritage Dictionary introduces the two types of symmetry.
Mirror symmetry is the more familiar, involving the exact reflection of a pattern on either side of a "mirror line" (or plane). The
letter "M" exhibits mirror symmetry; "R" does not (Fig. 4-14b).
Rotational symmetry specifies that a configuration can be rotatedsome fraction of 360 degrees (depending on the numerical type ofrotational symmetry) without changing the pattern. For example, asquare, exhibiting fourfold rotational symmetry about its center, canbe rotated 90, 180, or 270 degrees without detectable change. Similarly, an icosahedron has fivefold rotational symmetry about an axis
through a pair of opposite vertices (Fig. 4-14a). The letter "S"
exhibits twofold rotational symmetry; it looks the same after a
180-degree turn (Fig. 4-14b). In other words, in x-fold rotational
symmetry, constituents of a pattern are repeated x times about acommon center.
These concepts will prove useful as we proceed through Fuller's
discoveries.

5
Structure and "Pattern Integrity""I'm a little child and I've just found my mother's necklace."
Famous for his marathon lecture sessions, Fuller used to talkabout synergetics for days on end. Time constraints in later years
usually prohibited such extensive coverage, but he almost always
told the story of the necklace. Looking a bit like a small childhimself as he draped this ten-foot loop over his shoulders, Bucky
explained that the process of collecting "experimental evidence"starts with children.
Fuller was a remarkable teacher, particularly in his ability to
explain difficult concepts in simple terms. Not drawn to the formal
logic of proofs, his genius lay in his novel use of everyday experiences. Elaborately detailed descriptions, relying on familiar materials
and specific colors, were tailored to elucidate various complex phenomena. In one such scenario he is able to explain the intimidating
concept of precession, which is one of the mysteries of gyroscopic
motion, through a series of easily visualized events.l His images
materialize so vividly in the mind's eye that the underlying abstractstatements can be grasped effortlessly.
"I'm going to be a little child now." We are immediately in his
world, looking out. Even having heard this routine countless times,
one can forgive the simplicity of the story. This is stuff for five-yearolds, but it is riveting-and a welcome break in the often headylecture.
The necklace grows out of Fuller's insistence that every child isborn a genius-endlessly curious, probing, full of wonder about
everything. If a child's questions are rewarded with answers that feelright, that is, correspond to his experience, the inherent genius will
blossom. More often, not challenged creatively by tedious memorization that doesn't seem to relate to the world around him, a childsimply learns to play the game. Fuller's conviction that children
spontaneously leap at the chance to understand Universe when
excited by true and comprehensive information was a primary5. Structure and "Pattern Integrity"

55

motivating force behind synergetics. His aim was to supply models
to elucidate the wonders of science to adults and children alike.
It's an unusual necklace. Ten or more thin wooden dowels arelinked together with red rubber-tubing segments into a continuous
flexible loop. Bucky keeps taking it off his shoulders to remove
another one of the ten-inch dowels. One by one they drop to thefloor, as the necklace turns into recognizable polygons. Recognizable, that is, when he struggles to hold them out flat and round.
Soon, the "drapable necklace" resembles a hexagon, then a pentagon. Next, the four sticks that are easily persuaded to be a square,just as readily collapse into a bundle-four parallel sticks held in
one fist (Fig. 5-1).
His expression is utterly earnest, "You remember when the teacher
went to the blackboard and drew a square?" (Nods fill the lecturehall.) "Well, the only reason it stayed a square was that the blackboard held it there!" The shape collapses, dangling from one hand.
(Many laugh. Some look concerned; they have begun to sense that
he is deeply serious about this.)
One more stick is pulled out of the loop. Three are left dangling. If
he removed another, the necklace would disappear, for two sticks
alone cannot form an open loop. Connect the two ends of thethree-dowel string, and suddenly, "It holds its shape," he cries out,
astonished. Loudness underscores the importance of this fundamental truth, with enthusiasm undiminished by the repetition of athousand lectures. "Only the triangle is inherently stable" (609.01).
Bucky reminds us that the conditions and materials of the experiment did not change. That red tubing is still flexible, the sticks still

Fig. 5-1

56

A Fuller Explanation

Fig. 5-2. Leverage.

rigid. So what is responsible for the sudden change? Before going on,
he wants us to really understand why a triangle holds its shape.
Two sticks connected by a hinge create two lever arms. The farther
out a force is applied, the greater the mechanical advantage-whichmeans that forces of decreasing strength can accomplish the same
result (Fig. 5-2). Each flexible corner is stabilized with minimum
mechanical effort by a force exerted at the very ends of its two sticks,
or lever arms. A third stick, acting as a "push- pull brace," can be
attached to the ends of the other two sticks, to most efficiently
stabilize the flexible opposite angle. So each of the triangle's threesticks "stabilizes its opposite angle with minimum effort." Only atriangle has a built-in bracing device for each corner; therefore, onlya triangle is stable.
Pattern Integrity

The term" pattern integrity" is a product of Fuller's lifelong commitment to vocabulary suitable for describing Scenario Universe. He

5. Structure and "Pattern Integrity"

57

explains,
When we speak of pattern integrities, we refer to generalized patterns ofconceptuality gleaned sensorially from a plurality of special-case pattern experiences . .. . In a comprehensive view of nature, the physical world is seen as apatterning of patternings .... (505.01-4)

Let's start with his own simplest illustration. Tie a knot in a pieceof nylon rope. An "overhand knot," as the simplest possible knot, isa good starting point. Hold both ends of the rope and make a loop
by crossing one end over the other, tracing a full circle (360 degrees).
Then pick up the end that lies underneath, and go in through the
opening to link a second loop with the first (another 360-degreeturn). The procedure applies a set of instructions to a piece ofmaterial, and a pattern thereby becomes visible.
What if we had applied the same instructions to a segment ofmanila rope instead? Or a shoelace? Or even a piece of cooked
spaghetti? We would still create an overhand knot. The proceduredoes not need to specify material. "A pattern has an integrity
independent of the medium by virtue of which you have received theinformation that it exists" (505.201). The knot isn't that little bundlethat we can see and touch, it's a weightless design, made visible by
the rope.
The overhand-knot pattern has integrity: once tied, it stays put. In
contrast, consider directions that specify going around once (360
degrees), simply making a loop. This pattern quickly disappears withthe slightest provocation; it is not a pattern integrity. (Even though
the overhand knot depends on friction to maintain its existence, asingle loop will not be a stable pattern no matter how smooth orcoarse the rope.) Notice that it requires a minimum of two fullcircles to create a pattern integrity. 2 X 360 = 720 degrees, the sameas the sum of the surface angles of the tetrahedron (four triangles
yield 4 X 180 degrees). Minimum system, minimum knot, 720 degrees. A curious coincidence? Synergetics is full of such coincidences.
A similar example involves dropping a stone into a tank of water.
"The stone does not penetrate the water molecules," Fuller explainsin Synergetics, but rather "jostles the molecules," which in turn"jostle their neighboring molecules" and so on. The scattered jostling, appearing chaotic in anyone spot, produces a preciselyorganized cumulative reaction: perfect waves emanating in concentric circles.
Identical waves would be produced by dropping a stone in a tankfull of milk or kerosene (or any liquid of similar viscosity). A wave is58

A Fuller Explanation

not liquid; it is an event, reliably predicted by initial conditions. The
water will not surprise us and suddenly break out into triangularcraters. As the liquid's molecular array is rearranged by an outside
disturbance, all-embracing space permeates the experience. Because
liquids are by definition almost incompressible, they cannot react toan applied force by contracting and expanding; rather, the watermust move around. In short, the impact of any force is quickly
distributed, creating the specific pattern shaped by the interaction of
space's inherent constraints with the characteristics of liquid.
The concept thus introduced, Bucky goes on to the most important and misunderstood of all pattern integrities: life. "What isreally important ... about you or me is the thinkable you or the
thinkable me, the abstract metaphysical you or me, ... what communications we have made with one another" (801.23). Every humanbeing is a unique pattern integrity, temporarily given shape by flesh,
as is the knot by rope .
. . . All you see is a little of my pink face and hands and my shoes and clothing,and you can't see me, which is entirely the thinking, abstract, metaphysical me. It
becomes shocking to think that we recognize one another only as the touchable,
nonthinking biological organism and its clothed ensemble. (80l.23)

Our bodies are physical, but life is metaphysical. Housed in a
temporary arrangement of energy as cells, life is a pattern integrity
far more complex than the knot or the wave. Remember that all thematerial present in the cells of your body seven years ago has been
completely replaced today, somehow showing up with the samearrangement, color, and function. It doesn't matter whether you atebananas or tuna fish for lunch. A human being processes thousandsof tons of food, air, and water in a lifetime. Just as a slip knot tied ina segment of cotton rope, which is spliced to a piece of nylon rope,in turn spliced to manila rope, then to Dacron rope (and so on) canbe slid along the rope from material to material without changing its"pattern integrity," we too slide along the diverse strands supplied
by Universe-as "self-rebuilding, beautifully designed pattern integrities." No weight is lost at the moment of death. Whatever "life"
is, it's not physical.
The key is consciousness. "Mozart will always be there to any whohears his music." Likewise, "when we say 'atom' or think 'atom' weare ... with livingly thinkable Democritus who first conceived andnamed the invisible phenomenon 'atom'" (801.23). Life is made ofawareness and thought, not flesh and blood. Each human being
embodies a unique pattern integrity, evolving with every experienceand thought. The total pattern of an individual's life is inconceivably

5. Structure and "Pattern Integrity"

59

complex and ultimately eternal. No human being could ever completely describe such a pattern, as he can the overhand knot; thatcapability is relegated to the "Greater Intellectual Integrity of Eternally Regenerative Universe.,,2
If we seem to stray from the subject of mathematics, resist thetemptation to categorize rigidly. Synergetics does not stop withgeometry. Fuller was deeply impressed by a definition in a 1951
Massachusetts Institute of Technology catalog, which read
"Mathematics is the science of structure and pattern in general"(606.01): not games with numbers and equations, but the tools for
systematic analysis of reality. To Fuller this meant that mathematics
ought to enable the "comprehensivist" to see the underlying similarities between superficially disparate phenomena, which might bemissed by the specialist. Rope may not be much like water, but theknot is like the wave-is like the tetrahedron.
Our emphasis thus far has been on pattern. What about structure?
Let's go back to the regular polyhedra. On constructing the five
shapes out of wooden dowels and rubber connectors, it is immediately apparent that some are stable and others collapse. The"necklace" demonstrates that only triangles hold their shape, and so
the problem becomes quite simple.
Picture a cube. Better yet, make one out of dowels and rubber
tubing, or straws and string. Whatever material you choose, as longas the joints are flexible, the cube will collapse. Connectors with acertain degree of stiffness, such as marshmallows or pipec1eaners, aremisleading at first, because the cube appears to stand on its own.
However the shape is so easily rearranged by a slight push that theillusion does not last.
The six unstable windows must be braced in order for a cube to be
rigid. So six extra struts, inserted diagonally across each face, would
be the minimum number that could stabilize the system.
Six struts? Just like a tetrahedron! And not only are there the rightnumber of struts, but they can also be arranged the same way. A
regular tetrahedron fits inside a cube with its six edges preciselyaligned across the cube's faces (Fig. 5-3). We can therefore state that
there is an implied tetrahedron in every stable cube. Nothing in ourinvestigation thus far would predict the precise fit of a cube and atetrahedron. This and many other examples of shared symmetry
among polyhedra (as seen in the previous chapter) are powerfuldemonstrations of the order inherent in space.
If the cube is unstable without a scaffold of triangulation, what
about cardboard models? They stand up by themselves with notrouble. The key word is cardboard. A polyhedron constructed out

60

A Fuiier Explanation
Fig. 5-3. Inscribed tetrahedron stabilizes cube.

of stiff polygon faces, rather than edges and connectors, is effectively
triangulated. Cardboard provides the necessary diagonal brace. It
provides a lot of extra material too, but the untrained eye will notrecognize the redundancy at first.
Likewise, the stiffness of marshmallows or pipecleaners provides
triangulation, in the form of tiny web-like triangular gussets at thecorners, strong enough to stabilize the whole window. Furthermore,
stiff material is itself rigid because of triangulation on the molecularlevel.
Only when polyhedra are considered as vector systems is stability
an issue. Construction is one method of determining stability, but asimple formula utilized by Loeb can also be used to check: E =3V - 6. 3 If the number of edges is less than three times the numberof vertices minus six, the system will be unstable. But the criterion isreally even simpler: for polyhedra without interior edges, a stablesystem is always triangulated and a triangulated system is always
stable. 3V - 6 = E holds true for a polyhedral shell if and only ifthat system consists exclusively of triangles.
The other option is to establish internal triangles; an unstableshell can be stabilized by interior edges, or body diagonals. Insteadof triangulating the surface, the bracing members create triangles
inside the shell to maintain the system's stability.
Structure

Upstaged by the crowd of oversized polyhedral toys, Bucky again
resembles the small child we saw earlier, playing with his mother's
necklace. But the words this time are more ambiguous: "There are
only three basic structural systems in Universe."
Fuller long ago decided that science simply did not have adefinition of structure, and took it upon himself to remedy theoversight. Science, Fuller explains, did not feel the need for adefinition, because" structure" seemed to be self-evident. It holds its

5. Structure and "Pattern Integrity"

61

shape! Language, caught in the old world of "solids," did not keep
up with science's evolving understanding of the true nature ofmatter. In a universe consisting entirely of fast-moving energy, wemust ask how something holds its shape.
"Structure is defined as a locally regenerative pattern integrity of
Universe" (606.01). A good starting point. Structure is also "acomplex of events interacting to form a stable pattern." Similar, butmore specific: the pattern consists of action, not things.
"Regenerative" is an important qualification, because of the transient nature of energy. The pattern, not the energy flowing through,has a certain degree of permanence. A structure must therefore becontinually regenerating in order to be detected.
Structure is "local" because it is finite; it has a beginning and an
end. "We cannot have a total structure of Universe" (606.01).
"Interacting" signifies the emphasis on relationships.
The phrase "complex of events" suggests an analogy to constellations, whose components-though spectacularly far apart-areinterrelated for some cosmic span of time, creating a set of relationships, in other words, a pattern. The seven stars of the Big Dipper
are light-years apart-the epitome of nonsimultaneous energy events.
They are only perceived as a meaningful pattern from a special
vantage point, Spaceship Earth. The remoteness of individual atoms
in any structure or substance-not to mention the distance between
atomic constituents-prompted Fuller to write "one of the deeply
impressive things about structures is that they cohere at all." There isnothing" solid" about structure.
What do all structures have in common that allows their coherence?
Triangles. At the root of all stable complexes is nature's only
self-stabilizing pattern.
No Fuller study is complete without an "inventory": a list, not of
each and every "special case" example, but rather, of the types ofTable II"
Polyhedron
Tetrahedron
Octahedron
Icosahedron

Number of
Edges

VolumeVolume per 6 Edges

6
12
30

0.11785
0.471402.18170.117850.23570.4363'Comparison of results in third column:
0.2357 = 2 x 0.11785,
0.4363 = 3.70 x 0.11785.

62

A Fuller Explanation

categories. The task, then, is to combine our new working definitionof structure with the earlier one of systems. That means triangles
involved in a subdivision of Universe. The virtually unlimited varietyof irregular triangulated enclosures are not to be included in thisinventory; rather, we seek a list of symmetrical stable enclosures.
And now we can sit back, for our task is already finished.
Remember that only three systems can be made out of regulartriangles: tetrahedron, octahedron, and icosahedron. These are the"three prime structural systems of Universe."
What can be learned through this kind of simplification? As in
Euler's identification of vertices, edges, and faces, such categories
organize the otherwise indigestible data to reveal new importantfeatures. An example is seen in Fuller's "structural quanta": thetotal material used for each of the three structural systems (easily
measured in terms of number of edges) goes from six sticks to twelve
to thirty. Chapter 10 will cover Fuller's ideas on the subject ofvolume in detail, but for now we can demonstrate an interesting fact
while utilizing the traditional formulae of high-school geometry.

Fig. 5-4. "Dimpling."

5. Structure and "Pattern Integrity"

63

Going from the smallest to the largest structure, the volume
increases, not only absolutely, but relative to the number of edges. In
other words, the ratio of volume to structural investment is asignificant variable, increasing with additional structural quanta. The
same holds true for ratios of volume to surface-area, as will be seen
below. For clarity, we adopt unit edge lengths for all three polyhedra. Appendix B shows each step of the calculations, but therelevant results are displayed in Table II.
The implications of this information are suggested by Fuller'ssummarizing statement:
The tetrahedron gives one unit of environment control per structural quantum.
The octahedron gives two units of environment control per structural quantum. The
icosahedron gives 3.7 units .... (612.10)

Fuller referred to six edges as a "structural quantum" because thetotal number of edges in each polyhedron is a multiple of six.
"Environment control" simply refers to the ability to enclose andthereby regulate space.
Toward the goal of maximal enclosed space with minimal structural material (whether in terms of total strut length or surface area),
designs based on the icosahedral end of the spectrum are advantageous. Hence Fuller's geodesic dome. For resistance to externalloads, the tiny pointed tetrahedron is least vulnerable, for its concentrated structural elements resist buckling. The tetrahedron is alledges, enabling maximal structural resistance, and therefore highly
applicable to truss design. (See Chapter 9.)
The icosahedron "dimples" easily. Fuller's term means just what it
says. Push hard on one vertex and five triangles cave in, such that the
tip of the inverted pyramid reaches just beyond the icosahedron'scenter of gravity. (See Fig. 5-4.) The tetrahedron is unique in being

Fig. 5-5. Dimpling: one half of octahedron caves
in to nest inside other half.

64

A Fuller Explanation

impervious to dimpling. Push hard on any vertex and either thewhole system turns inside out (if the tetrahedron is made of rubber)or nothing happens; structural resistance prevails.
The octahedron, as expected, falls in the middle on both counts,that is, in terms of volume efficiency and load resistance. It will
"dimple," but in so doing one half caves in to "nest" exactly inside
the other half (Fig. 5-5).
Three of Fuller's inventions, the geodesic dome, the Dymaxion
Map, and the Octet Truss, stem directly from the above principles.
All three will be discussed in detail later, as other relevant geometricprinciples are revealed.

6
Angular Topology

Our study so far has primarily examined the conceptual foundationof synergetics. Except for occasional reference to volume and symmetry, the emphasis has been on numbers of elements rather than on
shape. It's now time to look at the rest of the picture. Fuller's
appreciation of the MIT definition of mathematics (" the science ofstructure and pattern in general") led him to ponder the appropriate
tools and methods. "Science" is a systematic endeavor, requiringexact procedures for its description of structure and pattern.
A coordinate system describes the shape and location of a body inspace by specifying the position of a sufficient number of that body's
components. But a position can only be specified by its relationshipto some other known location, or coordinate-system origin. In essence, mathematics functions by locating points relative to anagreed-upon frame of reference, such that the mathematician can say
there is a point here and a point there and they are related by thiskind of trajectory, and so on, until there is enough information todescribe the entire system. Fuller points out that this information
can be broken down into two aspects: shape and size.
What does "shape" consist of? "Shape is exclusively angular"(240.55): a simple but powerful observation. It's easy to envision
identical shapes of completely different scale: for example, an equilateral triangle is a precisely defined concept, yet it contains no
indication of size. It may be two miles or two centimeters in edge
length, but its angles must be 60 degrees. Shape is influenced only by
angle, and "an angle is an angle independent of the length of its
sides" (516.02). The word" triangle" by itself (without further modification) does not describe a specific shape but rather aconcept- three interrelated events without specific length or angle.
What does size consist of? Measurement, or dimension. In synergetics, these parameters are always expressed in terms of"frequency." The word is aptly applied, serving as a reminder of therole of time. Fuller dwells on the point: every real system (" special

66

A Fuller Explanationcase") involves time and duration. 1 Real systems are events, and it
takes time for an event to occur. He bases his objection to purely
static concepts in mathematics on the fact that they are incompatible
with twentieth-century scientific thought:
Since the measure of light's relative swiftness, which is far from instantaneous,the classical concepts of instant Universe and the mathematicians' instant lines havebecome both inadequate and invalid for inclusion in synergetics. (201.02)

Since Einstein, Bucky reminds us, we can no longer think in terms ofan "instant Universe," that is, a single-frame picture. Because evenlight has been found to have measurable speed, every aspect ofphysical Universe from the smallest tetrahedron to life itself involves
the passage of time. Quite simply, "it takes time to get from here to
there."
Frequency and Size

He insists upon nothing more adamantly than this distinctionbetween real ("experimentally demonstrable") phenomena and imaginary concepts. "Size" relates to real, time-dependent systems,
whereas "shape," influenced only by angle and therefore independent of time, is a factor in both real and conceptual systems.
[" Angles are ... independent of size. Size is always special-case experience" (515.14).]
But how does "frequency" apply to size and length? Frequency
connotes number: the number of times a repeating phenomenon
occurs within a specified interval-ordinarily an interval of time, but
Fuller extends the concept to include space. Length is measured insynergetics in terms of frequency to underline the fact that the"distance from here to there" involves time and can be specified interms of number: number of footsteps across the room, or number ofheartbeats during that interval, number of water molecules in a tube,
number of inches, number of photons, number of somethings. The
choice of increment depends on what is being measured, but
frequency (and hence size) is inescapably a function of time andnumber.
Units of Measurement

Fuller explains frequency as subdivisions of the whole, suggesting
another advantage of the term: it provides a built-in reminder that
there is no absolute or single correct unit of measurement; rather,6. Angular Topology

67

distance is measured relative to arbitrarily devised units. It is not aminor challenge to perceive distance this way; our conventional units-like inches or miles-are such an integral part of awareness thatthey seem a priori elements of size. The teacher in Fuller will not let
us accept such useful conventions blindly, and so he employs tools
such as "frequency" to keep us on our toes-aware of the nature of
distance.
Time and Repetition: Frequency versus Continuum

Just as length cannot exist without time, there also could be noawareness without time. Time, inseparable from all other phenomena, cannot be isolated. "Time is experience" (529.01).
The concept of time is inextricably tied to awareness; appropriately it is measured in terms of the frequency of detectablerepeating events. Periods of daylight reliably alternating with darkness gave us a unit we call a "day." Heartbeats might have definedthe "second," planting the awareness of that tiny increment inlong-ago human beings. The predictable repetition of days growing
longer and shorter with their accompanying weather changes defineda "year." To conceive of time requires repetition.
However, the limits of perception prevent recognition of theperiodicity in very high-frequency patterns such as light waves orrepeating molecules in a toothpick. If repetition is too frequent, weperceive a continuum rather than segmented events. Fuller's use of"frequency" to specify size draws attention to the nonexistence ofcontinuums. Here, as always, his goal is to develop a mathematicallanguage which accurately represents reality.
Shape and size are thus replaced by angle and frequency.
Fuller's principle of design covariables summarizes by stating thattwo factors are responsible for all variation. "Angle and frequency
modulation exclusively define all experiences, which events altogether constitute Universe" (208.00). In short, "structure andpattern in general" are described completely by only two parameters: angle and frequency-another way of saying that the differences between systems are entirely accounted for by changes inangle and length. Again the goal of such simplification is thedemystification of mathematics.
Remember that Fuller's overall goal was to isolate "nature's
coordinate system" - by which he meant the simplest and mostefficient reference system to describe the events of nature. We
gradually narrow in on his solution.

68

A Fuller Explanation

Topology and Vectors

Fuller has declared his scope: "Synergetics consists of topology
combined with vectorial geometry" (201.01). Topology in essence
analyzes numbers of elements. (Euler's law is topological, involving
neither symmetry nor size.) And now we must again think about
vectors, for they are the key to this combination.
Vectors provide an ideal tool for representing velocity, force, andother energetic phenomena. As you may recall, the concept is
actually quite simple-despite its lack of popularity among highschool students. A vector is a line with both specific length andangular orientation. It's the ultimate simplification of actions orforces, presenting only the two most relevant bits of information:
magnitude and direction. Mathematics defines this tool and theaccompanying rules for its manipulation, just as it defines the set ofreal numbers and the rules for addition and multiplication. The
mechanism as a whole enables us to predict the results of complex
interactions of forces and bodies in motion.
Surveying classical geometry, Fuller decided that" there was nothing to identify time, and nature has time, so I'd like to get that in
there." Vectors seemed to provide the solution, "I liked vectors. A
vector represents a real event of nature .... I wondered if I couldn't
draw up a geometry of vectors; that would mean having the elementsof experience." 2
Back to synergetics: What is meant by a combination of topology
and vectorial geometry? And how does it fit into the search for"nature's coordinate system"? By viewing polyhedra as vector diagrams, Fuller integrates the two subjects (vectors and topology) in adeliberate attempt to develop one comprehensive format to accommodate both the inherent shape of space and the behavior ofphysical phenomena. Polyhedra with vectors as edges necessarily
incorporate both shape and size.
Vector Polyhedra

The spectrum of possible forms of polyhedra is certainly informative
about the shape of space; polyhedra are systems of symmetry madevisible. Any configuration allowed by space can be demonstrated byvertices and edges, and, as noted earlier, experimentation quickly
reveals that the variety of possible forms is limited by spatialconstraints. Furthermore, the shape of space is fundamental to theevents of nature. Fuller believed that mathematics, the science ofstructure and pattern, should be based on these principles.

6. Angular Topology

69

So Fuller coined the rubric "angular topology" to express what hesaw as the principal characteristic of synergetics: integration of thestatic concepts of geometry with energetic reality. These may nothave been the words he used back then, but the desire for such asystem dates back to Bucky's early school days. Or at least that's
how the story goes. Such myths evolved over time to convey thespirit of the child's inquisitive confusion through concise anecdotes.
Fuller's lectures and writings incorporate a full repertoire of autobiographical moments in which the young Bucky has startlingly complete and rich realizations. The process is beautifully described by
Hugh Kenner, who relieves us of the burden of asking, "Did thatreally happen, just like that, one morning"?
Not that he deceives. He mythologizes, a normal work of the mind ... to embrace
multitudinous perceptions, making thousands of separate statements about different
things [into] summarizing statements ...
What a myth squeezes out is linear time, reducing all the fumblings and sortingsof years to an illuminative instant. We can see why Bucky needs myth. The visionthat possesses him eludes linearity ... The myth is anecdotal. 3

We return to the geometry lesson: the grade-school teacher has
put a drawing on the blackboard and said, "This is a cube." Young
Bucky wonders aloud, "How big is it, how much does it weigh, what
is its temperature, how long does it last"? The teacher says, "Don't
be fresh," and "You're not getting into the spirit of mathematics."
Again, the implication is that mathematics does not deal with realthings, but only with absurd constructs and arbitrary rules.
It's not hard to accept the message behind the story-that something about the teacher's lesson was profoundly disturbing to thechild. It seemed to Bucky that mathematics was a serious enterpriseand it should limit itself to "experimentally demonstrable" phenomena. That meant no fooling around in the fringe area of sizelesspoints, infinite planes, and weightless cubes. We reconsider theseearly musings in the new context of "angular topology" to see where
they led, and recall that Fuller was later to attribute mathematics'
lack of popularity to the perfectly natural discomfort people felt withthose elusive concepts. Explanations ought to be in terms of tangible
experience.
But then what are we to make of such claims as "Synergetics
permits conceptual modeling of the fourth and fifth arithmetic
powers; that is, fourth- and fifth-dimensional aggregations of points
or spheres in an entirely rational coordinate system that is congruent
with all the experimentally harvested data of astrophysics andmolecular physics ... " (202.01)? Under the heading "Angular Topol-

70

A Fuller Explanation

ogy," this statement is found too early in Fuller's Synergetics to beeasily understood. One might wonder if a page was left out of thatparticular copy; but the root of the confusion is not that easily
located. With some additional background material, we can begin tounderstand Fuller's assertion. The word "dimension" has been lurking behind the scenes in this entire discussion, and must now be
brought out into the open.
Dimension(1) A measure of spatial extent. (2) Magnitude, size, scope. (3) The
number of factors in a mathematical term. (4) A physical property,
often mass, length, time, regarded as a fundamental measure. (5)Any of the least number of independent coordinates required to
specify a point in space uniquely.
The above is a sampling of what you will find in English-language
dictionaries under "dimension." As you can see, there are a few
distinct meanings-essentially falling into three categories:(1) Physical extent or measurement, as in "what are the dimensionsof this room"?
(2) Orders of complexity, in the most general sense, as in "the many
dimensions" of an issue or problem. This meaning is as common
as it is widely applicable.
(3) The specifically mathematical application: the number of independent terms required to specify a point in space. Our conventional system utilizes three independent, mutually perpendicularaxes in space to accomplish this task. This is often the first
meaning to occur to people, especially when already thinking
about geometry. "Space is three-dimensional." However, thisassignment- treating the third category as an exclusive definition-seemed unacceptably limited to Fuller. Exposure to theordered polyhedra of mathematics and also to organic structuresand crystals found in nature makes an orientation toward perpendicularity seem quite arbitrary. Although right angles are
sprinkled throughout geometric shapes, they are by no means
dominant. And, more often than one might expect, ninety-degree
coordinates provide an awkward framework with which to describe both naturally occurring and conceptual formations.
Fuller viewed the Cartesian coordinate system with its three
perpendicular axes, conventionally labeled X, Y, and Z, as a rem-

6. Angular Topology

71

nant of "fiat-earth thinking." Early man, finding himself on a huge
fiat expanse, assumed that up-and-down and back-and-forth were
the fundamental directions of his universe. Ninety degrees was theobvious natural angle with which to segment and measure space.
Humankind has had more and more evidence of nature's radial andspherical bias throughout history-from the discovery of the shapeof planets to the behavior of radiation and cellular growth. But
neither Copernicus's spherical earth nor the vast array of biologicaland physical phenomena, all suggesting that angles other than ninety
degrees would provide more "natural" or convenient standards,
succeeded in reorienting the perpendicular bias of mathematics.
The "three dimensions" of mathematics-length, width, andheight-became part of an unshakable convention. That space cannot accommodate a fourth perpendicular direction is just one of its
many intrinsic constraints, and yet this limitation is too often seen as
the only characteristic of space. While mathematicians postulate
hypothetical "hypercubes" in their attempt to describe a spatial
fourth dimension, and physicists refer only to "time" as the fourth
dimension, Bucky preferred to call attention to the "four-dimensional" tetrahedron. Time is certainly a dimension, but the physicists'
progression "x, y, z, and t" seemed not to emphasize sufficiently
that time-permeating all space and all experience-is qualitatively
unlike the other" three dimensions."
As we develop an awareness that space has shape, right angles
gradually seem less "natural." The XYZ coordinate system often
serves to obscure rather than illuminate spatial characteristics. It is avaluable tool, which we can recognize as one alternative superimposed by human minds, not as a framework organic to the shape ofspace itself. The word "dimension" is used without contradiction todescribe the maximally symmetrical arrangement of three lines inspace; likewise it can be applied to time, but it's not the end of thestory.
One of the above dictionary definitions refers to the number ofcoordinates required to specify the location of a point in space.
Assuming the existence of a previously specified origin, the numberof coordinates happens to be three. Does this result reinforce theexclusive use of the XYZ axes? No, for the three coordinates
required do not have to be Cartesian; another option is sphericalcoordinates, in which the location of any point is fixed by specifying
two angles and a radial distance. Cartesian coordinates, on the other
hand, describe a location as the intersection of three lines originating
at given distances along three perpendicular axes. (See Figure 6-1 for

72

A Fuller Explanation

~

-",.

9<:'.:-t,",b,"Fig. 6-1. Cartesian versus spherical coordinates.

a comparison of the two methods.) The spherical approach is moresuited to Fuller's radial ("converging and diverging") Universe ; itsemphasis on angular coordinates encourages thinking in terms of"angle and frequency modulation."
Dimension is a widely encompassing term, and can legitimately
refer to numbers of factors in a variety of geometric phenomena.
Considerable time can be devoted to unraveling Fuller's different
uses of "dimension" in Synergetics, and we shall continue to cite
examples throughout our investigation.
Size

Fuller's book takes a firm stand in the opening sections: "Synergetics originates in the assumption that dimension must be physical"(200.02), meaning size. The declaration is soon reinforced: "There is6. Angular Topology

73

Fig. 6-2

no dimension without time" (527.01). Firmly imbedded in reality
now: it takes time to embody a concept. Everything ties together, so
far.
It would be uncharacteristically clear-cut if that were Fuller's only
use of dimension. Synergetics may start with dimension as size, butother applications of the multifaceted term are sprinkled throughout
the book. (Identification of space as three-dimensional is not one ofthem.) Keep in mind that this mathematical convention has a firm
hold; it's difficult to think otherwise about space-and consequently
not easy to view Fuller's material objectively.
Planes of Symmetry

Both the tetrahedron and the octahedron-two of the simplest
structures-incorporate four nonparallel planes. The faces of thetetrahedron present four distinct directions, just as the faces of the
cube provide three. The octahedron has four pairs of opposing
parallel triangles, and it can be demonstrated that they are parallelto the tetrahedral faces (Fig. 6-2). Fuller refers to this geometric trait
as "dimension," and through repeated observation places considerable emphasis on the inherent fourness of the" minimum system in
Universe." "The octahedron's planar system is four-dimensionallyreferenced, being parallel to the four symmetrically interacting planesof the tetrahedron ... " (527.31). The icosahedron, on the other hand,exhibiting various fivefold symmetries, embodies" five-dimensionality" (527.50) in Fuller's unorthodox appropriation of terms.

74

A Fuller Explanation

Other Applications of Dimension

Another twist: Fuller also refers to the three structuralparameters-vertices, edges, and faces-as different dimensions ofstructure. In a later section covering the concept of dimension, Fuller
reintroduces "constant relative abundance" (as explained in Chapter
4) under the heading "527.10 Three Unique Dimensional Abundances" -namely, vertices, edges, and faces. This and other ambiguous-if not contradictory-usages of certain terms can often obscure the mathematical statement being made. In this case, Fuller's
point about the consistent arithmetic relationships between vertices,
edges, and faces in closed systems is lost amid confusion about themeaning of "dimensional abundances."
Another unorthodox usage involves pairs of opposites. At one
point in Synergetics, for example, a magnet, with its positive andnegative poles, is called a two-dimensional system. Along the samelines, "Polar points are two dimensional: plus and minus, opposites"(527.21).
Finally, "dimensional aggregations" in the opening quotation ofthis section refers to numbers of layers in certain clusters of closely
packed spheres. We shall explore these patterns in Chapter 8.
Fuller's different uses of "dimension" may be confusing, but they are
not, strictly speaking, incorrect-at least not in terms of the dictionary. Mathematical convention is another issue.
Fuller does mention the historical precedent for conceiving ofspace as exclusively "three-dimensional," thereby explaining his
license to reevaluate our concept of dimension; however, the reference is too late in the book to clear up early confusion:... The Greeks came to employ 90-degreeness and unique perpendicularity to thesystem as a basic ... dimensional requirement for the ... unchallenged three-dimensional geometrical data coordination. (825.31)

So, while he does justify his usage with this reference to the word's
flexibility, the clarification is obscured by the book's sequence. The
reader seeking a quick reason to dismiss Synergetics might focus on
Fuller's extravagant citation of other dimensions early in the book.
His apparent familiarity with" the fourth dimension" provides justcause for suspicion; however, a simple change of article-from"the" to "a fourth dimension" -gives the term a very different
effect.
Like most subjects in synergetics, "dimension" cannot be neatly
presented in one complete package; boundaries are never that clearly6. Angular Topology

75

defined. In addition to the fact that different subjects overlap, there
can always be new twists. The trick is to leave ourselves open to
exploration, free to evaluate each new application without bias.

Angular Topology

Once in a long while, a "generalized principle" takes recognizable
shape and emerges out of the vast sea of man's cumulative findings.
For Fuller, these principles-characterized as true in every case-are
the real wealth of society. Applications may not always be immediately clear, but if an inventory of "generalized principles" ismade accessible, he reasoned, humanity will put them to use sooner
or later.
The "principle of angular topology" was recognized by themathematician and philosopher Rene Descartes (1596-1650), but
the title of course is Fuller's. Perhaps by giving Descartes's remarkable discovery a new title, Bucky hoped to excite the kind of attention
he felt it deserved.
In every polyhedral system, the sum of the angles around all thevertices is exactly 720 degrees less than the number of vertices times
360 degrees, or (360° X V) - 720°. True for the tetrahedron, true
for the crocodile. In Fuller's words, every system has exactly 720
degrees of "takeout."
If this principle seems complicated, it is only because the words
are hard to follow, but the following image should make it easier.
Picture a paper cone-the shape of an ice-cream cone without theice cream. Notice that a cone, having an opening at the base, is not a
closed system. Now, split the paper cone open by slicing a straightline from its pointed tip to the circular hole, and then spread thepiece of paper out flat on the floor like a rug (Fig. 6-3). There is now
an angular gap left by the paper, where the floor shows through.
That gap is the" takeout angle," the angular difference between a flat
map and a cone.
In the same manner, you can slice open some number of edges ofa polyhedron until its surface can be spread out like a rug. The
resulting map, similar to a dressmaker's pattern, is called in geometry a polyhedron's net. A net contains all faces of a polyhedron,some of them separated by angular gaps; it is a flat pattern which
can be folded along the edges, and taped together to generate its
polyhedron. Figure 6-4 shows nets of an icosahedron and an oc-

76

A Fuller Explanation

Fig. 6-3

Fig. 6-4

tahedron. The principle of angular topology states that the sum of
the angular gaps, no matter how simple or complex the system, will
be exactly 720 degrees.
Or go through these procedures in reverse: start with flat paperand cut out one pie-shaped segment to make a cone. Then keep
going, cutting out more slices, just until the point at which the paper6. Angular Topology

77

can be closed off into a system. This point is reached when you havetaken out exactly 720 degrees. It's a prerequisite to closure; there is
no leeway. To check, you can measure all the scraps that have beentaken out; the results are always the same: however irregular yourcuts or strange your resulting closed system, the total takeout mustbe 720 degrees. This consistent total presents a generalized principle
for closed systems.
The surface angles of any tetrahedron (regular or not) also happen
to add up to 720 degrees. (Four triangles: 4 X 180° = 720°.) Fuller
certainly isn't going to let that one slip by! The "difference between
the visibly definite system and the invisibly finite Universe [i.e.,plane] is always exactly one finite invisible tetrahedron ... " (224.10).
Consider once again the variety of systems. This principle-a firstcousin of Euler's law-describes an extraordinary consistency. The"720-degree excess" is an appropriate parallel to Euler's "constant2" in that there are 720 degrees in two complete revolutions. Bothare counterintuitive: Euler's law reveals that the number of edges isalways exactly two less than the vertices plus faces, no matter how
complex the system, just as the angular "takeout" is 720 degrees
whether the surface angles themselves add up to a total of 720
degrees, as in the tetrahedron, or to 57,600 degrees, as in the"four-frequency icosahedron." (Don't worry, that structure will be
explained below.) Table III shows the results for a few different
polyhedra, verifying the constant "excess" of 720 degrees.
Table III reveals another notable consistency: the sum of thesurface angles in every polyhedron is a multiple of the tetrahedron's
720 degrees (column 5). This calls to mind our earlier observationthat the number of edges in many ordered polyhedra is a multiple ofthe tetrahedron's six. (Refer to Chapter 4.)

Table ill

Polyhedron
Tetrahedron
Octahedron
Icosahedron
Cube
Pent. Dodec.
VE
4f Icosa

1
Number ofvertices,
V
4
6
12
8
20
12
162

2
V x 360°

3
Sum ofsurfaceangles4(Col. 2)-(Col. 3)

5(Col. 3)/720

1,440°
2,160°
4,320°
2,880°
7,200°
4,320°
58,320°720°
1,440°
3,600°
2,160°
6,480°
3,600°
57,600°

720°
720°
720°
720°
720°
720°
720°

1
2
5
3
9
5
80

78

A Fuller Explanation

Angular Takeout : An Example

A complicated system such as the four-frequency icosahedron provides an especially good illustration of this remarkable consistency.
The structure is an irregular polyhedron with 320 triangular faces,
and is based on the symmetry of the icosahedron. Each icosahedral
triangle is replaced by sixteen new smaller triangles, producing thetotal of 320 faces of this more or less spherical structure. Chapter 15
will describe the origin of high-frequency icosahedral enclosures indetail, but for now, we can understand that the faces are irregulartriangles. As shown in Figure 6-5, most vertices join six triangles,
and we recall from Chapter 4 that if six sixty-degree angles meet,they create a flat surface. Therefore, if six faces are to surround aconvex vertex of a polyhedron, their angular total must be less than
360 degrees-which produces the "angular takeout". Those interested in exploring how to calculate individual edge lengths andsurface angles can refer to Appendix A, "Chord Factors," and toAppendix C for a list of additional sources; other readers should
simply be aware that the values will be highly irregular numbers.
Having noted that, to assure convexity, the surface angles must add

Fig. 6-5. Four-frequency icosahedron.

6. Angular Topology

79

up to less than 360 degrees at each vertex, we can reflect on how
extremely small the gaps at each vertex in this structure will be.
If all 162 vertices are equally distant from the structure's center,
the average total of the surface angles at each will be 355 degrees, 3
minutes, and 20 seconds-or 355.5556 degrees. Thus, the" takeout"angle-instead of being concentrated at a few points, with 60
degrees removed from each of twelve vertices- is distributed among
many component triangles. Imagine cutting open this system andspreading out its multifaceted net. Each angular gap would be very
narrow, averaging 4 degrees, 26 minutes, and 40 seconds (4.4444
degrees.) If we were to draw this net, the pencil thickness itself
would be a nuisance. Supposing each edge is approximately one inch
long, the outermost or widest part of each gap will be less than
one-twelfth of an inch.
In addition to being miniscule, these numbers are typically quite
irregular, not at all simple fractions of degrees. Nevertheless, thesegaps, calculated (say) to six decimal places, add up to exactly
720.000000 degrees, no matter how many vertices in the system, or
how irregular the distribution.
One interesting implication of the principle of angular topology isfurther demonstration of the impossibility of the traditional sphereas defined by mathematics-an unreachable planar 360 degrees
around everyone of an infinite number of vertices. "The calculus
and spherical trigonometry alike assume that the sum of the angles
around any point on any sphere's surface is always 360 degrees"(224.11). Fuller goes on to point out that in order to achieve a closed
system, there must be no degrees taken out, distributed throughout
the vertices, thereby invalidating this assumption. "The demonstration thus far discloses that the sum of the angles around all thevertexes [sic] of a sphere will always be no degrees-or one tetrahedron-less that the sum of the vertexes times 360 degrees-ergo
one basic assumption of the calculus and spherical trigonometry isinvalid" (224.11). In other words, since the nO-degree takeout is aprerequisite to closure, even a sphere has to have infinitesimally lessthan 360 degrees around any given point on its surface. (We areforced to conclude that "infinitesimal" times "infinite" here equals
720 degrees.)
Angle Types

Finally, knowing the different types of angles in geometry will be
helpful. The nomenclature is straightforward. Surface angle is by

80

A Fuller Explanation

(a)

(c)

(b)

(d)

Fig. 6-6. (a) Surface angle; (b) dihedral angle ; (c) central angle ; (d) axial angle.

now familiar, referring to a corner angle of a polyhedral face (Fig.
6-6a).
Dihedral angles are the angles between adjacent faces, on theinside of a system. (Fig. 6-6b).
A central angle corresponds to a polyhedral edge and is measured
from the exact volumetric center of a system to each end of the edge
(Fig. 6-6c). Central angles provide a way to find exact edge lengthsof a given system for any desired radius.4 Central angles provide aneffective way to record relative lengths, for remember, an angle isindependent of the lengths of its sides. This means that we can list
the complete set of edge lengths for a complex polyhedral orgeodesic system in terms of the central angles corresponding to eachedge, and then directly calculate exact lengths for any desired radius,
with the help of a simple trigonometric equation. This process mightsound complicated at first, but is more expedient than the alternative, which is to specify one set of edge lengths, applying to only one

6. Angular Topology

81"special-case" system. Data for any different size would then have to
be completely recalculated, step by step, from scratch. Central angles
give us data for the general case, applicable to any particularrealization of the same shape. In architecture, this allows us to build
a geodesic dome of any size from one set of central-angle calculations. A pocket calculator is all that is required to simply multiply
the desired radius by twice the sine of half the central angle. 4
An axial angle is the angle between the edge of a polyhedron andan adjacent radius (Fig. 6-6d).
That finishes this chapter, but the subject of angle is never far
removed from any discussion in synergetics.

7
Vector Equilibrium

If you begin to suspect that the concepts hiding behind Fuller's
intimidating terminology are often easier to understand than theirtitles, you will soon find that" vector equilibrium" is no exception.
"The vector equilibrium is an omnidirectional equilibrium of forcesin which the magnitude of its explosive potentials is exactly matchedby the strength of its external cohering bonds" (430.03). If Fuller's
description doesn't make it crystal clear, read on! The VE, as it isusually called, is truly the cornerstone of synergetics.
Vectors are certainly familiar to us by now; but what is meant byequilibrium? The word is by no means esoteric; like "systems," itenjoys considerable popularity these days. That is no wonder, for theconcept is at the root of all phenomena, both physical andmetaphysical. Defined by The American Heritage Dictionary as "anycondition in which all acting influences are cancelled by others,
resulting in a stable, balanced, or unchanging system," equilibrium is
not inactivity, but rather a dynamic balance. 1 This balance is notnecessarily physical, but may be mental or emotional as well. In fact,
so much of experience is characterized by fluctuation in and out offragile balances, that it is easy to understand the word's frequent use-covering everything from structural to emotional to financial
equilibrium.
A simple mechanical model of equilibrium involves a ball and abowl. Allowed to roll around inside the open smooth surface, a ballwill finally come to rest at the bottom of the bowl, requiring renewed
force to set it back in motion. This state is called stable equilibrium.
On the outside of an inverted bowl (or dome) the ball might rest
briefly at the center, but the slightest disturbance will make it roll off-thus demonstrating metastable equilibrium. The third possibility isthat the ball sits on a flat table, in a state of neutral equilibrium (Fig.
7-1).
Nature exhibits a fundamental drive toward equilibrium. Scatteredpockets of varying temperatures will equalize at the mean tempera-

7. Vector Equilibrium

83oFig. 7-1. Stable, metastable, and neutral equilibrium.

ture; opposing forces of different magnitude naturally seek a state ofrest; these differences cannot remain imbalanced. Greater forces
overpower smaller forces, causing motion until they balance out.
Demonstrations of this universal tendency are provided by countlesseveryday experiences. For example, if a massive object sits on too
weak a shelf, the force exerted by gravity toward the earth's center
exceeds the strength of the shelf's restraining force and the object
comes crashing through. Motion continues until a new equilibrium isachieved by the object landing on a sturdy floor capable of matching
the gravitational force with an equal and opposite restraining force.
Apparent motion then ceases, as a stable equilibrium is maintained.
Invisible motion continues, however; atoms never stand still. The
systems approach encourages us to note that we can zoom in toobserve the same event on another level of resolution.
The front door is opened and quickly closed, allowing a rush ofcold winter air into the living room. Freezing temperatures dominate
the corner of the room near the door, while the other side by theradiator is cozy and warm. However, the imbalance quickly disappears; the temperature soon becomes more or less consistentthroughout the room.
Nature's tendency to seek equilibrium is a spontaneous reaction; itis the path of least resistance.
We have already seen that vectors model certain events andreactions of nature. In this discussion, we focus on one specific use
of vectors: to represent forces. The application is straightforward.
Forces push or pull on something. The strength or magnitude of aforce is represented by the length of the vector, and its direction is ofcourse specified by the orientation: frequency and angle, as Buckysays.
It follows, then, that a balance of forces is geometrically modelable. We can create a spatial diagram of the concept of equilibrium,
and in so doing learn more about space's inherent symmetry.
Bucky's love affair with vectors dates back to his World War I
Navy experience. Introduced to vector diagrams of colliding ships in84

A Fuller Explanation

the officer's training program, he discovered that the tiny arrows
contained all the necessary information about the ships' masses andspeeds and compass headings to predict the results of collisions or
the effect of tail winds and other influential forces. Bucky was
fascinated by the economical elegance of the system. These vectors
actually modeled the energetic events of reality-a pleasant contrast
to the mathematics teacher's "lines stretching to infinity." Bucky washooked. "A vector is an experience," he reminisces in a 1975
videotaped lecture, "so I thought, if I could only have a geometry ofvectors, that would be great." 2 This concept was introduced in theprevious chapter, but now we must discern the specific shape of theconfiguration generated by vector diagrams and models.
"Nature's Own Geometry"

We periodically remind ourselves of the purpose behind this geometric journey. Trying to faithfully trace Bucky's footsteps, we seek toisolate the "coordinate system of nature": how Universe is organized.
One of the essential parts of the mystery is how to account forstructural similarities between totally unrelated phenomena, vastlydifferent in both scale and material. The implication is that, rather
than being coordinated, things coordinate themselves. This selforganization occurs according to a set of physical forces or constraints, absolutely independent of scale or specific interactive forces.
In short, space shapes all that inhabits it.
But how? Through what vehicles does nature adhere to thisunderlying order? Let's look at Fuller's fundamental operating assumption:
It is a hypothesis of synergetics that forces in both macrocosmic and microcosmic
structures interact in the same way, moving toward the most economic equilibrium
packings. By embracing all the energetic phenomena of total physical experience,synergetics provides for a single coherent system of geometric principles .... (209.00)

Synergetics seeks to establish the natural laws through which theself-organization of systems in the most diverse fields of scienceoccurs. Science, as we noted earlier, acknowledges a fundamental
drive toward equilibrium, but what else can be observed about thistendency?
Gas molecules buzzing around in a closed-container are suddenly
allowed, by the removal of a barrier, into an adjacent empty compartment of the same size (Fig. 7-2). The molecules rapidly disperse,taking advantage of their new freedom by using the additional room

7. Vector Equilibrium

85

Fig. 7-2

to spread out and slow down. The reverse action-of all the gas
molecules suddenly gathering in one half of a container-has neverbeen observed, just as in the living room disparate temperaturesequalize, but that room will never spontaneously become warm on
one side while the other side suddenly cools off. The closed-container
experiment is the classic model for illustrating nature's en tropic
tendencies. The Austrian physicist Ludwig Boltzmann (1844-1906),
noted for his work in entropy theory, called the resulting dispersal
disorderly behavior. A geometer, however, might observe the individual gas molecules vying for the most room-accomplished of course
by a maximally symmetrical distribution-and not perceive such a
progression as disorder. Both perceptions call it equilibrium.
Spatial Considerations

A symmetrical distribution of "energy events" involves a large
number of equivalent separation distances. For a more tangible
image than provided by gas molecules, picture a large room full ofpeople asked to spread themselves out for stretching exercises. If theroom is sufficiently crowded, a more or less triangular pattern in the
distribution of people can be observed, as a result of individuals'
trying to maximize the area of their territory. Each person feels he or
she has more space when the distances between people are maximized, which is the case when all distances are as close to equivalentas possible. If it's hard to see why that equivalence implies atriangular pattern, read on.
Think about baking cookies on a tray. Intuition rather thangeometrical training tells you that the cookies in successive rows
should be offset to maximize the number which can fit on each tray
without spreading into each other. Observe in Figure 7-3 that a86

A Fuller Explanation

?

~

GOO

000
: Q0W

000

QOO

000
Fig. 7-3. Triangular pattern enables one more row of cookies than square patternwith the same minimum separation distance between cookies.

square distribution with the same minimum separation between
cookies wastes considerable tray area, resulting in fewer cookies thana triangular pattern.
In the same way, people in a room naturally (and quite unconscious of the advantages of triangular distribution) milling around
until each carves out a desirable comfort zone can end up by
increasing the overall symmetry. This organization does not require a
director at the head of the room. Nature behaves in the samemanner, seeking the most comfortable resting position. Forces continue to push or pull until counterbalanced, and in the absence ofother influences, symmetrical considerations dominate. (There is, inFuller's words, an "a priori absolute mystery" of why nature behaves
this way, which is beyond explanation. The question is thus how
Universe operates.)
The advantageous balances suggested by the term equilibrium canbe expressed in terms of symmetry. The properties of space arenecessarily behind all events and reactions in nature; hence Fuller's
assertion that forces in both "macrocosmic and microcosmic structures interact in the same way." Space is the same on every scale,embracing and molding the "most economic equilibrious packings."
With the conceptual foundation in place, we can now describe themodel proposed by Fuller to represent equilibrium. In his words, weseek the simplest "omni-accommodative system" able to model the
behavior of complex systems. Basically, we want to draw, or better

7. Vector Equilibrium87

yet, build that much-discussed balance of forces. To accomplish thedesired result, a model must incorporate two aspects of Fuller's
geometry: first it must consist of vectors, and secondly it must cover
all directions. In short, we want to illustrate an equilibrium ofvectors in space.
Planar Equilibrium

Spatial configurations tend to be difficult to visualize, whereas fiat
patterns are not, so we start with the page. Draw a vector of somearbitrary length-which we designate "unit length"-in any direction. To counteract that force, we position a second vector directly
opposite the first, head to head (Fig. 7-4a). They have the same
magnitude and opposite direction and are therefore balanced.

(a)

,,

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(b)

(d)

,
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(c)

Fig. 7-4. In search of equivalent radial and circumferential vectors.

88

A Fuller Explanation

It is an unstable balance however, easily knocked out of equilibrium by a force from any other direction. Suppose that theoriginal two forces push on a body with equal strength from east andwest. A force from the north or south, even if smaller than theeast-west pair, can easily destroy that unstable equilibrium. So how
can we most efficiently insure the stability of the body in question?
Suppose instead that the forces are directed outwardly from thebody (Fuller's "explosive potential," as quoted in the first paragraphof this chapter). To begin with, imagine four equal vectors, heading
north, south, east and west-that is, in the positive and negative
directions of the X and Y axes. To counteract the four explosive
forces, we need equivalent restraining forces ("implosive" or "embracing"). Fuller's "embracing" vectors are not technically part ofthe conventional language of vectors. Having neither head nor tail,their effect is simply restraining, like a net, and their magnitude isstill assumed to be represented by their length. As strength isgraphically depicted by vector-length, we soon find that there is noeasy way to draw four embracing vectors of unit length. In Figure
7-4b, the ends of our four explosive vectors are interconnected, butthese new lines are approximately 1.414 times as long as the outward
forces. The longer vector lines represent more powerful forces andthus overpower the explosive potential, meaning that the wholedisplay must collapse inwardly.
We might then choose to add more outward forces, maintaining
symmetry by an additional unit vector in between each of theoriginal four (Fig. 7-4c). Now we have eight equal forces emanatingfrom one point, and the resulting eight embracing lines are only
0.765 times as long as the unit length-too small to restrain the
explosive forces. This imbalance leads to outward dispersal of the
hypothetical system.
The sought-after balance will be achieved by "omnisymmetry",that is, maximum symmetry. The desired array must consist exclusively of unit-length vectors-both explosive and embracing.
One way to solve the problem is to picture a square grid of"energy events," interconnected by vectors. Squares provide an easy
starting point because they make up the basic framework of current
mathematics: the XY coordinate system. As before, the length ofvectors which connect "events" represents the strength of theirinterattraction. Because the distance between diagonal corners is
1.414 times the distance between adjacent loci, the attractive forces
represented are that much greater. Imbalance (or lack of equilibrium) in a diagram of forces represents motion. As a result of the7. Vector Equilibrium

89

attractive forces between neighboring energy events, they push andpull on each other until all the disparate separation distances became
equal. Again, like the cookie tray, a pattern of equilateral triangles isestablished. Every single energy event is a uniform distance apartfrom each neighbor, and there are 60-degree angles between all
vector connections. The forces are finally balanced, and the resulting
array informs us about the fundamental symmetry inherent in a fiat
surface.
Going back to the radial vector diagram, we now arrange six
vectors emanating from a point. The embracing vectors will be thesame length as the "exploding" group. An inescapable consequenceof this fact (obvious in retrospect) is that angles between all vectors
are also identical-not just those between radial vectors, but also
those between circumferential and radial vectors (Fig. 7-4d). No
other arrangement has this property, because 60-degree angles areintegral to equilateral triangles.
A diagram of radial vectors can be thought of as an apple pie
divided into some number of pieces. These pieces are always triangular, with two radial edges and one circumferential edge. Because of
the fact that the angular sum in every fiat triangle is 180 degrees, theonly way for a triangle to have all angles the same is with threesixty-degree angles. Therefore, the only angular measurement thatwill allow us to divide the vector pie "omnisymmetrically" is 60
degrees, requiring that unity (360 degrees) be divided six ways. The
procedure is straightforward so far. In the plane, equilibrium isdemonstrated by a hexagon (Fig. 7-4d).
Now we make the leap into space, with its accompanying leap incomplexity. It may be difficult to visualize a spatial array, especiallynoncubical configurations, but taking it step by step, we shall be able
to develop and understand Fuller's model.
In terms of vectorial dynamics, the outward radial thrust of the vector equilibrium is exactly balanced by the .;ircumferentially restraining chordal forces:
hence the figure is an equilibrium of vectors. All the edges of the figure are of equal
length, and this length is always the same as the distance of any of its vertexes fromthe center of the figure. (430.03)

A "geometry of vectors," Fuller reasoned, must be "omnidirectionally operative" - hence, radially oriented and omnisymmetrical.
Following the planar example, we want some number of vectors
emanating from an origin, situated so that the distances between
vector end points (vertices) are not only all equal to each other, butalso exactly equal to the length of the radial vectors.

90

A Fuller Explanation
Fig. 7-5. Vector equilibrium.

Cuboctahedron as Vector Equilibrium
We can understand the symmetry of the plane by observing that
although any polygon can be made to have equal edge lengths, only
the regular hexagon can have edges equal in length to the distance
between the polygon's center and its vertices. In the same way,although there are many regular and semiregular polyhedra withequal edge lengths, there is only one spatial configuration in which
the length of each polyhedral edge is equal to that of the radial
distance from its center of gravity to any vertex: the cuboctahedron(Fig. 7-5). 3 This shape therefore is the only one that allows therequisite arrangement of vectors to demonstrate equilibrium.
We first saw the cub octahedron as the degenerate truncation ofboth the cube and the octahedron, but at that point in our investigation we were only looking at surface topology. Now diving into theinterior shape, we discover this unique property of equivalence.
Table IV compares the radial lengths of various familiar polyhedra
given unit edge lengths. Only in the cuboctahedron-hereafter referred to by Fuller's term, vector equilibrium or VE-can the radius
be of unit length.
Again, in order for all vectors to be exactly the same length, theangles between them-both radial and circumferential-are necessarily equal. In Figure 7-5, the VE is shown with both radial andFig. 7-6. (a) Cuboctahedron; (b) twist cuboctahedron.

7. Vector Equilibrium

91

Table IV
Unit-edge polyhedron
Tetrahedron
Octahedron
Icosahedron
Cube
Pent. dodecahedron

VE

RadiusCentral
Angle

Axial
Angle

0.61240.7071
0.95110.86601.40121.0000109.47°90.00°
63.43°
70.53°
41.81 °
60.00°

35.26°
45.00°
58.28°
54.76°
69.04°
60.00°

edge vectors. Radial vectors connect the twelve vertices to thesystem's center, thereby forming twenty-four radiating equilateraltriangles, corresponding to each polyhedral edge and pointing inwardly. We should not be surprised to find an array of equilateraltriangles in the VE, for this is the only polygon with equal distances
and angles between all points. And, as vectors incorporate bothmagnitude and direction, an equilibrium of vectors must-in Fuller's
terminology-balance both angle and frequency. Sixty-degree angles
are inevitable.
What if we had anticipated the necessity of 60-degree angles? We
could then have started this part of the investigation by specifying
the angle between radial vectors and looking for the resulting implications of that choice. The discovery then would be that the necessary 60-degree gaps in a spatial array generate exactly twelve vectors,
just as six is the outcome in a plane. Had we started thus-with the
choice of angles-we would have had to check the resulting vector
lengths, to find out that indeed they are all the same. In either case,the end result is extremely satisfying.
VE: Results

Our first encounter with the vector equilibrium in Chapter 4- thenwe called it the "cuboctahedron" -illuminated its direct relationshipto the octahedron and the cube. We now elaborate on our description of the VE, and before the end of this investigation we shallknow almost everything about this extremely important shape. Let's
begin here with its major characteristics.
Above all, it is the "omnidirectional arrangement of forces." Thisequivalence is unique to the VE.
Secondly, this shape bears an interesting relationship to otherfamiliar polyhedra. Its twelve radii form eight symmetrically arrayed

92

A Fuller Explanation
Fig. 7-7. Eight radiating tetrahedra alternate with six
half octahedra.

regular tetrahedra-corresponding to the YE's eight triangular faces.
Figure 7-7 emphasizes the tetrahedra, which radiate outward edge toedge, creating six cavities in the shape of square-based pyramids.
Again, because of the uniform edge lengths everywhere, these cavities are actually perfect half octahedra, corresponding to the six
square faces of the YE, which in turn correspond to the six faces of
the cube, as was revealed by degenerate truncation in Chapter 4.
Thirdly, "the pattern of this nuclear equilibrium discloses four
hexagonal planes symmetrically interacting and symmetricallyarrayed ... around the nuclear center" (981.11). If you look closely at
Figure 7-8 the four hexagons are clearly visible: one parallel to thehorizon, one in the plane of the page, and two more, slanted to theright and to the left, at 60 degrees to the horizon. As we might haveexpected, the vector equilibrium consists-in a way exclusively-of
hexagons. The symmetrical properties of hexagons with respect tothe plane are evident (refer back to Fig. 7-4), and so the discovery ofintersecting hexagons in a spatial equilibrium of vectors is notsurpnsmg.
However, intuition cannot as easily predict the number of hexagons. An array of equivalent vectors (taking into consideration bothmagnitude and angular orientation) is achieved by exactly four
evenly spaced intersecting hexagons. Thus the existence of four
fundamental planar directions ("dimensions"?) describes one aspectof the inherent shape of space.

Fig. 7-8. Four hexagonal cross-sections of the YE.

7. Vector Equilibrium

93

These hexagons are exactly parallel to the four faces of thetetrahedron; having the same angular orientation, they are identicalmathematical planes. The only difference is that in the VE theyintersect at a common center, while in the minimum system they
together enclose space.
Also fascinating is the fact that each of the twelve radiating
vectors is perfectly aligned with an opposite vector-exactly 180
degrees apart. Thus the twelve can be seen as six intersecting lines
with a positive and negative direction (each line twice the length ofthe original unit vector)-just as the XYZ axes are three lines
intersecting to define six directions: three positive and three negative, evenly spaced with intervening angles of 90 degrees. Onceagain, these six intersecting lines are parallel to the tetrahedron'sedges. It was not at all obvious from our initial requirements for avector equilibrium display that the resulting radial lines would becollinear pairs, nor that these six (double-length) vectors would eachlie in the same plane as two others, producing four precisely defined
hexagons.
Our goal was to create a radial display of evenly spaced unit
vectors. In so doing, we arrive at two fundamental observations
about the order inherent in space: the existence of four distinct
planes of symmetry and six linear elements. Both aspects are firstexhibited in nature's choice of minimum system and secondly reinforced by her unique equilibrium configuration.
Degrees of Freedom

The subject of twelve fundamental directions of symmetry, with their
six natural positive-negative pairs, leads directly to a discussion ofthe "twelve degrees of freedom" inherent in space. The term isalmost self-descriptive, but can be best explained in reverse. That is,
we explore the number of degrees of freedom inherent in space (andthus effecting every system) in terms of how many restraining forces
are necessary to completely inhibit a system's motion. What is theminimum number of applied forces necessary to anchor a body inspace?
Again, we can start with a planar analogy. Imagine a fiat circular
disk, such as a coaster, lying on a table and held in place by two tautstrings pulling in opposite directions. The disk looks stable, but
actually is free to move back and forth, at 90 degrees to the line of
the two restraints (Fig. 7-9a). So, we try applying three tension

94

A Fuller Explanation

(a)

(b)
Fig. 7-9

forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's
position is fixed. Actually, it turns out that only the location of theexact center of the circle is fixed, for the disk is free to rotate slightly
in place. Because rotation involves motion directed at 90 degrees toall three strings, there is nothing to restrain the circle from twistingback and forth, as shown in Figure 7-9b. Three additional strings tocounteract each of the original restraints would have to be added toprevent all motion, for a total of three positive and three negative
vectors.
In space, a similar procedure involves a bicycle wheel. Supposethat our goal is to anchor the hub with a minimum of spokes. At firstglance this may appear to be the same problem as the previous
planar example; however, in this case, the hub has both width andlength. Both circular ends of the narrow hub-typically about a halfinch wide and 3 inches long-must be stabilized. With only six
spokes attaching the hub to the rim (three fixing the position of eachend, as shown in Figure 7-lOa), the system feels quite rigid; force canbe applied to the hub from any direction-up, down, back or forth-without budging it. However, the hub has no resistence to an
applied torque, the effect of which occurs at ninety degrees to the
spokes, and is therefore able to twist slightly about its long axis.
Three more spokes at each end, to counterbalance the original six,
remove the remaining flexibility. All twelve degrees of freedom are
finally accounted for, with a minimum of twelve spokes (Fig. 7-10b).
This experiment is quite rewarding to experience-well worth
trying for yourself. You don't need to go as far as dismantling abicycle wheel; just find a hoop of any material and size and a short

7. Vector Equilibrium

(a)

95

(b)

Fig. 7-10. Minimum of twelve spokes needed for stability.

dowel segment, and then connect the two with radial strings added
one at a time until the hub suddenly becomes rigidly restrained. It is
enormously satisfying to feel the hub become absolutely immobile(all "freedom" taken away), with the surprisingly low number oftwelve spokes. 4
What both the planar and spatial procedures indicate is that
degrees of freedom are both positive and negative. In anchoring the
hub of the bicycle wheel, there at first appear to be six degrees of
freedom; however, each has a positive and negative direction. In
conclusion, degrees of freedom measure the extent of a system's
mobility: how many alternative directions of motion must be impeded
before the body in space is completely restrained. 5
The twelve vectors needed to restrain a body can also be
omnidirectional, instead of the basically planar organization of the
spoke wheel. Fuller takes us through a similar sequence in Synergetics, which starts with a ball attached to one string. The ball is free toswing around in every direction; the only restraint is on the radial"sweepout" distance. The ball's motion is thus free to describe aspherical domain. The addition of a second string restricts the ball tomotion within a circular arc in a single plane (Fig. 7-11a). A third
string allows the ball to swing only back and forth, in a linear path.
The ball can always be pushed slightly out of place, no matter how
taut the three strings (Fig. 7-11b). And just as, in our search for theminimum system, a fourth event suddenly created insideness andoutsideness, by adding a fourth string to the ball, its position issuddenly fixed. (" Four-dimensionality" again.) In their most symmetrical array, the four strings go to the four vertices of an imaginary tetrahedron, and are therefore separated by approximately 109.47
degrees, the tetrahedron's central angle (Fig. 7-11c).

96

A Fuller Explanation

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7. Vector Equilibrium97

But of course that's not the end of the story; the ball is still free totwist in place. To prevent this slight rotation, three strings must be
attached to each location of the original four (Fig. 7-11d). This result
is related to the fact that three is the minimum number of coordinates needed to specify the location of a point in space, withreference to the origin of a coordinate system.
The whole picture is falling into place. Every physical body has
four basic sides, or four corners: two points alone are only collinear,and three are only coplanar; not until there are four corners can theproperty of spatial existence be recognized. As a result, any physicalbody must be held at four noncoplanar points, with three restraints
at each point, in order to be stabilized (Fig. 7-11d).
This result suddenly ties in to the earlier discussion of pattern
integrity. Triangles are necessary for stability. Therefore, while theball was seemingly held in place by four restraints, it could stillrotate locally because the locus of each individual restraint could not
be stable without triangulation-subsequently provided by the addition of three strings per locus.
Bucky explains the situation further. Consider the ball with theoriginal four restraints. The strings impinging on the ball create four
vertices without supplying the necessary six edges to stabilize theirposition with respect to each other. They essentially form an unstable quadrilateral rather than a stable tetrahedron. The lesson is thesame. There are four fundamental corners in every system, and eachmust be triangulated: 4 X 3 = 12. Thus there are twelve degrees of
freedom, tetrahedrally organized. Twelve is a frequently recurring
number in synergetics, a fundamental part of space and geometry, as
we shall see again and again.
The above procedure describes Fuller's own interpretation of"degrees of freedom," which must be distinguished from Loeb's
analysis of the concept, as briefly explained in Chapter 5.
The introduction to vector equilibrium is now complete except for
one philosophical consideration. It is important to realize that thewhole discussion is about conceptual-never actual-balance; equilibrium in any physical form can only be an approximation. No
matter how exactly centered the hub of the bicycle wheel seems andno matter how tight the spokes, gravity's pull on the hub will always
exert more tension on the upper spokes, leaving the lower spokes
ever so slightly slack and imperceptibly curved. The balance isimperfect. Moreover, energetic motion never ceases. The air molecules in the living room do not stop their vigorous motion once the98

A Fuller Explanation

temperature is consistent. The floor and fallen object press together
in a persistent dynamic exchange, encompassing furious activity at
the atomic and molecular levels.
There is always motion in real systems: some (however minute)residual springiness in tension materials, as well as ever-present
invisible bustling activity on the atomic scale. We cannot see theenergetic motion in most systems, and so our perception is that of astate of perfect equilibrium. And indeed, for all practical
purposes-that is, for a given level of resolution-we can have astable balance.
But Fuller cannot in good conscience leave it at that. He reminds
us that real equilibrium would mean an end to all, or "Universal
death." An end to aberrations and imperfections is an end to motionand energy. All physical reality-life and nonlife alike-consists
only of energy. Hence there is no absolute equilibrium:
Nature is said to abhor an equilibrium as much as she abhors a perfect vacuum
or a perfect anything.... The asymmetric deviations and aberrations relative toequilibrium are inherent in the imperfection of a limited life .... Despite theuntenability of equilibrium, it seemed to me that we could approach or employ it
referentially .... A comprehensive energy system could' employ the positive andnegative pulsations and intertransformative tendencies of equilibrium. (420.041)
The vector equilibrium is a condition in which nature never allows herself totarry. The vector equilibrium itself is never found exactly symmetrical in nature's
crystallography. Ever pulsive and impulsive, nature never pauses her cycling at
equilibrium: she refuses to get caught irrecoverably at the zero phase of energy.
(440.05)

All events, all systems exist as a result of their constant fluctuation in
and out of ideal equilibrium-far too rapidly for perception. Fuller'sgoal was to develop a model for what that theoretical ideal must look
like, in terms of spatial properties. But vector equilibrium is not a
structure; he is quick to point out the distinction: it is a system-tobe "comprehensively" grasped by "metaphysical minds":
Synergetics ... accommodates Heisenberg's indeterminism of mensuration inherent in the omniasymmetry of wavilinear physical pulsations in respect to the only
metaphysical (ergo, physically unattainable) waveless exactitude of absolute equilibrium. It is only from the vantage of eternal exactitude that metaphysical mind
intuitively discovers, comprehends, and equates the kinetic integrities of physical
Universe's pulsative asymmetries. (211.00)

The concept of imperfection can only be held relative to the mind's
grasp of theoretical perfection. In other words, "pulsative asymme-

7. Vector Equilibrium

99

tries" require a frame of reference in order to be defined andregistered.
Time is responsible for these asymmetries. Separate time out ofthe picture, and you are left with the absolute perfection of timelessness. Absolute equilibrium exists sub-time or meta-time; thepassage of the shortest instant of time will reveal "pulsative asymmetries." But metaphysical mind has an all important need fortimeless models, through which to understand Universe.

8Tales Told by the Spheres:
Closest Packing

Much has been written over the years by mathematicians andscientists about the problem of "closepacking" equiradius spheres.
It's not a subject that the rest of humanity has tended to get excitedabout; however, the orderly patterns revealed by these packings are
unexpectedly fascinating. Closepacking equiradius spheres might atfirst sound like the type of abstract mathematical game Fuller railed
against; after all, there's no such thing as a sphere. But if nature
exhibits no examples of pure spheres-that is, no perfectly continuous surfaces equidistant from one center-we can still discuss theconcept of a spherical domain. Imagine various approximations of
the model, such as a soap bubble or, less fragile, a Ping Pong ball.
The concept of multiple equiradius spheres turns out to be quite
useful, providing a superb tool with which to investigate the properties of space. Let's look into some of the reasons why.
Equilibrium: Equalization of Distances

The connection to equilibrium is perhaps the most important reason
to experiment with sphere packing. A sphere is defined as the locusof all points at a given distance from a central point; consequently,
in an array of tangent spheres, their centers will be separated by auniform distance. The configuration developed in the previous
chapter to represent vector equilibrium-requiring equal lengths inall directions-can be created quite simply with the aid of thismodel. If one sphere is completely surrounded by a number ofspheres of the same size, the distances between the internal sphere's
center and the centers of all surrounding spheres are necessarily thesame as the distance between the centers of adjacent external spheres,provided all spheres are in contact with each other. The resulting
cluster is shown in Figure 8-1, along with a cross-section of thepacking to illustrate that the distances are the same. Closepacked
spheres automatically set up an array of evenly spaced points. Equal
distances represent balanced forces: ergo, equilibrium.

8. Tales Told by the Spheres: Closest Packing

101

Fig. 8-1

Symmetry versus Specificity of Form

A sphere is the form of "omnisymmetry" in spatial reality. Symmetry describes the degree to which a system can be rearranged without
detectable change. The sphere's shape presents no corners, no angles- in short, no landmarks-by which to detect rotation or reflection.
Its very shapelessness enables us to explore the shape of space.
Furthermore, the total absence of angular form makes the precisely
sculpted shapes generated by packing the identical" shapeless" units
together all the more surprising. It is easy to see that individual
spheres, as omnisymmetrical forms with neither surface angles nor
specific facets to mold the form of clusters, cannot determine throughtheir own shape the overall shape of packings. In conclusion, we are
not so much interested in the (" nondemonstrable") spheres themselves, as in using sphere-packing as a medium through which spatialconstraints can take visible shape.

Organization of Identical Units

Finally, the standard model of an atom is spherical: packets ofenergy are spinning so rapidly about a tiny nucleus that the atom
can be considered occupying a spherical domain. (In fact, the orbit
of any object spinning in all directions defines a sphere.) We cantherefore pack spheres together in the hope of learning about atomic
and molecular aggregations. To state the problem more generally,
the organization of identical units is an important theme in biology
and chemistry. All sorts of units-such as atoms, molecules, cells,
DNA nucleotides-must be organized to function cooperatively in
structures far more complex than the individual units themselves.
Spatial constraints are responsible for much of the superb organiza-

102

A Fuller Explanation

tion of biological phenomena-allowing and encouraging certain
configurations while prohibiting others. Yet, despite the influential
role of space, scientific thought does not as a matter of course takethis into consideration.
Sphere-packing can be thought of as a method of blindly gathering evidence; we experiment with these identical units without
knowing the outcomes, and space enters in to direct traffic. The
resulting configurations are absolutely reliable. We are thereby able
to observe the shape of space, manifesting itself through the innocent
spheres.
New Level of Focus

Despite our discipline of viewing whole systems, we have reached apoint at which we must zoom in to look closely at certain details of
Fuller's Synergetics. The sections called" Closest Packing of Spheres"contain some of the most difficult passages in his book, rendering
Fuller's observations inaccessible without considerable perseverance.
Not only is the description hard to follow, but these patterns seem toelude application. It is therefore especially important to understandthe logic behind Fuller's use of sphere-packing in an investigation ofnature's coordinate system. Otherwise, it will be difficult to see howthese details fit back into the big picture. Even though the immediategoal of this text is to clarify the configurations described by Fuller, alist of results, no matter how clear, is not likely to be interesting
unless the premise behind the search is understood. At this point, thereader may even have thought of further reasons to add to the ones
stated above, for there are many dimensions of this issue. However,as it probably remains difficult to predict or visualize the patterns
themselves, we bravely proceed.
Background: Closepacking

Packing spheres together with a minimum of interstitial space is aproblem that still presents a challenge to mathematicians. l (Actually
the problem has been solved, but it turns out to be extremely
difficult to prove that the solution is indeed maximally dense.) For
our goal of exploring Fuller's studies, we only deal with one of thetwo types of closest packing described below.
Once again, we start with the plane in order to establish a firm
hold on the concept. Suppose we want to fit the largest possible

8. Tales Told by the Spheres: Closest Packing

103

Fig. 8·2

.",.-

....

,

,_ ..... /

\

I

\
I

number of pennies on the surface of a small table-another way ofsaying we want the pennies to lie as close to each other as possible.
As we saw in the previous chapter, a square grid of tangent pennies
wastes considerably more table space than a triangular array. Observe in Figure 8-2 that two tangent pennies create a valley thatnaturally embraces a third penny. All pennies are therefore allowed
as close together as physically possible if every penny is situated in avalley created by two others.
In the same way, there is only one closest-packing arrangement of
spheres in the plane: each sphere must be in contact with six others(Fig. 8-3). A second identical layer can be placed on top of the first,
with its spheres all landing in nests created by three neighbors on theFig. 8-3. Six spheres c10sepack around one.

104

A Fuller Explanation

first layer. To achieve our goal of packing spheres as close together
as possible, we have thus far had no choice as to the next step. A
third layer however can be superimposed on the second in one oftwo different ways to maintain maximum density. The spheres of thethird layer can either be placed directly above the spheres of the firstlayer or above the nests in the first layer. A schematic comparison of
the two packings is shown in Figure 8-4. The former is called

Fig. 8-4. Cubic packing (top) versus hexagonal packing (bottom), showing threelayers of spheres in each packing.

8. Tales Told by the Spheres: Closest Packing

105

hexagonal closepacking; the latter, cubic closepacking. In both cases,
every sphere touches exactly twelve others-as we might have anticipated from our VE studies. The difference between these two
packings is explained in the following description.
Instead of trying to imagine indefinitely large planar expanses, we
focus on a small portion of the closepacking-the arrangement
surrounding a single sphere. We start with one ball on a table; sixothers closepack around the first, and find themselves exactly tangent to each other. As with the pennies, there is no choice as to thenumber of spheres in the cluster. The fiat hexagon of spheres creates
six separate "nests" of three spheres each, as seen in Figure 8-3, but
additional spheres of the same size, sitting in anyone of the six

(a)

(b)

Fig. 8-5. Hexagonal (left) versus cubic (right) : twelve spheres pack tightly around
one.

106

A Fuller Explanation

nests, partially block adjacent nests. As a result, there is only room
for a ball in every other nest, allowing a total of three nesting
spheres on the hexagonal cluster. These three balls (by magic or else
by inherent spatial properties) rest exactly tangent to each other, aperfect equilateral triangle (Fig. 8-5a, top). The arrangement hasneither leftover space nor crowding; all three spheres on top aretightly shoved into nests, and-because the planar group (six aroundone) are clearly as close together as possible-all ten spheres areconvincingly closest-packed.
We can then flip the whole package over and repeat the procedureon the other side. We again have two choices: the second three-balladdition can be placed either directly above the three on the bottom(meaning both the top and bottom triangles are pointing the sameway), or it can be oriented the opposite way (Fig. 8-5a, b). The
former choice (hexagonal) outlines the vertices of a polyhedron inwhich the squares are adjacent to other squares (in three pairs
meeting at the "equator"), as are six of the eight triangles (Fig. 8-5a).
The latter choice (cubic) consistently alternates triangles and squares,
so that squares are entirely framed by neighboring triangles-andtriangles by squares (Fig. 8-5b).
The latter arrangement - "our friend the vector equilibrium" as
Bucky says-is the more symmetrical of the two choices, and istherefore used as the basis of Fuller's subsequent sphere-packing
studies. 2
Planes of Symmetry
The following observations pertain to an indefinite expanse of cubically closepacked spheres. For example, imagine a room full of Ping
Pong balls so tightly nested together that every ball touches exactly
twelve others as described above. The idea of a room full of balls at
first suggests such chaos that the precise organization arising out of
the requirements of closest packing is truly remarkable. The array
contains seven planes of symmetry, characterized by two different
types of cross-section. The first is obvious, because we generated the
cubic packing with successive layers of triangulated planes. However,
it is not necessarily obvious that there are four different orientationsof triangulated planes-parallel to the four faces of the tetrahedron(and therefore to the VE's four intersecting hexagons). Even though
we built this array by stacking triangular layers of balls in only one
direction, the result incorporates parallel triangulated layers in four
different directions. (We further note that all four hexagons of the8. Tales Told by the Spheres: Closest Packing

107

VE are preserved in cubic packing, whereas in hexagonal packing,hexagons are formed in only one orientation, the horizontal plane.)Secondly, there are three distinct planes characterized by a square
pattern of spheres. This discovery seems to contradict our expectations, for we have learned that squares are not closepacked.
However, the emergence of three mutually perpendicular square
patterns is an inescapable by-product of nesting triangular layers.
The square planes correspond to the VE's square faces, which consistof three mutually perpendicular pairs of parallel faces-just like thefaces of a cube.
In a space-filling array of closepacked spheres, these seven planes
extend indefinitely-with neither curve nor bend. A cross-section ofsuch a packing has a square or triangular arrangement, depending on
which way the packing is sliced. So although we started with only
triangulated layers (in order to create a maximally dense array ofspheres), square cross-sections automatically arose, just as octahedral
cavities automatically arose next to the radiating tetrahedra in thevector equilibrium. This is the shape of space.
It is interesting to note that, although when we stacked triangulated layers it was necessary to make a decision at the third layerthat led to two different packings, if we were to start out instead withthe square layers (unstable though they may be), there is only one
way to proceed. Successive square layers can be placed on top ofeach other so that each ball lands in a square nest (as opposed tobeing placed directly on top of another ball, creating an array ofcubes which would clearly not be closepacked). The internesting oflayers stabilizes the otherwise unstable separate layers. Once twolayers are packed together, every ball nests in a group of four ballson the adjacent layer-creating half-octahedral pyramids (Fig. 8-6)
separated by the inevitable by-product tetrahedra. Continuing tostack square layers in this way, cubic packing-rather than hexagonal-emerges. The vector equilibrium array is thus generated automatically, with no decisions along the way, by simply stacking
square layers. Each and every ball is surrounded by exactly twelve
others, in the more symmetrical of the two possibilities.
It is satisfying to reflect on the exquisite logic of this tradeoff:although balls arranged in square patterns are not as closely packed
as triangular planes, the nests are deeper. A ball placed in any
four-ball nest (to start a second layer) sinks deeply into the cluster;in comparison it seems perched on top of the tight triangular nest.
Therefore, successive square layers, although inefficient in themselves, fit more closely together than triangulated layers.

108

A Fuller Explanation

Fig. 8-6. Five oranges creating half-octahedron.

Part of the challenge to mathematicians in proving that thehexagonal and cubic packings qualify as the solution to minimizing
interstitial space is the fact that spheres in these two packings
occupy just over 74% of the available space, while the four-balltetrahedron alone is able to occupy 77.96% of its overall volume. l It
is easy to accept that four balls cannot be pushed any closer together
than the tetrahedral cluster and accordingly that the latter figure is
the maximum density. Therefore, 74% seems insufficient-not easy
to accept as the solution to the problem of closest packing. However,there is no getting around the fact that the six-ball octahedral cluster(Fig. 8-7) is less dense than the four-ball tetrahedron - as the former
has more leftover room in the middle-and that the constraints ofspace are such that tetrahedral sphere clusters simply cannot be
extended indefinitely by themselves. Attempting to fill space exclusively with tetrahedral groups, we quickly discover awkward leftovergaps-spaces not quite big enough to contain another sphere. In
order for spheres to be both consistently tangent and tightly nested
together, we have to allow the naturally alternating tetrahedral andoctahedral clusters. The problem would be remarkably easy if spheres

Fig. 8-7. Tetrahedron and octahedron.

8. Tales Told by the Spheres: Closest Packing

109

could pack tetrahedrally in an indefinite array, but they cannot. No

amount of force can change this constraint; space is simply notshaped that way. Twelve around one, creating 60-degree angles bothradially and axially, with alternating tetrahedra and octahedra, is theclosest packing.
Fuller Observations

Having settled on the most symmetrical and dense sphere packing,to faithfully present the characteristics of space, we are ready to
explore the shapes and "periodicities" observed by Fuller. The
reliable precision of these patterns indicates that they are molded by
space, not by imposed design.
What are these patterns that captured Fuller's attention so longago (and kept it for fifty years)?
We start very simply with the phenomenon of "triangular numbers," a sequence of numbers in which each successive term is equal
to the previous term plus the number of terms so far. These numbers
can be generated by triangular collections of balls, arranged as in a
rack of billiard balls. The total number of balls in each triangle, in aseries of progressively larger groups, is a triangular number. Figure
8-8 shows the first five groups. The first member of the sequence is"I"; the second is obtained by adding two to the first, to get "3"; thethird, by adding three to the other two, to get "6"; and so on. Eachsuccessive number is generated by the addition of a row with one
more ball than the last. The sequence of numbers thus generated(1,3,6,10,15,21,28, ... ) is specified by (n 2 - n)/2 for n =2,3,4, .... (n = 1 corresponds to 0, which is not-strictly speaking-modeled by a triangle).
1

2

3

4

Fig. 8-8. Triangular numbers.

5

110

A Fuller Explanation

The formula (n 2 - n) /2 might appear to be a more difficult way
to obtain these values (and certainly for small groups its easier justto count balls), but of course for the twentieth or even the ninth
triangular number, it's far more direct to subtract 10 from 10 2 anddivide that by 2 to get 45 than to draw rows and rows (nine rows) of
circles! Mathematics presents a shortcut-otherwise known as a
generalized principle.
So far we have a numerical progression (n 2 - n) /2 that happens
to be modeled by triangular clusters. We might have chosen to
discuss a variety of other sequences, for example, the numbers
generated by n 2 : 1,4,9,16,25, ... , which could be labeled "squarenumbers" because they are geometrically represented by squareclusters. But we have a specific motivation for paying attention to
triangular numbers, because of one especially significant characteristic: the nth term is the number of relationships between n items, forn = 1,2, 3,4, .... For example suppose that for any given number ofpeople, we wish to know how many telephone lines are required tolink everyone to everyone else by a private line. The answer is thetriangular number corresponding to a number of rows one less than
the number of people. Two people require only one line; threerequire three; four require six (as portrayed in Figure 3-1), and nrequire (n 2 - n) /2 private lines. It is no longer far-fetched toimagine applications for this formula. The real lesson from triangular numbers is that significant algebraic expressions, such as (n 2 n )/2, the number of relationships between n events, can be represented geometrically. That much established, we proceed to the next
develop mer t.
Tetrahedra
We can stack triangular arrays of decreasing size, creating tetrahedral clusters. (We could thereby isolate a sequence of "tetrahedral
numbers.") But let's go back to the very beginning.
One sphere alone is completely free to move, and closepacking isof course not an issue. Suppose we have two billiard balls tangent toeach other. If the only requirement is that the two balls stay incontact, we shall observe that they are free to roll around eachother's entire surfaces (Fig. 8-9a). We then introduce a third ball,allowing it to roll in any direction while touching at least one of thefirst two balls. It eventually rolls into the valley between the othertwo, establishing a naturally stable triangle (Fig. 8-9b). At this point
we require all three balls to stay in contact, and discover that they

8. Tales Told by the Spheres: Closest Packing

(a)

(b)

111

(c)

Fig. 8-9. "Four balls lock."are still able to roll, but only inward or outward (toward or awayfrom the triangle's center, in tandem) "like a rubber doughnut," touse Bucky's words. The freedom of motion of each sphere is thus
considerably more limited-spinning about one specific horizontal
axis instead of unrestrained motion in every direction.
A fourth ball rolls across the surfaces and lands comfortably inthe triangular nest. Suddenly, all four balls are locked into place,unable to roll or move in any direction (Fig. 8-9c). This is the firststable arrangement, with the requisite minimum of four. The tetrahedron is once again at the root of our investigation.
At this point, it may be illuminating to construct some of thesestructures, for example with Styrofoam balls and toothpicks, or
small plastic beads and glue. One particularly satisfying demonstration involves bringing four spheres together and trying to create asquare. It is easy to feel how unstable that arrangement is: the balls
gravitate naturally toward the tight tetrahedral cluster. Fuller placed
considerable emphasis on the benefits of hands-on construction to
gain thorough familiarity.
The theme of Fuller's tetrahedral sphere packings is the presenceor absence of nuclei. The word "nucleus" evokes the image of acentral ball spatially surrounded by other balls, which is exactly theway Fuller uses it for the VE packings. However, his observations
about tetrahedral patterns are based on a somewhat different approach. Most of the inpenetrability of the sphere-packing sections in
Synergetics can be removed with one simple clarification: a "nucleus"
in VE packings is defined as a ball at the geometrical center of thewhole cluster, whereas a "nucleus" in tetrahedral stacks is a ball at
the exact center of an individual planar layer.
Start with the four-ball tetrahedron developed above, which consists of a fourth ball added to a triangle of three others. Next, thesimple four-ball tetrahedron is placed on top of a flat six-ball

112

A Fuller Explanation

Fig. 8-10

"
Gi!IJ
- -:

-

- , -:: . ...;7 ,.

~-triangular base, creating a tetrahedron with three balls per edge (Fig.
8-10). There are three layers, with ten balls altogether- six plus
three plus one. Throughout his sphere-packing studies, Fuller uses
the number of tangency points per edge (in other words, the numberof spaces between spheres along an edge of the cluster, rather than
the number of spheres) for the assignment of frequency. The four-balltetrahedron is thus "one-frequency" (as is appropriate for the firstcase), and the next case, the ten-ball tetrahedron, is "two-frequency."
We can visualize that each sphere-cluster polyhedron corresponds toa line drawing (or toothpick structure) in which the spheres' centersloca te vertices which are interconnected by lines (or toothpick edges)
through tangency points. Recalling that "frequency" is defined as
the number of modular subdivisions, the justification for Fuller's
frequency assignment is evident from this translation, because thenumber of subdivisions (or line segments) per edge corresponds tothe number of spaces between spheres, rather than to the spheres
themselves, which correspond directly to the vertices (Fig. 8-10) .
Triangular clusters, each with one more row than the last, arestacked to create larger and larger tetrahedral packings. We place theten-ball (two-frequency) tetrahedron on top of the next triangularbase, which itself consists of ten balls, to get a three-frequency
tetrahedron, with twenty spheres altogether (Fig. 8-10). We have thus
begun a list of values that might be called tetrahedral numbers:
4, 10,20, followed by 20 plus the additional triangular layer of 15, tototal 35 (Fig. 8-11). The progression can be continued indefinitely.
Consider the different individual layers. There is a ball in theexact geometric center of some of the triangular groups, while others,
having three balls around the exact center instead, are left with acentral nest. Successive triangular clusters reveal a specific pattern :every third layer has a central ball, or nucleus. Fuller describes thisprogression as a "yes-no-no-yes-no-no" pattern. To see how it
works, let's look at the first few members of the sequence. One ball

8. Tales Told by the Spheres: Closest Packing

113

Fig. 8-11

alone is automatically central-"a potential nucleus" in Fuller's
words; it earns a "yes." The next layer, the three-ball triangle, has anest, but no nucleus; that merits a "no;" likewise for six ("no"). Not
until the ten-ball (three-frequency) group is there a nucleus-shownas the dark ball in Figure 8-12 ("yes" again). Each "yes" case (withnucleus) consists of a hexagonal arrangement with three additional
corners tacked on, to complete a triangle. The rest ("no" cases) areorganized triangularly from the center out to the corners-simple
threefold rotational symmetry, containing no central hexagon.
Fuller calls our attention to this periodicity (or pattern) within thesystem: starting from the top, the pattern is Y-N-N-Y-N-N-Y ...

Fig. 8-12. "Yes-no-no-yes-no-no."114

A Fuller Explanation(notice that "N" can stand for "nest" as well as for "no"):415.55 Nucleus and Nestable Configurations in Tetrahedra: In any number ofsuccessive planar layers of tetrahedrally organized sphere packings, every third
triangular layer has a sphere at its centroid (nucleus.)

Fuller's yes-no-no pattern describes the presence of nuclei in certain
layers of a pyramid; he does not ask which pyramids of gradually
increasing frequency contain an overall nucleus (at the center ofgravity.) This is a subject open for further exploration, which weleave for the time being as we continue to explore Fuller's material.
Vector Equilibrium

In cubic packing, twelve spheres surround one sphere, with eachsphere tangent to every neighbor and without any gaps. This perfectgeometric fit of the thirteen omnisymmetrical forms provides a basis
for understanding the fundamental directions inherent in space. But
we saw that the configuration is more specific than just a numericalconsistency; the spheres outline the vertices of the cuboctahedron, or
VE. This shape seems to appear out of nowhere. Created by thecluster of cornerless spheres shoved together, this result is as counterintuitive as it is reliable. The more we learn about the shape of space,however, the more natural the appearance of the VE-or any
manifestation of "twelve degrees of freedom" - becomes.
Frequency

The six squares and eight triangles outlined by the closepacked
spheres, although unmistakable even in the simplest case, become
more and more distinct as the frequency increases. As Fuller's
convention is to refer to the number of spaces (rather than spheres)along the "edge" of the cub octahedral cluster as the frequency of thesystem, the first twelve-around-one group is "one-frequency." The
next layer-in the VE case, a surrounding envelope rather than justanother layer added to the bottom-is two-frequency, having threeballs along each edge. The next is three-frequency, with four balls
per edge, and so on. It follows that the higher the frequency, thesmaller an individual sphere is in relation to its polyhedral face, andso these polygonal faces look progressively less bumpy, or more
sharply defined (Fig. 8-13). The precise planar organization of theclustered spheres becomes more obvious as the frequency increases.

8. Tales Told by the Spheres: Closest Packing

(a)

115

(c)

(b)
Fig. 8-13

The appearance of eight triangles and six squares was not an
accidental property of the first layer: the VE is here to stay. Thislesson is continually reinforced by additional layers.
Having established the shape of symmetrical nuclear sphere-packing we proceed to investigate numbers. Twelve balls fit tightly
around one; how many does it take to completely surround thetwelve with a second layer? By carefully placing balls in the "nests"
on the cluster's surface, we generate a two-frequency VE shell ofexactly forty-two balls. We might begin by placing one ball at thecenter of each of the six squares and then in each nest along thetwenty-four VE "edges", thereby superimposing six two-frequency
squares (nine balls each) over the simple four-ball squares (Fig.
8-13b). The edges of the six two-frequency squares supply thespheres for the adjacent two-frequency triangles (without adding any
more balls). These second-layer triangles cover the eight three-balltriangles of the first layer. The twelve vertices, or comer spheres, areeach shared by two squares, and so twelve must be subtracted fromfifty-four (nine spheres per square times six squares) to get the total
number for the second VE layer: forty-two spheres. (See Fig. 8-13.)
Envelop the whole package with a third layer, a three-frequency
shell with four balls per edge. Following the above procedure, we
count sixteen balls in each of the six squares, for a total of 16 X 6 =96, from which we must subtract the twelve vertices counted twice
due to overlap between squares. The edges of the eight triangles areagain already in place-provided by the edges of the squares-but
on this shell, a central sphere which belongs in each three-frequency

116

A Fuller Explanation

triangle must still be added; eight more spheres are therefore needed
to complete the enveloping layer. 96 minus 12 plus 8 yields a total of
92 balls.
The next shell (four-frequency) requires 162 balls; the fivefrequency layer consists of 252, six of 362, and so on. We are now
able to detect a pattern by looking carefully at these numbers:
12,42,92,162,252,362, .... It will come as no surprise to the observant student of numbers to learn that the next shell consists of
492 balls.
What exactly is going on? To begin with, we notice the consistent
last digit: every single number ends with 2, reminiscent of Euler's
law and its "constant 2." Fuller interprets this persistent "excess of
2" in radial sphere packing as further affirmation of the inevitable"poles of spinnability," inherent in the topology of singly closed
systems. And indeed, the temptation to embrace a single explanationis strong. The fact that the number of spheres per shell always ends
with the digit 2-even though those numbers increase drasticallywith each successive layer-seems too strange to ignore; we want an
explanation for nature's behavior. But at this stage, speculation as tosignificance is a sidetrack: our task is to fully describe the configurations. As soon as we fully understand the patterns and are thus
armed with the facts, such speculation will be appropriate andindeed inevitable.
After observing the reliable last digit, we can simplify our sequence-following Fuller's procedure-by subtracting the 2 fromeach term, removing the distraction to assist further analysis. Weareleft with 10,40,90,160,250,360,490, ... , all divisible by 10. So let's
divide by 10. This leaves 1,4,9,16,25,36,49, ... , and now thepattern is clear.
The latter sequence is generated by p, for / = 1,2,3,4,5,6,7, ....
We choose "/" in this case, to represent frequency, for it turns outthat the relationship between frequency and number of units pershell can be directly specified. Nature thus reveals yet another"generalized principle." This equation actually describes a straightforward edge-Iength-to-surface-area relationship, exactly what we
expect from geometry-in a slightly different format.
The next question is how to specify the relationship in precise
terms. We work in reverse from our final sequence (f = 1,2,3,4, ... )to generate the original sequence. First we must raise the frequency /to the second power, then multiply each term by 10, and then add 2:
10/ 2 + 2 therefore gives us the total number of spheres for any shell(specified by frequency) in nuclear sphere packings.

8. Tales Told by the Spheres: Closest Packing

117

Fig. 8-14. Removal of nuclear sphere.

Icosahedron

We return to the initial twelve-around-one cluster, to try a new twist(literally). Imagine that we have thirteen spherical balloons-insteadof Ping Pong balls-so that we can we reach through with a long pin
to puncture the nuclear balloon, causing it to slowly deflate. As thenuclear balloon disappears, the twelve outside balloons shift, closing
in symmetrically toward the empty space in the center. They can't go
far-only enough for the six unstable squares of the VE to "tighten
up" into two noncoplanar (or hinged) triangles. Squares were stablein the VE only because the nuclear sphere held them in place; thearray was thus stabilized by triangulation in radial directions, producing alternating tetrahedra and half octahedra. Now, without thenucleus, the surface must be stable by itself; it therefore must be
triangulated (Fig. 8-14).
The VE configuration has twelve" vertex" spheres. What shape dothese twelve spheres become when fully triangulated? The six squares
transform into two triangles each, which (added to the original eight)
make twenty triangles in all. Twenty triangles and twelve vertices?
That is none other than our friend the icosahedron, largest of theregular polyhedra, one of the "three prime structural systems in
Universe."
We can create higher-frequency versions of the icosahedron, butthey will always be single-layer shells. The icosahedral configuration
arose as a result of removing the VE nucleus; the remaining spheres
move in, partially filling the gap, and thus their positions no longer
allow a space-filling array of spheres. Because icosahedral clusters
are completely triangulated, they cannot be extended either inwardly
or outwardly; they lack the necessary alternation of tetrahedra andoctahedra.
Such omni-triangulated sphere-packing shells can have any
frequency, despite being restricted to single-layer construction. Let's118

A Fuller Explanation

see what happens. Consecutive higher-frequency icosahedral shells
cannot surround a previous layer as they do in the VE-which ofcourse starts with a nucleus and continually surrounds it with layers.
Icosahedral shells simply do not nest together. Instead, progressively
larger, or higher-frequency icosahedra must be built one by one, eachwith one more sphere per edge, and always single-thickness.
What will happen to the relationship between frequency andnumber of spheres on a given shell? It turns out that-while bothshape and volume change considerably-the number is unaffectedby this transition from the VE's fourteen faces to the twenty icosahedral triangles. We verify this fact through the following observations. Notice in Figure 8-15 that a square pattern of spheres can becompressed into a rhomboid (diamond) shape without changing thenumber of spheres. The resulting diamond is more tightly packed
than the square and consists of two triangles of the same frequency
as the original square, sharing one edge, that is, the row of spheresthat used to form the diagonal of the square. Figure 8-15 shows how
the spheres of two triangles on an icosahedral shell correspond to
one square face of a VE of the same frequency. And therefore,
because the spheres on an icosahedral shell are all as closely packed
as possible (as opposed to the VE, which alternates triangles with themore loosely packed square faces), a smaller, denser shell is produced, with the same number of spheres as a VE shell of the same
frequency. 10/ 2 + 2 therefore also applies to icosahedra.
The icosahedron contains as much interior volume relative tosurface area as is possible with only one type of face. Ever economical, nature therefore chooses icosahedral symmetry for the construction of a shell made of identical units; requiring a minimum ofeffort, this arrangement can arise automatically. Maximum volume,
minimum material. It is thus easy to account for the icosahedral
symmetry detected in the isometric virus capsid, the tough protein
shell created by nature to house and protect the more fragile genetic
material within, which is the source of the virus's instructions. 3
Nature consistently exhibits elegant solutions to design problems,
because she finds the most efficient, or least energetic, way tooperate. She has no choice but to adhere to the constraints of space.
The example of the spherical virus shell is worth our brief attention,for it provides an elegant illustration of the "design science" ofnature at work.
Let's examine the criteria: (1) A container must be constructed outof a large number of identical constituents (protein molecules), (2)8. Tales Told by the Spheres: Closest Packing

119

Fig. 8-15

for reasons which will be explained below, the shell must be able toself-assemble-that is, build and rebuild itself automatically, (3)maximum symmetry is advantageous to minimize the energy required for attractive bonds between the capsid molecules, and (4) thearrangement must be stable, which means triangulated.
The elegance of the relationship between structure and function iswell documented in modern biology, and the isometric virus is no
exception; its structure must be suited to its specific functions. A
tough shell is to completely enclose minute amounts of genetic
material-quantities necessarily insufficient for carrying detailed
bonding instructions-yet it must easily disassemble and reassemble
itself in order to release the viral genetic material into a host cell.
The overall structure must therefore be dictated by properties of thesubunits and by the constraints of space itself-both criteria alsoestablishing a built-in check system.
A sphere, which maximizes the volume-to-surface-area ratio, is the
key to an efficient solution. Interconnected molecules, which can
only approximate that theoretical sphere, will achieve a spherical
distribution most efficiently through icosahedral symmetry. Observations of isometric viruses have consistently revealed icosahedral
patterns, thus reconfirming that nature chooses optimal designs. 4 We
shall study this configuration in more detail in Chapter 15.

120

A Fuller Explanation

Further Discoveries: Nests

Throughout our investigation, we note the recurrence of a limited
inventory of polyhedral shapes. This is perhaps revealed mostdramatically by the behavior of closepacked spheres. A striking
example is found in the results of an extra sphere placed in thecentral nest of flat triangular clusters. The first case is already quite
familiar: a sphere placed in the minimum triangle of three spheresproduces a regular tetrahedron, assuming all four spheres are thesame size. A half octahedron is born out of a sphere nesting in asquare group of four spheres, and we might reproduce any numberof familiar shapes by putting equiradius balls together, but here we
confine ourselves to the triangular clusters, for the scientific methodtells us that the strength of an experiment often rests in drawing
boundaries. Results thus obtained may lead to broader generalizations about related questions.
Remember our observation that every "N," in Fuller's Y-N-N
pattern, represents a layer with a nest, since any triangular cluster
without a central ball has a central nest. The center of a ball added
to the first nest becomes the fourth vertex of a regular tetrahedron;
the next group, six balls, also has a central nest, so a seventh ball is
put in the space. Magically, the shape thus created is a significant
one: the semi symmetrical tetrahedral pyramid, which is exactly
one-eighth of the regular octahedron (Fig. 8-16). To understand what
is meant by "one-eighth" of an octahedron, imagine that we slice a
regular octahedron made out of "firm cheese," to use one of Bucky's
images, in half -creating two square-based pyramids (Egyptianstyle). Then, cut both halves into quarters, to get eight octants, eachwith an equilateral triangle base and three isosceles-triangle side
faces (45 °-45 °-90°). The octahedral central angle is 90 degrees, a factwhich will prove especially significant in the next chapter.
The seventh ball is simply placed in the nest of the six-balltriangular layer to complete the four vertices of an "octant." Nothing in our sphere-packing investigation thus far would lead us topredict the appearance of this important shape, which is already asignificant part of our polyhedral inventory. (Such unpredictability isgetting to be a pattern.) So we proceed to the next case.
The ten-ball triangle has a nucleus (1: yes; 3: no; 6: no; 10:
yes; ... ) and therefore no nest. So we skip to the fifteen-ball cluster
and, as before, drop a sixteenth ball in the central nest. We are nolonger surprised to discover that the resulting shallow pyramid is8. Tales Told by the Spheres: Closest Packing

121

Fig. 8-16

also a very special shape: one-quarter of a regular tetrahedron-a
portion encompassing the volume from the tetrahedron's center ofgravity out to anyone of its four faces (Fig. 8-16). The tetrahedron'scentral angle, 109.471 degrees (109°28'16"), seems so irregular thatthe sense of coincidence is underlined.
With fifteen balls in the plane and a sixteenth in the center, thispyramid is quite shallow-and in fact, as a section of the minimum
system, it is Fuller's terminal case. For each of the first five triangular numbers without nuclei, a sphere placed in the central nest forms
an important shape in the VE-octet framework, an idea that weshall explore in greater detail in the next chapter. We thus have cometo the end of this particular experiment, with the conclusion thatclosepacked spheres automatically yield many significant geometricshapes.
"I nterprecessi ng"

The essence of precession, to Fuller, is 90°. And indeed, the counterintuitive or mysterious thing about the behavior of gyroscopes (andother examples of precession in physics) is the resultant motion in a
direction ninety degrees away from that of an applied force. For
example if a downward force is imposed at the north point of a122

A Fuller Explanation

Fig. 8-17

gyroscope spinning clockwise, it will tilt toward the east: 90 degrees
away from the direction one intuitively expects. Fuller's "interprecessing" involves two systems "precessing" together, which heuses to mean oriented at 90 degrees with respect to each other. In his
sphere-packing studies, "interprecessing" reveals subtle facets ofsymmetry which might otherwise go unnoticed.
We start with the simplest case: two identical pairs of tangent
spheres, parallel to each other and separated by some distance.
Rotate one of the two-ball sets 90 degrees, and then move the twopairs toward each other until they meet in the middle, so that the
midpoints, or tangency points, of the two pairs are as close together
as possible. (Notice that without the 90-degree twist, the result ofbringing the parallel pairs together would be a square- unstable andnot closepacked.) What is the result of this simplest case of interprecessing? A tetrahedron, of course (Fig. 8-17).
In retrospect, the answer appears obvious, for the initial conditionof four spheres-the necessary ingredients of a tetrahedron-gives itaway. However, the experiment highlights the 90-degree symmetry of
the tetrahedron, which is otherwise obscured by the predominance of
triangles and 60-degree angles. Rather than elaborating on thetetrahedron's right-angle symmetry here, we shall allow subsequent8. Tales Told by the Spheres: Closest Packing

123

Fig.8-18a

demonstrations to further illustrate this orthogonal characteristic.
(See especially Chapters 9 and 10.)
Take two identical sets of sixty spheres, closepacked as shown in
Figure 8-l8a. Their irregular trapezoidal shape eludes immediate
identification. That they do not seem to be a part of our familiargroup of shapes is confirmed by numerous experiments in which
participants are given these two pieces and asked to put them
together in some way that seems correct. Countless false moves
involve bringing similar faces directly toward each other, and again
and again, the identical halves are put together in unsatisfying andincorrect ways. The correct solution is rarely discovered by theuninitiated-but once seen is unmistakable. This problem (simple,
after the fact) is initially challenging because it is so hard to get

Fig.8-18b

124

A Fuller Explanation

beyond the natural assumption that the two halves must approacheach other directly-as if one half were approaching its own reflection in a mirror. What actually has to occur of course is that one half
rotates 90 degrees with respect to the other (interprecessing) and thetwo rectangles mesh together perfectly, at right angles. "Wow!"
Bucky would exclaim, apparently as surprised as his audience. The
surprise is genuine in a sense, for the result is visually striking even if
one already knows the answer: a perfect tetrahedron, eight balls peredge, or seven-frequency (Fig. 8-18b.)
Along the same lines, we now look at 60-degree twists. An
especially pleasing example involves two simplest triangles, of threespheres each. The triangles face each other directly; then one rotates
60 degrees before pushing them together, and the result is anoctahedron. The six spheres are precisely situated as octahedronvertices, framing eight triangles. (Refer back to Fig. 8-7.)
Next, let's take two eighth-octahedron seven-ball sets (the six-balltriangle with a seventh ball in the central nest). The two triangularbases of each cluster face each other, and then one is rotated 60
degrees, allowing the triangles to come together as a six-pointed star,and suddenly the fourteen balls become a cube! This is the minimum
stable cube formed out of spheres (Fig. 8-19). Eight spheres alone,

Fig. 8-19

8. Tales Told by the Spheres: Closest Packing

125

positioned as the eight corners of the cube, are not closepacked, andthat configuration would therefore be unstable, as the spheres have atendency to roll into the unoccupied valleys.
In fact, just as a floppy toothpick cube needed six extra diagonal
sticks (the six edges of a tetrahedron) to stabilize the square faces,
eight balls also require an additional six, to complete a stable cube.
Thus we have fourteen balls altogether: a parallel to the fourteen
topological parameters (vertices plus edges plus faces) of the tetrahedron. The cube in every stable form seems to be based on an
implied tetrahedron.

A Final Philosophical Note

Fuller pointed out that sphere-packing models encourage us toconceive of area and volume in terms of discrete quanta instead of as
the physically impossible continuums promoted by traditional geometry:
Because there are no experimentally-known "continuums," we cannot concede
validity to the concept of continuous "surfaces" or of continuous "solids." Thedimensional characteristics we used to refer to as "areas" and "volumes," which arealways the second- and third-power values of linear increments, we can now identify
experimentally, arithmetically, and geometrically only as quantum units that aggregate as points, both in system-embracing areal aggregates and ... as volume-occupantaggregates. The areal and volumetric quanta of separately islanded "points" arealways accountable numerically as the second and third powers of the frequency ofmodular subdivision of the system's radial or circumferential vectors. (515.011)

He argued that sphere polyhedra, having the advantage of visibly
separate subunits, illustrate the otherwise invisible truth about physical reality. An awareness of the particles inherent in all physicalsystems (on some level of resolution) is nurtured, because in thesphere packings it is so logical to express volume and area in termsof number of units. The examples of the VE and icosahedron models
demonstrate how the terms for expressing length (frequency) andarea (number of particles) are actually related by a formula (10/ 2 +2)-as would be necessary in a new geometry.
Fuller attributed the precedent for thinking about volume in termsof quanta to Amadeo Avogadro (1776-1856) and his discovery that
equal volumes of all gases, under the same conditions of pressure
and temperature, contain the same number of molecules. Avogadro
thereby identified volume with number of molecules a long time ago.

126

A Fuller Explanaiion

We take this a step further, by remembering that, although we tend
to conceive of volume as a spatial continuum, our convention for
quantifying an amount of space uses number of imaginary
cubes-even if that quantity usually involves an extraneous partial
cube (or fraction) tacked on to a whole number. We shall discuss the
subject of volume more fully in Chapter 10. For now, we simply lay
the groundwork with the evidence that polyhedra can be constructed
out of a multitude of spheres, at different frequencies, and that theresulting models play an important role in satisfying Fuller's criteria
for a geometry consistent with Universe.

9Isotropic Vector Matrix

The isotropic vector matrix has already been introduced; we justdidn't know its name.
If you can visualize the space-filling array of spheres in "cubic
packing" described in the previous chapter, that's half the picture.
Now, imagine interconnecting the centers of all spheres-and theneliminating the spheres. Two collinear radii meeting at the tangency
point between adjacent spheres form one unit vector-the length ofwhich is equal to the sphere's diameter (Fig. 9-1). The resulting arrayof vectors is the "isotropic vector matrix," a space-filling network ofcontinuously alternating octahedra and tetrahedra. Reviewing thecharacteristics of cubic packing, we shall not be surprised to findthat all the newly formed vertices (the spheres' centers) are identically situated. Two types of cells, one type of vertex.
It's not hard to see how Fuller's search for a geometry of vectors
led him to the isotropic vector matrix. "Since vectors... produce
conceptual structural models of energy events, and since my hypothetical generalization of Avogadro's law requires that 'all the conditions of energy be everywhere the same,''' ponders Fuller, "whatdoes this condition look like as structured in vectorial geometry?"
His answer is ready: "Obviously all the vectors must be the samelength and all of them must interact [sic] at the same angles"(986.131b).
The isotropic vector matrix, or IVM, takes the VE a step further,
consisting of identical lengths and angles, not for vectors surrounding just one point, but surrounding every point in an indefinite
expanse. In Fuller's words, the IVM is "a multidimensional matrix
in which the vertexes are everywhere the same and equidistant fromone another" (222.25).
It is not correct to conclude that the IVM consists of many vector
equilibria packed together, for the VE by itself cannot fill space. To
understand why not, we look at isolated sections of the IVM. As
difficult as it is to visualize the overall matrix, a single row of128

A Fuller Explanation

Fig. 9-1. Unit vectors interconnect centers of adjacent unit-diameter closepacked
spheres.

alternating tetrahedra and octahedra, or even a planar expanse, canbe easily envisioned (Fig. 9-2a, b). Separate planar layers are thenstacked together in such a way that every octahedron is adjacent to atetrahedron and vice versa. Figure 9-3 shows three layers of theresulting matrix.
Every node in the IVM-as the origin of twelve unit vectors
radiating outwardly-is the center of a local vector equilibrium. The
ends of these unit vectors define the twelve vertices of the VE.

(a)

(b)

Fig. 9-2. (a) Single row of alternating tetrahedra and octahedra. (b) Planar expanseof alternating tetrahedra and octahedra.

9. Isotropic Vector Matrix

129

Fig. 9-3. Isotropic vector matrix and Octet Truss.

However, this does not mean that adjacent cub octahedra pack
together to produce a space-filling expanse. A symmetrical array canbe created by bringing the square faces of adjacent vector equilibria
together, but they are necessarily separated by octahedral
cavities- framed by the triangular faces of eight converging YEs.
The unavoidable octahedra between adjacent YEs provide yet anothermanifestation of the specificity of the shape of space. This array canbe readily understood by observing in Figure 9-4 that a packing ofvector equilibria is equivalent to a framework of cubes in which thecorners have been chopped off, thus automatically carving out anoctahedral cavity at every junction of eight boxes.
The above observations provide information about the shapes andangles of the IVM - the most symmetrical arrangement of points inspace-and therefore about the shape of space itself. These characteristics reveal the basis for the term "isotropic vector matrix": inFuller's words,"isotropic" meaning" everywhere the same," "isotropic vector" meaning" everywhere the same energy conditions." . .. This state of omnisameness of vectors .. . prescribes an everywhere state of equilibrium." (420.01-3)

He calls the IVM "multidimensional" because it "accommodates"(or occupies) all spatial dimensions, and-consistent with his unorthodox interpretation of dimension-space is " multi-" rather than130

A Fuller Explanation

Fig. 9·4. Octahedral cavities between adjacent VEs.

"three-dimensional." Vectors are directed in every possible direction, while deliberately maintaining equivalent lengths and angles.
This equivalence is necessarily determined by the symmetry of
space:
This matrix constitutes an array of equilateral triangles that corresponds with thecomprehensive coordination of nature's most economical, most comfortable, structural interrelationships employing 60-degree association and disassociation. (420.01)

As seen in the earlier development of vector equilibrium, spatial"omnisymmetry" incorporates four planes of symmetry: four unique
directions of equilateral triangles. Recalling the way cookies fit mosteconomically on a baking sheet, we can feel quite comfortable withthe triangular symmetry of the plane. The implication is that theshape of space can be described through four such continuous
planes.
A Quick Comparison : " Synergetics Accounting"Imagine one vertex within the IVM framework, which will be called
O-for origin. A unit vector (L = 1) pointing in any of the twelve
directions away from 0 ends at a vertex which we shall call A . A

9. Isotropic Vector Matrix

(a)

l31

(b)

Fig. 9-5. (a) 60-degree axes. (b) 90-degree axes.

second unit vector emanating from 0 is given a different orientation,
in a direction 60 degrees away from vector OA, and arrives at vertex
B. The distance between A and B is also unit length (Fig. 9-5a). As
simple and repetitive as this observation might seem, it is the essenceof "synergetics accounting," as opposed to "algebraic accounting."
The same procedure applied to a 90-degree framework, also using
unit vectors, places vertices A and B an irrational-number distanceapart. Unit increments along the x and y axes create points labeledsimply {(O, 1) and (1,0)}, which are separated by a (not so simple)
irrational Ii or 1.41421 ... units. Furthermore, if pathways between
vertices are to follow along the network of vectors, the square griddisallows the shortest route between A and B. Observe in Figure
9-5b that to get from A to B along the prescribed grid requires
traveling two units, despite the fact that they are separated by only
1.414. .. units. As energy always takes the shortest route, argues
Fuller, the XYZ system clearly does not serve to illuminate theevents of physical reality. In contrast, the most expedient route from
A to B in the triangular grid happens to be directly along theunit-length vector connecting the two points.
Irresolvable numbers do exist within the IVM, as there are squarecross-sections (corresponding to the square faces of the VE); however, the irrationals are not part of the fundamental orientation of
the system. To Fuller, a simple procedure, like the one describedabove, ought to yield simple ("omnirational") results. If a frame ofreference is itself convoluted, its ability to describe and measureother phenomena will be all the more so.
Cells: "Inherent Complementarity"

Let's back up and start again. The goal is to establish a symmetricaland complete spatial array of vectors, and one logical approach
might be to start with space's minimum system. We gather a number

132

A Fuller Explanation

(a)

(b)

I

'

(d)

(c)

Fig. 9-6. (a) Tetrahedra do not fit together. (b) Tetrahedra rearranged vertex tovertex. (c) A secord row of tetrahedra is placed above the first, such that all
tetrahedra meet vertex to vertex; the arrangement automatically creates octahedral
cavities. (d) Alternating tetrahedra and octahedra can fill space indefinitely.

of unit-vector tetrahedra and place them on the ground side by side,and it is immediately apparent that they will not pack together toproduce a continuous expanse (Fig. 9-6a). Awkward gaps between
adjacent tetrahedra cannot be filled by regular, or symmetrical,
shapes, precluding an isotropic array. This constraint is not new: we
saw in the previous chapter that tetrahedra cannot fill space; however, vector models display the shapes more clearly than spherepackings.
Still in pursuit of a space-filling array, we now rearrange thetetrahedra so that they meet vertex to vertex, each with one edge
along a continuous line, in multiple adjacent rows (Fig. 9-6b). Noticewhat happens when we interconnect the tetrahedral peaks (as would
be the result if a second layer of tetrahedra were placed on top of the9. Isotropic Vector Matrix

l33

first): precise octahedral cavities emerge in between all tetrahedra,
automatically completing the isotropic vector matrix with its alternating two shapes (Fig. 9-6c, d).
The reverse is also true: octahedra cannot themselves fill space,but when arranged edge to edge (not face to face or vertex to vertex)
the emergence of by-product tetrahedra reconfirms the persistentpairing. The developed matrix with its unit vectors and equivalentpoints of convergence thus depicts the inherent complementarity ofspace-meaning inevitable co-occurrence of octahedra and tetrahedra. Fuller draws a parallel between this and other inseparable
pairs such as electron-proton, concave-convex, male-female, andtension-compression. "Inherent complementarity of Universe" applies to the entire phenomenon of interdependent partners, whether
atomic constituents or polyhedral space-fillers. (Chapter 12 willfurther elaborate on Fuller's interpretation of the significance of"inherent complementarity.")
In conclusion, the development of both the VE and theIVM - whether through closest packing of spheres or by symmetricalarrangement of vectors-supports a sense of the balance of octahedral and tetrahedral symmetries inherent in space. Both configurations build themselves-in response to spatial constraints.

A Complete Picture

We now step inside the IVM to complete our investigation of thisornnisymmetrical network of vectors. The centers of closepacked
spheres constitute the vertices of most regular and semiregularpolyhedra. We looked at some of them in the previous chapter, andwith the use of toothpicks instead of Ping Pong balls, the outlines ofthese shapes can be more easily discerned.
We have already observed that vertices in the IVM fall into
triangular patterns in four distinct planar directions. Through ourexperience with cubic packing, we know to look for an additional
three planes of symmetry, characterized by a square distribution ofvertices. Neighboring octahedra share the edge between them, andthus the cross-sections of individual octahedra join together, forming
the square pattern of graph paper in three orthogonal directions.
Figure 9-7 highlights an IVM squared plane, by omitting certain
lines; half octahedra shown without tetrahedral edges clarify thesquare aspect of the omni-triangulated matrix.

134

A Fuller Explanation
Fig. 9-7. Half octahedra shown without tetrahedral edges, to emphasize a square crosssection of the IVM.

Angles

The combination of these two simple shapes in the IVM yields
surprisingly many different angles and potential shapes. Our attention tends to be focused on the surface characteristics of the tetrahedron and octahedron, and so we observe only triangles and
60-degree angles. However, the interior structure introduces distinct
new elements, such as the square octahedral cross-sections discussed
above. The next step is to list other interior angles, for a sense of therange of possible shapes contained within the matrix.
The dihedral angle (angle between two faces) in a regular tetrahedron is approximately 70°32'. The tetrahedron is unique in thatany two edges at a given vertex are part of a common face. Everyother polyhedron has interior angles in addition to surface angles
between edges, thus adding to the range of shapes incorporated into
each system. For example, any two nonadjacent edges at an octahedron vertex meet at 90-degree angles, thus forming squarecross-sections. The octahedron dihedral angle is 109°28', which-as
the supplement 1 to the tetrahedron's 70°32' - results in perfectly
flush surfaces shared by adjacent octahedra and tetrahedra, allowing
the continuous planes of the IVM (Fig. 9-8).
Both dihedral angles at first appear to be such irregular numbersthat this exact geometric fit is surprising-especially when we recall
our first encounter with the two shapes. Remember that we simply
surrounded vertices by three triangles, then by four, allowing thesystems to close off with as many triangles as necessary. (Refer to
Chapter 4.) The process provided no basis for predicting the exact

Fig. 9-8

9. Isotropic Vector Matrix

135

complementarity of the two polyhedral systems. The coincidence
continues with the addition of central nodes, as we shall see below.
Locating New Polyhedral Systems

The first new polyhedron consists simply of one octahedron with atetrahedron on two opposite sides (Fig. 9-8). The result, a rhombohedron, can be seen as a partially flattened cube. A toothpickmarshmallow model demonstrates the transition effectively, because
the marshmallow joints have sufficient stiffness to hold either inherently unstable shape. The rhombohedron's direct relationship to the
cube suggests a space-filling capability, which we shall explore in
greater depth in Chapter 12.
The next candidate, the VE, is too familiar to warrant further
description at this point. Twelve cub octahedral vertices can be

(a)

(b)

Fig. 9-9. (a) Truncation of three-frequency (3u) regular tetrahedron. (b) Truncationof 3u regular octahedron, showing only external surface of system.

136

A Fuller Explanation

located around every point in the IVM, thereby embracing eighttetrahedra and six half octahedra.
Furthermore, higher-frequency versions of any of the above polyhedra-tetrahedron, octahedron, rhombohedron, and VE-can beeasily located within the matrix, thus establishing the foundation fortruncated polyhedra. Subtract a half octahedron from each of the six
comers of a three-frequency octahedron to yield a symmetrical"truncated octahedron" with fourteen faces: six squares and eight
regular hexagons (Fig. 9-9b). A "truncated tetrahedron," with four
hexagons and four triangles, is left after a single-frequency tetrahedron is removed from each comer of a three-frequency tetrahedron (Fig. 9-9a). The same procedure applies to higher-frequency
versions of any of the above shapes, as well as further truncations oftruncated shapes. Such transformations can be plotted indefinitely.

Duality and the IVM

We now introduce a new level of flexibility with the addition of a
new set of vertices-in the exact center of each tetrahedron andoctahedron. These new vertices are connected to the original IVM
vertices, thereby introducing radial vectors into each of the original
cells. Figure 9-10 shows a single octahedron and tetrahedron withthese central nodes.
Angles

The central angles of a tetrahedron is approximately 109°28', exactly
equal to the octahedron's dihedral angle. No longer surprised bysuch relationships, we go on to look inside the octahedron and noteFig. 9-10. Central nodes of tetrahedron and octahedron.

9. Isotropic Vector Matrix

137

its central angle of 90 degrees, which is the surface angle of a cube.
The octahedron's three body diagonals, or six radii, thus form the
XYZ axes. Figure 9-10 highlights these central angles by showing
these two polyhedra with central nodes and radii. Right angles arethus integrated into the IVM system as by-products of the (stable)
triangulated octahedron, rather than by an arbitrary initial choice ofa network of unstable cubes. Since the IVM complex of octahedra
and tetrahedra emerges automatically as a consequence of its uniqueproperty of spatial omnisymmetry, the array is not the product of anarbitrary choice.

Polyhedra

We now observe considerable expansion of our inventory of generated shapes. Starting with the most familiar, we isolate the minimum
cube. Formed by a single tetrahedron embraced by four neighboringeighth-octahedral pyramids, or octants, the cube is once again basedon the tetrahedron. We first encountered this relationship in "Structure and Pattern Integrity" (using the tetrahedron to establish theminimum stable cube), and now we have determined the exact shapeof the leftover space: four eighth-octahedra. This observation indicates that "degenerate stellation" of the tetrahedron forms a cube.
The four vertices of the tetrahedron, together with the four centers ofneighboring octahedra, provide the eight corners of this basic building block. Its six square faces are created by two adjacent quarters of
the square cross-sections of single-frequency octahedra (Fig. 9-11).
As with other IVM systems, larger and larger cubes will be outlinedby more remote octahedron centers.
Next, we embrace a single octahedron by eight quarter tetrahedra,
thereby outlining the rhombic dodecahedron, whose twelve diamond
faces have obtuse angles of 109°28' and acute angles of
70 0 32'-generated by the tetrahedral central angle and two adjacent
axial angles, respectively. Its eight three-valent vertices are the centersof embracing tetrahedra, while its six four-valent vertices are theoriginal octahedron vertices (Fig. 9-12). Once again, we observe therelationship of duality between the VE and rhombic dodecahedron.
The former has fourteen faces (six four-sided and eight three-sided)
corresponding to the four-valent vertices and three-valent vertices ofthe latter. Likewise, the twelve four-valent VE vertices line up withthe twelve rhombic faces. (Refer to Fig. 4-14.)

138

A Fuller Explanation

Octa
Cross-section

Fig. 9-11. Relationship of cube and IVM.

DomainThe duality between VE and rhombic dodecahedron illustrates therelationship of duality and domain. Having already seen that spheresin closest packing outline the vertices of the VE, we now turn ourattention to the domain of individual spheres. 2 The domain of asphere is defined as the region closer to a given sphere's center than
to the center of any other sphere. This necessarily includes thesphere itself, as well as the portion of its surrounding gap that iscloser to that sphere than to any other. Imagine a point at the exact
center of an interstitial gap; this will be the dividing point between
neighboring domains, that is, a vertex of the polyhedron outlined by
the sphere's domain. This domain polyhedron happens to be therhombic dodecahedron. As each sphere in cubic packing is bydefinition identically situated, each domain must be the same. Therefore, the shape of this region consistently fits together to fill space.
Fuller's term for the rhombic dodecahedron is "spheric" because ofthis relationship to spheres in closest packing.

9. Isotropic Vector Matrix

139-

Fig. 9-12. Degenerate stellation of the octahedron is accomplished by affixing
quarter tetrahedra to each face.

We now have an experiential basis for the VE- rhombic-dodecahedron duality. Twelve vectors emanate from any point in the IVM,
locating the vertices of the VE, while poking through the center of
the twelve diamond faces which frame the point's domain. We wereintroduced to duality as exact face-to-vertex correspondence, andnow we see how duals can be instrumental in locating a system's
domain. Our investigation of space-filling in Chapter 12 will explore
this relationship more fully.
Returning to the IVM, we observe that four rhombic dodecahedra,or "spherics," come together at the center of each tetrahedron, suchthat the tetrahedron central angle becomes the obtuse surface angleof the spheric. In the same way, eight cubes meet at the center of
each octahedron, as allowed by the shared 90-degree surface andcentral angles, respectively.
For clarity, we shall refer to the new network, interconnecting thecenters of all octahedral and tetrahedral cells, as IVM', and we candraw the following conclusion. If the vertices of a given polyhedronare located in the IVM, then that system's dual will be outlined by
the IVM', and vice versa. Similarly, if a polyhedron is centered on a140

A Fuller Explanation

vertex of the IVM, its dual will be centered on a vertex in IVM'. For
example, we recall our first case of duality: the vertices of theoctahedron's dual, the cube, are supplied by octahedron centers,
which are nodes of IVM'.
This discovery leads us to another assumption. As truncation of
our familiar polyhedra yields shapes contained within the IVM, thedual operation, stellation, should produce polyhedra outlined by
IVM'. The assumption is valid: the additional IVM' vertices provide
the loci for the vertices of stellated versions of these basic shapes.
Actually, this observation is not new, for we have already seen thatquarter-tetrahedral pyramids affixed to octahedron faces produce
Fuller's spheric, or, in other words, that a degenerately stellated
octahedron becomes a rhombic dodecahedron. The three-valentvertices of this diamond faceted shape are tetrahedron centers, bydefinition nodes of IVM'.
Framework of Possibility

The isotropic vector matrix gives us a description of the symmetry of
space. We can think of this matrix as a framework of possible
directions and configurations of ordered space, or more simply, as aframe of reference. It is a network of vectors specifically situated tomodel nature's eternal tendency toward equilibrium. Lines are forces,
length is magnitude, and all is in balance. The IVM weaves together
a number of synergetics ideas: minimum system of Universe, vector
equilibrium (both exhibiting four planes of symmetry), twelve degrees of freedom, complementarity of octahedra and tetrahedra,
space-filling, and stability (exclusively a product of triangulation). In
so doing, it sets the stage for an energetic mathematics, and systematizes further investigation.
The IVM also provides an alternative to the XYZ system's absolute origin. Every vertex in the IVM can be considered a temporary local origin, which, as reinforced by Fuller's use of theconcept of "systems," is consistent with the requirements of describing Scenario Universe. ["All points in Universe are inherently centersof a local and unique isotropic-vector-matrix domain ... " (537.11).]
There can be no "absolute origin" in a scenario.
Finally, by describing such a wide variety of ordered
polyhedra-and thereby clarifying the relationships between different shapes-the IVM supports Fuller's concept of "intertransformability." Countless potential shapes and transformations can be9. Isotropic Vector Matrix

141

systematically represented within this omnisymmetrical matrix; it isa framework of possibility.
Invention: Octet Truss

Our familiarity with the IVM enables us to visualize and appreciate
Fuller's "Octet Truss." Awarded U.S. Patent 2,986,241 in 1961, thisstructural framework is so widespread in modern architecture that
one might assume buildings have always been constructed that way.
Again, as the story goes, the invention can be traced to 1899 when
Bucky was given toothpicks and half-dried peas in kindergarten. So
extremely farsighted and cross-eyed that he was effectively blind(until he received his first pair of eyeglasses a year later), Bucky
Fuller did not share the visual experience of his classmates andtherefore lacked the preformed assumption that structures were
supposed to be cubical. Thus, as other children quickly constructed
little cubes, young Bucky groped with the materials until he wassatisfied that his structures were sturdy. The result, much to thesurprise of his teachers (one of whom lived a long, long life, andperiodically wrote to Fuller recalling the event) was a complex ofalternating octahedra and tetrahedra. He had built his first Octet
Truss-also the first example of what was to become a lifetime habitof approaching structural tasks in revolutionary ways.
The experience had a great impact on the four-year-old, as herecounted in a 1975 lecture:
All the other kids, the minute they were told to make structures, immediately
tried to imitate houses. I couldn't see, so I felt. And a triangle felt great! I kept
going 'til it felt right, groping my way .... 3

The truss's omnisymmetrical triangulation distributes applied
forces so efficiently that the resulting strength of such an architectural framework is far greater than predicted by conventional formulae:
The unitary, systematic, nonredundant, octet-truss complex provides a total floor
system with higher structural performance abilities than engineers could possibly
ascribe to it through conventional structural analysis predicated only upon the
behavior of its several parts. (650.11)

Struts can be all one length, thus simplifying construction, while theminimal volume-to-material ratio inherent in the geometry of thetetrahedron4 maximizes resistance to external loads. The intrinsic
stability of triangulation together with efficient dispersal makes this142

A Fuller Explanation

system the most advantageous possible use of materials in a spaceframe configuration:
It is axiomatic to conventional engineering that if parts are "horizontal," they are
beams; and the total floor ability by such conventional engineering could be nostronger than the single strongest beam in the plural group. Thus their prediction
falls short of the true behavior of the octet truss by many magnitudes .... (650.11)

The octet truss takes the conceptual matrix into physical realization,and thus embodies Fuller's design science concept of using geometricprinciples to human advantage.
We can now appreciate the difference between diamond andgraphite. Both consisting of carbon atoms, the former is exquisitely
hard and clear, the latter soft and grey, and their differences are due
to geometry. Carbon atoms in the structure of diamond take advantage of the strength of tetrahedra; their organization can bethought of as a double octet truss, two intersecting matrices with thevertices of one overlapping the cells of the other. Stabilized by thehigh number of bonds between neighboring atoms, which also allow
forces to be distributed in many directions at once, the configuration
is supremely invulnerable. In contrast, carbon atoms in graphite areorganized into planar layers of hexagons-triangulated and stable inthemselves, but not rigidly connected to other layers. As a result,separate layers are able to shift slightly with respect to each other,which does not mean that graphite lacks all stability, but rather that
it is relatively soft. This softness enables graphite to leave visible
residue on the surface of paper, thus performing its useful functionin pencils. A more illustrative although less widely recognized application is that these sliding layers make graphite a powerful lubricant.
The comparison provides a spectacular example of synergy: rearrangement of identical constituents produces two vastly different
systems.
Thus we see that nature also employs design science.

10
Multiplication by Division:
In Search of Cosmic Hierarchy

It may seem that we have strayed from Fuller's "operational

mathematics" while investigating the symmetrical properties of various polyhedra in the previous chapter. Recall that "operational"
indicates an emphasis on procedure and experience: what to do todevelop and transform models or systems. "Multiplication by division" brings us back to experience, introducing an operational
strategy, which will add new meaning to Fuller's term "intertransformabilities." We thus elaborate on the shared symmetries amongshapes while discovering new transformations from one to another,and this time previous experience allows us to anticipate results.
Multiplication by division describes Bucky's journey through our
expanding polyhedral inventory. Previous exposure to both Loeb's
work and the IVM sets the stage, making us so familiar with theseshapes that additional results can be immediately placed in context.
The transformations explored in this chapter occur within the IVM
frame of reference, adding volume relationships to our accumulated
information about topology and symmetry. You may be surprised tofind that many statements seem obvious at this point; resist thetemptation to dismiss them as trivial. Appreciate instead the implication - which is that we cannot take a wrong turn. Each step isinherently tied to the shape of space; we can only uncover what isalready there.

Volume

The use of ratio is an inherent part of quantifying volume, and yet
not everyone is aware of the implicit comparison. As with measuring
distance, our conventional units can seem like a priori aspects ofvolume.
Once again, Fuller calls our attention to Avagadro's discovery thata given volume of any gas, subject to identical conditions of temper-

144

A Fuller Explanation

ature and pressure, always contains the same number of molecules.
"Suddenly we have volume clearly identified with number," declares
Bucky.
Actually, volume is intrinsically related to number. When we ask,
"what is the volume of that swimming pool"? we expect an answer
expressed in terms of some number of "cubic feet." What thisanswer tells us is how many cubes with an edge-length of one footcould fit into the pool. Whether the situation calls for feet, inches, or
centimeters, a cube of unit edge length is conventionally employed as
one unit. The word" volume" may evoke an image of a continuum;
however, it is quantified in terms of discrete quanta. We so uniformly express spatial quantity in terms of cubes that we are simply
not aware of the invisible framework of "ghost cubes" incorporated
into our concept of volume. We conceptualize space cubically:
length, width, and depth seem absolutely fundamental directions.
Again Bucky points out that this conceptual cube is a remnant of
fiat-earth thinking. Myopic in cosmic terms, humanity readily
adopted the orthogonal box as the correct shape with which tosegment space.

Results: Volume Ratios

Volume has to be measured relative to something, so why notexperiment with the tetrahedron? We are so used to using the cube,
the suggestion seems blasphemous-a violation of basic laws ofvolume. Nevertheless, given our growing list of the tetrahedron's
unique properties, such an experiment might be worthwhile.
Accordingly, we allow a tetrahedron of unit-edge length to becalled one unit of volume. The results are astonishingly rewarding.
Perfect whole-number values describe the volumes of most of thepolyhedra covered so far, and all of those contained within the IVM
and IVM' combined. (Some exceptions are found in transition
shapes-those that fall in between IVM vertices-as we shall see in
the next chapter.) In contrast, the volumes of these familiar polyhedra, relative to a cube as the unit shape, are strangely cumbersome
values-often irrational (never-ending) decimal fractions. Table V
displays the results, which we shall derive below. It compares thevolume ratios generated by three different polyhedra successively
adopted as one unit of volume. Five different systems are comparedfirst with the unit-edge cube, then with the unit-diagonal cube, andfinally with the unit-edge tetrahedron.

10. Multiplication by Division: In Search of Cosmic Hierarchy

145

Table V. Volume Ratios
Polyhedron Taken as Unit of Volume:
Polyhedron
Measured
Tetrahedron
Octahedron
Cube(unit diagonal)
Rhombic
dodecahedron
VE

Cube
Unit Edge

Unit Diagonal

Tetrahedron

0.11785
0.471400.33333
1.333331

0.353561

30.707102.35700

2
6.66666

620

4

Remember that these polyhedra arose as a consequence of spatial
symmetry; we simply located vertices within the unique isometric
array of vectors. Recalling this origin, it is again clear that thevarious shapes and sizes of the polyhedra in question are not the
product of deliberate design. Their whole-number volume ratios are
not contrived; we stumble onto them after the fact.
Why investigate two different cubes? Primarily, to demonstratethat neither choice yields the elegant results disclosed by the tetrahedron. All factors considered, the unit-diagonal cube is a better
choice, for it arises naturally out of the IVM network-that is, out of
the shape of space! Further justification for this choice will be
developed below.
Imagine building a cube out of the requisite twelve struts. The
topological recipe, which simply calls for three-valent vertices andfour-valent faces, in no way indicates precisely what the finishedproduct should look like. Without deliberate shaping into a perpendicular form by a knowing hand, the configuration does notfavor any particular surface angles. Which version of this hexahedronof quadrilaterals is the desired result? This ambiguity must somehow
be resolved. Chapter 5 revealed that an orthogonal "cube" -asdefined by mathematics-cannot be reliably created without diagonal braces. (Refer back to Fig. 5-3.) Compare this experience withbuilding an octahedron out of the same twelve structs. Following therecipe of four-valent vertices and three-valent faces. the octahedronbuilds itself. The interior shape is precisely specified by its topology;
in other words, the procedure leaves no room for choice. In order forthe cube to have that kind of integrity, or exactitude, six facediagonals must be inserted. An inscribed tetrahedron solves the
problem in a single step.

146

A Fuller Explanation

The above comparison reinforces our previous experience of how
the right-angled cube fits into spatial symmetry. Fuller's "operational mathematics" prescribes learning by procedure: pick up abox of toothpicks (pre-cut unit-vector models) and start building.
Space will let you know what works. The satisfaction gained fromfeeling the cube hold its shape draws attention to its supporting
diagonal members, and thus unit-vector diagonals seem an appropriate choice for comparing volumes. As volume is always amatter of ratio, we want to insure that our comparisons make sense,in this case, remain consistent with space's isotropic vector matrix.
That the vector-diagonal cube is contained within this hierarchy
adds to the advantages established by its stability.
Shape Comparisons: Qualities of Space

Once again, we take advantage of the ease of working with planar
configurations, before tackling space. We thus start by comparing
the characteristics of triangles and quadrilaterals, and then we shallattempt to apply our conclusions to tetrahedra and cubes. We begin
by drawing an irregular version of each polygon, and abserve thefollowing. If we bisect the edges of the two figures and interconnect
these points as shown in Figure lO-la, both shapes are divided into
four regions. However, the triangle and quadrilateral exhibit a strikingly different result. A triangle, no matter how irregular, automatically subdivides into four identical triangles-all geometrically similar to the original, that is, the same shape but a different size.
Observe in Figure lO-la that this is not true for the quadrilateral.
Excluding the special case of a parallelogram, the four small quadrilaterals will not be similar to their framing shape.
Now, on to space! A tetrahedron (of any shape or size) carved out
of firm cheese can be sliced parallel to one of its faces, removing aslab of any thickness, to produce a new smaller tetrahedron with
precisely the same shape as the original (Fig. lO-lb). This does notwork with the cube, or for that matter, with any other polyhedron,
regular or not. The ability to "accommodate asymmetrical aberration" without altering shape, observes Fuller, is unique to the minimum system of Universe. We add this observation to a growing listof special properties of the tetrahedron. (Appendix D.) Similarly, as
will be demonstrated below, an irregular tetrahedron can be subdivided to create smaller identical tetrahedral shapes, whereas anirregular hexahedron will yield dissimilar hexahedra, in the samemanner as its planar counterpart, the quadrilateral. Evidence thus
gradually accumulates to support using the tetrahedron (instead of

10. Multiplication by Division: In Search of Cosmic Hierarchy

147

(b)

(a)
Fig. 10-1

the cube) as the basic unit of structure-or mathematical starting
point.
Volume: Direct Comparison

Before looking more closely at volume ratios, we review the following mathematical generalization. No matter what unit of measurement is employed, the volume of any container is mathematically
proportional to a typical linear dimension raised to the third power.
This means that if we have two geometrically similar polyhedra, onewith twice the edge length of the other, the larger will contain exactly
eight times the volume of the smaller. (Having the same shape, thetwo systems will share a common "constant.")
To bring this mathematical law into experiential grasp, we consider two familiar shapes. It is easy to visualize that a cube of edge
length 2 consists of eight unit cubes (Fig. 10-2). Now imagine atetrahedron of edge length 2. Observe in Figure 10-3 that, just like its
cubic counterpart, the altitude of a two-frequency tetrahedron is two

Fig. 10-2. Two-frequency cube consists of eight unit
cubes.

148

A Fuller Explanation

2

1
Fig. 10-3

times that of a unit tetrahedron, and similarly that each face subdivides into four unit triangles. The latter observation indicates thatthe area of the large tetrahedron's base is four times that of the small
tetrahedron's unit-triangle base. Emulating the approach employed
by Loeb in his "Contribution to Synergetics,,,l we can deduce thefollowing, simply by utilizing traditional geometric formulae.
Let VOlT and Volt represent the volumes of the large (twofrequency) and small (unit-length) tetrahedra; AT and AI' the areasof their bases: and HT and HI' their altitudes.
According to the formula
volume of pyramid = constant X (area of base) X height.
So

VolT = KATHT

and Volt = KAtHI'

and since we observed in Figure lO-2b that the base of the large
tetrahedron is divided into four triangles, each of which is equal tothe base of the small one, it is clear that AT = 4A t • Similarly, thealtitude of the larger pyramid is twice that of the smaller, orHT = 2Ht • Substituting, we have
VOlT = K X 4A t X 2Hl'orand since
it follows that
VOlT = 8 Vol I'
or, in words, that the volume of the big tetrahedron is eight times
that of the little tetrahedron. The constant K cancels out of the10. Multiplication by Division: In Search of Cosmic Hierarchy

149

expression when the two equations are compared. This conclusion
will be useful in deriving the volume ratios displayed in Table V.

Multiplication by Division
Multiplication occurs only through progressive fractionation of the originalcomplex unity of the minimum structural systems of Universe: the tetrahedron.
(100.102b)
Instead of starting with parts-points, straight lines, and planes-and thenattempting to develop these inadequately definable parts into omnidirectional
experience identities, we start with the whole system in which the initial"point" ... inherently embraced all of its parameters ... all the rules of operational
procedure are always totally observed." (488.00)

Deeply impressed by Arthur Eddington's definition of science as"the systematic attempt to set in order the facts of experience,"
Fuller constantly sought meaningful organizations for groups ofexperiences or events. "Multiplication by division" is one such effort.
(" Events" of course includes structures and almost anything else; inenergetic Scenario Universe, things are events.)
Essentially, "multiplication by division" derives volume relations
through the straightforward logic of direct observation, rather thanby rote application of traditional formulae-which lead us throughawkward values before revealing the underlying simple relationships.
As will be seen below, this direct observation is accomplished bycomparing given polyhedra to the unit-length tetrahedron.
Tetrahedron as Starting Point

Fuller's organizing strategy begins with the tetrahedron, because asthe" topologically simplest structural system," it is a logical starting
point. Consistent with his emphasis on "whole systems," the ultimate reference point in synergetics is "Universe." The tetrahedron
thus acts as an appropriate "whole system" for the proceduredescribed below, in that it is "the first finite unitarily conceptual
subdivision of ... Universe" (987.011b). More complicated systems
are developed through subdivision of this tetrahedral starting point,
so that a progression is contained within (and organized by) thewhole:In respect to such a scenario Universe multiplication is always accomplished only
by progressively complex, but always rational, subdivisioning of the initially simplest structural system of Universe: the sizeless, timeless, generalized tetrahedron.
(986.048b)

150

A Fuller Explanation
Fig. 10-4

Onward! We seek to develop a variety of polyhedra throughsubdivision of the "whole" and in so doing provide the "experimental evidence" to verify the results shown in Table V. We imagine asingle regular tetrahedron, and then bisect each edge to create thetwo-frequency tetrahedron shown in Figure 10-3. One by one, weremove a single-frequency tetrahedron from each of the four corners,
unwrapping the hidden octahedron (Fig. 10-4). Choping off four unit
tetrahedra subtracts four units of volume from the initial total ofeight (the value determined earlier for a double-edge-Iength tetrahedron), indicating that the octahedral remainder has a volume ofexactly four. In other words, an octahedron has four times thevolume of a tetrahedron of the same edge length. We thus begin toderive the values in Table V.
Cube
Next, we split the octahedron in half, separating the two square-based
pyramids. Two additional perpendicular slices divide each pyramid
into quarters (Fig. 10-5), producing eight sections, or "octants," eachwith a volume of one-half (a volume of four, divided by eight, isequal to one-half). Each octant is an irregular tetrahedron with aunit-length equilateral base and three right-isoce1es-triangle sides.
The perpendicular corners of the eight octants meet at the octahedral
center of gravity, an orthogonal relationship first noted in theprevious chapter when IVM' vertices (body center) were added tothe cells of the IVM.

10. Multiplication by Division: In Search of Cosmic Hierarchy

151

1Fig. 10-5. Derivation of an octant: one-eighth of an octahedron.

To reconfirm the octahedron-tetrahedron volume relationship, weplace an octant (with its equilateral face down) next to a regulartetrahedron on a flat surface and observe that the altitude of theoctant is exactly half that of the tetrahedron. (A second octant canbe put on top of the first to check: the height of both octants
together is equal to the altitude of the tetrahedron, as shown in
F igure 10-6.) An octant, therefore, has the same base and half thealtitude as a regular tetrahedron, reconfirming that its volume ;<;exactly half the volume of the tetrahedron.
In "Structure and Pattern Integrity" we discovered that a regulartetrahedron fits inside a cube, and subsequently we learned that eachof the four "leftover" regions is equivalent to the portion of anoctahedron from one face to its center of gravity (Chapter 9). We
now take advantage of the recently disassembled octahedron for an
experiment. Having equilateral triangles in common, octants can besuperimposed on each face of a unit tetrahedron. One by one, fouroctants surround and thus obscure the tetrahedron. Lo and behold, a152

A Fuller Explanation

Fig. 10-6. The altitude of an octant is equal to one-half the altitude of thetetrahedron.

-Fig. 10-7. Four octants added to one tetrahedron produce one cube.

perfect cube emerges (Fig. 10-7). Four octants, with a combined
volume of two units, have been added to the unit-volume tetrahedron, for a total volume of three. Compared again with theirregular volumes listed in Table V, generated by the unit-edge cube,these whole-number ratios for the cube and octahedron, seem especially remarkable.
Vector Equilibrium

The volume of the VE can be quickly determined through direct
observation. Recall that twelve unit-length radii outline eight regulartetrahedra and six half octahedra. This fact, combined with our
newly generated volume data, provides a conclusive value for thevector equilibrium: six half octahedra, each with the tetrahedronvolume of two, plus eight unit-tetrahedra yields a total tetrahedron10. Multiplication by Division: In Search of Cosmic Hierarchy

153

volume of twenty: 12 + 8 = 20. This simple breakdown supplies
further evidence of a natural order of precise volume relationships,
and it is especially reassuring that a unit-length vector
equilibrium-the conceptual foundation of Fuller's energetic
mathematics- falls into place with its own whole-number volume
ratio.
Rhombic Dodecahedron

To begin with, our recently disassembled octahedron must be putback together. Octants are thus lifted away from the composite cube
shown in Figure 10-6, and their right-angled corners are again turned
inward to meet at the octahedron's center. Next, we divide twotetrahedra into quarters. Each tetrahedron yields four shallow
pyramids, encompassing the region from an outside face to thecenter of gravity (Fig. 10-8). As before, the equilateral base of eachshallow pyramid allows the quarter tetrahedra to fit directly onto an
octahedral face, and as soon as eight quarter tetrahedra are attachedto the octahedron, a rhombic dodecahedron emerges (Fig. 9-12).
Two units of volume have been added to the octahedral four, for a
total of exactly six.
Looking at the shape of the rhombic dodecahedron, with its
strange angles and facets, the perfection of this whole-number volume

Fig. 10-8

154

A Fuller Explanation

relationship seems particularly remarkable. Because both shape andtopological characteristics of the tetrahedron and rhombic dodecahedron appear utterly dissimilar, the discovery of this orderly relationship between the two polyhedra adds significantly to our growing
sense of an underlying spatial order.
Again, the considerable flexibility of the IVM framework enables
us to plot all of the above polyhedra with IVM and IVM' vertices.
The rationale for using the unit-diagonal cube also applies to therhombic dodecahedron, which arises naturally out of the interactionof IVM and IVM' cells. Recall that every octahedron in the matrix issurrounded by quarter tetrahedra, thereby defining rhombic
dodecahedra with unit-length diagonals.
We discovered many shared symmetries in the previous chapter bydissecting the IVM, and we now develop the significance of theseobservations with the discovery of the rational volume relationships
inherent in this framework. Bucky was not the first to discover theseratios, but he may have been their most visible spokesman. He
brought this esoteric information to the attention of countless packed
lecture halls, as one of the more satisfying indications of the fallibility of our coordinate system. (Such sublime disclosures by naturemust not go unheralded!) These volume ratios provided Fuller with apowerful source of confidence in the legitimacy of pursuing synergetics, and indeed their significance is worth our serious consideration. What accounts for the lack of attention paid to these simple
mathematical facts? Loeb offers the following explanation:
When these relations are derived with the aid of the usual formulae for thevolume of a pyramid, V = t Ah, a good many irrational numbers are involved, andthe simple integral ratios emerge almost incidentally. Somehow, these simple integral values of the volume ratios of common solids are not part of our scientificculture, and a lack of familiarity with them frequently leads to unnecessarily
cumbersome computations. It appears that a bias of our culture to orthogonal
Cartesian coordinates has obscured these relations. [My italics.]2

Multiplication by Division

Bisecting the edges, as before, we take special note of the tetrahedron's square cross-section. This fourfold symmetry was a significant factor in previous discussions, notably in the sphere-packing
demonstration, in which two sets of spheres came together at 90
degrees and unexpectedly produced a tetrahedron. However, this
aspect of the tetrahedron is easily overlooked; as a triangular pyramid, with its preponderance of 60-degree angles, this shape is easily
perceived as a completely triangular affair.

10. Multiplication by Division: In Search of Cosmic Hierarchy

155

(b)

(a)
Fig. 10-9Suppose we ask the following question: how much of our cheese
tetrahedron would be chopped off when the newly exposed surface(created by the slice) is a perfect square? Without a certain amountof previous exposure to the tetrahedron, your reaction would probably be that the slicing-plane could never be square. It might be avery small triangle, or any number of larger triangles as you position
the knife closer to the base. But a square?!
Wait. The tetrahedron has four faces. Do they somehow outline asquare? Anyone who has read this far knows the answer, butcountless students challenged to find that hidden square have been
stuck. Handicapped by the perpendicular bias of mathematics, they
are unable to find the square cross-section in the exact center of thetetrahedron, which-once seen-is unmistakable (Fig. IO-9a).
The square in Figure IO-9a is parallel to and between two opposing edges, which themselves are perpendicular to each other. Delineating the square cross-sections corresponding to each of the threesets of opposite edges, this aspect of the tetrahedron's symmetry is
exhausted, and the two-frequency subdivision is complete. A total of
twelve new edges outline the octahedron, and by now this relationship is quite familiar (Fig. IO-9b).
We continue inward. This time, bisect and interconnect the edgesof the octahedron. The process is equivalent to Loeb's "degenerate
truncation" and outlines the edges of a vector equilibrium hiding
inside the regular octahedron (Fig. IO-IOb). We could continue, by
joining the midpoints of VE edges, to produce a "rhombicuboctahedron"; however, Fuller's sequence comes to an end at the vector
equilibrium. The final lines, which are one-quarter the length of theedges of the original tetrahedron and the smallest vectors in themodel, are thus designated as unit vectors.
We quickly run through the sequence in reverse to review thegeometric relationships. A four-frequency tetrahedron, in which each

156

A Fuller Explanation

(a)

(c)

Fig. 10-10

Fig. IO-lledge is equivalent to four unit vectors, is the smallest tetrahedron tocontain a complete VE in its center, and so it acts as the ultimate"whole system" (Fig. 10-11). The review starts in the center: aunit-length half octahedron is tacked onto each square face of thenuclear VE, thereby forming a two-frequency octahedron, which inturn has two-frequency tetrahedra added to four of its eight faces tocreate the large tetrahedron. Figure 10-10 illustrates this transition,
and Figure 10-11 shows the complete four-frequency tetrahedronlO. Multiplication by Division: In Search of Cosmic Hierarchy

157

and its implied hierarchical system, employing progressively thicker
lines to emphasize the three different polyhedra.

Cosmic Hierarchy (of Nuclear Event Patternings)"The Cosmic Hierarchy is comprised of the tetrahedron's intertransformable interrelationships" (lOOA03b). Fuller's curious description
is now clear, for we have become familiar with most of these"intertransformable relationships" and how they fit into the IVM
context-as well as with the simple operations that transform one
shape into another.
The order of this polyhedral hierarchy is determined by complexity, from least to most. It is worth noting that the ladder can extend
inward indefinitely by progressive subdivision; edges can be continually bisected to generate higher frequency systems. Notice how
convenient it is to have "conceptuality independent of size." We do
not have to specify the size of the initial tetrahedron, for Fuller's useof frequency to designate length provides a means to specify thesystem's geometric characteristics precisely, without recourse to"special case" examples. Ratios remain consistent; like conceptuality, they are independent of size.
In summation, "cosmic hierarchy" pertains to volume ratios aswell as to complexity and frequency, and its relationships are uncovered through "multiplication by division." In this way, Fuller
describes the order inherent in space.
Volume Reconsidered

Fuller attaches considerable significance to volume ratios and cosmic
hierarchy. The subject suggests a number of philosophical implications, concerning both reasons why these orderly relationships go
unnoticed and also the potential benefits of a mathematics thatemphasizes systems and the relationships of parts to wholes. The
first aspect is best summarized by Loeb in his "Contribution":
Uncritical acceptance of geometrical formulas as fundamental laws, particularly
in systems that do not naturally fit orthogonal Cartesian coordinates, frequently
leads to unnecessarily clumsy calculations and tends to obscure fundamental
relationships. It is well to avoid instilling too rigid a faith in the orthogonal system
into students of tender and impressionable age!3

Overdependence on the cube is the culprit. Fuller attributes muchof our attatchment to this building block to an understandable desire

158

A Fuller Explanationfor" monological" solutions, or single answers to complex questions.
Blissfully unaware of the "inherent complementarity" of Universe,
he cautions, humanity naturally sought one "building block" withwhich to understand space, a single unit to be the basis of allmathematics. Even without considering the concept of "inherent
complementarity," there are significant advantages to using thetetrahedron instead of the cube as the basic unit in quantifying
volume. Bucky would complete his argument by reminding us thatthe cube is inefficient. Having demonstrated the respective volumesof this traditional shape compared to nature's minimum system, hewould summarize by saying, "if you use cubes, you use three times
as much space as necessary." And, once again, "Nature is always
most economica1."
Finally, he hypothesizes that the irrational volumes of simple
polyhedra, inherent in cubic accounting, tended to reduce the importance of these basic systems in the eyes of mathematicians. Shapeswith such troublesome volumes could not possibly be relevant tonatural order:
Though almost all the involved geometries were long well known, they had
always been quantized in terms of the cube as volumetric unity ... : this method
produced such a disarray of irrational fraction values as to imply that the otherpolyhedra were only side-show geometric freaks or, at best, "interesting aesthetic
objets d'art." (454.02)

The exclusive adoption of the cube thus served to inhibit sustained
serious attention to the other polyhedra.
Back out to the big picture. To understand Universe, Fuller
argued, we must think in terms of the synergetic principles governing
the relationship of parts to whole systems. Multiplication by division
is one of many exercises to encourage the development of a habitualorientation toward solving problems in context. If our early mathematics training encourages us to isolate and consider parts separately, rather than as components of a larger system, then, Fuller
thought, our natural inclination throughout life would be to viewproblems myopically.
In Fuller's view, we have been blinded to a whole family ofrational order by an initial (90-degree) wrong turn-itself a result of
humanity's early perception of an up-down platform Earth.

11
Jitterbug

Synergetics can be described as dynamic geometry. Its treatment ofpolyhedra as vector diagrams and emphasis on the changes andtransformations in systems distinguishes Fuller's work from thetraditional geometric approach. His conviction that mathematics
ought to supply dynamic models-in recognition of dynamic Universe-led to a number of interesting discoveries. "Jitterbug" is themost striking example.
Twelve equiradius spheres pack tightly around one, as noted
earlier, and if the nuclear sphere is removed the other twelve canshift slightly inward. A vector equilibrium thereby contracts into thetriangulated icosahedron. What would this transformation from
Chapter 8 look like in terms of vectors?
Following the example set by the transition from closepacked
spheres to the isotropic vector matrix, we replace spheres with
vectors. Twenty-four wooden dowels and twelve four-way rubber
connectors are put together to make a vector equilibrium (Fig.
ll-la). The model consists of eight triangles and six squares flexibly
hinged together, and intentionally lacks the VE radial vectors, which
would correspond to the now missing nuclear sphere. Therefore,
since square windows collapse, we are back to the issue of stability.
In Chapter 5, we asked how many additional sticks were needed tostabilize each unstable system. For the VE, the answer was simple:
six, one to brace each unstable square window. This time around wetake a more open-minded approach to unstable systems and see
where it leads. Suppose we don't stabilize the VE?
Instability enables motion. But what kind of motion? This new
outlook inspires us to explore the ways in which flexible vector
models change. The discoveries are remarkably satisfying (and theprocedures are somewhat playful). In the case of the VE, the result isan elegant dance of symmetry that Bucky called the "jitterbug.""Jitterbug" describes a transformation of the stick-model VE, inwhich all twelve vertices move toward the system's center at the160

A Fuller Explanation

(a)

(b)

(c)

(d)
Fig. H · tsame rate. The advantages of having an actual model on hand at thispoint are greater for the jitterbug than for any of the previous
concepts. A verbal account of the transformation, no matter how
precise, is inadequate; likewise for drawings. The wonder of thejitterbug lies in its motion-from the unique equilibrium arrangement through various disorderly stages and on to new order. Jitterbug models tend to capture the attention of the most disinterested
bystander; their dance is fascinating to watch. So find twenty-four
sticks and twelve connectors and put them together; the model isguaranteed to intrigue.
Folding a Polyhedron

The model fascinates because so much seems to be happening at
once. The vector-equilibrium starting point looks simple enough.
Unstable if left to its own devices, the VE must be deliberately held
open-with one triangle fiat against a table top and two hands
holding the opposite (top) triangle. Notice that the two triangles
point in opposite directions, together forming a six-pointed star if
you peer into the system from above (Fig. ll-la).

11. Jitterbug

161

Now, simply lower the top triangle toward its opposite triangle(i.e., toward the table) without allowing either one to rotate. The first
surprise is that the equator seems to be twisting, despite your careful
avoidance of rotation. If you pull the triangle back out and try it
again, you will see that this equatorial twist can go in either direction; in fact the system can oscillate back and forth, going throughthe zero point, or equilibrium, every time.
Secondly, although you are only pushing on one direction (forcing
the triangle toward the table) the entire system contracts symmetrically-like a round balloon slowly deflating. Apparently, the effectsof your unidirectional force are omnidirectional. You can see andfeel that although you push and pull along a single line the contraction and expansion are both uniformly spherical.
Focus on the square windows, because only squares can change;
triangles hold their shape. As the top triangle approaches the bottom
triangle, each square compresses slightly, becoming a fat diamond,and the dance has begun. The radius of the system is now slightly
shorter than the length of its twenty-four edges (vector equilibrium
no longer). The dance continues as the top triangle approaches thebottom triangle and the diamonds grow slightly narrower, reaching
the point at which their width is exactly equal to the edge length(Fig. ll-lb). The VE has thus turned into an icosahedron-at least
the shape of an icosahedron-but we don't stop to add the six extra
sticks across the diamond windows to complete the picture, for thatwould turn our jitterbug into a stable structure.
Instead, keep going: past the icosahedral stage, the diamonds, everthinner, soon become narrow slits and finally snap shut (Fig. ll-lc, d).
The dance comes to a halt in the form of an octahedron. Twenty-four
sticks have come together into twelve pairs, creating a double-edge
octahedron (Fig. ll-ld).
This contraction is continuous, and so there are countless slightly
different stages, but only three are significant geometric shapes. The
jitterbug is of interest primarily because of its surprising flow fromone polyhedron into another; the emphasis is on its motion. However, we do want to be familiar with its geometric check-points.
First, the vector equilibrium. Unit-vector edges are balanced byunit-length radii. When the system contracts to the icosahedral
position, the distances between each vertex and its five neighboring
vertices are suddenly the same-unlike the VE, in which each vertex
has only four nearest neighbors, each one the unit distance away,
while two additional neighbors are approximately 1.414 units away,

162

A Fuller Explanation

(a)

(b)

Fig. 11-2

i.e., the length of the square's diagonal. To accommodate this surface
equivalence, the radius must decrease from 1.0 to 0.9511. Nature will
not compromise on these numbers: for all neighboring vertices to be
separated by equal lengths, the radius must be shorter than that
length. A perfect static balance is impossible; hence the dynamic,
eternally fluctuating events of Universe.
The twelve vertices continue their inward journey until they landat the six corners of an octahedron; sets of dowels clamp together,grouping the twelve vertices into six pairs (Fig. 11-1d). The radius
has decreased to 0.7071 times its original length, and that's the end.
But wait! There's another twist left in the jitterbug. Hold on tothat top triangle, which has been so carefully kept from rotating
until now, and deliberately start to twist it. (The triangle will only
yield in one direction, depending on the direction of the jitterbug's
initial twist.) If you turn it far enough (180 degrees) the entire system
collapses into a flat two-frequency triangle spread out on the table(Fig. 11-2a). Then, fold in the three corner triangles, like petals of aflower, bringing their edges together to create the fourth and final
stage: the minimum system of Universe (Fig. 11-2b).
It's a dense tetrahedron, with four parallel sticks for each of its six
edges, three converged vertices at each of its four corners, and aradius of 0.6124. "Quadrivalent," says Bucky, and hence full of
explosive potential-ready to spring back out into "our friend thevector equilibrium."This folded model also demonstrates the tetrahedron "turning
itself inside out," for the three petals can be opened and flattenedout again, and then folded back in the opposite direction, creating
the mirror-image or "negative" tetrahedron. Fuller then reminds usthat" unity is plural and at minimum two" and every system has aninvisible negative counterpart. "Negative Universe is the complementary but invisible Universe" (351.00). Such digressions areunavoidable; for Fuller the implications of a model are always11. Jitterbug

163

multifaceted, one observation plunging into another, layers uponlayers, intertwined.
Volume and Phase Changes

The jitterbug exhibits a total transformation of shape and size
without any change in material. Nothing is added or taken away, but
the system's characteristics are profoundly altered by rearrangementof the parts. The concept is reminiscent of the differences between
ice, water, and water vapor, all consisting exclusively of H 20 molecules. When a child-whose model-making experience is limited to"building blocks" - first learns that rearrangement of the constituents is responsible for these profound changes, the idea is noteasily accepted. Fuller maintains that experience with models likethe jitterbug would better prepare a child for the lessons of science.
Chemistry's invisible phase changes would seem perfectly logical, as
they would be consistent with first-hand experience. And indeed, thejitterbug's floppy, flexible behavior as VE is parallel to that of a gas:
the dense tetrahedral configuration with its motion totally restrained
is more like the "solid" phase. Same stuff, radically different properties. He has a point. These dynamic models inspire a different kindof conceptualizing.
At the zero point, twenty-four wooden dowels and twelve rubber
connectors embrace a volume of twenty, as we recall from "Multiplication by Division." After contracting and twisting through thejitterbug, the bundle of sticks encloses a single unit of volume, one
tetrahedron. The system has thus gone from twenty tetrahedronvolumes to one, with a stop at four, in the octahedron.
Icosahedron

The icosahedron however refuses to cooperate. Its volume of approximately 18.51 is not as appealing as the whole-number ratios
shared by the other stopping points. litterbug now introduces arationale. The icosahedron is a phase that falls in between octahedron and vector equilibrium, rather than a definitive stopping
point in the flow. The jitterbug is a continuous transformation
through countless transitional stages, both regular and not, and at
certain intervals an ordered polyhedron emerges. Found when thejitterbug is simply open as far as possible, the cuboctahedron isdefinitive, absolute zero. The octahedron clicks into place when six

164

A Fuller Explanation

pairs of vertices suddenly come together. No ambiguity at either
point. The icosahedral stage on the other hand is always approximate; we have to eye the distances between vertices, guessing whether
or not they are equal to one. The dance does not stop naturally atthis point; we just recognize the familiar shape along the way from
VE to octahedron. It is thus a transient phase of the jitterbug-withno reason to stop and rest, no choice but to continue.
Similarly, the icosahedral vertices fall in between nodes of theIVM, the omnisymmetrical framework that outlines most of thesymmetrical geometric shapes. One result of being out of phase withthis matrix (which is also defined by c10sepacked spheres) is that the
icosahedron's frequency cannot be increased by surrounding it withadditional layers of spheres, or vectors. It cannot grow modularly;
the initial choice of size, or frequency, is final. To change thefrequency, a new model must be built from scratch. The icosahedronis thus restricted to single-layer construction. "The icosahedron must
collapse to exist," explains Bucky; it always "behaves independently"of the other polyhedra:
The icosahedron goes out of rational tunability due to its radius being too little
to permit it having the same-size nuclear sphere, therefore putting it in a different
frequency system. (461.05)

Accordingly, its volume does not fit into the "cosmic hierarchy" ofrational systems.
Single Layer versus IVM

What are the consequences of the icosahedron's "independence" ofthe cosmic hierarchy? As a collapsed VE, it is always a shell, a
single-layer construction:
The icosahedron, in order to contract, must be a single-layer affair. You could
not have two adjacent layers of vector equilibria and then have them collapse to
become the icosahedron .... So you can only have this contraction in a single-layerof the vector equilibrium, and it has to be an outside layer, remote from otherlayers .... It may have as high a frequency as nature may require. The center isvacant. (456.20-1)

Accordingly, as we recall from Chapter 8, the design chosen by
nature for many protective shells involves icosahedral
symmetry-from the microscopic virus capsid to larger (visible withan ordinary microscope) radiolaria, the cornea of an eye, and aplethora of other elegant creations.

11. Jitterbug

165"Trans-Universe" versus "Locally Operative""The vector-equilibrium railroad tracks are trans-Universe, but the
icosahedron is a locally operative system" (458.12). Fuller's ambiguous and somewhat mystical declaration becomes almost straightforward after the jitterbug demonstration. Vector equilibrium is incorporated into an infinitely extending network. Conceptual and timeless, VE is everywhere; it is the balance of forces at the root of all
phenomena. The icosahedron, on the other hand, is always aspecial-case collapse, an aberration in the omnisymmetrical frame ofreference. Aberrations are finite, local, inescapably stuck in time, as
we recall from "Angular Topology." The icosahedron is thus fundamentally different from VE, which-with its timeless perfectionpermeates all of Universe.

Fives
Fivefold symmetry dominates the icosahedron, distinguishing it once
again from the cosmic hierarchy, with three-, four-, and sixfold
rotational symmetries. This is another sign of the icosahedron's
nonconformism. It's full of fives: to begin with, the obvious five
triangles around each vertex, determining the symmetry about eachof its long axes. Then, its thirty edges fall into five sets of six
orthogonal edges, that is, three parallel pairs of mutually perpendicular edges. Figure 11-3a highlights the five distinct sets of orthogonaledges. l (The edges can also be grouped into sets of five parallel edges

(a)

(b)

Fig. 11-3. (a) Five sets of six edges: each set of six consists of three mutuallyperpendicular pairs. (b) Connecting midpoints of one set of six edges outlines aregular octahedron.

166

A Fuller Explanationembracing the equator, in six different directions.) Joining the midpoints of the six edges of one set displayed in Figure 11-3a, wediscover an octahedron hiding inside-implicit in the icosahedral
symmetry-in one of five possible orientations (Fig. 11-3b). The
icosahedron may be out of phase with the rest of the IVM family,
but it displays many significant relationships to these other shapes,
which, being unexpected, are all the more fascinating to uncover. A
few examples will be given below, and there's always room forfurther exploration. Just as in our earlier development of the cosmic
hierarchy, we investigate how various shapes fit inside each other,and thereby learn about similarities in shape, volume, and valency.
Whereas the cosmic-hierarchy relationships are consistently
straightforward and balanced (just bisect edges and connect midpoints to generate the next shape), whenever the icosahedron is
introduced, more intricate connections emerge. We therefore have tolook somewhat harder to find these new relationships which highlight the icosahedron's transitional role in the hierarchy.
We saw how the octahedron emerges out of the arrangement of
icosahedral edges, on the inside, and now we reverse the situation.
The icosahedron can be oriented so that eight of its twenty faces are
coplanar with and flush against the eight faces of a surroundingoctahedron, while the twelve icosahedral vertices are located on its
twelve edges. However, the icosahedron must sit in a skew (ortwisted) position, with its vertices intersecting the octahedral edges
off center, dividing each edge into two segments, the longer l.618
times the length of the shorter. This asymmetry means that there are
two distinct orientations of the icosahedron inside the octahedronpositive and negative, as shown in Figure 11-4a, b.
The ratio l.618 to 1, known as the "golden section," might haveplayed a prominent role in synergetics, for it shows up frequently

(a)

(b)
Fig. 11-4

11. Jitterbug

167(especially in relation to the icosahedron); however, Fuller rarely
mentions this intriguing number. Accordingly, this text will not
spend time exploring the famous ratio, which-as a source offascination to geometers for millenia-enjoys considerable press
already. (Specifically, for the role played by the golden section in the
icosahedron, see Loeb's "Contribution to Synergetics" and its "Addendum" in Synergetics 2, both Section G. 2 ) Fuller has a different
method of coping with such relationships; rather than describing
certain comparisons and their numerical values, he employs geometric "modules" -a holistic way of describing geometry with geometry.
liS-Modules"

The icosahedron's crooked position within an inscribing octahedrondefines specific leftover space-six pockets of empty territory between the icosahedron and its octahedral embrace (Fig. 11-5). The
symmetry of the six identical pockets is such that each can be split inhalf, producing two equivalent irregular tetrahedra, which then further divide into two mirror-image halves. This final thin tetrahedron
is Fuller's "S-quanta module." It is a volumetric unit that describes
the degree to which the icosahedron is out of phase with the IVM.
Just as grade-school "long division" introduces the arithmetic remainder, the process of dividing an octahedron by an icosahedron
requires a geometric remainder-the S-module. Chapter 13 will
describe Fuller's A- and B-modules, volumetric counterparts of the
S-module; taken altogether these quantum units comprise Fuller'sfinite accounting system. Finally, Figure 11-5 indicates thegolden-section ratio between different edge lengths of the S-module.
Icosahedron and Rhombic Dodecahedron

A pattern emerges. The out-of-phase relationship between
icosahedron and different IVM polyhedra appears to involve thegolden section. We test the pattern on the rhombic dodecahedron.
Do its twelve faces correspond to the twelve vertices of the icosahedron? It turns out that the two shapes exhibit an interesting
relationship, with the icosahedron fitting inside the rhombic
dodecahedron, predictably in a skew position. Its vertices impingeon the rhombic faces slightly off center, dividing the long diagonal ofeach diamond into two unequal segments-the longer again 1.618

168

A Fuller Explanationr-

1.618 . ..

f
1

1

Fig. 11-5. S-module.

times the length of the shorter (Fig. 11-6). Ever reliable, the golden
section reinforces our awareness of the underlying order in space.
Pentagonal Dodecahedron

Finally, recall the pentagonal faces of the icosahedron's dual; thefivefold symmetry of this dodecahedron is right out in the open.
Furthermore, the pentagon is a prime source of golden section ratios(see Loeb's "Contribution to Synergetics"). The pentagonal

k--

,--\

,

_ _ ' .6'8 ~.

Fig. 11-6. The golden-section ratio isrevealed in the relationship between
icosahedron and rhombic dodecahedron.

11. Jitterbug

169

dodecahedron is of course also out of phase with the IVM, for its
symmetry scheme is the same as that of its dual, the icosahedron.

Four Dimensions

Back to the jitterbug. Fuller proposes that this fluid transition fromstage to stage is best described as four-dimensional:
The vector-equilibrium model displays four-dimensional hexagonal central crosssection ... . (966.04)
Four-dimensionality evolves in omnisymmetric equality of radial and chordal
rates of convergence and divergence . . .. (966.02)

First of all, radial and chordal equivalence produces four distinct
planes, and secondly, the jitterbug contraction operates around four
independent axes. Let's see how this works.
Triangles hold their shape, and therefore an equivalent model tothe stick jitterbug described above can be built out of eight cardboard triangles hinged together with strong tape. The advantage inthis case of "solid" triangles is that it makes Fuller's" four-dimensional" assignment easier to understand (Fig. 11-7).
The eight triangles operate in pairs. Diametrically opposite triangles remain aligned, rotating synchronously about a common axis as
they together approach the jitterbug center at a constant rate. The
separate pairs thus rotate around four different axes, displaying
simultaneous motion in four distinct directions.
"Push straight toward the table; don't let either triangle rotate" :Bucky emphasizes the simplicity of the task. He then feigns surpriseat the subsequent twisting at the equator. The entire system converges symmetrically, despite the unidirectional force. He calls thisbehavior four-dimensionality, referring to the four independent directions of rotation.

Fig. 11-7. Four independent axes of rotation.

170

A Fuller Explanation

Complex of Jitterbugs

In an isotropic vector matrix, adjacent vector equilibria create octahedral cavities; to fill space the two polyhedra must occur in equal
numbers. Contemplating this alternating array in light of the VE's
jitterbug behavior, Fuller suspected that a complex of jitterbugs
could be synchronized to twist and contract, while their octahedral
counterparts simultaneously expand, twisting open into YEs. It is an
extraordinarily difficult vision to conjure up in the mind's eye: anarray of synchronized twisting triangles- transforming from orderto chaotic inscrutability and then back into order-but with all theplaces switched. YEs become octahedra; octahedra become YEs.
The ease of confirming the jitterbugging of one VE does little
toward answering the question of whether a number of interconnected jitterbugs can coordinate to create this dynamic labyrinth.
Would additional YEs packed around a single one serve to lock it inplace? The question is difficult to answer without actually putting itto the test with the aid of a model-an awesome task. Thanks to an
ingeniously designed four-valent universal joint 3 to accommodate
the intricate twisting of adjacent triangular plates, a magnificent
sculpture has emerged after considerable speculation. A movable
complex of stainless steel and aluminum triangles hinged together
effectively demonstrates that the convoluted transformation is possible (Photo. 11-1). This translation from abstract mathematical concept to physical manifestation of the motion is both an engineeringand an aesthetic feat.
Fuller proposes that this complex of pulsing jitterbugs demonstrates the effects of a force propagating through space-a tangible
display of otherwise invisible energy events:
1032.20 Energy Wave Propagation: '" You introduce just one energy
action-push or pull-into the field, and its inertia provides the reaction to yourpush or pull; the resultant propagates the ... omni-intertransformations whose comprehensive synergetic effect in tum propagates an omnidirectional wave. (1032.20)

In other words, the unique symmetry of the VE combines with this
newfound jitterbug property to produce a model of omnisymmetricalmotion, a radiating wave of activity. Just as the IVM is a staticconceptual framework-describing the symmetry of space-thismodel illustrates the concept of dynamic, "eternally pulsating" energy events in space. It causes the IVM to come to life.
A model can elucidate a concept without being an exact duplicateof the phenomenon in question. In fact, considering the oddly11. Jitterbug

171

Photo. 11-1. Complex of jitterbugs: An array of alternating octahedra and vector
equilibria is shown on the left; on the right, the display is undergoing the simultaneous transformation of all cells, such that octahedra are opening up to become YEs
while the YEs are contracting into octahedra. The action is frozen in mid transformation, making it possible to see the icosahedral phase. Photo courtesy of Carl
Solway Gallery, Cinncinnati, Ohio.

mystical language that creeps into modern physicists' description of
atomic and subatomic behavior, we can conclude that invisible
reality does not readily submit to large-scale reenactment with"solid" materials. The intention of Fuller's models therefore is toprovide a consistent analogy-a tangible display that parallels andthereby explains invisible behavior:
Dropping a stone in the water discloses a planar pattern of precessional wave
regeneration. The local unit-energy force articulates an omnidirectional, spherically172

A Fuller Explanation

expanding, four-dimensional counterpart of the planar water waves' circular expansion. (lO32.20)

The expanding concentric waves made by a stone dropped in alake are directly visible on the water's surface. It is therefore easy to
picture the image of a wave propagating across a plane. It is moredifficult to visualize a corresponding situation in space, which isprecisely the territory Fuller set out to conquer. He strove to clarify
invisible aspects of reality through models that can be seen and felt.
The complex of jitterbugs makes the concept of an expanding
spherical wave of energy visible.
Whatever the analogous events in Universe, the model is intricate
and phenomenal. Fuller's argument is that nature depends uponsuch dynamic orderly coordination. The complex of jitterbugs demonstrates a complicated but organized operation, and if nature
permits this transformation of her omnisymmetrical framework, shemight use the same trick elsewhere. At the moment, jitterbugs
therefore merely hint at possibilities. It is worth reflecting on theextraordinary intuition required to have discovered this subtle andmagnificent geometric phenomenon.
Other Dynamic Models

Topology and Phase
Physical Universe differentiates into three categories: liquid, crystalline, and gaseous phases. These distinct states of matter arise as aresult of changing temperature or pressure, which induce different
types of bonds. Fuller proposes a simple geometric analogy to makethese invisible changes easy to comprehend. Appropriately, the model
is completed with an exhaustive enumeration of the ways in which
two tetrahedra can be connected to each other. The three arrangements of the minimum conceptual system model the three phases ofphysical matter.
First, two tetrahedra are triple-bonded, sharing one face between
them (Fig. 11-8a). Because the relative positions of the two tetrahedra are completely fixed, the arrangement qualifies as a stable setof relationships and represents a crystalline structure: "The closestpacking, triple-bonded, fixed-end arrangement corresponds withrigid-structure molecular compounds" (93l.60). Once again, theintention is not to create a large-scale duplicate of a particular"solid" compound, but rather to display the different characteristics
of each chemical phase. The model is a kind of visual shorthand.

11. Jitterbug

(a)

(b)

173

(c)

Fig. 11-8

One of the three bonds is now released, leaving a double bond
between two tetrahedra. The shared edge acts like a hinge (Fig.
11-8b); the tetrahedral pair can swing back and forth, but theycannot be moved closer together or farther apart. The configurationis thus noncompressible-one of the distinguishing characteristics ofliquids. It bends any way you desire-malleable just like aliquid-but the double bond persists:
The medium-packed condition of a double-bonded, hinged arrangement is still
flexible, but sum-totally as an aggregate, all space-filling complex is noncompressible-as are liquids. (931.60)

Finally, we break another of the bonds. "Single-bonded" tetrahedra are joined by a vertex, and their behavior is analogous to thatof a gas. The vertex bond acts as a universal joint; the two halves canswing freely with respect to each other, moving together and apartwithout disrupting the type of bond (Fig. 11-8c). The arrangement iscompressible, expandable, and completely flexible-short of dissociation. Perpetual connectedness indicates that both tetrahedra continue to participate in the same substance; they exhibit a consistent
relationship, but lack structural definition:
Tetrahedra linked together entirely by ... single-bonded universal jointing use
lots of space, which is the openmost condition of flexibility and mutability characterizing the behavior of gases. (931.60)

The analogy is complete. All the while, Bucky holds the simple
structures in his hands, and explains the different basic properties ofsolids, liquids, and gases. With any luck, a small child is present,
forcing him to keep his discourse simple. How can the same type ofmolecule produce such radically different substances? Bucky offers atangible explanation through geometry. This model is perhaps usefulas a mnemonic device-an easy way to remember the chemistry174

A Fuller Explanation

lesson by relating the different characteristics to their analogous
stage of the model-rather than as a true demonstration of phasechanges in a substance.
It is worth noting that this model of interconnected tetrahedra ismore appropriate, and, in fact, quite accurate, in connection with thebonding of carbon atoms within molecules. In conclusion, it isunfortunate that this and other polyhedral characteristics and relationships are generally overlooked in educational curricula.
Fuller developed many dynamic models, and readers who go on tofurther study will find a variety of examples in Synergetics. The
transformations discussed in this chapter set the stage to explore
other examples. Appropriate parallels in nature may well arise fromsuch efforts.

12"All-Space" Filling:
New Types of Packing Crates

Fuller in his characteristic drive for verbal accuracy updates geometry's conventional term "space filling" with his own more descriptive(and predictably longer) "all-space filling." Here's the puzzle: whichof the polyhedra introduced so far can pack together in such a waythat all available volume is occupied without any gaps?
The concept is not new; ever since closepacking equiradius spheres,
we have danced around the issue of space filling, and in the process
made most of the discoveries that this chapter will expand upon. The
IVM disclosed certain space fillers, while making it clear that otherpolyhedra did not share this ability. However, filling all space now
becomes the focus of our investigation, calling for the systematic
analysis that enables new insights and a more thorough understanding.
Despite its obvious applicability, space filling is not emphasized in
Fuller's work. An ability to "fill all space" is generally mentioned as
further description of a given polyhedron rather than providing an
investigative starting point for synergetics. Fuller's "operational"
approach encourages more experimental exercises, such as packing
spheres together, which then lead to space fillers after the fact.
In view of our overall goal of researching the characteristics ofspace, what could be more logical than to ask what fits into it? Whatshapes are accommodated by space? The notion that space is not apassive vacuum gradually becomes second nature; experience haschanged our awareness. Now we want to become ever more exact
about these active properties. The existence of an extremely limitedgroup of polyhedra that can pack together to fill all space is one of
the more direct illustrations of the specificity of spatial characteristics.
The puzzle is quite challenging. Without actually making a hordeof tiny cardboard models of the polyhedra in question, these spatialconfigurations are extraordinarily difficult to visualize-with the sole
exception of the obvious space filler, an array of cubes. Once again,

176

A Fuller Explanation

to gain more experience with the concept, we revert to the plane, orto be more accurate, the page. In deference to Bucky's strict precision, we must acknowledge that the theoretical "plane" is a non demonstrable concept, but we can certainly (and quite appropriately)
discuss filling up a page.
Plane Tessellations

The mathematical title may be somewhat intimidating, but plane
tessellations are actually quite familiar. Derived from the Latin wordfor" tiling," tessellation, in mathematics, refers to planar patterns ofpolygons. Because of the ease of working with flat patterns, we begin
our study of space filling with the analogous situation in the plane.
Which regular polygons fit together edge to edge to fill a page?
We can fill up a page with squares, as anyone who has ever seengraph paper knows, and the pattern created by equilateral triangles
is almost as familiar. Another successful tessellation is found on
many bathroom floors covered by tiny hexagonal tiles (Fig. 12-1a).
With these immediately apparent examples, we might begin tosuspect that any regular polygon can fill a page. However, a little
experimentation quickly reveals that we have already exhausted thepossibilities.
Three regular pentagons (equilateral and equiangular) placed side
by side leave a 36-degree angular gap-not nearly wide enough toaccommodate a fourth pentagon. Five-sided tiles are thus disqualified. Three heptagons simply cannot fit around a single point, as
we saw in Chapter 4; octagons meet a similar fate, as of course do
any polygons with more sides. We are suddenly confronted with avery limited group of plane fillers: triangles, squares, and hexagons.
Opening up the field to allow combinations of regular polygons
while maintaining equivalent vertices expands the inventory only
slightly. Eight "semiregular" tessellations join the three patterns
above (Fig. 12-1b). Still an impressively small group. These eleventilings can be categorized in terms of the different rotational symmetries exhibited by each, and within this group every category of
repeating planar pattern is represented. It is fascinating to reflect on
the implications of these results: for example, a wallpaper designer
can only create what the limitations inherent in the plane will allow.
These cumulative experiences-especially those as straightforward
as plane tessellations- nurture our growing awareness of spatiall2. "All-Space" Filling: New Types of Packing Crates

177

(a)

(b)

Fig. 12-1

constraints. And with each step, our knowledge of the elegant
precision of this order expands synergetically.
Filling Space

Cubes stack neatly together to fill space. What other polyhedra
exhibit this property? Attempts to fit regular tetrahedra together are
quickly frustrated; likewise for octahedra. However, working to-178

A Fuller Explanation

gether, the two shapes can fill space indefinitely. None of this is new
information: we discovered the complementarity of octahedra andtetrahedra while exploring isometric arrays of both spheres andvectors. Subsequently, multiplication by division uncovered the octahedron hiding inside every tetrahedron. The octahedron- tetrahedron marriage is clearly an eternal bond.
Neither icosahedra nor pentagonal dodecahedra can fill all space.
[" Icosahedra, though symmetrical in themselves, will not close-packwith one another or with any other symmetrical polyhedra" (910.01).]
The cube thus stands alone among regular polyhedra.

Complementarity
But let's reevaluate our apparently simple array of cubes. As theobvious solution to filling all space with a single polyhedron, thispacking seems to provide the most straightforward information
about the shape of space. But look further. What if you could see the
cubes' face diagonals? An implied tetrahedron awaits visibility. Now
imagine filling in the necessary diagonals, so that the inscribed
tetrahedron-surrounded by four eighth-octahedra-appears in eachcube. At every junction of eight cubes, the octahedral parts cometogether and form one complete octahedron around each cubicalcorner (Fig. 12-2). As rectilinear boxes are unstable without diagonalbracing, a stabilized packing of cubes turns into an Octet Truss.
Whether visible or not, the octet symmetry is implicit in the configuration.
The Greeks failed to get at the triangulated heart of their stack of
cubes, philosophizes Bucky, for "like all humans they were innately
intent upon finding the 'Building Block' of Universe." Had theyexperimented with arranging tetrahedra vertex to vertex and beenFig. 12-2. Octahedron at junction of eight cubes.

12. "All-Space" Filling: New Types of Packing Crates

179

confronted with the inescapable octahedral cavities, he continues,
theywould have anticipated the physicists' 1922 discovery of "fundamental complementarity." ... But the Greeks did not do so, and they tied up humanity's
accounting with the cube which now, two thousand years later, has humanity in alethal bind of 99 percent scientific illiteracy. (986.049b)

Fueled by the developments of twentieth-century physics, Fuller
spoke frequently and emphatically of the "inherent complementarity" of Universe. He cites two examples in particular out of the many
provided by quantum physics. First, the "complementarity principle"
announced by Niels Bohr (1885-1962) in 1928, which goes hand inhand with Heisenberg's indeterminism. Bohr summarizes the basic
feature of quantum physics by stating that experimental evidencecannot be comprehended within a single frame, but rather must be
understood as "complementary," or partial, information. The totality of a phenomenon must therefore be represented through morethan one complementary part, for all aspects cannot be accurately
measured simultaneously. 1
Fuller also calls our attention to a later Nobel-winning development made by two Chinese physicists working in the U.S. In 1957,
Tsung Dao Lee and Chen Ning Yang were honored for theirdiscovery that "parity" is not conserved in weak interactions, for thesubatomic particles involved show" handedness." 2 In other words,
simplifies Bucky, fundamental complementarity does not consist of
mirror-image pairs. Concave-convex, tension-compression,
proton-neutron, male-female: Universe is always plural, supportinginterdependent, inseparable pairs which are not simply mirror-image
halves. It is thus reassuringly consistent that the essence of structureand space should also exhibit this fundamental dualism. There is no
single building block of Universe.
Other Space Fi lIers

The role played by tetrahedra and octahedra in an array ofcubes-not to mention in the IVM-demonstrates that these two
structures combine to create a variety of polyhedra. This observationsuggests an operational strategy: experiment with various combinations in the hope of finding other space fillers.
Two tetrahedra are affixed to opposite faces of an octahedron withthe same edge length. Supplementary dihedral angles cause adjacent180

A Fuller Explanation

triangular faces to be coplanar, thereby merging into six rhombic
faces (Fig. 9-8). The resulting slanted structure, introduced in Chapter
9, is called a rhombohedron and can be thought of as a partiallyflattened cube. Six squares have simply been squashed into diamonds. It's not hard to imagine that an entire array of toothpickcubes could lean over in unison, transforming into an array of
distorted cubes. Rhombohedra therefore fill space. The simplicity ofthis development hints at a starting point.
In his "Contribution to Synergetics," Loeb analyzes various polyhedra for their divisibility into tetrahedra and octahedra, anddemonstrates that a shape will fill space if it consists of two tetrahedra
for every one octahedron. 3 Suddenly out of randomness an orderemerges, and trial-and-error is replaced by a generalized principle.
The ability to fill all space can be added to the list of descriptive
properties (such as stability, symmetry, duality) by which we categorize the scattered cast of polyhedral characters. And now, armed
with Loeb's conclusive analysis, we will continue to explore thepuzzle from a slightly different frame of reference.
The Search Continues

The IVM simplifies our task, providing a frame of reference whichitself fills space. All of the polyhedra covered so far, with the
exception of the icosahedron and pentagonal dodecahedron, areoutlined within the IVM and IVM' framework by various combinations of octahedral and tetrahedral components (both parts andwholes). We can therefore survey the matrix to ascertain which ofthese systems are able to meet face to face without intervening
cavities.
Polyhedra that fit together without gaps must completely surrounda common vertex. IVM vertices therefore provide a good starting
point; we can systematically investigate the different types of nodes
in the matrix, checking the surrounding cells for space-filling polyhedra. Our first node reveals an already familiar space-filling team:each IVM vertex joins six octahedra and eight tetrahedra-as well as
polyhedral combinations of the two. The most obvious of these
combinations is the vector equilibrium, and our study of the IVM
already made it clear that YEs do not fill space, but rather mustcooperate with octahedra to create an uninterupted array, providing
yet another example of complementarity. The necessity of thispairing follows directly from octahedron-tetrahedron interdepen-

12. "All-Space" Filling: New Types of Packing Crates

181

dence. (Chapter 13 will elaborate on such space-filling teams, whicharise out of the two-to-one ratio mentioned above.)
Our next step is to investigate other vertices. Interconnecting thecenters of octahedra, we trace the array of minimum cubes. That
one's easy. The fact that cubes fill space is not new; what else can welearn?
If octahedron centers provide the meeting point for eight cubes,
what happens at the centers of tetrahedra? To begin with we observethat four quarter tetrahedra convene. The base of each shallow
pyramid is the face of a neighboring octahedron, so we ask whatshape is created by an octahedron framed by eight quarter tetrahedra. As seen in Chapter 9, this is equivalent to Loeb's "degeneratestellation"; the thin triangular faces of adjacent quarter tetrahedra
become coplanar when surrounding an octahedron, and thereby
merge into one diamond face. The result: a rhombic dodecahedron(Fig. 9-12). In IVM context, this means that every octahedron
reaches out, incorporating eight neighboring quarter tetrahedra, sothat the tetrahedra are completely used up (no leftovers). The entire
matrix is thus involved, meaning of course that rhombic dodecahedra fill space. By interconnecting the centers of every tetrahedron in the IVM we automatically generate an array of rhombic
dodecahedra.
The space-filling puzzle becomes more enticing with this new
addition. With its many diamond faces and irregular surface angles,
this shape is not, like the cube, an obvious space filler. A casual
observer would not suspect this intricate polyhedron of fitting sobeautifully together. In fact, without the advantage of the IVM, it isdifficult to picture how rhombic dodecahedra manage to fill space.

The Dual Perspective

Recalling from Chapter 4 that the rhombic dodecahedron is also a"degenerately stellated" cube, it is interesting to observe the relationship between the two packings. A cube with four body diagonals
dividing the inside into six pyramids is shown in Figure 12-3a. Thesesquare-based pyramids are the exact shape required to degenerately
stellate a second cube. We can "unwrap" that subdivided cube andplace its components on the six faces of a second intact cube-only
to arrive once again at our diamond-faceted friend (Fig. 12-3b). We
therefore have a new way to visualize the rhombic-dodecahedron

182

A Fuller Explanation

(a)

(c)

Fig. 12-3

packing. Start with an array of cubes, in which every other cube is
subdivided by a central node, while the rest remain empty. We havejust described an array of rhombic dodecahedra (Fig. 12-3c). A
framework of cubes is so familiar and readily imagined that thisexercise brings the rhombic dodecahedron's space-filling capability
into easy grasp.4

12. "All-Space" Filling: New Types of Packing Crates

183

Duality and Domain in Sphere Packing

Imagine that our closest-packed spheres are actually perfectly roundballoons. Supposing they are all packed tightly into a closed container which insures that their positions are fixed, we then try to
picture what would happen if more air were steadily pumped into
each balloon. Remember that they are unable to move-preventing
the natural reaction of collectively spreading out and taking up more
space. Instead each individual balloon expands and presses more
tightly against its neighbors so that the points of tangency merge
into planes of tangency:
A bubble is only a spherical bubble by itself. The minute you get two bubbles
together, they develop a plane between them. (536.44)

We allow the balloons to grow to the extent that all available spaceis occupied. What is the shape of the balloons once they merge
together? Remember that twelve spheres pack tightly around one,and so the tangency points between spheres, which were located at
the twelve (four-valent) vertices of vector equilibria, must now be
replaced by the same number of four-valent faces. We are thus
reminded that the domain of a sphere in closest packing-in Fuller's
words "the sphere and the sphere's own space" - is a rhombic
dodecahedron.
The above sequence provides an important insight into the spacefilling ability of rhombic dodecahedra. By renaming this diamondfaceted polyhedron "spheric," Fuller places considerable emphasis
on its relationship to closest packing. The spheric is thus presented
as a cosmically significant shape, the domain of the generalizedenergy event, and consequently, the domain of every intersection in
the omnisymmetrical vector matrix.
Synergetics thus arrives at its all-space fillers through investigationof nature's omnisymmetrical framework. IVM provides the context:
A "spheric" is anyone of the rhombic dodecahedra symmetrically recurrent
throughout an isotropic vector-matrix-geometry ... (426.10)
426.20 Allspace Filling:... Each rhombic dodecahedron defines exactly theunique and omnisimilar domain of every radiantly alternate vertex ... as well as theunique and omnisimilar domains ... of any aggregate of closest-packed uniradius
spheres ... "In a later section of Synergetics, Fuller expands upon the above
observations with a more general statement about any point inspace.

184

A Fuller Explanation

The most complete description of the domain of a point is not a vector
equilibrium but a rhombic dodecahedron, because it would have to be allspacefilling and because it has the most omnidirectional symmetry. The nearest thing you
could get to a sphere in relation to a point, and which would fill all space, is arhombic dodecahedron. (536.43)

Truncated Octahedron

The" truncated octahedron," or tetrakaidecahedron, is also a known
space filler (Fig. 12-4a). A model of this semi regular polyhedron canbe constructed by taping together eight cardboard hexagons and six
squares of the same edge length. But Fuller is wary of that approach,a geometry which supports the illusion of" solids." Instead, synergetics prescribes that we view the system in context. Therefore, togenerate this shape we start with a "three-frequency octahedron."With its edges divided into three equal segments, this octahedronsubmits quite naturally to vertex truncation, as shown in Chapter 9,
Figure 9-9b. A (single-frequency) half octahedron can be chopped
off from each corner, creating six square faces and converting thethree-frequency triangles into equilateral hexagons.
How do they fit together? Hexagonal faces of two truncatedoctahedra come together after rotating 60 degrees with respect toeach other, so that square faces alternate (rather than landing next to-Fig. 12-4

12. "All-Space" Filling: New Types of Packing Crates

185

each other) and begin to frame a cavity in the exact tetrakaidecahedral shape (Fig. 12-4b).
Four tetrakaidecahedra fit together around one IVM vertex. To
ascertain the exact relationship of this packing to the IVM will
require some investigation: how remote are the vertices involved,
and how many cells are incorporated in between? We will answer
these questions below, but first we go back to the simpler cases, for asense of the whole progression.
Two to One: A Review

We can now appreciate Loeb's development of the requirement thata space-filling polyhedron can be broken down into twice as manyoctahedra as tetrahedra. 3
The cube consists of one tetrahedron plus four eighthoctahedra-or a total of one half octahedron. This ratio of 1 to ~certainly qualifies. Eight quarter tetrahedra (for a total of two)embrace one octahedron to create the rhombic dodecahedron. With
this confirmation, we start a list of results, displayed in Table VI.
The VE consists of eight tetrahedra and six half octahedra: 8 to 3.
Lacking one octahedron, the VE must not be a space filler, aconclusion that is consistent with our earlier observation of the IVM.
However, pair a VE with the missing octahedron and all space canbe filled. The two-to-one ratio also serves as a prescription forcomplementary pairing; if an IVM system is not a space filler, theratio tells us what's missing.
The rest? All-space-filling rhombohedra are utterly straightforward; with two tetrahedra on either side of an octa, this shape gave
Table VI. Octahedron-tetrahedron ratio in space filling
Polyhedron
Tetrahedron
Octahedron
Cube
Rhombic dodecahedron

VE

Rhombohedron
3 v octahedron
Truncated octahedron
3 v tetrahedron
Truncated tetrahedronaS pace filler.

Ratio (tetrahedra: octahedra)
1:0
0: 1
1:~a

2: 1a
8:3
2: 1"
32:19
32: 16 a
11: 4
7:4

186

A Fuller Explanation

us our starting point for the space-filling ratio. Icosahedra cannot be
broken down into tetrahedra or octahedra; the icosahedron is eternally out of phase. That brings us to the truncations, which can beanalyzed in terms of the IVM. As a representative example, we
explore the truncated octahedron.
The three-frequency octahedron incorporates nineteen singlefrequency octahedra and thirty-two tetrahedra. It can be a frustrating experience to count these individual cells, but it can be done!
Skeptics are encouraged to get out a box of toothpicks and somemini-marshmallows. Figure 12-5 illustrates the dissection of one halfof a three-frequency octahedron, to make it easier to count the unit
cells in individual layers. Sixteen tetrahedra and nineteen half octahedra are clearly visible in the three layers shown. These results are
simply doubled to verify the totals in the first sentence of thisparagraph.
The ratio of 32: 19 does not qualify. Not surprisingly, the threefrequency octahedron-like any regular octahedron, for size does
not affect shape-is not a space filler. The truncation process removes half of a small octahedron from each of the six corners, or
three octahedra altogether. Nineteen minus three leaves sixteen,
while the total of thirty-two tetrahedra does not change. 32: 16 isindeed two to one, and the rule holds true once again.

= 2x

+

Fig. 12-5. Individual layers of three-frequency octahedron.

12. "All-Space" Filling: New Types of Packing Crates

187

The truncated tetrahedron can be dissected in the same manner as
its octahedral counterpart. The IVM starting point, a three-frequency
tetrahedron, consists of eleven unit tetrahedra and four octahedra,
and is clearly not a space filler-as we well know. Figure 12-6
enables the unit cells to be counted, by separating the layers of afour-frequency tetrahedron. In the process of counting, we can also
finally verify the claim introduced in Chapter 2 that third-power
numerical values (2 3, 33, 4 3, etc.) are represented by the volumes oftetrahedra of increasing frequency. Recall that the expression "xcubed" is derived from the fact that cubes of progressively higher
frequency consist of 1,8,27,64, ... unit cubes. Fuller points out thatexactly the same volumetric increase is exhibited by tetrahedra, as=++

Fig. 12-6. Volume accounting in tetrahedra of increasing frequency: third-power
model

188

A Fuller Explanation

portrayed in Figure 2-1b. We now take advantage of the disectedfour-frequency tetrahedron to verify these values. Adding up thevolumes of the unit tetrahedra (each with a volume of one) and unit
octahedra (each with a volume of four) in successive layers, wequickly find that the "cubic" numbers are confirmed, as shown in
Figure 12-6. A two-frequency tetrahedron has a volume of 8; athree-frequency, 27; a four-frequency, 64; and so on. "Nature is not'cubing,' she is 'tetrahedroning,''' announces the triumphant Bucky.
Back to the space-filling investigation, our next candidate is the(semiregular) truncated tetrahedron, which in synergetics is derived
from a three-frequency tetrahedron. Subtracting a unit tetrahedron
from each of the four corners yields a truncated tetrahedron consisting of seven small tetrahedra and four octahedra (Fig. 9-9a).
Seven-to-four does not correspond to the desired ratio, and we thus
learn that this shape will not fill space either; it lacks one unit
tetrahedron. As a final illustration of the disection method, we notethe new totals when a three-frequency tetrahedron is paired with athree-frequency octahedron. The resulting inventory consists of 43
tetrahedra and 23 octahedra, which does not quite reach the desired
double ratio. What if we add a second three-frequency tetrahedron?
The final total is then 54 tetrahedra and 27 octahedra, and that's it.
Two 3v tetrahedra paired with one 3v octahedron are able to fillspace. Once again, shape proves independent of size.
The above examples provide first-hand confirmation of the spacefilling hypothesis, allowing us to feel confident that a new truth has
been revealed.
Out of all the accumulated data, a consistent finding emerges.
With this generalized principle, we are equipped to determine whether
any polyhedron that submits to analysis in terms of octahedral andtetrahedral components is a space filler. We just have to break it
down into these constituent parts and check the ratio for therequisite two-to-one. The ratio replaces the arduous task of trying topush shapes together and twist them around to see if they pack; it isanother shortcut. IVM provides the framework; two-to-one is theratio; all criteria supplied.
Coincidence? Magic? Or rules of spatial order?

13
The Heart of the Matter:A- and B-Quanta Modules

Minimum system, triangulation, equilibrium of vectors, closestpacked spheres, and space-filling: the path toward the isotropic
vector matrix has many origins, any of which can yield the uniqueomnisymmetrical array. The result of traveling all of these routes is a
powerful awareness of the interplay of octahedral and tetrahedral
symmetries in space. However, the more thoroughly we search theIVM, the greater the intricacy of these "intertransformabilities"calling for a new level of analysis to keep track of our discoveries.
IVM got us down to the basics. Even the cube, mathematics'
conventional building-block, reduces to octahedron~tetrahedroncomponents. However, tetrahedra and octahedra are not true structural quanta, for it was often necessary to break them apart into
subcomponents in order to build other polyhedra. Our task istherefore still unfinished. We have yet to get to the heart of thetetrahedron and octahedron.
Necessity thus leads us to Fuller's A- and B-quanta modules. 1
The missing element in our IVM analysis is a way to handle
redundancy. Symmetry-the degree to which a system looks exactly
the same in different orientations-is a kind of structural redundancy. The tetrahedron, the octahedron, and their various combinations all have a high degree of symmetry, and now we intend to
get to the root of it. How? By subdividing symmetrically until we cango no further. As long as a system exhibits some degree of symmetry,it can be divided into identical subunits, which can be put together
to recreate the original system. Ergo, the system was redundant.
Through progressive subdivision, we can locate the minimum subunit of any system. This final asymmetrical module-or "least
common denominator" (LCD)-contains the geometrical dataneeded to reconstruct the whole system. We find the LCD by
subdividing a polyhedron until we reach the limit case, that is, a
module that can no longer be split into similar units.

190

A Fuller Explanation

A-Quanta Modules

Let's start again with our highly symmetrical friend, the tetrahedron.
The fact that its four faces are equivalent presents the first opportunity for subdivision-resulting in four equal parts. Each quartertetrahedron encompasses the region from the center of gravity (cg) toa face (Fig. 13-1a). It is evident from the threefold symmetry of theseshallow pyramids that each can be sliced into three identical pieces,
as if it were a triangular pie (Fig. 13-1b). The resulting pie pieceslong thin tetrahedra stretching from the apex of the quarter tetrahedron (cg of regular tetrahedron) out to an original unit-length edge-are quite irregular, and so the process is almost complete. However, one type of symmetry conspicuously remains. Each sliver canbe split in half to produce two mirror-image parts: a right- and aleft-handed version with identical angles and lengths (Fig. 13-1c).
And suddenly we have come to the end. There is no possible way to
divide that final product into equal parts; the shape thus generated isthe limit case. Fuller calls this asymmetrical tetrahedron the "Aquanta module."A-quanta modules contain the complete geometric ingredientsneeded to create a regular tetrahedron. We need twenty-four A-modules (twelve positive and twelve negative) to make a tetrahedron, but
one module alone supplies the information. The A-module, representing the volumetric essence of the tetrahedron, introduces a newkind of building block.
B-Quanta Modules

Having ascertained the minimum unit of the minimum system, wemust not forget about inherent complementarity. The above analysis
is repeated. A regular octahedron splits into eight equivalent pyramids (octants) (each of which divides into six equal pieces), as did
the quarter tetrahedra, yielding forty-eight asymmetrical minimum
units (LCDs) of the octahedron (Fig. 13-2a). That would have been
the end of the story except for a subtle catch, an overlooked
redundancy.
The eighth-octahedron has twice the altitude of the quarter tetrahedron, but both pyramids have the same (equilateral triangle) base.
The quarter tetrahedron therefore fits right inside the octant, occupying exactly half the available volume (Fig. 13-2b). The remaining
volume is pure octahedron-a hat-shaped wedge that accounts forthe shape difference between the two pyramids. This concave trian-13. The Heart of the Matter: A-B Quanta Modules

191

(a)

(b)

(c)

Fig. 13-1. Development of A module.

gular lid can be subdivided into six equal irregular tetrahedra (threepositive, three negative) and these are called B-quanta modules. They
are generated by the LCD of the octahedron after that of thetetrahedron is taken away. As long as the octahedron's asymmetricalunit contained a complete A-module within its boundaries, the unit
was redundant. By removing the A-module which had occupied half

192

A Fuller Explanation

(a)

(b)

(c)

Fig. 13-2. (a) One-sixth of an "octant" equals -is of an octahedron. (b) The quartertetrahedron fits inside the octant and has half the altitude. (c) Subtracting an A
module fromof an octahedron defines the B module.

*

the volume of the octahedron's LCD, we finally achieve a second
true modular quantum (Fig. 13-2c). Thinner and more pointed thanthe A's, the B-quanta have a very different shape but precisely thesame volume. Neither A nor B can be made from the other; they are
fundamentally distinct, complementary equivolume modules.
The LCD of the octahedron (*) consists of one A and one B,
while the LCD of the tetrahedron (14) is simply an A-module. The
analysis is complete: we have broken down our IVM constituents
into their essential characteristics. We can go no further.
However, the field is now wide open for experimentation. Armedwith the IVM quantum units, we can anticipate a tremendous rangeof combinations and permutations, or rearrangements. Having systematically analyzed our basic systems, we now start to put theiressential quanta back together in order to further understand therelationships between polyhedra. We have thus developed a far moresophisticated (and specific) framework with which to explore polyhedral intertransformability. Fuller takes it a step further. (Why stopwith polyhedra?)If you are willing to go along with the physicists, recognizing complementarity,then you will see that tetrahedra plus octahedra~and their common constituents,
the unit-volume A- and B-Quanta Modules~provide a satisfactory way for both

13. The Heart of the Matter: A-B Quanta Modules

193

physical and metaphysical, generalized cosmic accounting of all human experience.
(950.34)

We also observe considerable multiplication of complexity with the
new framework created by subdivision. As Bucky would have it:
multiplication by division.
Energy Characteristics

Progressive subdivision has left us with legitimate geometric quanta:
final packages which cannot be split into equal halves. In a general
sense, the model parallels science's search for the ultimate aspects ofreality. Physics probes deeper and deeper into matter, breaking it
down into ever-smaller constituents-cells, molecules, atoms, subatomic particles-ultimately seeking a package of energy (quarks?)
which cannot be split apart. Fuller probes similarly into his geometry, hoping to gain insights about Universe itself.
Fuller's profound faith in the significance of reliable
patterns-coupled with his unflagging determination to find nature's
coordinate system-led him to draw many parallels between synergetics and nature. Most of these are both suggestive and undeveloped; he planted his seeds, left the cultivation for posterity, andwent on with his search. Certain that science can be modeled, Fuller
felt a great responsibility to pursue what he saw as an ever more
relevant investigation; if patterns are to emerge, sufficient data must
be collected. Toward the goal of clarifying the patterns observed by
Fuller, our strategy might be to forge ahead: cover as much groundas possible and worry about significance later. However, in hiscoverage of intriguing geometric properties, Fuller often immediately
assigns connections to physical phenomena. We can neither ignore
nor confirm such speculation, and so for now, we merely record andfile away. The respective energy" valving" (ability to direct, store,
control) properties of A- and B-modules is a typical example.
Once again, the issue is based on "operational" procedure. Modelsof Fuller's two tetrahedral quanta are constructed out of paper. The
process starts with a planar" net" (a flat pattern piece which can befolded up and taped together to make a specific polyhedron). Consider for example the regular tetrahedron. Its four equilateral-triangle faces can be generated by folding one (double-size) triangle alonglines that connect mid-edge points (Fig. 13-3a). This ability to form atetrahedron by folding one triangle is not to be taken for granted;
the situation that allows all four faces of a generic tetrahedron to fit

194

A Fuiler Expianation

A

\

\

\

\

/
/

\

\

\1

1

/

.....

,

1

I

B

(b)

(a)

Fig. 13-3

inside a triangular frame-requiring exactly supplementary angles
out of the infinite possibilities-is an exception. We are not surprised
by Figure 13-3a for we expect such exceptional cooperation from theuniquely symmetrical regular tetrahedron.
The surprise is that the asymmetrical A-module unfolds into one
planar triangle: "an asymmetrical triangle with three different edge
sizes, yet with the rare property of folding up into a whole irregulartetrahedron" (914.01). This unusual property makes it a kind of pureform. The B-module, on the other hand, will always fold out into
four separate planar triangles no matter which vertex you start with;
its net will never fit into one triangular frame. Figure 13-3b compares the two nets.
Fuller connects this geometric property with "energy." A-modules
are thus said to concentrate or hold energy, while B's release or
distribute. This conclusion is based on the fact that "energy bounces
around in A's working toward the narrowest vertex," only able toescape at a "twist vertex exit" (921.15). Comparing pattern pieces,
A's planar net offers three escapes; the jagged four-triangle complexof B offers twice that number (Fig. 13-3b). It is a somewhat bizarre
observation to begin with, and characteristically it led to the following ambiguous assignment of meaning: a vertex, or non-180-degree
junction, in a planar net represents disorderly energy-escaping properties. Hence in A-modules, energy is seen as contained, able to13. The Heart of the Matter: A-B Quanta Modules

195

bounce around inside the net without many available exits, whereas
in B-modules, energy is quickly released.
We can conceive of this energy in many ways-as light beams, asbouncing electrons, or even as billiard balls for a more tangible
image. All three qualify as "energy events," and having a specificimage in mind makes it easier to think about the different "energyholding" characteristics. To understand and evaluate Fuller's assertion, we go along with his use of "energy," for the word covers agreat deal of territory already and his usage is internally consistent.
Fuller calls our attention to a geometric property that we may nothave otherwise noticed, and with respect to this phenomenon ofplanar nets there is no doubt as to the difference between A- andB-modules. What is the significance of this distinction? What are weto conclude about the orderly contained A versus the disorderly
sprawling B? In terms of physical Universe, a judgement probably
cannot be made. However, for the purposes of this text and ofcontinuing to explore the geometric interactions of the two quanta,
we adopt Fuller's energy assignment: A's conserve; B's dissipate. It
provides a consistent reference system with which to classify the twoquanta and their subsequent interactions. Furthermore, two basic
modules exhibiting the same volume and different energy characteristics provide an even more attractive model of "fundamental complementarity": equivalent weight or importance, opposite charge.
Sound familiar? The parallels are tantalizing.
Mite

Next we apply our LCD analysis to the IVM. The procedure-calling for progressive subdivision in search of the minimum repeating
unit-comes to an end with a unit that can no longer be symmetrically divided. As before, we seek the smallest system that can be
duplicated to recreate the whole IVM-a microcosm, containing all
the ingredients of the macro-array.
Having already split tetrahedra into equal quarters and octahedra
into eighths, we skip directly to a unit consisting of a quartertetrahedron and an eighth-octahedron back to back, sharing anequilateral-triangle face. This unit, which connects the geometricalcenters (or cg's) of any adjacent tetrahedron and octahedron in the
IVM, exhibits the same threefold symmetry as its two triangularpyramids taken separately. Final subdivision thus yields six equivalent asymmetrical tetrahedra: three positive and three negative (Fig.
13-4).

196

A Fuller Explanation

Fig. 13-4. LCD of IVM: "Mite."As the smallest repeating unit of the IVM, this system is theminimum space filler, thus inspiring Fuller's term "Mite" (MInimum
space-filling TEtrahedron.) A collection of A-modules cannot fillspace; they can only make regular tetrahedra. The skinny irregular
B's cannot even create a symmetrical polyhedron by themselves-letalone fill space. The Mite is the LCD of the omnisymmetricalspace-filling matrix, and therefore is the minimum case, the firstall-space filler:
954.09 We find the Mite tetrahedron . . . to be the smallest, simplest, geometricallypossible (volume, field, or charge), allspace-filling module of the isotropic vector
matrix of Universe.

Knowing that the Mite encompasses the asymmetrical units ofboth the tetrahedron and octahedron, we can identify its constituent
A- and B-modules. The tetrahedron contributes an A-module, while
the adjacent octahedron adds both another A (the mirror image ofthe first) and a B, for a total of three equivolume Modules. The
positive and negative A's are in balance, and the solo B may beeither positive or negative, thereby determining the sign of the whole
Mite. Like A- and B-modules, Mites come in one of two possible
orientations (Fig. 13-5a).
We cannot fail to comment on another 2 : 1 ratio just displayed by
the minimum space-filler. As suggested by this discovery, it turns outthat there must be two A-modules for every B-module in any
space-filling polyhedron. We shall see how this rule applies to ourfamiliar candidates below, and once again, Loeb's "Contribution"
provides a more thorough analysis of the phenomenon. Appropriate

13. The Heart of the Matter: A-B Quanta Modules

197

B'
(a)

p•

.,
:1

•

· 1

: r

*'--_ _~--..... rq(b)Fig, 13-5, (a) Orientations of Mite. (b) Unexpected mirror plane in Mite.

examples will be cited throughout this chapter, but readers areencouraged to turn to the back of Synergetics for Loeb's report. 2
Mirrors
There's more to the Mite than meets the eye. Figure 13-5a shows acorresponding face of both a positive and a negative Mite. As every
Mite incorporates both a positive and a negative A-module, it is theunpaired B-module-slanting off to one side or the other-that
determines the charge of the overall system.
From the above recipe, it appears evident that this minimum
space-filling tetrahedron is either right- or left-handed. This was also
evident from our initial generating procedure, subdividing the IVM
to carve out the smallest repeating unit with mirror symmetry, whichis then split into equivalent mirror-image (positive and negative)
Mites.
However, a surprising and easily overlooked result of joining two
A's and a B modifies the above conclusion. An unexpected mirror
plane runs through the middle of this irregular tetrahedron that wehave assumed must be either right- or left-handed. This means that a198

A Fuller Explanation

Mite is actually its own mirror image. The positive and negative
versions are identical. The slanting orientation depicted in Figure
13-5a obscures this fact; however, it turns out that the Mite has two
isoceles faces, a fact that indicates that the remaining two faces mustbe identical (but mirror-image) triangles. Therefore, the outsidecontainer of the Mite (ignoring the arrangement of its modular
ingredients) incorporates a subtle mirror plane and is actually exactly the same shape as its mirror image. A +A -B+ equals A +A -B(Fig. 13-5b). This extraordinary fact means that we don't needpositive and negative Mites to fill all space; one or the other version-or both in random combinations-will suffice. As its own mirror
image, any Mite can fill either position, positive or negative.
We can conclude therefore that the two versions are identical;
however, Bucky points out that different internal configurations
cannot be ignored, for they point to different energy characteristics:
Though outwardly conformed identically with one another, the Mites are always
either positively or negatively biased internally with respect to their energy valving(amplifying, chocking, cutting off, and holding) proclivities ... (954.43)

Cubes into Mites

Once the A- and B-game gets going, significant relationships are
uncovered at every move. The game thus becomes more and more
fascinating as it is played. For example, we might carve open a cube,
generating six square-based pyramids, one from each face to the cg.
Then slice each square pyramid into quarters, like a peanut-butter
sandwich. We thereby rediscover the Mite, observing that the cube
consists of twenty-four Mites (Fig. 13-6a). Oriented with one of its
isosceles triangles (45 °-45 °-90°) on the cube's surface, the Mite
simply rotates to transform Octet Trusses into boxes.
Rhombic Dodecahedra

Next, we apply the LCD procedure to the rhombic dodecahedron.
Its twelve faces each frame a diamond-shaped valley ending at the cg
point. Each of these inverted pyramids can be split into four (twopositive, two negative) asymmetrical tetrahedra (Fig. 13-6b). This is
the LCD of Fuller's "spheric," and it is none other than the Mite,
the LCD of the isotropic vector matrix. Forty-eight Mites in yet
another orientation make up this space-filling shape. Not surprisingly, space-filling and Mites have an important connection.

13. The Heart of the Matter: A-BQuanta Modules

199

(a)

(b)

Fig. 13-6. The LCD of both cube and rhombic dodecahedron is the Mite.

No longer caught off guard by such interconnectedness, we conclude by observing that the Mite-faithfully representing octet symmetry-is also an integral component of the cube and the rhombic
dodecahedron.
Coupler

Fuller's coupler is an irregular octahedron made of eight Mites-or
sixteen A-modules and eight B-modules (Fig. 13-7). Given thiscomposition, the "semisymmetrical" coupler is clearly a space filler.
Because of the Mite's newfound mirror symmetry, the coupler does
not have to have equal numbers of positive and negative B-modules;
any eight Mites will make a coupler. This special octahedron has two

200

A Fuller Explanationz

Fig. 13-7. Each Mite is one octant of the coupler.

equal-length axes (which will be referred to as x and y) and a third
shorter axis (z). The point of intersection of the three axes will becalled K.
The equal x and y axes outline a square equatorial cross-section,
which we can now identify as the face of a cube. In fact, the coupler
is two sixth-cube pyramids back to back. 3
The other cross-sections (xz and yz) are diamonds- in fact, theexact shape of a rhombic dodecahedron's face. At this point we shallnot be surprised to learn that splitting the coupler in half along
either diamond cross-section isolates one-twelfth of the rhombic
dodecahedron. Six half couplers make a cube; twelve half couplers(split the other way) make a rhombic dodecahedron.
"Couplers literally couple 'everything'" (954.50). Aptly named,
the new octahedron joins together both pairs of cubes and pairs ofrhombic dodecahedra. Fuller's nomenclature proves quite logical:
We give it the name the Coupler because it always occurs between the adjacently
matching diamond faces of all the symmetrical allspace-filling rhombic dodecahedra, the" spherics" .. . . (954.47)

The coupler's different pairs of opposite vertices reach to the geometric centers of two adjacent polyhedra (cube or spheric, depending on
the orientation), incorporating their shared face as a cross-section(Fig. 13-8a, b). Half a coupler belongs to one cube (or spheric), andthe other half to its neighbor. So the coupler literally couples-well,not "everything" but-a couple of space-fillers.
Fuller continues: The coupler's role is cosmically relevant, for"rhombic dodecahedra are the unique cosmic domains of theirrespectively embraced unit radius closest-packed spheres" (954.47).
The coupler therefore connects the centers of gravity of adjacent13. The Heart of the Matter: A-B Quanta Modules

201

(b)
Fig. 13-8. "Coupler."

spheres, and its domain includes both the spheres and the intervening (dead air) space. This observation explains why the coupler's
volumetric center was labeled K: it marks the exact "kissing point"
between tangent spheres in a closepacked array. Now we have thecomplete story behind Fuller's somewhat dense explanation of his
coupler:. . . The uniquely asymmetrical octahedra serve most economically to join, or
couple, the centers of volume of each of the 12 unit radius spheres tangentiallyclosest packed around every closest packed sphere in Universe, with the center ofvolume of that omnisymmetrical, ergo nuclear, sphere. (954.48)

Perhaps the most intriguing aspect of the coupler is its similar role inthe cube and in the rhombic dodecahedron, thereby linking (orcoupling) the two space fillers in a new partnership.
Volume and Energy

The inventory of twenty-four modules (sixteen A, eight B) indicatethat the coupler has the same volume as the regular tetrahedron with
its twenty-four A-quanta. Both have a tetrahedral volume of one.
(Recall also that a coupler consists of two sixth-cube pyramids back

202

A Fuller Explanation

Fig. 13-9. Different rearrangements of eight Mites in the coupler.

t

to back, orof a cube, and that in Chapter 10 the volume of atetrahedron was shown to be t that of a cube.) Dissimilar insymmetry and shape, they are related by their shared unit volume,
inviting comparison. The coupler is a different sort of minimum
system: a semi symmetrical space filler, in contrast with the tetrahedron's origin as the minimum system of any kind, i.e., the first caseof insideness and outsideness.
Finally, we have to comment on the coupler's internal flexibility.
The number of different ways to arrange eight Mites is greatly
increased by the unexpected mirror symmetry. Since positive andnegative Mites can switch places, we actually have a pool of sixteen
from which to choose for each of the eight positions. A coupler
might consist of four positive and four negative Mites, or all positive,
all negative, or any of the possible combinations in between: (0-8,
1- 7, 2-6, 3-5, 4-4, ... ,8-0). Then, within each of these nine possible groups, the Mites can be switched around into four different
arrangements. (A few of these combinations are shown in Figure13-9.) The resulting 36 varieties of couplers all have the sameoutward shape, but in Fuller's view, their internal variations represent important distinctions in energy behavior:
When we discover the many rearrangements within the uniquely asymmetric
Coupler octahedra of volume one permitted by the unique self-interorientability of
the A and B Modules without any manifest of external conformation alteration, we
find that under some arrangements they are abetting the X axis interconnectings

13. The Heart of the Matter: A-B Quanta Modules

203

between nuclear spheres and their 12 closest-packed ... spheres, or the Y axis
interconnectings. . .. (954.58)
When we consider that each of the eight couplers which surround each nuclear
coupler may consist of any of 36 different AAB intramural orientations, we
comprehend that the number of potentially unique nucleus and nuclear-shell interpatternings is adequate to account for all chemical element isotopal variations ... aswell as accommodation ... for all the nuclear substructurings, while doing so byomnirational quantation. . .. (954.54)

In other words, "energy" travels in one direction or another depending on the arrangement of the (oppositely biased) A's and B's.
Synergetics thus accounts for nature's incredible variety and complexity despite its small number of different constituents. The secondary level of organization involves grouping the different couplers
together and results in an explosion of potential variations. A- andB-modules can thus be rearranged into orderly octet configurations
in myriad ways; clusters might appear quite chaotic in some locations and precisely ordered into whole octahedra and tetrahedra inothers, while the overall space-filling matrix remains intact. As in
genetics, a small number of simple constituents are able to generate
a virtually unlimited repertoire of patterns.

Review: All-Space Fillers

Dismantling the cube, we saw that each of its six inverted pyramids
breaks down into four Mites. What does this tell us about A- andB-modules? 24 Mites yield a total of 72 modules, with twice as many
A's and B's (or 48 : 24). This two-to-one ratio gets to the modular
heart of the octahedron-tetrahedron prerequisite for space filling, as
developed in the last chapter.
Neither tetrahedra with only A's nor octahedra with equal numbers of A's and B's can qualify as space fillers. So far, so good.
Thinking only in terms of A- and B-modules, what proportion oftetrahedra and octahedra would we need to satisfy the recipe of two
A's for every B? The octahedron's 48 B's must co-occur with 96 A's;
the octahedron itself supplies half of them, but 48 A-modules arestill missing. Two tetrahedra will provide exactly the right number ofA-modules to complete the formula, thus reconfirming the one-to-two
octahedron-tetrahedron ratio discovered in the previous chapter.
The 48 Mites in the rhombic dodecahedron are easily disected into
96 A's and 48 B's, or a total of 144 modules. With twice as many A's

204

A Fuller Explanation

as B's, Fuller's "spheric" does not contradict the growing evidence ofa general rule.
A- and B-modules in the truncated octahedron-also a space filler-can be counted by recalling the numbers of internal singlefrequency tetrahedra and octahedra determined in the last chapter.
32 tetrahedra, with 24 A-modules each, contribute 768 A's, while 16
octahedra consist of 768 A's and 768 B's, for a total of 1536 A's and
768 B's altogether (768 X 2 = 1536). We can begin to have confidence in the reliability of our 2 : 1 ratio, especially in view of thejump to much larger numbers. With a total of 2304 modules, this is
the smallest truncated octahedron outlined by IVM vertices. Fuller's
inventory of all-space fillers (954.10) lists the truncated octahedron
with twice the linear dimensions and eight times the volume (consisting of 18,432 quanta modules) rather than the smaller version. The
reason for this choice is not clear; however, the magnitude of thesequantities hints at the complexity of his modular system-in termsof both the variety of systems that can be made from the modules
and the number of rearrangements within those systems. From thesenumbers it is evident that, although the quanta modules help us tounderstand the conceptual essence of many polyhedral intertransformabilities, they are very impractical for hands-on experimentation.
So many modules are needed to make complete polyhedra that thissystem does not offer an ideal strategy for model making. Instead,
we might utilize the analysis to work out relationships on paper.
From the above examples, we can see that the A- and B-modules
get to the root of the two-tetrahedron-one-octahedron rule developed in the previous chapter. Because the earlier analysis depended
on disassembling its two basic units, it was necessary to probe
further to isolate the real quanta. A- and B-modules, which cannot
be symmetrically subdivided, were isolated as the true quanta withwhich to measure and analyze related polyhedral systems.
Impressed by the geometric significance of these modules, Fuller
proposes that somewhere within this discovery lie secrets with far
greater applicability than just to geometry:
The A and B Quanta Modules may possibly quantize our total experience. It is aphenomenal matter to discover asymmetrical polyhedral units of geometry that are
reorientably compositable to occupy one asymmetrical polyhedral space; it isequally unique that, despite disparate asymmetric polyhedral form, both have thesame volume .... Their unit volume and energy quanta values provide a geometry
elucidating both fundamental structuring and fundamental and complex intertransformings, both gravitational and radiational. (920.01)

13. The Heart of the Matter: A-B Quanta Modules

205

From their energy associations to their remarkable symmetry, thesemodules synthesize much of Fuller's research. Significant relationships to physical phenomena may well reward continued investigation, for nature also deals with discrete quanta, creating endlessvariation through synergetic recombinations. Fuller reasoned that his
geometric quanta-the end result of a systematic and logical progression of steps-must relate to physical phenomena. The approachis typically Fuller's: assume significance until proven otherwise. In
essence he suggests that tiny whole or discrete systems should
replace irrational unending digits-somehow providing a comprehensive rational coordinate system.

14
Cosmic Railroad Tracks: Great Circles

Any planar closed loop drawn on the surface of a sphere is necessarily a perfect circle, as a result of the sphere's steady curvature.
Such loops qualify as either "great" or "lesser" circles, and thedistinction is defined in mathematics as follows. A great circle isformed by the intersection of a plane passing through the center of asphere with the surface of that sphere. Any other circle, no matter
what size, is lesser. The center of a great circle coincides with thesphere's center. In short, a great circle is an equator-found in any
angular orientation, but always around the fattest part of its sphere.
What's so great about a great circle? Above all, it provides theshortest route between any two points on a sphere. This geometric
fact is not obvious in many cases; for example, looking at a globe it
appears that the logical route between two points situated somedistance apart on the Tropic of Cancer involves traveling along theirshared "lesser circle" band (Fig. 14-1a). However, the principle ismade more obvious by Fuller's juxtaposition of two extreme cases.
He describes a small lesser circle near the North Pole of an imaginary globe, and labels two points A and B (Fig. 14-1b). As with thelarger Tropic of Cancer, the eye naturally travels from A to B along
their mutual lesser-circle path, without suspecting that this represents the "long way around." Fuller then redraws the same lesser
circle in a new location-superimposed over the globe's equator sothat A and B both fall on the horizontal great circle. The shortest
route between A and B is suddenly obvious. Likewise, between any
two points on a sphere the most expedient route will be a great-circle
segment; Fuller's example makes it easy to see that a lesser-circle
path will always present a detour.
Other special characteristics: Any two great circles on a spheremust intersect twice-specifically, at two points 180 degrees apart.
There is no other possibility: unless they are actually the same circle,
two great circles can neither avoid each other altogether, nor intersect only once, nor intersect more than twice. Finally, the junction of
two great circles inscribes two pairs of equal and opposite angles on

14. Cosmic Railroad Tracks: Great Circles

207

Tropic of Cancer
A

Equator

B

(b)

(a)
Fig. 14-1

the sphere's surface, whose two angular values add up to 180
degrees; the statement is equivalent for the intersection of two
Euclidean "straight lines." Lesser circles do not share this property;
in fact, their intersection produces opposite angles which are necessarily unequal. In conclusion, on the surface of a sphere, only great
circles have the geometric characteristics of "straight lines."
As we examine different aspects of great circles, you will noticethat much of this material is quite complicated; the patterns are too
intricate to be readily visualized in the mind's eye. Furthermore,connections drawn between the geometric models and physical phenomena are unusually speculative. However, as Fuller's Synergetics
devotes considerable attention to great circles, anyone who has tried
to decipher these sections will welcome full coverage. Referring backand forth from the text to the drawings will be essential.
Why Are We Talking About Spheres?

The vertices of regular and semiregular polyhedra lie on the surface
of an imaginary sphere, which is to say that all vertices are equidistant from a polyhedron's center. Given this fact, we can picture
spherical versions of each polyhedron, in which the polyhedral edges
have stretched outward to become great-circle arcs and the faces
have expanded into curved surfaces, as if each shape had been blown
up like a balloon. Figure 14-2 shows a spherical tetrahedron, octahedron, and icosahedron as examples. Comparing these systems
with their planar counterparts, it is clear that polyhedral edges are
actually chords of great-circle arcs. We can conclude that the shortest distance between two events of a system always involves a great
circle.

208

A Fuller Explanation

Fig. 14-2. Spherical polyhedra.

The concept (rather than the reality) of a sphere-i.e., an omnisymmetrical container-acts as a frame of reference for polyhedral
systems. Spherical polyhedra thus introduce new versions of familiar
characters. The topological information (that is, the numbers andvalencies of vertices, edges, and faces) of any polyhedron are displayed on a spherical canvas. An obvious consequence of this type of
display is that shape is no longer a variable. Shape similarities, which
are so rigorously accounted for by A and B modules, are thus
ignored; our investigation now focuses in on topological, or surface,characteristics. Transforming polyhedra into balloons temporarilyequalizes shape and size, providing a "common denominator" forother comparisons. The process develops a somewhat unorthodox
chart.
New Classification System

However, the chart is not yet complete. Simply projecting edges andfaces onto a spherical surface does not teach us anything new. We
have yet to exploit the nature of the sphere.
Spheres suggest spin. That's how synergetics initially arrives at theomnidirectional form. Spin any system in all directions, and ultimately the action will have defined a circumscribing spherical envelope. Fuller places considerable emphasis on the" spinnability" ofsystems, arguing that as everything in Universe is in motion, thedifferent axes of spin inherent in systems are worthy of investigation.

14. Cosmic Railroad Tracks: Great Circles

209

All polyhedra have three sets of topological aspects: vertices,
edges, and faces. These sets correspond to three types of axes ofrotational symmetry (or" spin") which connect pairs of either polaropposite vertices, mid-edge points, or face centers. As a polyhedron
spins about anyone of these axes, an implied great circle is generated midway between the two poles, in other words, at the equator.
Equators corresponding to all existing axes of symmetry can be
simultaneously represented on a spherical surface, creating an exhaustive chart of the topological symmetries of a given system. Eachsymmetrical polyhedron has its own great-circle diagram, which
incorporates all its axes of rotational symmetry-or axes of spin, in
Fuller's terminology. The patterns generated by related polyhedra
may include some of the same circles, as determined by common
symmetries; however, the complete chart of a polyhedron is exactly
shared only by its dual, as will be shown below.
Great circles reveal a new aspect of polyhedral "intertransformability," a novel (if obscure) means of detecting symmetrical relationships among systems. We can anticipate the emergence of decidedly unfamiliar patterns in this game for their resemblance to thesource polyhedron is sometimes subtle. Finally, in pursuing thisstudy we discover new variations on upper limits and minimum
cases. Great circles thus provide another tool with which to detect
inherent spatial constraints.
Great-Circle Patterns

Let's begin with a representative system, the octahedron. We interconnect polar opposites, starting with vertices, followed by mid-edge
points and finally face centers. Six paired vertices are connected bythree mutually perpendicular lines, the familiar XYZ axes meeting at
the octahedral center of gravity (Fig. 14-3a). These axes define threeorthogonal great circles, which divide the sphere's surface into eight
triangular areas, or octants. Three symmetrically arranged great
circles will always form the edges of a spherical octahedron (Fig.
14-2, middle). That much is straightforward.
On to edges. The octahedron's twelve edges consist of six opposing
pairs, the midpoints of which can be connected by six intersecting
axes (Fig. 14-3b). The same number of great circles are thereby
generated, delineating another facet of the octahedron's symmetry(Fig. 14-4). Unlike the previous case, the pattern made by six great
circles does not look like an octahedron. Its twenty-four right210

A Fuller Explanation

(a)

(b)

(c)

Fig. 14-3. Different sets of axes of symmetry. (a) Paired vertices yield 3 axes offour-fold rotational symmetry. (b) Paired edges yield 6 two-fold axes. (c) Paired
faces yield 4 three-fold axes.

Fig. 14-4. Six great circles of the octahedron.

isosceles triangles 1 outline the edges of both the spherical cube andrhombic dodecahedron, as well as the edges of two intersecting
spherical tetrahedra (otherwise known as the "star tetrahedron"),thus highlighting the topological relationship between these foursystems. This exercise demonstrates a new aspect of "intertransformability": great circles generated by a given polyhedron often delineate the spherical edges of its symmetrical cousins.

14. Cosmic Railroad Tracks: Great Circles

211

Fig. 14-5. Four great circles.

Finally, the centers of opposite faces are joined together (Fig.
14-3c). The octahedron's four pairs of triangles define four intersecting axes, and in tum four symmetrically arrayed great circles (Fig.
14-5). As the spherical edges of Fuller's vector equilibrium, thispattern is particularly significant, as will be developed below.
The sets of three, six, and four great circles can all be superimposed on one sphere to exhaustively display the "unique topological aspects" of the octahedron. With a grand total of thirteen, we
have located all the great circles that correspond to the octahedron'ssymmetry. We can go no further.
Now consider the cube. It is quickly apparent that its total
great-circle pattern will be identical to that of the octahedron. The
cube's eight vertices generate the same four great circles as theoctahedron's faces, its six faces correspond to the three orthogonal
great circles, and the twelve edge midpoints are identically situatedto those of the octahedron. Such is the result of duality. The same
sets of circles are generated by different elements, and the end results
are equivalent.
Next, we tum to the VE. With two kinds of faces the situation
might seem more complicated; however, the above procedure still
applies. The VE's eight triangles define the same four axes as thefaces of the octahedron, while its squares contribute three orthogonal(XYZ) axes. Seven great circles altogether are generated by the axesof the fourteen VE faces. Twelve vertices correspond to the same six
great circles as those of the octahedron (or cube) edges, and finally
24 edges spin out the unfamiliar pattern of twelve great circles. Witha total of 25 great circles, the topological parameters of the VE are
exhausted (Fig. 14-6).

212

A Fuller Explanation
Fig. 14-6. Twenty-five great circles of the
VE.

The tetrahedron presents a slightly different situation in that its
vertices do not group into polar opposites, but rather are positioned
directly across from the centers of faces. However, four axes ofsymmetry (all going through the center of gravity) can be created byconnecting the vertices with their opposite faces. In having axes ofrotational symmetry that connect faces and vertices, the tetrahedronis again unique; as we recall from Chapter 4, only tetrahedra havethe same number of vertices as faces. (Only the tetrahedron is its
own dual.) The four great circles generated by these unorthodox axes
produce (once again) the spherical vector equilibrium. This is not
surprising if we recall the lesson from "multiplication by division,"
which first uncovered this shared symmetry between the tetrahedron,octahedron, and VE: By simply interconnecting mid-edge points, allthree polyhedra were found to be inherent in the topological makeupof the starting-point tetrahedron. They share the same four axes ofsymmetry.
The axes of symmetry associated with the tetrahedron's six edges
are more orthodox: mid-edge points of opposite edges are simply
joined to reestablish the XYZ axes. Familiar by now with evidenceof right angles hiding within this triangular shape, we can no longer
be caught off guard by this discovery. The corresponding three great
circles are the edges of the spherical octahedron, once again illustrating the depth of the octahedron-tetrahedron relationship. And now
all topological aspects are used up ; the tetrahedron has seven great
circles, the minimum number possible for symmetrical polyhedra.
Next, we look at the maximum case. It's not easy to accept theconcept of an upper limit on the number of symmetrically positioned
great circles that can be imposed on a sphere; common sense
suggests that we should be able to keep adding new rings inde-14. Cosmic Railroad Tracks: Great Circles

213

Fig. 14-7. Six great circles of the icosahedron.

finitely. However, we recall from Chapter 3 that all systems arepolyhedral-that is, everything that divides inside from outside canbe described in terms of some number (four or more) of "event
complexes" and their relationships-and from Chapter 4 that thesystem with the greatest number of identical regular polygons andequivalent vertices is an icosahedron. This tells us that the number ofgreat circles allowed by the topological aspects of the icosahedron is
the maximum for these symmetrical patterns.
The axes defined by the icosahedron's twelve vertices introduce six
great circles. We already have a pattern with six circles(octahedron-VE, Fig. 14-4); however, this is an entirely new set,"out of phase" with the earlier group, as defined by the "jitterbug"
transformation (Fig. 14-7). Outlining 12 pentagons and 20 triangles(as opposed to 24 isosceles triangles), these arcs present the mostsymmetrical arrangement of six great circles. When a vector equilibrium contracts into an icosahedron in the jitterbug transformation, the radius-edge-length equivalence is lost, but the distances
between adjacent vertices on the surface are suddenly all equal. As
an icosahedron produces the most symmetrical distribution of twelve
vertices on a closed system, it follows that the same is true for theircorresponding six great circles.
Next, the axes of symmetry connecting the centers of icosahedronfaces generate ten great circles, while the thirty edges spin out fifteen
more: 6 + 10 + 15 = 31, the total number of great circles in thelimit-case pattern (Fig. 14-8). These circles provide the sphericaledges of a pentagonal dodecahedron as well as those of an
icosahedron, and (less predictably) also include the octahedron.
These relationships will be explored below.
Least Common Denominator

The spherical icosahedron divides the surface of a sphere into thegreatest number of completely regular domains (with both equal arc
lengths and equal surface angles). We can therefore subdivide one of214

A Fuller Explanation
Fig. 14-8. Thirty-one great circles.

(a)

(b)

Fig. 14-9. Maximum of 120 equivalent domains: LCD.

these symmetrical triangles to find the least common denominator of
surface unity-in much the same way we isolated A and B modules.
Each triangle is split into six equal parts by perpendicular bisectors,
to obtain a final non-symmetrically-divisible unit (Fig. 14-9a). The
result is 120 asymmetrical triangles (60 positive, 60 negative), themaximum number of equivalent domains on the surface of a sphere.
These perpendicular bisectors are the icosahedron's fifteen great
circles (Fig. 14-9b).
Imagine that some number of hypothetical creatures are to inhabit
the surface of a large sphere and it is considered necessary that each14. Cosmic Railroad Tracks: Great Circles

215

one sit in the middle of an identical plot. The consequence of thisstipulation is that no more than 120 creatures can fit on the sphere,no matter how large it is. This result is certainly counterintuitive, forassuming a large enough sphere, it seems that we should be able toaccommodate as many creatures as we want. However, the unyielding laws of symmetry limit the population to the unexpectedly low
number of 120.
We go back to the 31-great-circle diagram to observe the 120
triangles in context. These asymmetrical triangles are true LCD
units; anyone of them contains all the geometric informationnecessary to reconstruct the entire pattern. This is an importantfactor in the development of geodesic domes, as will be seen in thenext chapter, which discusses the relationship of great circles togeodesic domes.
LCD: "I ntertransformabil ity"

The following exploration is similar to the transmutations of A andB modules and uncovers some of the same relationships; however,consistent with great-circle limitations, this study deals only withsurface characteristics. Each symmetrical great-circle pattern has aleast common denominator. For example, the spherical octahedron
has eight equilateral faces, which can be split into six asymmetricalof the whole surface. These triangles
triangles, each one i of t, or
are LCD units, because they cannot be further subdivided to yield
equivalent shapes.
To isolate the LCD of 25 great circles, we must take into consideration that the spherical VE has two types of faces. Therefore, thesmallest unit that can be reproduced to generate the whole pattern;8'

Fig. 14-10. LCD unit.

216

A Fuller Explanationi

requires of a VE triangle joined to i of an adjacent square. In thisway, both aspects of the VE pattern are incorporated into the LCD,
and the result is an asymmetrical triangle that covers iR of the sphere(Fig. 14-10).
By changing the boundaries, LCDs of a given spherical polyhedron often can be caused to make up the faces of various otherpolyhedra. Interesting transformations are found between the great
circles of the VE and the icosahedron, for the shift from 25 to 31 isanother result of the icosahedron's "out of phase" role in the"cosmic hierarchy."
LCD of 31 Great Circles
Each triangle of the icosahedron consists of six LCDs, for a total of
120 asymmetrical triangles. That much is straightforward, as are thefirst two transformations.
Instead of the standard groups of six LCDs making up icosahedronfaces, we change the boundaries. Four of these units form thediamond face of the spherical rhombic triacontahedron, so that wehave thirty groups of four triangles instead of twenty groups of six.
Figure 14-11a shows that each icosahedron edge is the long diagonalof one of the thirty diamonds; the planar rhombic triacontahedron isthus a "degenerately stellated" icosahedron.
Secondly, we can recombine the 120 units into twelve groups often with each triangular unit radiating out from an icosahedron
vertex, to highlight the pentagonal dodecahedron (Fig. 14-11b).
N ext, we discover a few asymmetrical transformations. A sphericaltriangle of fifteen LCD units, incorporating a complete icosahedron
triangle radially framed (like a pinwheel) by nine extra LCDs, is one
face of the spherical octahedron. With this observation, we isolate
the octahedron face just by looking at the pattern, so it's worth
checking the arithmetic: multiply fifteen units per face by eight faces
to get 120, or the whole system. Figure 14-11c shows this skew
relationship, again reminiscent of the jitterbug transformation. And
lastly, we observe the edges of a spherical VE. At first this seems topresent a contradiction, given the un synchronized relationship of the25- and 31-great-circle patterns. However, the VE's four great circles
are included among the 31 icosahedral equators, although asymmetrically positioned with respect to its vertices (Fig. 14-11d). We now
recall from Chapter 11 that it was possible for all the faces of anoctahedron to be aligned with eight of the icosahedron's twenty (Fig.
11-5). This skew correspondance, which defined the S module, shows14. Cosmic Railroad Tracks: Great Circles

(a)

(c)

217

(b)

(d)

Fig. 14-11. Revealed in the 31-great-circle pattern: (a) rhombic triacontahedron;(b) pentagonal dodecahedron; (c) octahedron; and (d) YE.

how a subset of four out of the ten circles generated by icosahedronfaces will be correctly situated to create the spherical VE.
The number of different polyhedra hiding within the 31 great
circles reemphasizes the existence of significant relationships between the "out of phase" icosahedral fantily and the IVM group.
VE's 25 Great Circles

The intertransformability displayed by the VE's least common denontinator is straightforward, in contrast to the skew relationships
demonstrated above. Groups of four units create diamond faces

218

A Fuller Explanation

(a)

(b)

(c)

(d)

Fig. 14-12. Revealed in 25 great-circle pattern: (a) rhombic dodecahedron ;(b) octahedron; (c) cube; and (d) tetrahedron.

exactly centered over each VE vertex, thereby defining the twelve
faces of its dual, the rhombic dodecahedron (Fig. 14-12a). Six LCD
triangles come together to create octahedron faces (Fig. 14-12b),
while the cube's six squares each consists of eight asymmetrical units(Fig. 14-12c). And a spherical tetrahedron uses a dozen LCOs per
face, distributing the 48 units among only four faces (Fig. 14-12d).
This brief look at various regroupings of LCD units shows how
great-circle diagrams provide a new way to classify certain polyhedral characteristics, and thereby discover shared symmetries between systems.

14. Cosmic Railroad Tracks: Great Circles

219

Operational Mathematics

Fuller made a remarkable discovery about great-circle patterns thatis responsible for their great significance in his mathematics. This
discovery involves an intricate relationship between central andsurface angles and could so easily be missed that one cannot help
reflecting on the intuition that led Fuller to such an insight. As withother aspects of his "operational" method, the demonstration relies
on readily available materials, but its significance extends to structuring in nature. This is a particularly satisfying exercise, and readers
are encouraged to make Bucky's discovery themselves by following
the simple instructions. Rather than a "plane," Bucky starts with areal system, a "finite piece of paper" (831.01). Using a compass,draw four circles with diameter of approximately 6 inches, and thencut them out with scissors. Fold each one. Then fold the resulting
semicircles into thirds, as shown in Figure 14-13. The section labeled
A is folded toward you, while C is folded back, producing a
Z-shaped cross-section.
Making sure that all creases are sharp, unfold the paper to obtaina circle with three intersecting diameters clearly marked by fold lines(Fig. 14-13). The circle is thus divided into six equilateral triangles.
The process of sweeping out a circle with a compass insures that all
radii are equal, and because at one point in the procedure all six
pie-slices are piled up together, we know that the central angles must--Fig. 14-13

220

A Fuller Explanation
Fig. 14-14. Four bowties create the four
great circles of the VE.

all be exactly the same. 360 degrees divided among six equal segments yields 60-degree angles. First-hand experience has confirmed
both the constant radius and sixty-degree angles, and therefore thepresence of equilateral triangles with arc segments at their outeredges is experientially proved.
One fold (line AB) faces you; the other two folds are facing away.
Bringing point A to point B, we create one of Fuller's "bowties"(Fig. 14-13, bottom). A bobby pin straddling the seam keeps thebowtie together: two unit-length regular tetrahedra joined by anedge. We repeat the procedure three times, producing four bowties inall. It is then apparent that two of them can be placed seam to seamand pinned together with two more bobby pins, to produce four
tetrahedra surrounding a half-octahedral cavity. The other two bowties are similarly paired, and finally the two pairs are connectedalong their congruent fold lines with four more bobby pins. A
complete paper sphere emerges (Fig. 14-14). This strange procedure
has created a very familiar pattern: four continuous great circles, ora spherical VE. The only materials required are four paper circles
and twelve bobby pins.
What has happened? Four separate paper circles have been folded,
bent, transformed into bowties, and pinned together without paying
any attention to converging angles. No special jig is required to line
up adjacent bowties and insure that consecutive great-circle arcs arecollinear. Folded edges are simply brought together, and four continuous great circles magically reappear, as if the original paper
circles were still intact. Looking only at the finished model, it
appears that we had to cleverly cut slits in the paper circles in justthe right places to allow the four circles to pass through each other.
The procedure is reminiscent of the magic trick in which a hankerchief14. Cosmic Railroad Tracks: Great Circles

221

is cut into many tiny pieces and thrown randomly into a hat, only toreappear intact.
A spherical VE can be constructed through this simple folding
exercise because of the specific interplay of its surface and central
angles. Remarkable numerical cooperation is required to allow adjacent central angles to fold out of flat circular disks, while automatically generating correct surface angles. That four great circles willsubmit to this bowtie operation is not at all obvious from studying
the whole pattern, and even less so in the case of other, far more
complicated models.
Conservation of Angle

Physics tells us that a beam of light directed toward a mirror at someangle from the left will bounce away from the mirror making thesame angle on the right. The angle of incidence is equal to the angleof reflection. Fuller points out that the same is true for the great-circle
models, if we think of the paths as trajectories.
Great circles maintain the illusion of being continuous bands,argues Fuller; however the bowtie procedure reveals the truth about
these patterns, and in so doing illustrates an aspect of energy-event
reality. A great-circle path may look continuous, but what really
happens is that as soon as a trajectory (or great-circle arc) meets an
obstacle, in the form of another great-circle event, they collide andthe course of both trajectories is necessarily altered. Both paths areforced to bounce back, just like a ball bouncing off a wall.
Here's the fun part. Because the intersection of two great circles
provide two pairs of equal angles, their paths mimic the classicalcollision in physics. The same angular situation results from twooverlapping great circles as from an idealized "energy-event" collision; these symmetrical patterns can therefore be thought of as thepaths taken by billiard balls on the surface of a spherical pool table.
If you didn't see the collisions, you might think that two balls went
through the same point at the same time; however, their true paths
are bowties.
Imagine a great-circle wall constructed vertically out from thesurface of a sphere. If a ball traveling parallel to the sphere's surface(describing another great circle) hit the wall and bounced back," ... it would bounce inwardly off that wall at the same angle that it
would have traversed the great-circle line had the wall not been
there ... " (901.13; Fig. 14-15a). Adjacent bow-ties therefore produce
collinear great-circle arcs, because the angle made by the paper disk

222

A Fuller Explanation

,
:

fi

~\

"

,

;'

'-

3. -, " - .. - ,,-. 4~.
..-, .' ... '..~
..
/I
2")(,
"
.~" 'II,, , /-, ""

...

,4

'Il.

o

f

•

-.

~

I

1:

~ , ."\".
Ii-.5 .'' .' :", -·-·1t...· ...... .......... ·· ~···
"
; ,
,
,
/
I

(a)

(b)

Fig. 14-15. (a) Ball bounces back from the wall at the same angle it would havemade wi th the wall on the other side had the wall not been there. (b) "Local holding
pattern:" figure-eight loop.

"bouncing back" from a bobby-pin collision point is the same angle
the circle would have made on the other side if the bobby pin had
not been there. Fuller continues, "and it would bounce angularly off
successively encountered walls in a similar-triangle manner . .. "(901.13). Hence the completed bowties. The image of the collidingball and great-circle wall makes it easier to understand why Fullerinterpreted great circles as the trajectories of energy events, andexplains why he was convinced that these models have physicalsignificance. His bowtie analogy is consistent with the classicalcollision model.
Consider the paper model described above. Soon after an energy
event meets its first collision, the new trajectory meets a second
obstacle and its course is again altered. Conservation of angle
determines the new heading once again, and the process repeats
until, at the sixth collision, the great-circle path comes back uponitself, completing the bowtie loop (Fig. 14-15b).
Successive arc segments of one energy event form figure-eightloops, or "local holding patterns" (455.05), which lie side by side
and appear to be continuous great circles. Construction paper shows
us how it works, and Fuller tells us that this is what happens inphysical reality as well. Discrete energy events form local circuits,
just as discrete paper circles form bowties; they only look continuous.

14. Cosmic Railroad Tracks: Great Circles

223

Foldable Systems

Just as we needed at least four vertices to make a system, "four isalso the minimum number of great circles that may be folded into
local bow ties and fastened corner-to-corner to make the whole
sphere ... " (455.04). An interesting characteristic of this minimum
model is that the sum of the areas of its four separate circles, which
fold up to create the model, is equal to the surface area of the spherethey define, or 4wR2. A system with less than four great circles
cannot be directly constructed out of that number of bowties, but insome cases the pattern can be simulated using more than theprescribed number of paper disks and doubling the polyhedral
edges:
You cannot make a spherical octahedron or a spherical tetrahedron by itself ...
lO9°28' of angle cannot be broken up into 360-degree-totalling sphericalincrements .... (842.02-3)

A spherical tetrahedron can only be created out of "foldable" paper
circles by constructing the six-great-circle spherical cube, which
produces two intersecting tetrahedra. See Fig. 14-16: six great circles
will submit to a bowtie construction, similar to the four-great-circle
model described above:
842.04 Nor can we project the spherical octahedron by folding three great circles.
The only way. .. is by making six great circles with all the edges double ....

In this construction, six circles are folded in half and then the
resulting semicircles are each folded in the middle at a right angle.
The six bent semicircle pancakes (rather than three open bowties)
are then simply pinned together to simulate the three great circles ofthe octahedron (Fig. 14-17).
Fuller attributes these discoveries to a "basic cosmic sixness":
There is a basic cosmic sixness of the two sets of tetrahedra in the vector
equilibrium. There is a basic cosmic sixness also in an octahedron minimally-greatcircle-produced of six great circles; you can see only three because they are doubled
up. And there are also six great circles occurring in the icosahedron. All these arefoldable .... This sixness corresponds to our six quanta: our six vectors that makeone quantum. (842.05-6)

Both six-great-circle patterns can be constructed out of foldable
circles. The star-tetrahedron version involves six bowties with tworight isosceles triangles on the surface (60° -60° -90°) created bycentral angles of 70°32' (a) and 54°44' (b), as labeled in Figure

224

A Fuller Explanation

6x

Fig. 14-16. Six bowties create the six great circles of the cube or octahedron.

14-16. The six bowties are pinned together at the seams to create the24 surface triangles of the spherical cube and star tetrahedron. And
it works: the illusion of six continuous circles is maintained.
The icosahedral six great circles fold into pentagonal bowties.
Fold lines divide each circle into ten equal slices of pie, carving out
the 36-degree central angles. Each circle is then pinched together at
one point to form a double-pentagon figure-eight. Just like thetriangular predecessors, each circle folds into a "local circuit" and isconnected to other local circuits to create the illusion of continuous
great circles (Fig. 14-18).
Fuller experiments with the "foldability" of considerably more
intricate patterns. Larger numbers of great circles intersect more
frequently, and arc segments have correspondingly smaller central
angles. The number of folds and the precision required for each of
the tiny irregular angles make these higher-frequency models extraordinarily difficult to build, and even more difficult to visualize
without a model. Each of the ten great circles of the icosahedron canfold into winding chains of six narrow tetrahedra, which theninterlink to reproduce the ten-great-circle pattern. 2 Fuller claims that14. Cosmic Railroad Tracks: Great Circles

ED

225

~
::--1-\ -, L.- . --"i' -I .......

,,
I

Fig. 14-17. Three-great-circle model requires six foldable circles.

Fig. 14-18. Six pentagonal bowties create the six great circles of the icosahedron.

the fifteen-great-circle pattern (outlining the 120 LCD units) can bereconstructed with fifteen four-tetrahedron chains. 3 While a paper
circle will fold into four consecutive LCD tetrahedra, it is not clear
how they fit together to recreate the whole sphere. We note howeverthat this pattern can be easily generated by using thirty paper circles
and doubling the edges. Each circle folds into four adjacent LCD
tetrahedra, to form one self-contained diamond-a face of therhombic triacontahedron (Fig. 14-19).

226

A Fuller Explanation

Fig. 14-19. Fifteen-great-circle model, created by doubling the edges.

For the purposes of this chapter, we want to understand the basics
of how foldable great circles work; further experimentation withconstruction paper is left to curious (and ambitious) readers. We
proceed to look at Fuller's interpretation of the significance of thisbehavior in terms of physical reality.

Energy Paths
Here's the basic premise: First, the concept of a sphere provides amodel of the generalized system, and on the surface of a sphere theshortest route is a geodesic path. Secondly, Universe breaks down
into discrete systems, which consist exclusively of energy events andtheir relationships. And finally, energy is always in motion, perpetually transferred between finite local systems along most direct routes,
and therefore, energy must be traveling through Universe via greatcircle paths.
Gas Molecules
We consider a specific example:
A vast number of molecules of gas interacting in great circles inside of a sphere
will produce a number of great-circle triangles. The velocity of their accomplishmentof this structural system of total inter triangulation averaging will seem to beinstantaneous to the human observer. (703.14)

14. Cosmic Railroad Tracks: Great Circles

227

Fuller's hypothetical molecules are bouncing around so fast that theoverall pattern of their activity seems instantaneous. But in reality,
he reminds us, there is no "instant Universe"; time is always afactor. Molecules collide and bounce back; equal angles created by
the symmetry of their collisions create the illusion of full great circles
interweaving, but their trajectories are really local loops:
The triangles, being dynamically resilient, mutably intertransform one another,
imposing an averaging of the random-force vectors of the entire system, resulting inangular self-interstabilizing as a pattern of omnispherical symmetry. (703.14)

To Fuller the ability to fold individual paper circles and automatically generate an entire symmetrical pattern gives great-circle models
important physical relevance. They demonstrate his statement thatno two lines can go through the same point at the same time andillustrate the concept of "interference patterns." The model, says
Fuller, is consistent with physical behavior.
Although physical systems are always imperfect, the result of avast number of interactions is approximate symmetry. With enoughdata or time, all possible paths can be tried, and the properties ofspace come into play. What does a maximally symmetrical distribution on a spherical surface look like?
The aggregate of all the inter-great-circlings resolve themselves typically into aregular pattern of 12 pentagons and 20 triangles, or sometimes more complexedly,
into 12 pentagons, 30 hexagons, and 80 triangles described by 240 great-circle
chords. (703.14)

The pattern is always icosahedral, some version of the maximallysymmetrical shell. (Refer to Chapter 11 for a comparison of spacefilling versus shell symmetries.) Icosahedral symmetry fits the mostgreat circles on a closed system. One notable characteristic of thispattern, no matter how high the frequency of subdivision, 4 is thepresence of exactly twelve pentagons evenly distributed around thesystem-one at the location of each five-valent icosahedral vertex:" ... the 12 pentagons, and only 12, will persist as constants; also thenumber of triangles will occur in multiples of 20" (703.15). This
constant number of pentagons, together with" twelve spheres aroundone" and "twelve degrees of freedom," suggests a fundamental
twelveness inherent in space. We shall return to this pattern in thenext chapter, looking at geodesic structures.
Fuller uses molecules bouncing around in a spherical system as amodel, because molecules are energy events occurring in large enoughnumbers to describe the symmetrical patterns developed as poly-

228

A Fuller Explanation

hedral abstractions. In conclusion, energetic behavior is subject tosymmetrical constraints, and in order to adhere to logic or theory,
probability calls for very large numbers.
Great-Circle Railroad Tracks of Energy

Chapter 8 examined the closest packing of spheres. The next question in Fuller's investigation is how "energy" will navigate throughthese clusters. The idea that closepacked spheres present a sort ofconceptual model of physical Universe is at the root of Fuller's
great-circle studies. As a space-filling array of discrete systems,
whose omnisymmetrical qualities recommend them as a general
representation of eternally spinning energy-event systems, closepacked spheres do provide a tantalizing model. Chapter 8 described
the closest packing, which places every sphere in contact with twelve
others, and this chapter mapped out the complete network of shortest-distance paths around a spherical system described by thesetwelve contact points. The "cosmic railroad tracks" thus described
were the 25 great circles of the VE:
The 12 points of tangency of unit-radius spheres in closest packing, such as is
employed by any given chemical element, are important because energies travelingover the surface of spheres must follow the most economical spherical surface
routes, which are inherently great circle routes, and in order to travel over a series ofspheres, they could pass from one sphere to another only at the 12 points oftangency of anyone sphere with its closest-packed neighboring uniform-radius
sphere." (452.01)

In an effort to describe a symbolic model of atomic and molecular
activity, Fuller allows the omnisymmetrical (or perhaps "omni-spinnable" is more appropriate) form of spheres to represent atoms andexplains that energy, or charge, can only travel from system tosystem through points of tangency. (Fuller is also quick to point out
that tangency is actually "extremely close proximity", for in physicalreality nothing" touches.") Energy is thus found either in finite local
circuits (bowties) on one system, or jumping over to a neighboring
system through VE vertices:
These four great-circle sets of the vector equilibrium [i.e. sets of 3, 4, 6, and 12]
demonstrate all the shortest, most economical railroad "routes" between all the
points in Universe, traveling either convexly or concavely. The physical-energy
travel patterns can either follow the great-circle routes from sphere to sphere or goaround in local holding patterns of figure eights on one sphere. Either is permitted
and accommodated." (455.05)

14. Cosmic Railroad Tracks: Great Circles

229

These "universal railroad tracks" are specifically along the 25 great
circles because of the relationship of vector equilibrium to thecosmically significant closepacked spheres.

Icosahedron as Local Shunting Circuit"The vector equilibrium railroad tracks are trans-Universe, but the
icosahedron is a locally operative system" (458.12). This distinction,
introduced by the jitterbug transformation, has particular significance in Fuller's great-circle theories. The VE is integral to ourspace-filling network of equivalent vectors; however, once the VE
contracts into the icosahedron with its slightly shorter radius, it is
disconnected from that universal IVM network. It loses its contacts.
"Energy" is free to travel endlessly throughout the railroad tracks of
Universe, sliding from sphere to sphere along the economic greatcircle paths, until it runs into an icosahedron. The icosahedral 31
great circles are not" trans-Universe" lines of supply; their function
is to disconnect energy from the closest-packing railroad tracks anddirect it into local orbits. The icosahedron throws the switch:
The icosahedron's function in Universe may be to throw the switch of cosmic
energy into a local shunting circuit. In the icosahedron energy gets itself locked up
even more by the six great circles-which may explain why electrons are borrowable
and independent of the proton-neutron group. 458.11

Fuller suggests that there might be a meaningful connection between
the icosahedron and the electron, because the tiny negative charge isreadily transferred from atom to atom in molecules and crystals. The
icosahedron's independent role, in Fuller's view, "shunting" energy
into local circuits (that is, able to disconnect energy charges from abigger matrix) is suggestive of the electron's role:
458.05 The energy charge of the electron is easy to discharge from the surfaces ofsystems. Our 25 great circles could lock up a whole lot of energy to be discharged.
The spark could jump over at this point .... If we assume that the vertexes arepoints of discharge, then we see how the six great circles of the icosahedron-which
never get near its own vertexes-may represent the way the residual charge will
always remain bold on the surface of the icosahedron.

Lacking contact points, the icosahedron is a free-floating unit in
Universe; so is the electron. The suggestion of a relationship remains
just that, a tantalizing parallel, seeds perhaps of future investigation,
but certainly among Fuller's more abstruse parallels.

230

A Fuller Explanation

Inventory: Seven Unique Cosmic Axes of Symmetry

The VE's 25 great circles incorporate those of the rhombic dodecahedron, octahedron, cube, and tetrahedron. The icosahedral 31 belong to a different family of symmetries. Both groups together
constitute seven sets of axes of symmetry: four contributed by the
VE's vertices, edges, and two types of faces, and three by theicosahedron's vertices, edges, and faces.
1042.05 The seven unique cosmic axes of symmetry describe all of crystallography. They describe the all and only 6reat circles foldable into bow ties, which may
be reassembled to produce the seven, great-circle, spherical sets ....

We have a list of symmetrical possibilities. With this inventory,
Fuller integrates specific information about inherent spatial characteristics with concepts of energy behavior, to gain insights about
structuring in nature.
Excess of One
It is interesting to note that the number of great circles associatedwith each polyhedron is always one more than the number of its
edges. For example:
Edges
Tetrahedron
Octahedron
Cube

VE

Icosahedron

6

12
12
24
30

Great Circles
7

13
13

25
31

Fuller quickly identifies this constant "excess of one great circle"(and its implied two poles) with the "excess two polar vertices
characterizing all topological systems" (1052.31). However, this discovery is not a new bit of magic, but rather follows directly from
Euler's law. Recalling the way in which great circles are generated,
we realize that each circle corresponds to a pair of either vertices,
edges, or faces. Therefore, the number of great circles can be tallied
by counting half the total number of topological aspects, or ~(E +F + V). We can now write an equation stating that one less than thenumber of great circles is equal to the number of edges:

HE + F + V) - 1 = E.
Multiplying both sides by 2, we have
E+F+V-2=2E,

14. Cosmic Railroad Tracks: Great Circles

orand finally231

F+V-2=E,
F+ V= E + 2,

which is Euler's law.
Even if Fuller's cosmic railroad tracks leave you skeptical, great
circles provide fascinating geometric patterns, which introduce a new
system of classifying and comparing the topology and symmetry ofvarious polyhedra. Part of their fascination lies in the surprisingly
limited number of variations among the great-circle sets generated
by our cast of polyhedra. But perhaps most importantly, experimentswith great circles-building wire models-provided the impetus andthe clues for Fuller's subsequent journey into geodesics.

15
From Geodesic to Tensegrity:
The Invisible Made Visible"A geodesic is the most economical relationship between any twoevents" (702.01). Fuller's definition immediately calls to mind great
circles, which provide the shortest routes between two events on aspherical system. Actually, clarifies Fuller, the general case of "mosteconomical relationship" is necessarily a great circle. "It is a special
case in geodesics which finds that a seemingly straight line is theshortest distance between any two points in a plane." In otherwords, a given area may be such a small portion of a sphericalsystem that it appears fiat; however, because all identifiable experiences belong to systems, "great-circle segment" and "geodesic" areinterchangeable in synergetics.
Already familiar with the theory of great circles and their polyhedral symmetries, we can apply theory to practice. A little experimentation uncovers two important discoveries, demonstrating again
why the discipline of building models was essential to Fuller'smathematical exploration.
The first discovery is easily visualized without actually building amodel. Imagine a metal sphere and a wire ring just large enough to
fit around the widest girth of the sphere. This circular ring qualifies
as a great circle and therefore can delineate the shortest route
between any two points on the sphere. However, there is a practicalproblem; the ring slides off the sphere. It may seem like a strange
observation, but Fuller tended to investigate unusual aspects of his
subject matter, and frequently such seemingly whimsical sidetracks
have proved fruitful.
Bring in a second wire circle of the same diameter. As we recall
from the last chapter, a pair of great circles intersect at two points
180 degrees apart. To keep the two wire loops on the sphere, they aretied together at both crossing points. The effort fails, for the twocircles are free to spin around their common axis, and as soon as
they line up both circles slide off the sphere together. Try again. A
third great circle is placed anywhere on the sphere except through15. From Geodesic to Tensegrity: The Invisible Made Visible

233

Fig. 15-1

the intersection of the other two. Whether arranged randomly orsymmetrically (as a spherical octahedron), when intersections aretied together all three circles are immobilized. Triangulation createsa stable cage. "Not until we have three noncommonly polarized,
great-circle bands providing omnitriangulation... do we have thegreat circles acting structurally to self-interstabilize ... " (706.20).
Three differently oriented great circles is the minimum for a stablemodel. The discovery may seem trivial at first, but we have no
indication from design that the stability of a "three-way grid" iswidely understood in our culture. On the other hand, Southeast
Asians have utilized this principle for thousands of years. Fuller
points out that a vital need for strong baskets led them long ago to
discover that a triangulated weave stays rigidly in place, whereas thetwo-way weave used by other cultures is easily distorted.l Southeast
Asian children today still play with a reed sphere consisting of six
interwoven great circles-perhaps the oldest known toy (Fig. 15-1).
Characterized by icosahedral symmetry, the ancient design utilizes
the inherent stability of a three-way grid to make a lightweight andvirtually indestructible ball out of delicate reeds. A modern toy has
yet to improve upon its simplicity and durability.
The second discovery, which should be experienced to be fully
appreciated, will nevertheless not come as a surprise at this point.
Experimenting with wire models, Fuller found that the more great
circles, the stronger the sphere. That much is self-evident, but thedegree to which their strength increased far exceeded his expectations. This was not the kind of linear relationship exhibited byordinary structures; rather the increasing rigidity of his great-circle
models (from the minimum three to the icosahedral 31) could only
be called synergetic.
We skip directly to the strongest model, provided by last chapter's
limit case, 31 great circles. Triangulated geodesic arcs produce an234

A Fuller Explanation

Fig. 15-2. LCD of the icosahedron is asymmetrically subdivided by the 31-great-circle
pattern.

extremely sturdy, lightweight enclosure out of thin wire. Let's lookclosely at the pattern; our study is made considerably less complicated by isolating the LCD as described in the preceding chapter.
Accordingly, we study only one of the 120 triangles framed by theicosahedron's 15 great circles, and observe that it is asymmetricallysubdivided by the 6- and 12-great-circle patterns (Fig. 15-2). Notice
the variations in arc length and surface angle- made more obvious
by viewing this small region out of context. The overall three-way
grid incorporates longer and shorter arcs, thereby subdividing thesphere's surface into triangles of very different shape and size.
The load distribution and resulting strength of these wire models
is a function of symmetry; the longer the arc, the more vulnerable it
is to stress. 2 It is clear that the most advantageous system would
have all arcs as close to the same length as possible, but how can we
improve upon the symmetry of the 31-circle pattern?
A second problem with the arrangement is that as a limit case, itdoes not present a logical course for further subdivision. To build
progressively larger models with sufficient strength, we must find away to generate more and more great-circle segments, or "higher
frequency" in synergetics terminology. Both problems are solved by
Fuller's next step.
To explore other methods of developing large multifaceted enclosures, we go back to the system that already has the greatest
number of equivalent regular faces, the icosahedron. A three-way
grid of evenly spaced lines (imagine a triangular checkerboard)
divides the icosahedron's equilateral triangles into as many smaller
triangles as desired (Fig. 15-3). However, this subdivision does not
yet lead to an effective design strategy, for if neighboring triangles ofa structural system lie in the same plane, the enclosure will deflect inand out, like a trampoline, in reaction to an applied load. Unlessvertices are reinforced using rigid joints-in which case, the system

15. From Geodesic to Tensegrity: The Invisible Made Visible

235

Fig. 15-3. Four-frequency (4v) triangles
superimposed on each face of the icosahedron.

is functionally equivalent to the original icosahedron-the advantages of triangulation are lost. It is thus clear that a convex
polyhedral enclosure with more than twenty triangles cannot consist
exclusively of equilateral faces. Adjacent triangles must differ slightly
in shape and size to allow angles around each six-valent vertex to
add up to less than 360 degrees, as necessary for continuous convexity.3 Fortunately, this unavoidable variation among chord lengths
can be far less than that of the 31-great-circle pattern. To understandhow these irregular triangles are generated, we back up and review
the problem as a whole.
Theory Behind Geodesic Structures: Summary

We can think of Fuller's task as a mathematical game confined to
the rules of synergetics, in which the goal is to enclose as muchvolume as possible with the least amount of material. A solution
must combine two geometric principles-integrating overall shapeconsiderations with the requirements for local stability. Geometrytells us that the shape with the greatest volume/surface-area ratio isa sphere; synergetics challenges that sphere to materialize. "Since
physics has found no continuums, we have had to clear up what wemean by a sphere," writes Fuller (1023.11). This game calls for asolution we can actually build and touch, which means the best
structural approximation of that elusive sphere.
We thus have redefined the problem in terms of "operational
mathematics." To get started, we might experiment with toothpicks,
thus limiting the game to enclosures with identical struts. This brings

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A Fuller Explanation

us back to Fuller's" three prime structural systems in Universe" - theinventory of self-stabilizing systems with equal vectors. Of the three- tetrahedron, octahedron, and icosahedron - the thi rd has the mostvolume per toothpick (Chapter 5). An icosahedron is thus the best
we can do with toothpicks, but of course it is not a satisfactory
solution.
How can the enclosure become more spherical and still be stable?
The geometric logic continues: through systematic symmetrical subdivision of our best approximation-taking full advantage of thestability of triangles. It hardly needs to be restated: "If we want tohave a structure, we have to have triangles" (610.12).
In short, two simple principles taken together lead to a solution to
the stated problem. Combine the advantageous shape of the icosahedron with the stability of triangles, and the geodesic dome almost
materializes. The logic is as exquisite as it is simple.
But we're missing a step. How can we develop the "checkerboard"
tessellation illustrated above into a functional structure? The icosahedral edges can be divided into any number of segments. The
greater the number, or "frequency," the more spherelike the end
result will be. Then, to complete the transformation, each new vertex(superimposed on icosahedron faces) must be projected out to thesurface of the imaginary sphere defined by the icosahedral
vertices-and then interconnected by great-circle chords (Fig. 15-4).
The nature of this projection accounts for the slight variation in

Fig. 15-4. Each new vertex is projected outward to the surface of an imaginary
sphere defined by the original icosahedron vertices.

15. From Geodesic to Tensegrity: The Invisible Made Visible

237

Fig. 15·5. 4v icosahedron: transformation of Figure 15·3.

shape and size of triangles; the farther away a chord is from an
icosahedral vertex, the more it is stretched during the transformationfrom planar to spherical.
The resulting omnitriangulated polyhedron has six triangles at
every vertex except for those located at the twelve vertices of theoriginal icosahedron, which continue to join five triangles (Fig. 15-5).
The system can have indefinitely many six-valent nodes, but theexistence of exactly twelve five-valent vertices is a prerequisite toclosure. Happily, there is also a considerable engineering advantage
in having five or six struts leading out from each vertex: forces are
instantly distributed in many directions-omniradially, Bucky might
say-producing structures with unprecedented strength in relation
to their weight.
This is the basic strategy behind the geodesic dome.
Geodesic Design in Nature

Fuller points out that an extremely high-frequency geodesic polyhedron provides the true model of physical systems which we interpret as spheres, as for example a soap bubble. The notion of acontinuous surface equidistant from a central point is scientifically238

A Fuller Explanation

unacceptable, that is, inconsistent with physical reality; on somelevel of resolution all "spheres" consist of discrete quanta-untold
numbers of energy events interconnected by an even greater numberof vector-relationships, or forces. He has clarified this particularmisconception countless times:
535.11 Because spherical sensations are produced by polyhedral arrays of interferences identified as points approximately equidistant from a point at the approximate center, and because the mass-attractive or -repulsive relationships of all
points with all others are most economically shown by chords and not arcs, thespherical array of points produces ... very-high-frequency, omnitriangulated geodesic structures ....

Our eyes cannot see individual molecules in the delicate transparent
soap bubble, nor can we detect the chordal chemical attractions
between molecules. Nevertheless they exist, explains Fuller, and it is
our responsibility to understand and teach the truth about Universe.
Once again, his goal is to provide tangible models of otherwise
invisible phenomena.
Of all possible solutions, a high-frequency triangulated shell with
icosahedral symmetry provides the most efficient method of enclosing space with a minimum of material and effort. Accordingly,
nature relies on this elegant design in many situations calling forprotective enclosures, regardless of scale. Examples include the small
sea creatures called Radiolaria, 4 the fibrous web of the eye's cornea,
and the protein shell of many viruses. 5 We looked into the structureof spherical viruses in Chapter 8 and can now go into greater detail
based on our increased familiarity with geodesic theory.
The design problem is familiar by now: tiny amounts of genetic
material must be protected by a tough protein shell. As nature is
scrupulously efficient, the choice is clear. A "spherical" distributionof protein molecules will satisfy the basic criteria in terms ofconserving material relative to volume, while icosahedral symmetry
will provide the most even distribution. The natural balance soughtby chemical forces leads to approximately equivalent spans, which isgeometrically accomplished by high-frequency icosahedral systems.
In short, the structure of viruses is a product of nature's eternaltendency toward equilibrium. Geometry imposes the rules.
Reassuringly, observations (with the electron microscope) of isometric virus shells have consistently revealed icosahedral designs. Dr.
Aaron Klug, who first observed the geodesic structuring of viruses,
wrote to Fuller in 1962 telling him of this discovery. Bucky, delighted by the news, immediately wrote back with the formula forthe number of nodes on a shell (lOF + 2, varying according to15. From Geodesic to Tensegrity: The Invisible Made Visible239

frequency) as confirmation of Klug's hypothesis. Klug answered thatthe values obtained from this equation proved consistent with thevirus research, and thereby provided Fuller with one of his mostvalued anecdotes-a prime example of nature's economic elegance-which enriched many lectures in subsequent years.
Insufficient awareness of spatial constraints causes these structuralsimilarities to be perceived as "coincidence." As geodesic domes
were utilized worldwide 15 years before electron microscopy enabled
detection of virus capsids in 1962, the resemblance of the tiny
biological forms to large architectural structures seemed to manyquite extraordinary and improbable. Science does not as a rule take
into consideration the active role of space; however, such awarenesscan assist the prediction of unknown structures based on theirfunctional demands.
Another fascinating example was contributed to the inventory ofgeodesic structuring by scientists at General Dynamics working on
the problem of rocket reentry, who wrote to Fuller describing theirresults and enclosing photographs. The experiment involved twohemispheres of thin-sheet titanium, precisely machined to achieve
consistent shell thickness. The diameter of one was exactly an inch
greater than that of the other, so that when the larger was placed
over the smaller, a half-inch hemispherical cavity separated the twoshells. Their bases were sealed together to create a double-shelldome, and the air was then pumped out of the intervening space tocreate a vacuum. Atmospheric pressure outside the dome caused its
thin titanium sheet to buckle in toward the vacuum. The hemisphere"dimpled" in a "pure icosahedral pattern," as Fuller recalls in a
1975 lecture. Like the virus, it had no choice! The titanium sheet
experienced an automatic reaction based on the shape of space;caving in most efficiently required a symmetrical distribution of
dimples. The frequency of this pattern, that is, the number of
dimples per icosahedral "edge," was found to be inversely proportional to the shell thickness: a thicker shell produced fewer dimples
and vice versa.
Consider the above progression. We started with a mathematical
puzzle; geometry laid out the rules and led to a solution, and it
turned out that nature had been playing the same game all along.
That is essentially how Fuller describes his experience in developing
the geodesic dome:
I did not copy nature's structural patterns. ... I began to explore structure anddevelop it in pure mathematical principle, out of which the patterns emerged in pureprinciple and developed themselves in pure principle. I then ... applied them topractical tasks. The reappearance of [geodesic] structures in scientists' findings at

240

A Fuller Explanation

various levels of inquiry confirms the mathematical coordinating system employedby nature. (203.09)

The principles behind the geodesic dome are not new; they are
eternal laws of nature. The application of these geometric facts to abuilding system is new. Fuller is quick to explain,
Though ... similar in patternings to ... flies' eyes, geodesic structuring is true
invention. .. Flies' eyes do not provide human-dwelling precedent or man-occupiable ... structures. (640.01)

Invention can be defined as the novel application of generalized
principles. Chapter 16 will explore the concept in more detail.
Geodesic Domes: Design Variables

The study of great-circle patterns seems at first like a completely
abstract endeavor. However, the construction of models demonstrates that spheres with a greater number of shorter arc-segments
have a significant structural advantage over simpler structures, andsuggests potential applicability. Fuller's early models, based directly
on great-circle patterns showed considerable strength but did not go
far enough. A new method of generating geodesic structures wasneeded to produce higher-frequency structures-with less variation
among chord lengths-than the 31-circle pattern. This progression
led Fuller to concentrate years of attention on icosahedral geodesic
designs. He discovered that the inherently self-stabilizing geodesic
polyhedron could be truncated as the basis of a stable dome (Fig.
15-6). These domes, which could be any fraction of a geodesic
sphere, must sit on the ground to complete a "system." Dependingon the situation, a half-sphere truncation, ~, t, or any number ofother fractions might be desirable. Fuller saw that the possibilities
were endless; geodesic domes could be designed to be built out ofalmost any material at any frequency.
The geodesic pattern is also a variable. In addition to the above"checkerboard," Fuller developed a number of other "geodesic

Fig. 15-6. 5/8th truncation of 4v icosa-as basis of a geodesic dome.

15. From Geodesic to Tensegrity: The Invisible Made Visible

(a)

241

(b)
I~

?hv
/

\

/

\

/

\

/

\

/

\

/

\

/

/

/-

(c)

-

-

-- - - (d)-\- - ~\

Fig. 15-7

breakdowns." An icosahedral face can be subdivided by a variety ofpatterns-each one yielding a different design for potential domes.
The most common breakdown subdivides each triangular face withlines parallel to its three edges as explained above (Fig. 15-7a). The
number of segments along each icosahedral edge specifies thefrequency of the resulting geodesic dome, and the number of triangles per icosa face will always be F for an "f- frequency" structure.
A geodesic dome might be 2- or 32-frequency depending on size andmaterial; there's no inherent upper limit, but the exponentially
increasing numbers of different strut types become prohibitively
complicated at very high frequencies.
The second breakdown subdivides icosahedral faces into triangles
with lines perpendicular to icosahedral edges, producing slightly
fewer triangles on the overall structure than the parallel version (12per face for a 4-frequency breakdown, as compared to 16) and issomewhat less symmetrical (Fig. 15-7b). Fuller's diamond pattern is
yet another choice (Fig. 15-7c). This design is unstable unless constructed out of panels rather than struts. If the situation allows mass
production of panels, the diamond pattern will have an advantage
over the first breakdown because of its fewer different types of faces.
A number of aluminum domes have been built according to thisdesign.
Finally, the deliberate omission of certain chords in Figure 15-7a
produces Fuller's "basket-weave" design, characterized by hexagons

242

A Fuller Expla..'l.ationand pentagons entirely framed by triangles (Fig. I5-7d). This patternlends itself to building domes out of bamboo for example: struts in athree-way weave are simply tied together at crossings.
There are many other potential designs. The field is as wide open
as the number of ways to symmetrically subdivide an icosahedron
with great-circle chords. The fundamental characteristics of theresulting enclosures are the same. Geometric principles are exploited
to gain unprecedented structural efficiency, just as nature, in response to the interplay of physical forces and the constraints ofspace, produces icosahedral geodesic patterns for many enclosures.
Geodesic domes of virtually unlimited size can be built by increasing the frequency as needed. The mathematics behind this undertaking is cumbersome but not conceptually difficult. Calculations are
simplified (or at least kept under control) by an understanding ofsymmetry: complete information for an entire dome of any frequency
is contained within its LCD triangle. Using the formulae of spherical
trigonometry, we can manipulate the central and surface angles ofspherical triangles to derive values with which to obtain strut lengths.
Struts are great-circle chords, and each one is subtended by a specificcentral angle (Fig. I5-8a).
Depending on construction methods and materials, we might
strive to keep struts as close to the same length as possible, orinstead try to develop triangles with maximally similar shapes, orwork toward any number of other preferred solutions. Frequency isitself an important variable; greater numbers of shorter struts may
be more efficient in terms of supporting loads, but construction iscorrespondingly more difficult. Such tradeoffs must be carefully
weighed, and fortunately trigonometry allows enormous flexibility interms of potential solutions. Taking advantage of symmetry, we areable to experiment with different strut lengths in the LCD triangle,
thereby only manipulating angles and lengths for a very small
portion of the whole system, while developing the mathematicalspecifications for an entire geodesic dome (Fig. I5-8b). Finally,geodesic domes can be elongated or pear-shaped, or (theoretically)
even shaped like elephants. The constant curvature of a sphere
produces the greatest strength, but these other options do exist andcan be developed with no more than a pocket calculator and a lot of
paper.
As a final note, it is important to realize that while the theory
behind geodesic domes is strikingly simple-and previously contemplated by others before Fuller was granted U.S. Patent 2,682,235 in
I954-the actual translation from theory to practical structures
involved fantastically intricate mathematical development. As with

15. From Geodesic to Tensegrity: The Invisible Made Visible

243

(b)
(a)

Fig. 15-8. (a) Use of LCD in geodesic-dome calculations. (b) Examples: LCD for 6v
and 4 v icosahedral geodesic domes.

most invention and design, the initial insight was not enough to
produce a 250-foot-diameter clear-spanning structure overnight. The
subsequent calculations required enormous aptitude and perseverance. Consider the precision necessary to have six struts meeting at
the same point, at thousands of different vertices; minute errors in
strut lengths at only a few points will accumulate and produce vast
discrepancies elsewhere on the dome. Dealing with tolerances similar
to that of the aircraft industry rather than the relatively crude
building world, Fuller had to develop absolutely reliable trigonometric data to enable the construction of extremely large domes.

Tensegrity

Before geodesic domes appeared on the scene in 1948, tbe dome of
St. Peter's Cathedral in Rome, with a diameter of 150 feet, wasunchallenged as the largest architectural clear span. 150 feet musthave been seen as a fundamental upper limit, a sort of divine zoning

244

A Fuller Explanation

law. Ultimately, the inherent stability of triangles cannot alone
account for the geodesic dome's ability to span unlimited distanceswith no interior supports-nor for its unprecedented strength-toweight ratio.
The rest of the explanation lies in an understanding of tensegrity,
Fuller's contraction of the two words tension and integrity.
"Tensegrity describes a structural-relationship principle in which
structural shape is guaranteed by the ... continuous, tensional behaviors of the system and not by the discontinuous exclusively local
compressional member behaviors" (700.011); Fuller thus introduces
a discussion of the interplay of tension and compression forces in
Universe. His term also refers to the inescapable co-occurrence oftension and compression, while its first syllable emphasizes the too
often overlooked role of tension.
All systems consist of some combination of tension and compression forces. The two are inseparable.
All systems? What about a simple piece of nylon rope pulled atboth ends between two hands? Isn't that pure tension?
Try to visualize the experiment: pull as hard as you can on bothends and notice what happens to the thickness of the rope. Its
diameter shrinks slightly, betraying the invisible compression force
around the rope's circumference, or perpendicular to the applied
tension. Even in this simple example, unplanned compression isinevitable.
The same is true in reverse. Applying compression to a column
introduces surface tension around its girth. Pushing from both sides
along the long axis of a strut causes the outside surface to stretch,albeit slightly. Tension and compression go hand in hand, as simultaneous complementary functions; however one or the other usually
dominates a given situation.
641.01 No tension member is innocent of compression, and no compression
member is innocent of tension ....
641.02 Tension and compression are inseparable and coordinate functions ofstructural systems, but one may be at its" high tide" aspect, i.e., most prominent
phase, while the other is at low tide, or least prominent aspect. The visibl}' tensioned
rope is compressively contracted in almost invisible increments of its girth dimensions .... This low-tide aspect of compression occurs in planes perpendular to its
tensed axis . .. .

We thereby draw the distinction: our rope is a tension element; theloaded column, a compression member. 6 Effective design must balance the two interdependent forces in preferred ways.

15. From Geodesic to Tensegrity: The Invisible Made Visible

245

The apparently insurmountable limit to the clear-span of a structure existed because the interdependence of tension and compression
was not fully understood. In general, the history of construction
reveals an overwhelming dependence on compression. Our conceptof building has been inseparably tied to that of weight; early
humanity piled one stone on top of another, and we continue toemploy the same single strategy, fighting gravity with sheer mass.
But compressional continuity has its limits, such as the impossibilityof achieving spans greater than 150 feet. Any larger dome would
collapse under the force of its own weight. Moreover, althougharchitects may not have reflected on the principles of tensional
integrity, necessity apparently forced them to add a powerful iron
chain around the base of St. Peter's dome; the outward thrusts fromall that compression needed further restraint.
Fuller decided that a better approach was needed than that ofslapping on a bandage at the end. Following nature's example,
tension must be designed into the structure at the start. In fact,
tension must be primary.
Nature's Example

Look around; nature's been using tensegrity all along. Humanity wasable to overlook this structural truth for thousands of years because
tension tends to be invisible. Seeing rocks sitting on the ground andbricks piled upon bricks, we have developed a virtually unshakable"solid-things" understanding of how Universe works. The ubiquitous
tension forces, from gravity to intermolecular attraction, tend to be
more subtle:... at the invisible level of atomic structuring the coherence of the myriad atomicarchipelagos of a "single" pebble's compressional mass is provided by comprehensively continuous tension. This fact was invisible to and unthought of by historicalman up to yesterday. .. there was naught to disturb, challenge or dissolve his "solid
things" thinking ... 7

The inseparable partnership of tension and compression does notmean that the two forces are the same. On the contrary, fundamental
differences are the basis of their successful interdependence.
Compression is local, discontinuous, says Bucky. When we load acolumn, we push it together. If we push a thin column too hard, thecolumn will buckle like a banana; there is no other way for the
stressed member to yield. For this reason, compression members are
subject to an inherent limit to their length relative to their cross-sec-

246

A Fuller Explanation

tional area, called the "slenderness ratio." The development ofstronger materials has increased that ratio only slightly over thousands of years: from a maximum of 18 : 1 for stone columns inancient Greece to approximately 33 : 1 for modern steel. The limit is
not as much a result of inadequate materials as of geometry:compression is directed inward, and hence eventually forces theoverstressed column to buckle. Compression fights against the shapeof a strut.
Tension, on the other hand, pulls apart. The direction of tension
serves to reinforce the shape of the stressed member. Pullingstraightens; pushing bends. As a result, assuming stronger andstronger materials, there is no inherent geometrical limit to thelength of a tension component. While the capability of compression
members has remained more or less the same, tension materials have
improved by leaps and bounds, and significant advances continue
today.
Compression was the sole basis of man-made structures until thetensile strength of wood was exploited, enabling for the first time theconstruction of structures light enough to float on water: simple
rafts, initially, followed by progressively more sophisticated rowing
and sailing vessels. But at 10,000 pounds per square inch, the tensile
strength of wood was still overshadowed by the 50,000 psi compression strength of stone masonry. Not until 1851 saw the first mass
production of steel-with a tensile strength equal to its
compression-resisting capability of 50,000 psi-was tension finallybrought into parity with compression, explains Fuller; "so tension isa very new thing." This development enabled the Brooklyn Bridge in1883 and ushered in a whole new era of tensional design. Scientists
rapidly created metal alloys of greater and greater tensile strengthwith less and less weight, ultimately leading to jet airplanes andother previously inconceivable miracles. A new material called carbon
fiber, with an unprecedented 600,000 psi, was responsible for a
recent "miracle". In 1979, Paul MacCready pedaled the firsthuman-powered aircraft, which he called the "Gossamer Albatross,"over the English Channel; two years later he repeated the journeywith a completely solar-powered plane. The only reason he was able
to accomplish this feat, emphasizes Fuller, was that the extraordinary tensile strength of carbon fiber allowed the plane to have awing span of 96 feet, while weighing only 55 pounds; you could holdit up in one hand. But the newspapers didn't mention the carbon
fiber, he declares sternly; "nobody talks about this invisible capabil-15. From Geodesic to Tensegrity: The Invisible Made Visible

247

ity." This beautiful example of "doing more with less" was a favorite
for Bucky, who never forgot being told by well-meaning adults
during his first eight years, "Darling, it is inherently impossible for
man to fly."
Driving toward a specific observation about Universe, Bucky
describes theoretically ideal structural components for each of thetwo forces, as suggested by their different characteristics. Why will ashort fat column not buckle under a compressive force that easily
breaks a tall thin column of the same material and weight? Geometry governs the situation as follows. A system is most resistant tocompression in one direction, namely along an axis perpendicular to
its widest cross-section, in which case, its vulnerable girth is asstrong as possible. This is the neutral axis. The wider its girth, themore impervious a column will be to compression. Therefore, a short
fat column is better able to resist buckling than a tall thin column ofthe same material and weight, because the latter lacks sufficientresistence perpendicular to the line of force. Similarly, if a compression force that will easily break a long column (a pencil, forexample) is applied perpendicular to its length, that column will be
unharmed (Fig. 15-9a, b). That much is common sense, but in lessextreme situations an understanding of the "neutral axis" is necessary to enable the prediction of exact results.
Next, imaging loading a slightly malleable cigar-shaped column;
compression causes the girth to expand, forcing the column tobecome progressively more spherical. This transformation suggests a
candidate for the ideal compression component (Fig. 15-9c).
A sphere is the only shape in which every axis is a neutral axis,
which is to say, a sphere's width is the same in every orientation.
Therefore, this shape resists compression from any direction; itcannot buckle. Hence the ball bearing. This tangible example illustrates the ideal design for compression.
What about tension? Evidence of longer, thinner, and ever more
resilient tension materials suggests that there's no inherent limit tolength. Fuller takes this a step further: "May we not get to where wehave very great lengths and no cross-section at all?" He answers his
own question, "This is just the way Universe is playing the game."
Gravity is that invisible limitless tension force. "The Earth and theMoon are invisibly cohered ... "; the tension cable has reached thelimit case in thinness: it's nonexistent. "You have enormous tension
with no section at all." A splendid design! The solar system is thus amagnificent tensegrity: discontinuous compression spheres (i.e.,

248

A Fuller Explanationol77/Y/7(a)//'7/11017 II 77
(b)

-0
(c)

Fig. 15-9planets) are intercoordinated-never touching each other- by a seaof continuous tension. "Every use of gravi ty is a use of ... sectionlesstensioning," Fuller continues, observing that" this is also true within
the atoms: true in the macrocosm and true in the microcosm"(645.03 - 5).
By mid-twentieth century, it was clear that the design plan ofUniverse involves islanded compression and continuous tension, but

15. From Geodesic to Tensegrity: The Invisible Made Visible249

up until thenman had been superficially misled into [thinking] that there could be solids orcontinuous compression .... Only man's mentality has been wrong in trying toorganize the idea of structure. (645.04)

New Concept of Construction

That humanity can learn from the principles of nature is the essenceof Fuller's message. We must abandon our building-block concept ofstructure in favor of comprehensive solutions which take advantageof the inherent qualities of tension and compression. The latter tends
to do the local, isolated structural tasks in nature, while the former
specializes in cohering systems over great distances. While we understand that Universe is not structured like a stack of bricks, thatawareness has not affected our approach to construction. A "building-block" approach has persisted more or less unchanged for thousands of years, pitting structural bulk against gravity's vigilant force.
Instead, argues Fuller, we must think in terms of whole systems inequilibrium, omnidirectional forces interacting in self-stabilizing patterns. If, emulating nature, structural design capitalizes on the integrity of tension, these "whole systems" will prove far stronger thananalysis of their separate parts could predict.
Additionally, we can learn from nature's structuring method:
converging and diverging, she produces bubbles, explosions, stars,
and the radially expanding sound and light waves. Energy pushes
out, and its expansion is countered by tensional restraints such as
the pull of gravity and molecular forces. Eventually, the two dynamics reach an equilibrium, a tentative balance such as a soap
bubble.
Fuller heralded an "era of thinking and conscious designing interms of comprehensive tension and discontinuous compression"(640.42). He saw an unmistakable change taking place, largely going
unnoticed. This "new era" began with the spoke wheel, which Fuller
pinpoints as man's first breakthrough into tensegrity thinking:I saw that his structural conceptioning of the wire wheel documented his
intellectual designing breakthrough into such thinking and structuring. The compressional hub of the wire wheel is clearly islanded or isolated from the compressional "atoll" comprising the rim of the wheel. The compressional islands areinterpositioned in structural stability only by the tensional spokes .... This reverses
the historical structural strategy of man. (640.42)

The wheel's use of tension enables a far more efficient and lightweight250

A Fuller Explanation
Fig. 15-10. Wheel with compression
spokes.

structure than could be produced with compression spokes. Tension
materials are inherently smaller and lighter than compression materials carrying equivalent loads.
The wheel was originally an exclusively compression structurestarting with the cave man's stone cylinder and progressing toslightly more sophisticated designs like "the old artillery wheel"cited by Fuller in Synergetics. 8 It continues to be perceived as such.
(A version of that wheel is sketched in Figure 15-10.) Widespreadawareness of the tensional integrity responsible for the spoke wheel's
lightweight efficiency has not been reached. The load on a bicycle
wheel is therefore often seen as "sitting" on the lower spokes-likecolumns-rather than hanging from the top spokes. Despite design
breakthroughs, humanity as a whole is still caught in "solid-things"
thinking.
Many other structures that rely on tensional integrity can be cited,such as suspension bridges and sailboats. However, Fuller points outthat tension was usually incorporated as a "secondary accessory ofprimary compressional structuring." In other words, ancientman-habitually relying on compressional continuity-inserted a"solid" mast into his hull, but finding that the wind kept blowing his
mast over, he added a set of stabilizing tension wires, or stays, innautical terminology. Somehow he failed to learn from his accidental
design, in which the extraordinarily thin, lightweight tension adjuncts withstand the same forces as the heavy solid mast.
Tension has been secondary in all man's building and compression has been
primary, for he always thought of compression as solid .... Earth and ship seemedalike, compressionally continuous. (640.50)

Modeling the Invisible

Tensional integrity is certainly how Universe works, pondered Bucky
in the 1940s, but how can I illustrate this invisible phenomenon? Can
the co-occurrence of discontinuous compression and continuous
tension be modeled in such a way as to bring this structural principle

15. From Geodesic to Tensegrity: The Invisible Made Visible

251

into easy grasp? He was determined to display the invisible truths ofscience in a scale that can be perceived by human senses.
Could it be done? Could discontinuous reality be modeled? Science in the twentieth century feels exempt from modelability, philosophized Bucky; ever since the isolation of the electron in 1898,
scientists have felt increasingly more sure of their lack of responsibility to explain their work to the layman. The hypothetical scientist of
Fuller's lectures declares, "I am sorry to say, reality is both invisible
and unmodelable."In the summers of 1947 and 1948, Fuller taught at Black Mountain College, and spoke constantly of "tensional integrity." Universe
seems to rely on continuous tension to embrace islanded compression elements, he mused; we must find a way to model this structuralprinciple. Much to his delight, a student and later well-known
sculptor, Kenneth Snelson, provided the answer. He presented his
discovery to Fuller: a small structure consisting of three separated
struts held rigidly in place with a few strings. This was the birth ofan explosion of geometric tensegrity structures. Inspired by Snelson's
discovery, Fuller went on to create tensegrity versions of countless
polyhedra.
Tensegrity Polyhedra

Tensegrities can be derived from all polyhedra, whether regular,
semiregular, high-frequency geodesic, or irregular, typically with one
strut representing each edge of the polyhedron. Struts do not comein contact with each other, but instead are held in place by a
network of tension elements, or strings-producing completely stable sculptural systems. The complexity of molecular interactions
aside, the two types of components are characterized by axial-force
states, thereby using materials most efficiently, because components
can be far lighter than would be required to withstand bending. The
technical sound of the words in no way prepares you for theexquisite appearance of these structures. Photograph 15-1 shows atensegrity icosahedron and tetrahedron, while Photograph 15-2 displays a 3v tensegrity icosahedron, as representative tensegrity polyhedra. However, there's no substitute for actual models.
At last, says Bucky, we are able to experience at first hand thetruth about structure: systems cohere through tensional continuity,and nothing in Universe touches anything else:... Tensegrity structures satisfy our conceptual requirement that we may nothave two events passing through the same point at the same time. Vectors [i.e.,

252

A Fuller Explanation

Photo. 15-1. Tensegrity icosahedron and tensegrity tetrahedron. Photograph by Amy
C. Edmondson.
struts] converge in tensegrity, but they never actually get together; they only get into
critical proxirnities and twist by each other. (716.11)

A tensegrity icosahedron, therefore, is more honest than a toothpick-marshmallow structure which seems to have five edges touchingat each vertex. In the tensegrity, edges all come within criticalproximity of the location of a "vertex" and" twist by each other."
Figure 15-11 illustrates the relationship between the tensegrity
icosahedron and its Platonic counterpart. Instead of a five-valent"point" with the illusion of continuity, each convergence is marked
by a pentagon of string, thus illustrating the fact that individualenergy events do not touch but instead hover in a state of dynamic
equilibrium. Tensegrity structures provide a visible, tangible illustration of an invisible truth. Forces and their interactions are brought
out in the open.
Perhaps the most significant lesson from tensegrity structures liesin their unexpected strength. A tensegrity's apparent extreme fragility is completely deceptive. The uninformed observer will usually
approach such a structure with great caution, touch it hesitantly andgently so as not to break the delicate model, and (if persistent)
ultimately realize that a tensegrity can be thrown around the room
without harm. The erroneous initial assumption is a result of adeeply ingrained bias in favor of compression as the reliable sourceof structure. We perceive string and cable as flimsy, but actually the15. From Geodesic to Tensegrity: The Invisible Made Visible

253

Photo. 15-2. 3v tensegrity icosahedron with 90 struts. Photograph courtesy ofThomas T. K. Zung, Buckminster Fuller, Sadao and Zung Architects, Cleveland,Ohio.

magnitude of a force that can be carried by "delicate" tension
materials can far exceed the corresponding capacity of compression
elements. Humanity's perception is in need of retuning.
The other aspect of a tensegrity's remarkable strength is the rapid
omnidirectional distribution of applied forces. One of the advantagesof a network of tension elements is efficient dispersal of loads
around a structure, enabling the whole system to withstand forces far
greater than could be predicted by engineering analysis of theseparate components. Welcome back, synergy:
This is not the behavior we are used to in any structures of previous experiences ...
Ordinary beams deflect locally .. . The tensegrity "beam" does not act independently but acts only in concert with "the whole building", which contracts only
symmetrically when the beam is loaded ... . The tensegrity system is synergetic . . ..
(724.33 - 4)

This" whole system" behavior can be detected by pushing or pulling

254

A Fuller Explanation

Fig. 15-11. Relationship of tensegrity icosahedron to its Platonic (planar) counterpart.

on two opposite struts of certain tensegrities. The entire system willcontract or expand symmetrically like a balloon, and also will springback to its equilibrium configuration when the applied force isremoved. (Photo 15-3 shows the simple six-strut tensegrity, which isarguably the most elegant illustration of this uniform contraction or
expansion in response to a unidirectional force.) Similarly, thissynergetic behavior insures a balanced distribution of stresses:
If you ... tauten one point in a tensegrity system, all the other parts of it tighten
evenly. If you twang any tension member anywhere in the structure, it wilI give thesame resonant note as the others. (720.10)

Fuller felt that this dispersal is not really understood by mostengineers, and as a result, they have been unable to predict oranalyze the extraordinary capabilities of tensegrities and geodesic
domes. Note that discontinuous compression and continuous tension
also characterize geodesic domes, but as these structures lack thevisually legible quality of the geometric models, one cannot at a
glance identify the operative forces:
Structural analysis and engineering-design strategies ... were predicated upon the
stress analysis of individual beams, columns, and cantilevers as separate components ... [and] could in no way predict, let alone rely upon, the synergetic behaviors
of geodesics .. .. Engineering was, therefore, and as yet is, utterly unable to analyzeeffectively and correctly tensegrity geodesic structural spheres in which none of thecompression members ever touch one another and only the tension is continuous.
(640.02)

The ultimate result of this conceptualizing and model building isthat the barrier to ever-larger clear-spanning enclosures has been

15. From Geodesic to Tensegrity: The Invisible Made Visible

255

Photo. 15-3. Six-strut "expanded octahedron" tensegrity. Photograph courtesy of
the Buckminster Fuller Institute, Los Angeles, Calif.

removed. By understanding the crucial role of tension, we can learn
to manipulate it in preferred ways. "We are able to reach unlimited
spans because our only limitation is tension, where there is noinherent limit to cross-section due to length" (764.02). Cosmic zoning laws are repealed.

Pneumatics
We recall Fuller's great-circle description in which a vast number ofgas molecules are bouncing around inside a sphere, with theirgreat-circle chords ultimately describing an icosahedral pattern as aresult of spatial constraints. Tensegrity now completes the image.
Fuller explains that tensegrity provides a tangible demonstrationof what happens inside a balloon. We tend to think of the balloon's
skin as a continuous surface; however, a more accurate picture is256

A Fuller Explanationthat of a network of molecules in close proximity, such that the
spaces between them are smaller than air molecules, allowing the
network to act as an effective cage:
The balloon is indeed not only full of holes, but it is in fact utterly discontinuous.
It is a net and not a bag. In fact, it is a spherical galaxy of critically neighboringenergy events. (761.03)

Gas molecules push out against the tensed rubber net and a dynamic
equilibrium is maintained; compression and tension are in balance.
Our eyes cannot see the bustling molecular activity of the balloon,
and Fuller sees synergetics as a way to help us tune in to thisinvisible behavior. A high-frequency tensegrity icosahedron does justthat:
In the geodesic tensegrity sphere, each of the entirely independent, compressional
chord struts represents two oppositely directioned and force-paired molecules. The
tensegrity compressional chords do not touch one another. They operate independently, trying to escape outwardly from the sphere, but are held in by thespherical-tensional integrity's closed network system of great-circle connectors ....
(703.16)

He elaborates on the parallel, holding up his icosahedral tensegritythat has traveled with him to hundreds of lectures: "This is aballoon, except that the tension components are only placed rightwhere they're needed."9 It's a balloon with all the excess tension
taken out; strings are located only where the strut (molecule) wants
to impinge on the sphere's surface.
We are thus led back to the necessity of three-way great-circling:
A gas-filled balloon is not stratified. If it were, it would collapse like a Japaneselantern .... Once we have three or more ... push-pull paths [of paired kinetic
molecules] they must inherently triangulate by push-pull into stabilization of opposite angles. Triangulation means self-stabilizing; which creates omnidirectional
symmetry; which makes an inherent three-way spherical symmetry grid; which is
the geodesic structure. (766.02-4)

The analogy is complete. Pneumatics are dynamic high-frequency
tensegrity geodesic configurations.
The principle of tensegrity, perhaps more than any other single
aspect of synergetics, has yet to be exploited to its real potential interms of design advantage. But the elegant simplicity of these
remarkable structures hints at the nature of a future design revolution-toward innovative designs with unprecedented performanceper pound. Finally, the tensegrity system, besides suggesting structural applications, has also provided a useful model in science.

15. From Geodesic to Tensegrity: The Invisible Made Visible

257

Case in Point: Donald Ingber

One striking example was contributed to the growing list in 1983 by
Donald Ingber, then a doctoral student at Yale University working
on the biology of tumor formation and malignant invasion. Ingber
had been exposed to tensegrity structures in an undergraduate design
course-a playful option on the other end of the academic spectrum,
which was to influence his vision as a scientist in profound ways.
While pursuing his research in cell biology, Ingber began to observe
some fundamental similarities between the subjects of his two seemingly opposite investigations. It appeared that tensegrity theory wasapplicable to biological systems; that is, Fuller's structures exhibitedcertain dynamic characteristics that were analogous to cell and tissue
behavior. Significant structural parallels between the behavior ofcellular and tensegrity systems led Ingber to powerful insights about
the regulation of cell shape, differentiation, and growth, and thereby
suggested a strategy for further investigation. lO
Ingber proceeded to build, test, and study tensegrity structures inan effort to understand the implications of his proposed model, andwas excited by the results of the comparison. His revolutionary
approach has led to significant breakthroughs in his research into
cancer formation, which he now continues at Harvard University
Medical School. Without a detailed description of Ingber's research,
we can still profit by the theme of his radical theory. He proposes a"whole system" approach in which "the architectural form of atissue may itself serve to coordinate and regulate the shape, orientation, and growth of its individual cells through transmission of thephysical forces of tension and compression characteristic of a given
three dimensional configuration."llThere is no more appropriate conclusion than that provided by
Ingber's own description in his 1983 letter to Fuller:
The beauty of life is once again that of geometry with spatial constraints as the
only unifying principle. It is of interest to note that, as presented in the accompanying paper, cancer may then be viewed as the opposite of life resulting from a
breakdown of this geometric hierarchy of synergetic arrangements. II

16"Design Science""I did not set out to design a geodesic dome," Fuller once said, "I
set out to discover the principles operative in Universe. For all I
knew, this could have led to a pair of flying slippers." This playful
declaration stands as a concise summary of the philosophy behind
Fuller's life's work and introduces the relationship of synergetics todesign. "Design science," in the most general terms, maintains thatfaithful observation of Universe is the basis of successful invention.
The idea therefore is not to invent some strange new gadget, hoping
there will be a market for it, but rather to tap into the exquisite
workings of nature. While the significance of scientific discoveries is
not always immediately understood, the accumulated "generalized
principles" have been applied in innovative ways throughout history,
producing artifacts which have gradually transformed the physicalenvironment. Therein lies the key to humanity's success aboard
Spaceship Earth, explains Bucky Fuller.
"Comprehensive Anticipatory Design Science"

Characteristically, his title expands- to embrace the full significanceof this vital human endeavor.
Fuller defines design as the deliberate ordering of components. Thus
distinguished from randomness, design implies the presence of intellect. His definition is worth our serious attention, for the word is too
often associated with the concept of decoration-secondary or superficial embellishment as opposed to thoroughly developed systems.
Fuller was quick to point out that Universe overflows with evidence
of design; unimaginably intricate and reliable energy patterns reveal"eternal design interrelationship principles." Technology, to Fuller,
is principle in action, and so "Universe is nothing but incredible
technology." Its awesome complexity is the inspiration for Fuller's
phrase "Intellectual Integrity of Eternally Regenerative Universe"-a16. "Design Science"259

weighty title attempting to convey a nonanthropomorphic respect fora greater (in fact all-encompassing) divine intelligence.
Combining this newly defined word with" science," to describe a
new discipline or field, further enriches its significance. "Science"hints at the necessary rigor, suggesting a systematic new study.
Fuller thereby expands the realm of "design"; the scientific method
is essential, for "design science" involves the application of principle.
He points out that "generalized principles" are eternal truths, as
opposed to special-case statements or transient facts, and as such are
inherent aspects of reality waiting to be discovered. Only humanbeings are able to discern such truths (science) and thereby participate in their own evolution (design). "Design science" is thus
saturated with meaning: humanity alone has access to the design
laws of Universe, and that has determined our unique evolutionary
function. Just as bees are meant to cross-pollinate, we are meant tosolve problems. Without specialized long beaks or wings or otherrole-specific physical traits, human beings have learned to exploit
mechanical advantage, discipline the electron, travel more quickly
than the fastest leopard, and fly farther than the strongest bird. Our
unique advantage is a faculty called "mind," which can integrate
disparate facts of experience. We are therefore deliberately designed
to be "comprehensivists" while all other creatures are specialists.
This brings us to the next aspect of Fuller's wordy title.
"Comprehensive ...

1/

Fuller was profoundly impressed by the danger of overspecialization.
He was once asked to speak at a convention of the American
Association for the Advancement of Science, and the experience
provided one of his best parables. Whether by luck or throughFuller's characteristic genius for detecting significant patterns, hehappened to encounter two papers with a striking similarity-presented at different sections of the conference. The reports, on biology
and on anthropology, both happened to discuss the phenomenon of
extinction: the former investigating various extinct species, and thelatter, extinct human tribes. Both papers concluded that the cause of
extinction was overspecialization, which, taken to an extreme, precludes general adaptability. Fuller took the message to heart.
He says his predilection for thinking comprehensively began with
his World War I Navy experience. Belonging to the last generationof sailors that preceeded ship-to-shore radiotelephone equipment,

260

A Fuller Explanation

officers in 1917 were still trained as "comprehensivists." Everyone
had to be capable of handling any job on the ship, for voicecommunication with land was not yet possible. The need for all
members of a crew to act quickly in an emergency demanded quick"comprehensive" thinking and the flexibility to take over any job
without instructions from superior naval powers. A second lesson
was that a sailing vessel is itself a managable whole system, andevery member of her crew is working toward the success of the wholeship. It seemed to Bucky a very desirable way to operate. He began
to see that our entire planet is one system and deliberately set out tounderstand the interrelatedness of human affairs. He later coined thefamous" Spaceship Earth" to symbolize this approach, as it became
ever more apparent that effective design had to recognize the existence of a finite and inescapably connected whole-system world.
Lack of such awareness leaves us stranded on a ship with the
starboard side short-sightedly using much of its time and energy inan effort to sink the port side, and vice versa.
Fuller then calls our attention to a subtle irony implicit in thecompartmentalization of the sciences, which require an ever narrower focus as one pursues a given scientific discipline more deeply,
as if to deny the relatedness of various aspects of scientific knowledge. At the beginning of this century, he recalls, chemistry andbiology, for example, were totally separate fields, the former encompassing chemical elements and their reactions, the latter pondering
the classification of species and the mystery of evolution. Moreover,all branches of science involved philosophical speculation. Time wasgiven to questions such as "what is life?" and the difference between
life and nonlife could not have seemed more self-evident.
In this century, however, science faced an unpredicted development. As a necessary reaction to new discovery, new fields emerged,
such as biochemistry, defying the rigid boundaries between disciplines. (Science responded by making individuals that much more
specialized within such new categories, laments Fuller.) Moresophisticated equipment had revealed the chemistry of life, forcing
scientists to integrate their fields. The helix of nucleotides in DNA,the magnificent chemical transitions in photosynthesis, and hundredsof newly observed reactions belonged inarguably to both chemistry
and biology, and precipitated the birth of "biochemistry." Andgradually the clear boundary between life and nonlife was dissolving,for both consist ultimately of electrochemical process. But scientists
were no longer the" natural philosophers" characteristic of the turnof the century, maintains Fuller, and so this astonishing evolution

16. "Design Science"

261

went largely unheralded. Specialization removed the burden of asking what a scientific development means in terms of the whole
picture.
Fuller has a reason for pointing out such historical trends: thedesign scientist will be maximally effective as a comprehensive
thinker. Once again, he redefines and thereby expands the concept ofdesign. The design scientist is not to be concerned with an attractive
handle for refrigerator doors, but rather with the whole concept ofthe distribution and preservation of food for humanity. Such subjects, he emphasizes, are not too large to think about. Only throughsystematic comprehensive planning does humanity have a chance tosurvive its growing crisis.
A "comprehensivist," he continues, may periodically have to"plunge very deeply" into a narrow subject or specialized project;however, such activity is always part of a larger plan. "Local
problem solvers" can function with a global perspective. Our emphasis must shift from "earning a living" to accomplishing vital tasks if
humanity is to survive, cautions Fuller; moreover, the "living" willtake care of itself if we concentrate on doing what needs to be done.
These sweeping statements at first may seem difficult to apply;
however, Fuller's philosophy is backed up by a lifetime of revolutionary invention and research into world patterns and trends, which
stem directly from his 1927 decision to think about the whole system
of "Spaceship Earth." He explains that his accomplishments were
only possible because he gave himself the license to be a generalist: it
is feasible to conceptualize humanity's food production and distribution (for example) as a whole system; it is a complicated study, but
one with clearly defined boundaries: just food-where it is grown,
where and when it is eaten, how and at what cost to the environment, consumer, etc. 1" ... Anticipatory ... "Finally, the design scientist must think ahead. In each industry, there
are specific "gestation rates" that determine the length of time
between invention and widespread practical application. These inherent lags vary according to the nature of a design: in the electronics industry, for example, it is only a matter of months before a newinvention can be incorporated into commercial production; carmanufacturers might require five years to bring a new idea to theconsumer; and housing presents the slowest evolution of all, Fuller'srather optimistic estimate of the 'gestation rate' being fifty years.

262

A Fuller Explanation

Psychological resistance to change, absence of urgency, and ignorance
keep our approach to housing many generations behind our technological capability. The design scientist must take these lags into
consideration, explains Fuller; an invention often must wait until its
time, but the designer has a responsibility to anticipate long-term
developments. Finally, a necessary implication is that we can glean
important clues through the study of trends, and thereby determine
what needs to be done.
So let us look at the relationship between "comprehensive anticipatory design science" and synergetics. Invention, as stated above,is the novel application of one or more "generalized principles." In
the previous chapter, we discussed two inventions, the geodesic dome
and tensegrity structures, in light of this statement. Briefly, the dome
combines the inherent stability of triangles with the advantageousvolume-to-surface-area ratio of spheres. A variety of structures basedon geodesic chords could satisfy the above requirements, but icosahedral symmetry approximates the spherical distribution most efficiently. Both geodesic domes and tensegrities are direct applicationsof the principle describing the specific interdependence of tension
and compression.
The theory of geodesic domes is taken a step further by anadditional principle: the varying rates of geometric expansion. As
discovered and exploited long ago by clipper-ship owners-as well
as by today's shipping industry-a ship with twice the length ofanother has eight times the volume and four times the surface area.
Translated into practical advantage, the cargo (i.e. payload) of larger
vessels increases rapidly with respect to the amount of material anddrag, which together determine the effort and cost of building ships
and driving them through the sea. Larger ships are therefore morecost-effective. Despite the pivotal role of this geometric principle inshaping the historical direction of shipping, the flow of resources,
and the ultimate mobility of humankind, these varying rates are not
popularly recognized. With respect to geodesic domes, this means
doubling the diameter increases the material fourfold and encloses
eight times the volume. As both cost and temperature control of an
enclosure are directly determined by surface area, the efficiency of ageodesic dome increases drastically with size. The implication is that
their true design-science advantage is yet to be realized and may
involve very large structures in novel environment control applications. Whether utilized for shelter, food production, recreation, orother functions, geodesic domes can enclose so much space with solittle material that unprecedented future applications are not un-16. "Design Science"263

likely. In conclusion, the geodesic dome embodies design science atwork.
Dymaxion Map2

Another of Fuller's inventions, in response to a very different problem than architectural design, is based on similar geometric principles. The problem is to draw a flat map of the world without thegross distortions inherent in the Mercator projection. In the early
1940s, dismayed by this widely accepted map's inaccurate depictionof our world-a visual lie presenting Greenland as three times thesize of Australia, when exactly the reverse is true- Fuller wasdetermined to discover a better solution.
Let's consider the design problem. Visual data must be reliablytranslated from the surface of a spherical "whole system" onto a flat
display with only one side. To understand how the Mercator projection attempts to accomplish this task, imagine wrapping a large
rectangular piece of paper around a transparent globe, forming acylinder that touches only the equator. The geographical outlines arethen projected directly outward to the cylindrical paper, as if by alight source inside the globe casting omnidirectional shadows. As aresult, the visual information is accurately translated to the paper
only at the equator; some distortion exists slightly above and below
the equator, and it increases radically as one goes farther north andsouth on the map. In many versions Antartica is left out altogether,even South America is smaller than the gigantically distorted Greenland, and certain land areas-depending on which country has
produced the map-must be split in half to turn the cylinder into aflat poster.
Unwilling to accept such distortion as necessary, Fuller started
from scratch. If a spherical system is to be translated onto a flat
surface, what is the most efficient and direct solution? It's a geometry
problem; the relevant "generalized principle" involves the polyhedral system which best approximates a sphere with only one typeof face. (The latter consideration insures an evenly distributed projection.) That system, which is of course an icosahedron, is the basisof a reliable and simple solution.
Imagine a globe with the edges of a spherical icosahedron superimposed on its surface by thin steel straps. Chapter 14 describedplanar polyhedra expanding into spherical polyhedra, as if drawn on
balloons; we now visualize the reverse process. The steel arcs slowly
unbend into straight-edge chords, while the curved triangle faces

264

A Fuller Explanation

flatten out into planar equilateral triangles. The overall shape change
is relatively slight (consider for comparison a spherical tetrahedron
undergoing the same operation, or a sphere turning into a cylinder asin the Mercator projection), and the global "whole system" ispreserved. As the globe transforms into an icosahedron, twenty
spherical triangles with 72-degree comers become planar triangles
with 60-degree comers; 12 degrees are squeezed out of each angle.
(With three angles per triangle, 12 degrees times 60 angles equals
none other than our old friend the "720-degree takeout.") Because
each triangle of the spherical icosahedron covers a relatively small
portion of the sphere and is thus fairly flat, the distortion during thistransformation is minimal-and in fact invisible to the untrained
eye. Moreover, the polyhedral projection automatically distributes
the distortion symmetrically around the globe's surface and thereby
insures that the relative sizes of land masses are accurate. (This is
why a regular polyhedron is a preferred vehicle; different types offaces would distort slightly different amounts during the transition
from spherical to planar.) Finally, all geographical data are contained within the triangular boundaries; there is no "spilling" ofinformation or need to fill in gaps with "extra" land as in the
Mercator.
The next step is straightforward. Unfold the icosahedron to display its twenty triangles on a flat surface. The result is a map of theentire world with little distortion of the relative shape and size ofland masses, and no breaks in the continental contours (Fig. 16-1).
No nation is split and shown on two opposite sides of the map as if
separated by 20,000 miles.
That last step required more work than is immediately apparent,
however. It took Fuller two years of experimenting to find an
orientation in which all twelve icosahedral vertices land in the ocean-an essential requirement if land masses are not to be ripped apart.
Observe in Figure 16-1 that many of the vertices are extremely close
to shore. One can imagine the frustrating task of searching for twelve
water locations; moving a vertex away from land on one side of the
globe would instantly result in a number of vertices bumping into
land somewhere else. Contemplating the five or six angular gaps
which are precariously close to land masses, one suspects that
Fuller's final arrangement may be a unique solution to the problem.
The Dymaxion Map, awarded U.S. Patent 2,393,676 in 1946, is an
unprecedented cartographic accomplishment, which was made possible by a straightforward application of geometry. This map istherefore another superb example of the design-science approach.
Fuller considered the problem outside the context of traditional

16. "Design Science"

265

nster Fuller
Fig. 16-1. Dymaxion Map@, used with pernusslOn of the Buckmi.)
Institute
the
about
tion
informamore
forC
Institute. (See Appendix

266

A Fuller Explanation

(a)

(b)

Fig. 16-2. Hanging bookshelf, U.S. Patent 4,377,114 (1983).

map-making; rather than attempting to work with and refine history's
previous best solution, he started over. He sets the example ofconsidering a design problem as a whole system.
Suspended Storage Systems
A third synergetic design application is Fuller's 1983 hanging
bookshelf. Conceived as a space-saving storage device, the narrow
shelving unit is suspended from the ceiling by six wires. The original
design was a tall hexagonal column of wood, with compartments on
all sides creating omnidirectional access, as the unit is suspended
rather than leaning against the wall (Fig. 16-2). Hanging by its few
cables, the column appears ready to swing back and forth at theslightest push from any direction. People would approach this structure-on display for a while in a Philadelphia bookstore-and touch16. "Design Science"267

it hesitantly and gently, only to be astonished by its resistance. The
more daring will ultimately lean against the column with all theirweight and discover that it does not move. Just as for the tensegrity
sculptures, this rigidity is especially surprising because we do not
expect "delicate" tension elements to be capable of the same strength
as "solid" columns; however, even those who anticipate the great
capability of tension are caught off guard by this display. In spite ofrecognition that thin cables can support a massive block of wood,
the fact that the shelf does not swing even slightly out of placeremains astonishing.
This invention utilizes one geometric principle: Fuller's "twelve
degrees of freedom." Twelve vectors, or independent forces (sixpositive, six negative), must be applied to a body in space tocompletely restrict its mobility. In Chapter 7 we looked at theapplication of this inherent spatial characteristic to a bicycle wheel,
and observed that a minimum of twelve spokes was required torigidly restrain its hub. Fuller's bookshelf presents a similar design
problem; why then is it anchored with only six "spokes"? The
answer lies in the massiveness of this gigantic" hub"; the hexagonal
column is heavy enough to pull firmly against the six wires andremove the remaining, or negative, six degrees of freedom. The
cables are symmetrically arranged-as are the six spokes anchoring
one half of our hypothetical bicycle hub-but in this case gravity
takes care of the other six spokes, pulling in the opposite direction
from each cable by taking advantage of a heavy object's considerable
attraction to the earth. If the column were made of lightweightplastic, the design would not work; six degrees of freedom would
still be unaccounted for.
This design represents a remarkably simple application of a synergetic principle, and produces a startling piece of furniture, themajor significance of which-if saving floor space is not a consideration-may be educational. It is a profoundly reorienting experienceto feel the precariously suspended shelf's refusal to budge.
The above examples suggest that design science can be considereda science of spatial order. As such, this study is necessarily comprehensive; space is everywhere. We also learn that invention does not
spring fully formed out of principle, but rather requires a little work.
First a need is ascertained, as for example for an efficient shelter
system. Then, relevant principles are gathered. From the jumble ofknown truths, one or two might apply to a problem. The next step isto pull them in, experiment, twist them around, and not give up:
seek that innovative application of an age-old principle.

268

A Fuller Explanation

More with Less

The three designs described above were chosen as straightforward
examples of design science. However, Fuller's main purpose is to call
our attention to an invisible design revolution already taking place,to inspire our active participation in guiding this progression in
preferred ways. He points out fantastic technological advances, suchas new communications satellites, each one weighting a quarter-ton
and outperforming 175,000 tons of transoceanic copper cable. 3 Anyone who remembers the shaky transatlantic telephone connections of
the past can appreciate the qualitative improvement as well. Similarly, ever stronger metal alloys enhance humankind's structural
capability. The average person is not aware of this metallurgicalrevolution, says Bucky, because it cannot be seen. An invisible
reality is quietly taking over, accomplishing so much more with somuch less material and other resources, that the logical extension is
sufficient and sustainable life support for all humanity.
To further explain this potential, Fuller discovers that he is forced
to redefine "wealth." Too long associated with money and othertangible and limited resources, wealth is actually the organizedcapacity of society to apply its resources to take care of lives. A
computer is not worth much in terms of its content of precious
materials; its value is in the processing of information and knowledge. Wealth involves energy and knowledge; the former is neither
created nor destroyed, and the latter is constantly increasing. Therefore, humanity's true wealth is constantly increasing and has noinherent upper limit. There is a fixed amount of gold in the world,
but the currency of our emerging era is knowledge and its creative
application.
Finally, Fuller points out that Thomas Malthus could not haveforseen this technological revolution. His epochal conclusion in 1805that population increases geometrically while its resources-ability
to feed, clothe, and house itself-expand arithmetically at best, is
now obsolete. Malthus did not anticipate the phenomenon of more
effective performance using less resources. His declaration predates
refrigeration, let alone the information and communications revolution, emphasizes Bucky, and yet humanity's social and economic
institutions are still based on the assumption of fundamental scarcity. Malthusian thinking has controlled human affairs for so longthat we have mistaken it for absolute truth. The only barrier to a
successfully sustainable planet is ignorance, Bucky declares. Fundamental scarcity is a remnant of the dark ages.

16. "Design Science"

269

The essential message of Fuller's design science is that humanbeings have access to the design laws of Universe, and a responsibility to use the extraordinary phenomenon of mind to discover andapply such principles. Our function is problem-solving. Synergetics,
the discipline behind Fuller's more-with-Iess philosophy, above all
encourages us to experiment. This material is superbly suited tonurture and enhance creativity, demanding both numerical rigor andintuitive leaps. The systematic study of spatial complexities is still
young, and its significance and utility as yet undeveloped. The futureis wide open, but we must probe and step beyond our fragile
equilibrium, if Fuller's vision is to be tested.
A design science revolution is imperative.

Appendix A: Trigonometric Calculations

Finding Interior Angles of Polygons

Pentagon
360 0 -;- 5 = f3
72 0 = f3

f3 + 2(ta) = 1800a = 180 0

f3-

a = 108 0

Hexagon360 0 -;- 6 = f3 = 60 0

f3 + a = 1800a = 120 0......

/'...... 4I""""f3 ....,I
I
I

Heptagon
360 0 -;- 7 = f3 = 51.43 0

a = 180 0 - 51.43 0
a = 128.57 0

272

A Fuller Explanation

Chord Factors~ Polyhedraledge...--------'

RadiusCentral angle(corresponds toa polyhedral edge)1./
sin 1. a = 2....

R

2

R sin ~a = ~/2R sin ~a = /R [2 sin ~a 1 = /R X [chord factor] =/For every central angle a, there is a numerical value called a "chord
factor," generated by the equation [2 sin ~a], which can be multiplied by thedesired radius of a polyhedral system to calculate the exact length of thechord (edge) subtended by a.

Appendix B: Volume Calculations for
Three Prime Structural Systems
Tetrahedron

Using traditional formula:v = t AbH,
where A b = area of base and H = height.
(1) Area of base:
Ab=

Bh

2 =

1 x sin 60°
21(0.8660)

2= 0.4330.
(2) Height of tetrahedron :H~

H 2 + (0.5774)2 = 12,

H = 0.8165.

~
0.5774(3) Volume:V =Ab X H3= 0.11785.

~74
~

0.5

A Fuller Explanation

274

Octahedron

Two square-based pyramids:
Ab H ] X 2 = V.
[ -3-(1) Area of base:

"

""
"x
/

/

/

/

//
/

"-

/

"-

=1

""

H

1110~.8~6~60\.'---v.- - - 'o.s(2) Height of pyramid:

H2 + (0.5)2 = (0.8660f,
H = 0.7071.

(3) Volume of half octahedron:(0 .7071)(1)---- =

3

0.2357.

(4) Volume of octahedron:2 X 0.2357 = 0.47140.

Icosahedron

Twenty pyramids with equilateral bases and side-edge length equal to the
icosahedron radius.

Appendix B: Volume Calculations for Three Prime Structural Systems(1) Pentagonal cross-section:x = 2[sin 54 0 X 1]=

1.61803/(2) Radius R:(2R)2 = (1.61803)2 + (1)2,
R = 0.95106 .

~74

0.5774

0.5(3) Pyramid height H:(0.95106)2 = (.5774)2 + H2 ,H = 0.7557.

275

276

A Fuller Explanation(4) Area of pyramid base is equal to area of base of regular tetrahedron:Ab =1(0.866)2= 0.4330.

(5) Pyramid volume:t(AbH) = 0.1091.

(6) 1Volurne of icosahedron:
20 X (0.1091) = 2.1817.

Appendix C: Sources of
Additional Information

All books, unless noted by an asterisk, are available from the Buckminster
Fuller Institute, 1743 S. La Cienega Boulevard, Los Angeles, CA 90035. A
number of versions of the Dymaxion Map can also be obtained; write theInstitute for a complete list.

GeneralSome of the other books by R. Buckminster Fuller (in addition to thoselisted in the bibliography) recommended for further reading:Operating Manual for Spaceship Earth (Carbondale, Ill.: Southern Illinois
University Press, 1969).
No More Secondhand God (Carbondale, Ill.: Southern Illinois University
Press, 1969). Poems and essays.

Biographical Material
Robert Snyder, Buckminster Fuller: An Autographical Monologue Scenario,(New York: St. Martin's Press, 1980). A great introduction, written by
Fuller's son-in-law using Fuller's own words, with hundreds ofphotographs from the Fuller archives.
E. J. Applewhite, Cosmic Fishing (New York: Macmillan, 1977). Especiallygood for its description of the collaboration between Bucky and his
long-time friend E. J. Applewhite in the process of writing Synergetics.
Hugh Kenner, BUCKY: A Guided Tour of Buckminster Fuller (New York:
William Morrow, 1973).

Geodesic Mathematics
Hugh Kenner, Geodesic Math: And How to Use It (Berkeley, University ofCalifornia Press, 1976).

278

A Fuller Explanation

World Game

"50 Years of Design Science Revolution and the World Game," historical
documentation (articles, clippings) with commentary by Fuller, 1969.
Available only through the Buckminster Fuller Institute.
Medard Gabel, Earth, Energy and Everyone (Garden City, N.Y.:
Doubleday, 1975).
Medard Gabel, HO-ping: Food For Everyone (Garden City, N.Y.:
Doubleday, 1976).
Write for newsletter, information about seminars and workshops, and list ofavailable material to The World Game, University City Science Center,
3508 Market Street #214, Philadelphia, PA 19104.

Appendix D: Special Properties of
the Tetrahedron(1) Minimum system: the tetrahedron is the first case of insideness andoutsideness.
(2) The regular tetrahedron fits inside the cube, with its edges providing
the diagonals across the cube's six faces, and thereby supplying the six
supporting struts needed to stabilize the otherwise unstable cube.
Furthermore, two intersecting regular tetrahedra outline all eight vertices of
the cube.
(3) The tetrahedron is unique in being its own dual.
(4) The six edges of the regular tetrahedron are parallel to the six
intersecting vectors that define the vector equilibrium.
(5) Similarly, the four faces of the regular tetrahedron are the same four
planes of symmetry inherent in the vector equilibrium and in cubic
closepacking of spheres. The tetrahedron is thus at the root of anornnisymmetrical space-filling vector matrix, or isotropic vector matrix.
(6) When the volume of a tetrahedron is specified as one unit, otherordered polyhedra are found to have precise whole-number volume ratios,
as opposed to the cumbersome and often irrational quantities generated by
employing the cube as the unit of volume. Furthermore, the tetrahedron has
the most surface area per unit of volume.
(7) Of all polyhedra, the tetrahedron has the greatest resistance to an
applied load. It is the only system that cannot "dimple"; reacting to an
external force, a tetrahedron must either remain unchanged or tum
completely "inside out."(8) The surface angles of the tetrahedron add up to 720 degrees, which isthe" angular takeout" inherent in all closed systems.
(9) The tetrahedron is the starting point, or "whole system," in Fuller's"Cosmic Hierarchy," and as such contains the axes of symmetry thatcharacterize all the polyhedra of the isotropic vector matrix, or face-centeredcubic symmetry in crystallography.
(10) Packing spheres together requires a minimum of four balls, toproduce a stable arrangment, automatically forming a regular tetrahedron.
The centers of the four spheres define the tetrahedral vertices. In Fuller's
words. "four balls lock."

280

A Fuller Explanation(11) It has been demonstrated that many unstable polyhedra can befolded into tetrahedra, as in the jitterbug transformation.
(12) Fuller refers to the six edges of a tetrahedron as one "quantum" ofstructure, because the number of edges in regular, semi regular, and highfrequency geodesic polyhedra is always a multiple of six.

Appendix E: Glossary

Includes specific terminology coined and/or used in an unusual way by
Buckminster Fuller (in italics) as well as terms from conventional
mathematics and geometry that may be unfamiliar.
A-module: Asymmetrical tetrahedron, which encompasses one twenty-fourthof the regular tetrahedron.
acute angle: Angle less than 90 degrees.
angle: Formed by two lines (or vectors) diverging from a common crossing.
An angle is necessarily independent of size.
angular topology: Fuller's term, intended to describe a principal function ofsynergetics, that is, description of all structure and pattern by variationof only two variables: angle and frequency. (See "frequency.")arc: A segment of a curve.
B-module: Asymmetrical tetrahedron, which is the result of subtracting an
A module from one forty-eighth of the regular octahedron.
complementarity, or inherent complementarity of Uniwrse: Necessary
coexistence, inseparable pairs. Fuller emphasizes that Universe consistsof complementary teams, such as concave-convex, tension-compression,
positive-negative, and male-female, and concludes that "unity isinherently plural."concave: Curved toward the observer, such as the surface of a sphere, orother enclosure, as seen from the inside.
convex: Curved away from the observer, such as the surface of a sphere as
seen from the outside.
coupler: Semisymmetrical octahedron, which consists of sixteen "AModules" and eight "B-Modules," or eight "Mites."degrees of freedom: This term is used by Fuller to mean the number of
independent forces necessary to completely restrain a body in space, andby Loeb in reference to the overall stability of systems.
design: Deliberate ordering of components.
design science: See Chapter 16.
dimension: See Chapter 6. Used by Fuller to include spatial extent, orders ofcomplexity, and distinct facets of symmetry.
dimpling: Yielding inwardly to produce local indent in structural system,
caving-in.

282

A Fuller Explanation

Dymaxion: Fuller's trademark word, created by Marshall Fields Department
Store in the late 1920s, for use in promotion of exhibit featuring Fuller's
revolutionary house design. Also used by Fuller to mean "doing morewith less."
edge: Geometry term: connection between points (or vertices) and boundary
between two faces, line. Fuller prefers to substitute" vector" or else referto the specific material of construction, such as toothpick or straw.
energy event: Fuller's substitute for geometric term" point." Also descriptive
term for natural phenomena, discrete constituents of Universe. Replaces"obsolete" vocabulary such as "solid," "point," "thing," etc.
ephemeralization: Doing more with less, via design science and technologicalinvention.
face: Geometric term: polyhedral window, polygon, area.
finite accounting system: Describes concept of physical reality consisting, onsome level, of discrete indivisible particles, as opposed to continuous
surfaces or masses. See Chapter 2.
frequency: Used by Fuller to specify length and size in general. The
intention is to employ a more precisely descriptive term for both geometricsystems and events in nature, than provided by specific units ofmeasurement. Fuller points out that frequency never relates to thequantity "one," for it necessarily involves a plurality of experiences.
high-frequency energy event: Describes most tangible structures, which might
be popularly thought of as "solids." A good example to illustrate theconcept of "high-frequency energy event" through a visible image isfound in the white foam of breaking ocean waves. Upon closer inspection,
the apparent continuum of whiteness is a result of an enormous numberof tiny clear bubbles, which appear continuous because of their close
proximity. This punctuated consistency is analogous to all matter,although most examples are not visible to the naked eye.
frequency of modular subdivision: Number of subdivisions per edge, in apolyhedron, or number of discrete subdivisions per module in any
system.
generalized principle: Rule that holds true without exception. Eternal law ofnature. See Chapter l.
great circle: A circle on the surface of a sphere, which lies in a planeintersecting the center of that sphere. Equatorial ring.
great-circle arc: Segment of great circle.
Greater Intellectual Integrity of Universe: Sum total of all "generalized
principles," complex "unknowable" totality of Universe. Divine intellect.
Perfection of eternally regenerative pattern integrities of Universe.
in, out: Seen by Fuller as appropriate replacements for up, down, to
describe the directions toward the earth's center and away from theearth's surface, respectively.

Appendix E: Glossary

283

inherent complementarity of Universe: See "complementarity."

interior angle of polygon: Angle between two edges measured through theinside.
interprecessing: See precession.
intertransfonnability: Phenomenon of significant relationships between
systems, allowing transformations from one to another. Applies to both
polyhedra and natural structures. Examples include shared symmetry
among polyhedra and common constituents arranged differently to
produce different substances.
isotropic vector matrix, IVM: Space-filling array of unit-length vectors, inwhich all vectors are identically situated. The "omnisymmetrical" matrix
consists of an indefinite expanse of alternating tetrahedra and octahedra,
with 60-degree angles between adjacent vectors. This conceptual IVM
framework can be actualized with building materials to create an Octet
Truss, which is an extremely efficient and lightweight architectural spaceframe.
jitterbug transfonnation: Transformation of a cuboctahedron model, in
which flexible joints allow the unstable polyhedron to contract in aradially symmetrical manner, and thereby take on the shape of various
other polyhedral systems.
Mite: Two mirror-image "A-modules" and one "B-module" combined tocreate an asymmetrical tetrahedron, in which three right angles surround
one vertex. Trirectangular tetrahedron.
nature's coordinate system: The mathematically expressible system thatgoverns the coordination of both physical and metaphysical phenomena.
Set of generalizations about the way systems are structured and able tocohere over time. Interplay of the principles describing spatial complexity
with the requirements of minimum energy in the organization of natural
structures.
nest: Valley, or local indentation, between adjacent closepacked spheres.

net: Planar array of adjacent polygons which can be folded along sharededges and closed together to create a specific polyhedron.
obtuse angle: Angle greater than 90 degrees.
omni-accommodative: Able to accommodate all spatial directions, or model
all transformations; omnidirectional.
omni-interaccommodative: Describes cooperative relationship between
noncontradictory principles or evidence, which are thus more significant
considered together than separately.
omnisymmetrical: Symmetry in all spatial directions.
pattern integrity: Reliable or consistent arrangement of "energy events" (orconstituent parts) in dynamically regenerative system. A pattern with

284

A Fuller Explanation

structural integrity, that is, a pattern that coheres for some period oftime.
precession: Two or more systems in motion with respect to each other
involving 90-degree tum. In addition to its meaning in physicsdescribing a complex motion of a rotating body in response to an applied
torque-Fuller employs this word (as well as his own "interprecessing")
to refer to two geometrical systems which, oriented perpendicularly toeach other, reveal a new system or geometric relationship.
quanta: Used by Fuller to mean indivisible discrete units, limit-case particles,
isolated energy events. See finite accounting system.

regular polyhedron: Polyhedron composed exclusively of one type ofpolygonal face meeting at identical vertices.
right isosceles triangle: Triangle which includes two equal edges (or arc
lengths) with a right-angle between them. Planar version has angles of
45 0 -45 0 -90 0 ; one particular spherical version has angles of 60 0 -60 0 -90 0 •similar: Geometrical term: having the same shape, but not necessarily thesame size.
size: Dimension, extent. Relates to actual constructs, or "special-case"systems.
Spaceship Earth: Coined by Fuller to convey a sense of a finite, wholesystem planet, in which the lives of all human beings (or passengers) are
interrelated. The idea is to encourage thinking of Earth as a single systemwith a common interest in successful survival.
special-case: Relates to specific example rather than generalized system.
Used especially to refer to specific manifestation of generalized concept.
spheric: Rhombic dodecahedron. Fuller's term, derived from the fact thatthe rhombic dodecahedron is the domain of each sphere in closest
packing.

spherical triangle: A curved area bounded by three connected great-circle
arcs. The result of interconnecting three points on the surface of aspherical system.
structural system: Triangulated system.

supplementary angles: Angles that add up to 180 degrees.
synergy: Behavior of whole systems not predictable from the behavior of
separate parts.
system: Four or more interrelated "events."
tune-in-ability: Possibility of isolating or focusing on specific phenomena as
independent systems despite interrelatedness of such phenomena to manyother systems. (See Chapter 3.) Boundaries can be constantly redefined,
according to the particular goal of an investigation. Also refers to limitsof resolution in defining systems.

Appendix E: Glossary

285

Universe: Paraphrasing Fuller, Universe is the aggregate of all experience.
(See Chapter 1.) The role of the observers, or humanity, is an essential
component of the definition, for awareness is a prerequisite to defining
and understanding. Experience consists of dynamic, regenerative patternsof energy-perhaps "omnidynamic" would be an appropriate term.
-valent: Number of connections or elements, such as number of edges
meeting at a vertex, or number of sides of a polygonal face; e.g.,"four-valent" vertex of an octahedron, and "three-valent" face (triangle)of the same, or three-valent vertex (and face) of a tetrahedron.
valving: Deliberate channeling of energy and resources in preferred ways.
vector: Represents energy event, consists of magnitude and direction,
represented on paper by an arrow with specific length (or frequency) andangular orientation, and used by Fuller instead of "edges" in polyhedra.
Vectors also represent relationships between energy events.
vector equilibrium: Fuller's term for the cuboctahedron, in recognition of its
equal radial and edge lengths.
vertex: Geometrical term: crossing, convergence of lines or edges, joint,
point, polyhedral comer.
wealth: Organized capacity of society to apply generalized principles toward
present and future life support.
whole number: In mathematics, positive integer, without fractional part.

Bibliography

E. J. Applewhite, Cosmic Fishing (New York: Macmillan, 1977).
Buckminster Fuller, Synergetics: The Geometry of Thinking (New York:
Macmillan, 1975).
Buckminster Fuller, Synergetics 2: Further Explorations in The Geometry ofThinking (New York: Macmillan, 1979).
Buckminster Fuller, Utopia or Oblivion (London: Allan Lane, The Penguin
Press, 1979).
Buckminster Fuller, Critical Path (New York: St. Martin's Press, 1981).
Buckminster Fuller, Inventions (New York: St. Martin's Press, 1983).
Buckminster Fuller, Intuition (Garden City, N.Y.: Anchor Press/
Doubleday, 1973).
Buckminster Fuller and Robert Marks, The Dymaxion World of Buckminster
Fuller (Garden City, N.Y.: Anchor Press/Doubleday, 1973).
Hugh Kenner, Bucky: A Guided Tour of Buckminster Fuller (New York:
William Morrow, 1973).
Arthur L. Loeb, Space Structures: Their Harmony and Counterpoint(Reading, Mass.: Addison-Wesley, Advanced Book Program, 1976).
Peter Pearce, Structure in Nature Is a Strategy for Design (Cambridge,
Mass.: MIT Press, 1978).

Notes

Chapter 1

IBuckminster Fuller, Synergetics: The Geometry of Thinking (New York:
Macmillan Publishing, 1976), p. 30, section 216.03: "Comprehension ofconceptual mathematics and the return to modelability are among the mostcritical factors governing humanity's epochal transition from bumblebee-likeself's honey-seeking preoccupation into the realistic prospect of aspontaneously coordinate planetary society." (My italics-A.E.) As statedin the introductory "Note to Readers," quotations from Synergetics will
hereafter simply be followed by a numerical section reference in parenthesis.
Quotation from Synergetics 2: Further Explorations in the Geometry ofThinking will be followed by the section number and the letter" b."2These statements were included in many of Fuller's lectures. I am
quoting from memory and from personal notes made over the years.
Hereafter, assume this to be the source if not otherwise referenced.
3See later chapters, especially Chapter 3 for explanation of alloys, andChapter 16 for discussion of Malthus.
4See "Spaceship Earth" in the Glossary (Appendix E).
5Sir Arthur Eddington, (1882-1944), English astronomer and physicist.
6Quoted from epic videotaped lecture session, recorded in January 1975,
in Philadelphia, over a ten-day period. The total length of these lectures,
which are part of Fuller's archives, now located in the Buckminster Fuller
Institute in Los Angeles, is 43 hours. (Hereafter referred to as "43-hour
videotape. ")7See e.g. Powers of Ten, by Philip and Phyllis Morrison and The Office ofCharles and Ray Eames (New York: Freeman, Scientific American Books,
1982).
8See Appendix C for biographical references.
9Arthur L. Loeb, Space Structures (Reading, MA: Addison-Wesley,
Advanced Book Program, 1976), p. xvii.
1°F = GMmjr 2 : The attractive force due to gravity between two objects
is equal to the "gravitational constant" (G) times the product of theirmasses and divided by the separation-distance raised to the second power.
Chapter 2

lSee Glossary, "whole number."

290

A Fuller Explanation

2See Appendix C.
3It must be noted that tetrahedra cannot fit together face to face to form
the larger tetrahedra shown, but rather must alternate with octahedra, aswill be explored at length in later chapters (8, 9, 10, 12, 13). The values
given for the volume of each tetrahedron are based on using a unit-length
tetrahedron as one unit of volume, in exactly the same manner that aunit-length cube is conventionally employed. Despite the tetrahedron's
inability to fill space, the relative volumes of tetrahedra of increasing size
are identical to those exhibited by cubes of increasing size.
Chapter 3

lSee Loeb's preface to Fuller's Synergetics: "In rejecting the predigested,
Buckrninster Fuller has had to discover the world by himself" (p. xv).
2Buckrninster Fuller, Intuition (Garden City, N.Y.: Anchor Press/
Doubleday, 1973), p. 39.
3See Glossary, "spherical triangle," "concave," "convex."
443-hour videotape.
5Quoted several times by Fuller, in 43-hour videotape.
6Hugh Kenner, Bucky: A Guided Tour of Buckminster Fuller (New York:
William Morrow, 1973), p. 113. See Glossary (Appendix E) for definition of"Dymaxion. "Chapter 4

1" Contribution to Synergetics" is a 52-page supplement by Arthur Loeb
included in Fuller's Synergetics, pp. 821-876.
2See Glossary, "-valent".
3Arthur Loeb, Space Structures, Chapter 6.
4Loeb, p. 11.
5Chapter 5 explains why "structural system" implies triangulated system.
6Loeb, p. 40.
7Loeb, p. 63.
8See Glossary, "acute" and "obtuse."
Chapter 5

1The best renditions of the precession sequence are found in videotaped
lectures, because Fuller's gestures are as important as his words; the43-hour videotape contains an especially good version. Original written
document was published in Fortune, May, 1940. Fuller wrote the two-pagepiece in response to a request (or challenge!) from the Sperry Gyroscopecompany.

Notes291

2See Glossary, "Greater Intellectual Integrity of Universe."
3Loeb, pp. 29-30. The derivation of the equation 3V - E = 6, which isthought to have originated with Maxwell, can be derived from degree-offreedom analysis of molecular spectra, and was applied by Loeb topolyhedral systems.
Chapter 6

ISee Glossary, "special case."
243-hour videotape.
3Kenner, pp. 129-131.
4Appendix A, "chord factors."
Chapter 7

IWilliam Morris, Editor, The American Heritage Dictionary (New York:
American Heritage Publishing Co., 1970), p. 442.
243-hour videotape.
3It must be noted however that if the cub octahedron is sliced in half and
one "hemisphere" is rotated 60 degrees with respect to the other, theresulting " twist cuboctahedron" (Fig. 7-6b) maintains the radialcircumferential equivalence. With its asymmetrical arrangement of faces,
however, this shape is not similarly suited to model equilibrium. The
desired balance of vectors is therefore achieved through the straight
cuboctahedron (Fig. 7-6a).
4Amy Edmondson, "The Minimal Tensegrity Wheel," Thesis forcompletion of B.A. degree requirements, Harvard University, 1980. Refer to
VES Teaching Collection, Carpenter Center, Harvard University.
5Fuller's use of the term "degrees of freedom" must be distinguishedfrom the conventional treatment of the subject which specifies that a rigid
body has six degrees of freedom (three translational and three rotational)
which have two directions each, thus requiring twelve unidirectional
constraints.
Chapter 8

IN. J. A. Sloane, "The Packing of Spheres," Scientific American (January1984), p. 116.
2Cubic packing thus corresponds to the VE (cuboctahedron), while
hexagonal packing corresponds to the "twist" VE. See Footnote 3 in
Chapter 7 for the difference.
31. Rayment, T. S. Baker, D. L. D. Caspar, and W. T. Murakami,"Polyoma Virus Structure at 22.5 A Resolution," Nature, 295 (14 January1982), p. 110-115.

292

A Fuller E'Cplanation

4Donald Caspar and Aaron Klug, "Physical Principles in theConstruction of Regular Viruses," Cold Spring Harbor Symposia on
Quantitative Biology, XXVII (1963), pp. 1-3.
Chapter 9

lSee Glossary, "supplementary angles."2A. L. Loeb, "Contribution to Synergetics," in Synergetics: The Geometryof Thinking (New York: Macmillan, 1976), pp. 860-875; "A Systematic
Survey of Cubic Crystal Structure," J. Solid State Chemistry, 1 (1970), pp.
237-267.
343-hour videotape.
4Refer to Chapter 5 and Appendix D.
Chapter 10

lA. L. Loeb, "Contribution to Synergetics," pp. 821-876. Volume ratios
are derived by comparing geometrically similar polyhedra so that "shape
constants" cancel out of the formulae. Credit for this approach belongs to
Loeb and Pearsall, and it must be noted that this section (H) of Contribution
to Synergetics is adapted from an article by Loeb in The Mathematics
Teacher.
2Loeb, p. 836.
3Loeb, p. 832.
Chapter 11

lLoeb, p. 829.
2A. L. Loeb, "Addendum to Contribution to Synergetics," published in
Synergetics 2: Further Explorations in the Geometry of Thinking, (NewYork: Macmillan Publishing, 1979), pp. 473-476.
3Dennis Dreher of Bethel, Maine designed an omnidirectional hinging
joint, which allows the necessary twisting of adjacent VE triangles in thejitterbug. This joint can be seen in Photograph 11-1.
Chapter 12l"Bohr, Niels" Encyclopaedia Britannica, 1971.
2Isaac Asimov, Asimov's Guide to Science (New York: Basic Books,1972), pp. 334-5.
3Loeb, "Contribution to Synergetics," pp. 836-847.
4Loeb, "Coda," Space Structures, pp. 147-162.

Notes293

Chapter 13

Discovered independently by Loeb and called" moduledra."
2Loeb, "Contribution to Synergetics" pp. 847-855;
3Space Structures, "Coda," pp. 147-162.
1

Chapter 14

lSee Appendix E, "right isosceles triangles."
2Fuller, Synergetics, Figure 455.20, p. 178.
3Fuller, Synergetics, Figure 458.12, p. 189.
4See Appendix E and Chapter 15.
Chapter 15

IBuckminster Fuller, Critical Path (New York: St. Martins Press, 1981),
p.13.
2This fact was demonstrated in the" tensegrity" bicycle wheel experimentmentioned in Chapter 7. In tests to determine the minimum number oftension (Dacron string) "spokes" required to stabilize its hub, the wheel's
eventual structural failure originated with buckling of its rim. Long arc
spans (which were a consequence of the low number of radial spokes) were
too thin to withstand the compression force created by loading the hub andtransmitted to the rim through the tension spokes. The usual arrangement,
which consists of a fairly large number of spokes (36 or more), therefore
turns out to be advantageous; despite the fact that there are many more
spokes than necessary to restrain the hub, this large number does serve to
subdivide the otherwise vulnerable arc segments of the compression-element
rim. See also Edmondson, "The Minimal Tensegrity Wheel."
3Recalling the principle of angular topology in Chapter 6, we also knowthat the" angular takeout" is 720 degrees. That is, if we subtract the sum of
the surface angles at each vertex of a convex polyhedron from 360 degrees,
the sum of all these differences will be exactly 720 degrees.
4Ernst Haeckel, Art Forms in Nature (New York: Dover, 1974), plate 1(Radiolaria) and plate 5 (calcareous sponges).
5See Chapter 8, notes 3 and 4.
6Note that the engineering term "pure axial force" is thus a convenient
simplification, which is effective in terms of structural analysis, rather thanan accurate scientific description. A strut which is to carry either axial
compression or tension can be significantly lighter than one which is subject
to bending or torque.
7Buckminster Fuller, "Tensegrity," Creative Science and Technology, IV,
No.3 (January-February, 1981), p. 11.

294

A Fuller Explanation

RSynergetics, p. 354.

9See note 2 in Chapter 1.
lOThis very general summary is based on Dr. Donald Ingber's paper," Cells as Tensegrity Structures: Architectural Regulation of Histodifferentiation by Physical Forces Transduced over Basement Membrane,"published as a chapter in Gene Expression During Normal and Malignant
Differentiation, (L. C. Anderson, C. G. Gahmberg, and P. Ekblom, eds.;
Orlando, Fla.: Academic Press, 1985), pp. 13-22.
llTaken from a letter from Ingber to Buckminster Fuller, April 5, 1983.
Chapter 16

IThis approach characterizes Fuller's "World Game" studies. See
Appendix C for sources of information about this research.
2Buckminster Fuller and Shoji Sadao, Cartographers; Copyright R.
Buckminster Fuller, 1954. "Dymaxion Map" is a trademark of the
Buckminster Fuller Institute.
3 Fuller, Critical Path, p. xxiii.

Index(Bold print indicates page number which includes illustration of entry.)
A-module, 167, 189-192, 191, 199, 208,
214, 215
defining new framework, 193, 203-205
energy characteristics, 194-195, 198
net, 194, 195
volume, 201-204
see also Mite, Valving
Alloys, 4, 33, 246, 268
chrome-nickel steel, 34
Angle types, definition of, 79-81, 81
axial, central, dihedral, surface, 79
Angular topology, 65-81 (Chapter 6),
69
principle of, 75-77"720-degree excess," 77-79"takeout angle," 75-79, 264
Architecture, 1, 18, 81, 141, 239, 243,
245
housing, 261-262
Avogadro, Amadeo, 125, 127, 143
B-module, 167, 189-192, 192, 199,208,
214,215
energy characteristics, 194-195, 198,
202-204
net, 194, 195
volume, 201-204
see also Mite, Valving
Bicycle wheel, 94,95,249-250,267
Black Mountain College, 251
Bohr, Niels, 179
Boltzmann, Ludwig, 85
Bookshelf, see Hanging bookshelf
Bowties, see Great circles
Brain-Mind distinction, l3, 259, 269
Buckminster Fuller Institute, 255, 265

Carbon atoms
carbon fiber, 246-247
diamond versus graphite, 29, 142
general bonding of, 174
Chinese physicists, Tsung Dao Lee andChen Ning Yang, 179
Closepacked spheres, sphere packing, 9,18, 100-126 (Chapter 8), 127, 128,l32, l33, l38, 154, 164, 175, 178,
183, 189, 201, 227
cubic versus hexagonal, 104, 105, 106,
lO7
formula for total numbers of spheres,
115, 116, 118, 119, 238
icosahedron and sphere packing, 117,
118, 159; see also Shell systems
mathematical challenge, 102-106, 108
nests, 103-107, 111, 115, 120, 121
planes of symmetry, 106-107
tetrahedral clusters, 108, 110-114, 120
triangulation, 117
vector equilibrium and spherepacking, 101-107, 114-116, 159
see also Nuclear spheres
Complementarity l31, 178-180
complementary quanta: A- and Bmodules, 192, 193, 195
concave-convex,
tension-compression, and otherpairs, 30, l33, 179, 244
fundamental complementarity inphysics, 179, 192, 195
inherent complementarity of Universe,
l33, 158, 178, 179
of octahedron and tetrahedron, l31,
132,133-135,137,140,178,180
Compression, see Tensegrity

296

Index

Consciousness, defined by Fuller, 11,12
Constant relative abundance, 44, 74
Cookies, 85, 86, 130
Coordinate system, 2, 65, 97, 205, 240
Cartesian, 70, 71, 72, 88, 97, 131, 137,
140, 154, 157, 209, 211
origin, 12, 33, 65, 97, 140
spherical, 71, 72
Copernicus, 71
Cosmic hierarchy, 9, 143, 156-158, 165,
166,216
Cosmic Railroad Tracks, see Great
Circles
Coupler, 199, 200, 201, 202; see also
Mite
Cube, 40, 46, 211
employment as basic unit ofmathematics, 7, 8, 14, 20, 21, 71, 73,
144,157,158,179,190
ghost, 8, 65, 144
inherent tetrahedron in, 46, 59, 60,
137, 138, 145, 146, 178, 185
instability of, 59, 60, 107, 124, 135,
137,141,145,146,178
used as unit of volume, 144, 145, 152,
158
tetrahedral volume of, 151, 152
see also Isotropic vector matrix, cube
and IVM; Mite, cubes and Mites
Cuboctahedron, 48, 49
duality of, 51"twist," 90, lOS
see also Vector equilibrium
Degrees of freedom, see Twelve degrees
of freedom
Democritus, 58
Descartes, Rene, 75
Design Science, 1, 13, 118, 142, 258-262,
264, 267, (Chapter 16)
comprehensive thinking, 259-261
design-science revolution, 268-269
see also Invention
Dictionary as inventory of experiences,19
Dimension, 41, 65, 69- 75
definition of, 70
four-dimensional, 71, 73, 74, 92, 95,
169,172

multidimensional, 127, 129, 130
other applications of, 74, 75
three-dimensional, 70, 71, 73, 74, 130
Dodecahedron, see Pentagonal
dodecahedron, Rhombic
dodecahedron
Domain, 133, 213-215,
and closepacked spheres, 138, 201
and duality, 138, 139, 181-184
Duality, 45, 49, 180, 183, 211
dual operations, 51, 140
dual polyhedra, 45, 46, 50, 51, 168,
212
IVM and, 136, 137, 139, 181, 182
see also Domain
Dymaxion, 34
Map, 64, 263, 264, 265
Eddington, Arthur, 7, 33, 149
Edmondson, Amy c., 5,252
Einstein, 11, 13
E = Mc 2 , 13
Empire State Building, 8Energy-event
descriptive term to replace" solid," 7,
8, 27, 195
discrete energy events, 18, 125, 126,
193, 222, 228, 256; see also Finite
accounting system"energy-event Universe," 14, 17, 27,
51,66,221,226,228
Engineering, 141, 142, 237, 253, 254
Environment control, 63, 262
Equilibrium, see Vector equilibrium
Euler, Leonhard, 42, 43, 62
Euler's Law, 43-45, 77, 116, 230, 231
Extinction, 259"Fake bubbles," 15-17
Flat-earth thinking, 14, 19, 20, 71, 144,
158
Finite accounting system, 18, 19, 125,
126, 208; see also Energy-event
Food system, 261, 262
Four planes of symmetry, 93, 106, 130,
133, 140, 169
Frequency, 65, 112, 227
angle and, 67, 72, 83, 91

Indexversus continuum, 67, 235
formula to relate frequency andnumber of spheres, 116, 118, 125,
238,239
geodesic domes and, 235-243
higher-frequency polyhedra, 135, 136,
148, 155-157, 184, 186-188, 227,
235, 237, 256
and size, 66, 67
sphere-packing and, 112, 114-117, 125
and time, 67
Fuller Institute, see Buckminster Fuller
Institute
General Dynamics, titanium shell
experiment, 239
Generalized principle, 1, 12, 13, 33-35,
110, 116, 180, 188, 240, 259, 262,
263
principle of angular topology, see
Angular topology
see also Principle of design covariables
Geodesics, 227, 231, 235, 237, 233-257(Chapter 15)
four-frequency icosahedron, 77, 78,
237,240
geodesic dome, 1, 5, 63, 236, 239-243,
254, 258, 262, 263
geodesic mathematics, 3, 232
geodesic polyhedra, 28, 77, 78, 227,
231,236, 240, 256
great-circle chords, 236, 242, 262
variable geodesic patterns, 240-242
see also Frequency, Shell systems
God, 11, 14, 59,258,259
Golden section, 18, 166-168
Gravity
Fuller anecdotes, 2, 247, 248
mass interattraction, 13, 33, 204, 245,
249, 267
Great circles, 26, 206-231 (Chapter 14)
bowties, 219-228, 230
cosmic railroad tracks, 228, 229, 231
definition of, 206
energy paths, 226-229
great-circle arc, 207, 221, 232, 233
great-circle patterns, 209-213
icosahedral, 212-217, 224-227, 230,
234, 235, 255

297

icosahedron as local shunting circuit,229
minimum models, 223, 224
shortest path, 206, 207, 226, 228, 232
Greece, ancient, 7, 36
geometry of, 37, 74, 178, 179
Greek, use in nomenclature, 25, 27,
34, 39see also "Pi"Hanging bookshelf, 266, 267
Harvard University, 3, 9, 37, 257
Sever Hall, 3"Hands-on" mathematics, 2, 24
Heisenberg, indeterminism, 179
Icosahedron, 26, 39, 49"dimpling," 63and geodesics, 236-242
jitterbug and, 161, 163, 164
net, 76"out-of-phase" role, 164, 167, 169,
178, 180, 186, 216, 217
structural system, 61-63
symmetry of, 73, 164, 165, 166,
213-215, 233, 262, 263
volume of, 163, 164
see also Closepacked spheres;Dymaxion Map; Great circles,
icosahedron as local shunting
circuit; Shell systems; Universe,"trans-Universe" versus "locallyoperative system"
Icosadodecahedron,49
Infini ty, 23
infinite straight line, 6, 84
infinite number of vertices in sphere,
16,17,79
see also Straight line
Ingber, Donald, 4, 257
In-Out versus Up-Down, 19,20
Interprecessing, 121, 124Intertransformabilities, 46, 50-52, 140,
143, 157, 189, 192, 204, 209, 210,
215-218
Invention, 141, 240, 258
gestation rates, 261, 262
inventor, 1

298

Index

Invention (cont.)
see also Design science
Invisibility
invisible reality, 2, 14, 34, 163, 170,
172, 232, 246, 247, 250-252, 256,
268
invisible Universe, 162, 250
Isotropic vector matrix (IVM), 10,
127-142 (Chapter 9),143,150,159,
164, 189, 192, 195-197
alternating octahedra and tetrahedra,
127,128, 131-134, 137, 146, 178,189
cube and IVM, 137, 138,139
equilibrium, 129
frame of reference, 140, 143, 144, 146,
154,157,164,165,167,170,175,
180, 181, 183, 185, 188, 204, 217,
229
IVM', 139, 140, 143, 150, 154, 180,
181
omnisymmetry, 129, 130, 137, 141,
165, 183, 189
omnitriangulation,133
square cross-section of IVM, 133, 134
VE and IVM, 135; see also Vector
equilibrium and space-fillingsee also Space-filling
IVM, see Isotropic vector matrix
Jitterbug
complex of jitterbugs, 170, 171
flexible VE model, 159, 169
solid-triangle model, 169
transformation, 159-163, 169, 170,
174, 213, 216
Kenner, lIugh, 69, 290
Klug, Aaron, 238, 239
Least common denominator (LCD),
189-192, 195, 196, 198, 213-218,
225, 234, 242, 243
Lee, Tsung Dao, 179
Lesser circle, 206
Tropic of Cancer, 206, 207
Leverage, 12, 56

Life, see Pattern integrity
Life-support, 24, 268
Local holding patterns, 222, 228
Loeb, Arthur L., 3-5, 10, 37, 44-48, 51,
52,60,143,147,155,167,180,185,
196, 197; see also Space Structures

MacCready, Paul
Gossamer Albatross, 246, 247
Malthus, Thomas, 4, 268
Massachusetts Institute of Technology(MIT), 4, 5, 59, 66
Mercator projection, 263, 264
Mite, 195, 196, 199, 200, 202
cubes and, 198,200
mirror symmetry of, 197, 198, 202
rearrangement of, 202, 203
rhombic dodecahedron and, 198, 200More with less, 268, 269
Mozart, 58
Multiplication by division, 143, 149, 154,
156-158,162,178,193,212

Nature's coordinate system, 2, 9-11, 16,
17,24,34,67,68,84,102,193
Nests, see Closepacked spheres
Net, polyhedral, 75,76, 79, 193, 194
Neutral axis, 247
Nuclear spheres, 100, 101, 105, 111, 201,
203
removal of, 117, 118, 159, 164
tetrahedron and nuclei, 112-114, 120
VE and nuclei, 114-116, 118, 159
see also Closepacked spheres

Octahedron, 38, 73, 139closepacked spheres and, 107, 108, 120
jitterbug and, 161-163
net, 76octahedral cavities, 92, 107, 129, 132,
133,150,155,166,170,178,185
structural system, 61-64, 137
symmetry of, 73, 93, 209-211
truncated see tetrakaidecahedron
see also Isotropic vector matrix, BmoduleIndexOctant, 120, 137, 150-153, 151, 178, 185,
190
Octet symmetry, see Symmetry
Octet Truss, 1, 63, 64, 141, 142, 178, 198
Operational mathematics, 6, 8, 10, 24,
29,30,143,146,175,219,235
operational procedure, 9, 18, 179, 193
Pattern integrity, 54, 56-59, 97
life, 9, 58, 59
knot, 57-59"thinkable me," 58wave, 57, 58, 171
Pentagonaldodecahedron,41,49
symmetry of, 41, 49, 168, 216
Phase changes, see Solids
Philosophy (of Buckminster Fuller), 1,
32, 97-99, 260, 269
philosophy and geometry, 32
Pi, 15-17
Platonic polyhedra, 34, 45, 252, 254
derivation of, 37-43
see also Regular polyhedra
Poles of spinnability, 44, 116, 208
Principle of angular topology, see
Angular topology
Principle of design covariables, 67
Quantum, 28, 124, 125
discrete quanta, 144, 205, 238edges (six) as one, 28, 59, 62, 63, 77,
124
physics, 179
units, 167
see also A-module, B-module
Regular polyhedra, 28, 37-43, 52; seealso Platonic polyhedra
Rhombic dodecahedron, 28, SO, 51,52,
218
duality of, 51, 181, 182
icosahedron and, 137
IVM and, 137
space-filling property, 181, 182, 185
spheric, 138, 139, 140, 183, 198, 200,
204
volume, 153, 154299

see also Mite, rhombic dodecahedronand Mites
Rhombic triacontahedron, 26, 216, 225
Rhombicuboctahedron, 155
Rhombohedron, 135, 180, 185, 186
S-module, 167, 168, 216
Saint Peter's Cathedral Dome, 243, 245
Schlaefli's formula, 44
Science of spatial complexity, 9, 23, 267,
269
order inherent in space, 23, 71, 84, 86,
100, 106, 154, 157, 168, 175, 230,267
shape of space, 10, 11, 36, 68, 92, 101,
102, 107, 109, 114, 129, 130, 143,
144, 145, 146
spatial constraints, 9, 10, 36, 41, 42,
68,84,101,119,131,133,176,177,
209,239,242,257
Semiregular polyhedra, 28, 49, 52
Shell, or single-layer, systems, 117-119,
164, 227, 238, 239; see also Virus
Sixness
cosmic, 223
six positive-negative linear directions,93,114,267
Slenderness ratio, 246
Snelson, Kenneth, 251
Sodium chloride, 33
Solar system as tensegrity, 247-248
Solidsgeometric, 7, 17, 34, 154
impossibility of, 7, 16, 27, 61, 124,
125,171,184,245,249,250
phase changes in chemistry, 163, 172,
173,174
solid-things thinking, 250, 267
Solway, Carl: Carl Solway Gallery, 171
Southeast Asian basketry, 233
Space-filling
all-space filling, 175, 183, 196, 200,
203-205
complementarity, 170; see also
Isotropic vector matrix, alternatingoctahedra and tetrahedra
complex, 173, 196, 203, 229
cubes, 175, 177, 181
domain of sphere, 138

300

Index

Space-filling (cont.)
filling space with closepacked spheres,
107,108,109,228
formula for space filling, 180,
185-188, 203-205
IVM and, 127, 132, 139, 140, 180,
189; see also Isotropic vector matrix
octet symmetry, 121, l78, 199, 203
rhombic dodecahedron, 181, 182
rhombohedron, 135, 180, 185
space fillers, 175, 179-181, 185, 188,
196,200,201,203
teams, 180, 181, 185
truncated octahedron, 184, 204; seealso Tetrakaidecahedron
see also Mite
Spaceship Earth, 5, 20, 61, 258, 260, 261
Space Structures, 10,47
Special case, 65, 66, 259
special-case experience, 13, 28
special-case system, 66, 81, 157
Sphereimpossibility of, 15-18, 235, 237, 238;
see also Infinity
omnisymmetrical form, 101, 114,208,228
surface area of, 17,223, 235, 262
Spheric, see Rhombic dodecahedron
Spherical polyhedra, 207, 208, 209, 210,
212, 213, 215, 220, 223, 233, 263
Spherical triangles, 29, 30, 210, 214, 216,
223, 226
Spherical trigonometry, 79, 242, 243
Star tetrahedron, 46, 210, 224
Stellation, 47
definition of, 47, 48
degenerate stellation, 48, 50, 51, 52,
137, 139, 140, 181, 216
Straight line
chord, 17, 238, 263
Euclidean, 207, 323
imaginary straight line, 7
impossibility of, 4, 6, 8
vector as replacement for, 8, 38, 68
Structural stability
applied loads, 63, 64
necklace, 54-57
prime structural systems, 60-63, 117,
236
stability and jitterbug, 159-161

stability formula, 60
structure defined, 61
triangulation, 59-63, 97, 117, 119, 140,
141, 189, 226, 233, 235, 236, 242,
256
see also Triangles, stability ofSunsetFuller anecdotes about, 2, 4, 20
sunclipse, 20
sunsight, 20
Symmetry
defined, 52, 53, 101, 189
mirror symmetry, 53, 101, 190
octet symmetry, 121, l78, 199,203
omnisymmetry, 88, 89, 91, 93, 101,
114, 140, 141, 228; see also Isotropic
vector matrix
planar symmetry, 85-87
polyhedra as symmetry patterns, 68,
168, 180
rotational symmetry, 53, 101, 113, 165,
166, 169, l76, 195, 209, 210
seven unique axes of symmetry, 209,
210, 211, 213, 230
spatial symmetry, see omnisymmetry
see also Closepacked spheres, Greatcircles, Interprecessing, Isotropic
vector matrix, Sixness, Four planesof symmetry
Synergetics accounting, 130, 131
cosmic accounting, 193
Synergetics: The Geometry of Thinking,4, 6, 13, 24, 28, 29, 33, 34, 44, 49,
70, 72, 74, 95, 102, Ill, 174, 183,
197, 207, 250"Contributions to Synergetics," 37,
148, 157, 167, 168, 180, 196, 197
Synergetics 2: Further Explorations in
the Geometry of Thinking, 167
System, definition of, 25, 26, 38, 44

Takeout angle, see Angular topology
Tensegrity, 3, 244, 245, 247
interplay of tension and compressionin Universe, 244-249, 250, 251, 256,257
models, 251-255
pneumatics, 255, 256
tensile strength, 34, 246, 252, 253

Indextension materials, 98, 246, 247, 253,
267
use of tension in construction, 249,
250
Tessellations; 39-42, 176, 177, 236
Tetrahedronbasic unit in synergetics, 28, 38, Ill,
147,149,150,172,173,212
central angle of, 95, 121, 136, 137
cheese, 147, 155
four-dimensional, 71, 73, 93
inside-out, 63, 162
isotropic vector matrix and, 134-141
jitterbug and, 162
minimum system of Universe, 26, 27,
31, 32, 73, 97, Ill, 131, 140, 146,
149,158,172,189,190,202,223
net, 193, 194
pattern integrity, 59
perpendicular symmetry of, 122-124,
154, 155
rigidity of, 63, 142
sphere-cluster tetrahedra, see
Closepacked spheres
subdivision of tetrahedron, 150, 153,
155, 189, 190; see also A-modulesurface angles of, 57, 77
tetrahedroning, 21, 22, 187, 188
truncation of, 46, 47, 135
topology of, 43, 212
unit of volume, 144, 145, 148, 149,
150,152,158,163,201
Tetrakaidecahedron, 48, 135, 136, 184,
185
Three-way grid, 232, 243, 256; see also
Structural stability, triangulation
Titanium shell experiment, 239
Thinking, Fuller's explanation of, 31-33
Triangles
equilateral triangles in vector
equilibrium, 91, 117
similar, 146, 147, 148stability of, 26, 55, 56, 61, 97, 161,
244, 262; see also Structural
stability, triangulation"triangling" instead of squaring, 21
triangular numbers, 109, 110
see also Closepacked spheres,
Isotropic vector matrix
Tropic of Cancer, see Lesser circles

301

Truncation, 46, 184
definition of, 46, 47
degenerate, 47, 51, 52, 90, 92, 155
isotropic vector matrix and, 135, 136,
140, 184-188
Truss, see Octet Truss
Tune-in-ability, 30, 31
Twelve degrees of freedom, 93-97, 114,
227, 267
degrees of freedom in space, 94, 95,
96
freedom of motion in sphere packing,III
planar analogy, 93, 94
tetrahedron and degrees of freedom,
95-97
see also Bicycle wheel
United States Navy, Fuller's experience,
16, 83, 259, 260
United States Patents, 141, 242, 264, 266
Unity is plural, 30, 162
Universe
defined by Fuller, 11, 12differences in Universe as modulationof angle and frequency, 67
product of interaction between
principles and energy, 13, 37, 56-58,
61,162,258"scenario Universe," 11, 12, 33, 61,
140, 149"single-frame" Universe, 11, 12, 66"trans-Universe" versus" locallyoperative," 165
see also Complementarity
Valency, 29, 47, 166, 170
definition of, 29, 47
in duality, 51, 52, 137, 139, 140
polyhedral parameter, 38, 39"quadrivalent," 162
Valving, 193, 198
Vectorand energy events, 68, 127
and forces, 68, 83, 88, 238
interactions, 68, 83, 84, 87
as replacement for line, see Straight
line

302

Index

Vector ( cont.)
sphere packing and, see Isotropic
vector matrix
topology and, 68, 69
see also Isotropic vector matrix,
Synergetics accounting, Vector
equilibrium
Vector equilibrium (VE)closepacked spheres and, 100, 106,114~ 116, 164conceptual versus actual equilibrium,
97~99, 162, 165
cuboctahedron as, 90
equilibrium defined, 82
equilibrium in nature, 249
hexagons and equilibrium, 87, 89, 90,
92, 101, 103, 105, 106, 113, 169
hidden VE inside octahedron, 155, 156
model of, 86, 87,90,93, 98, 170
space filling and, 127, 130, 170, 180,
185
tetrahedron and, 92, 93, 152, 153
time and equilibrium, 99
see also Isotropic vector matrix,
Jitterbug

Videotape, 43-hour, 12, 31, 84, 118, 141
Virus, 9, 118, 119, 238, 239
Visual literacy, 22
Volume ratios, 143, 145~147, 149, 150,
157, 163
use of volume constant, 147, 148
volume-to-surface-area ratio, 17,
61~63, 118, 119, 141, 235, 262

Wealth,268
Whole numbers, 29, 30, 33, 35, 37, 149,
156, 158, 189, 249, 257, 260, 263,
264,266
whole-number relationship, 17, 18, 28,
144

Yale University, 257
Yang, Chen Ning, 179

Zung, Thomas T.K., Architects, 253