Joseph D. Clinton, Page 1

Clinton’s Equal Central Angle Conjecture
JOSEPH D. CLINTON, PolyModular, Ltd.

BACKGROUND
In 1937 Michael Goldberg introduced “a class of multi-symmetric polyhedra” consisting oftwelve pentagons, eight quadrilaterals or four triangles and all additional faces being
hexagons. Thus he introduced the fact “trihedral polyhedra which posses the same number ofhexagonal faces in addition to 12 (8, or 4) regularly and symmetrically disposed pentagons(quadrilaterals, or triangles) can be topologically different.” Ref 1

Figure 1. Goldberg’s polyhedra
Buckminster Fuller recognized the same patterns of hexagons
being clustered around the twelve pentagons of his
icosahedron based geodesic forms, as can be seen in his 1965
“Geodesic Structures” patent US 3,197,927. Ref 2 He took
advantage of this characteristic to provide circular openings
inside the pentagons and hexagons while retaining thetriangular structuring between. The cylindrical holes
provided additional stiffness and the elements between theholes would nest easily for packing and shipping during
disassembly. Fuller applied his concept to structures hereferred to as the “Fly’s Eye” structures. Ref 3
Figure 2. Buckminster Fuller’s geodesic structure

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After Harold Kroto and Richard Smalley published their discoveryof the Carbon molecule C60 in 1985, Ref 4, P.W. Fowler suggested
that “icosahedral carbon clusters with pentagonal and hexagonal
faces are Goldberg polyhedra. They have 20(b2 + bc + c2) atoms,
where b and c are non-negative integers, and obey an electroncounting rule similar to the famous Hückel (4n + 2) prescription.
When b – c is divisible by 3 the cluster has a multiple of 60 atoms
and is closed-shell.” He also suggested “a tetrahedral structure…isthe best candidate for C120.” Ref 5
Figure 3. Fowler’s C120
Another application of the Goldberg polyhedra came from the work of John P Snyder when
he published “an equal-area map projection for polyhedral globes” in 1992. Ref 6

Figure 4. Snyder’s EMAP grid
As a further advancement of the Snyder EMAP grid a class of spatial data structures called
Geodesic discrete global grid systems (Geodesic DGGS) has been studied. Ref 7, 8 One suchgrid, the Icosahedral Snyder Equal Area grid (ISEA), subdivides the face of an icosahedron
into hexagons and uses the Snyder equal area projection to transform the hexagon grid to thesphere. The resulting Goldberg polyhedron consists of 12 pentagons with clusters of hexagons
covering the remaining area of the sphere. The coarseness of the hexagonal grid determines
the number of overlapping Goldberg polyhedra.

Figure 5. Geodesic DGGS-ISEA Ref 9
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In the late 1960’s under a NASA research project thirteen geometrical methods of describinga three-way grid for tessellating the surface of a sphere after Fuller’s methods were studied.
Ref 10 Many others have been studied and reported in the literature since.
Class IIIClass IIClass IIIClass IClass IFigure 6. Spherical three way grid
If select vertices are removed from such grids the Goldberg polyhedra emerges. The
topologies of these many geometrical solutions remain the same within each group.

Figure 7. Seven geometries of a Goldberg icosahedral polyhedron
GOLDBERG’S POLYHEDRA’S TOPOLOGY“Topologically, the arrangement of the hexagons in a triangular patch is the same as in a 30º
sector of a regular honeycomb arrangement of hexagons…Using a, b as the inclined
coordinates (60º between axes) of the vertex of a patch, the square of the distance from thecenter of the patch to the vertex is equal to a2 + ab + b2…the total number of faces boundingthe ‘polyhedron’ is 10(a2 + ab +b2)+2;” for the icosahedralsystem, 4(a2 + ab +b2)+2 for the octahedral system, and 2(a2+ ab +b2)+2 for the tetrahedral system. Ref 11, 2

It can be shown that for the each Goldberg polyhedron system
the number of:
Vertices = 2(F - 2) and Edges = 3(F - 2)
Where: F = the number of faces for the polyhedron sought.

Figure 8. Goldberg’s polyhedra where a = 1 and b = 2
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The topological groups have been separated into three classes: Ref 10, 11, 12.
Class I
“Subscript a is an integer and b is always zero or b is an integer and a isalways zero. The lines of the triangular segmentation on the polyhedron face run parallel tothe edges.”
Class II
“Subscript b is an integer and is equal to a. The lines of the triangularsegmentation on the polyhedron face run perpendicular to the edges.”
Class III
“Subscript a and b can be any pair of integers as long as a ≠ b and a or b ≠zero. The lines of the triangular segmentation on the polyhedron face run oblique to theedges.”
Class IClass IIIClass IIIClass IIClass IIFigure 9. Goldberg polyhedra classes
THE CONJECTURE
A sphere may be tessellated in such a manner that it will be made up of a group of sphericalpolygons, with edges having equal central angles and having the topological characteristics ofthe Goldberg polyhedra.
THE GEOMETRY
A mathematical solution has not been found to prove the conjecture, however, enoughGoldberg polyhedra having equal edge central angles have been found to convince this authorthat it has validity.
Twenty seven spherical tessellations of the Goldberg type have been found; nine each basedon the icosahedron, octahedron and the tetrahedron. In each group 5 are of the Class I type, 3
are of the Class II type and, 1 the Class III type.
Tables 1 - 3 give their properties, and Figures 10 - 12 illustrate one face of the icosahedron,octahedron, or tetrahedron as a tessellated unit that would be repeated to develop the entire
sphere. It should be pointed out that for the Class III types, an enantiomorphic of the one
illustrated does exist for each one, but, are not shown here. One should also note thatsymmetry advantages of the Schwarz, Ref 13, triangle within each regular spherical triangular
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face can be used to an advantage when generating the forms. Each form in the figures will beidentified by its base polyhedron, I, O, or T with a subscript a, b.
Table 1. Icosahedral System
22

Where: F=10(a +ab+b )+2, V = 2(F - 2), E = 3(F - 2) and r = 1
abFaces Vertices Edges Edge arc length Central angle
10122030
0.729741.8103113260900.4063
23.28152042801200.3509
20.1025
21721402100.2648
15.1704
3092
180
270
0.233513.3763221222403600.201911.568840162320480
0.175010.0277
50252
500
750
0.14008.0237
33272
540
810
0.13477.7201
Table 2. Octahedral System
22

Where: F=4(a +ab+b )+2, V = 2(F - 2), E = 3(F - 2) and r = 1
abFaces Vertices Edges Edge arc length Central angle
1068
121.2310
70.5288111424360.6435
36.8699201832480.5564
31.8807213056840.4248
24.34173038721080.376721.5813
225096144
0.329118.8588
4066
128
192
0.283316.2336
50102
200
300
0.226913.001433110216324
0.218512.5176
Table 3. Tetrahedral System
22Where: F=2(a +ab+b )+2, V = 2(F - 2), E = 3(F - 2) and r = 1
abFaces Vertices Edges Edge arc length Central angle
104
4
61.9106
109.4712
118
12180.8810
50.4788201016
240.7778
44.5639211628
420.6369
36.49053020
36540.5698
32.6499222648720.4959
28.411040346496
0.428724.5645
5052100150
0.341019.53593356108
1620.3291
18.8551

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I 1, 0I 1, 1

I 2, 0I 2, 1

I 3, 0I 2, 2

I 4, 0I 5, 0I 3, 3Figure 10. Icosahedral Goldberg equal central angle spherical tessellations

O 1, 0O 1, 1Figure 11. Octahedral Goldberg equal central angle spherical tessellations
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O 2, 0Joseph D. Clinton, Page 7

O 2, 1

O 3, 0O 2, 2

O 4, 0O 5, 0O 3, 3Figure 11, continued. Octahedral Goldberg equal central angle spherical tessellations

T 1, 0T 1, 1

T 2, 0T 2, 1

T 3, 0T 2, 2

Figure 12. Tetrahedral Goldberg equal central angle spherical tessellations
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T 4, 0T 5, 0T 3, 3Figure 12, continued. Tetrahedral Goldberg equal central angle spherical tessellations
ADDATIONAL RESEARCHHopefully one will find a mathematical solution to the problem of tessellating a sphere in sucha manner that it will be made up of a group of spherical polygons, with edges having equal
central angles and having the topological characteristics of the Goldberg polyhedra. In
addition to this problem it may also be useful to study other spherical tessellated forms withedges having equal central angles. The diamond tessellations on the surface of a sphere appear
to have similar characteristics. Ref 14 The Fullerenes described by Fowler Ref 15 using thespiral conjecture may also be a subject for investigation.
Since first proposing the conjecture stated herein in a paper to the 5th International Conferenceon Space Structures, held at the University of Surrey, Guildford, UK on 19-21 August 2002, it
has come to my attention that Dick Fischbeck Ref 16 has been working on similar problem.
He randomly arranges cones of consistent diameter on the surface of a sphere based on thespherical excess angle of a polyhedron. He has also proposed a variation where the central
angles of the cones may consistently generate random tetrahedral pyramids that will tessellatethe sphere. Perhaps a careful study of his RanDomes will provide a solution to the Goldberg
polyhedra equal central angle problem.
REFERENCES
1.
GOLDBERG M, A Class of Mult-Symmetric Polyhedra, Tôhoku Mathematical
Journal, vol 43, June 1937, pp 104-108.
2.
FULLER B, Geodesic Structures, US Patent 3,197,927, August 3, 1965.
3.
FULLER B, Inventions: The Patented Works of R. Buckminster Fuller, St Martin’s
Press, New York, 1983, pp 214-226.
4.
KROTO H W, HEATH J R, O’BRIEN S C, CURL, R F, SMALLEY, R E, C60:
Buckminsterfullerene, Nature, vol 318, no 4, 1985, pp 162.
5.
FOWLER P W, How Unusual is C60? Magic Numbers for Carbon Clusters, Chemical
Physics Letters, vol 131, no 6, November 1986, pp 444-450.
6.
SNYDER J P, An Equal-Area Map projection for Polyhedral Globes, Cartographica,vol 29, no 1, 1992, pp 10-21.
7.
CARR D, KAHN R, SAHR K, OLSEN T, ISEA Discrete Global Grids, StatisticalComputing & Statistical Graphics Newsletter, vol 8, no 2/3, 1997, pp 31-38.
8.
SAHR K, WHITE D, Discrete Global Grid Systems, to appear in: Computing Scienceand Statistics, (1998).
9.
Discrete Global Grid Research at Terra Cognita, http://bufo.geo.orst.edu/tc/firma/gg/

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10.
11.
12.
13.
14.
15.
16.

CLINTON J, Advanced Structural Geometry Studies: Part I – Polyhedral Subdivision
Concepts, NASA, Washington D C, NASA CR-1734, 1971.
COXETER H S M, Virus Macromoleculs and Geodesic Domes, A Spectrum of
Mathematics, 1972, pp 98-107.
WENNINGER M J, Spherical Models, Cambridge University Press, Cambridge,
1979, pp120-124.
SCHWARZ H A, Zur Theorie der hypergeometrischen Reihe, J. reine angew. Math,vol 75, 1873, pp 292-335.
CLINTON J, A Method of Tessellating a Surface, Patent Application WO 02/03121
A2, 10 January 2002.
FOWLER P W, MANOLOPOULOS D E, An Atlas of Fullerenes, Clarendon Press,
Oxford, 1995, pp 23-27.
Fischbeck, Dick, RanDomes, SNEC, Tivoli, NY, 25–26 January 2003. see
http://snec.cjfernley.com

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