Definitions
Topology/Geometry of Geodesics

Joseph D. Clinton

SNEC-04
28-29 June 2003

Magnus J. Wenninger

Introduction• Definitions• Topology– Goldberg’s polyhedra– Classes of Geodesic polyhedra– Triangular tessellations– Diamond tessellations– Hexagonal tessellations• Geometry– Methods

Definitions• Parent Polyhedra

Definitions• Planer form of the
Parent Polyhedra• Spherical form of the
Parent Polyhedra

Definitions• Polyhedron Face (F)– Any of the plane or
spherical polygons making up
the surface of the polyhedronFF

Definitions• Principle Triangle (PT)– One of the triangles of theparent polyhedron used indevelopment of the three-way
grid of the geodesic form.
– It may be planer (PPT) or it
may be spherical (PST)

PST

PPTPPT

Definitions• Schwarz Triangle (ST)– “A problem proposed andsolved by Schwarz in 1873: tofind all spherical triangles
which lead, by repeatedreflection in their sides, to aset of congruent triangles
covering the sphere a finite
number of times.”– There are only 44 kinds ofSchwarz triangles.

STDefinitions
Great Circle• Great Circle– Any plane passing
through the center of asphere its intersection
with the surface of thesphere is a Great Circle
and is the largest circle
on the sphere.
– All other circles on thesurface of the sphere arereferred to as Lesser
Circles.

Circle Centerand Sphere CenterDefinitionscfαΩ• Geodesic Geometrical
Elements– Face angle (αα)– Dihedral angle (ββ)– Central angle (δδ)– Axial angle (ΩΩ)
– Chord factor (cf)βδSphere Center(0,0,0)

Definitions• Face Angle (Alpha α)– An angle formed by two chords
meeting in a common point and lying
in a plane that is the face of thegeodesic polyhedron.
– For the spherical form of the face
it is the dihedral angle between two
great circle planes with a common
intersection and containing the twochords meeting in a common pointthat lies on the line of intersection of
the two great circle planes.
Sphere Center(0,0,0)αDefinitions• Dihedral Angle (Beta β)– An angle formed by two planes
meeting in a common line
– The two planes themselves are
faces of the dihedral angle, and theircommon chord is the common line.
– To measure the dihedral angle
measure the angle whose vertex is on
the chord of the dihedral angle andwhose sides are perpendicular to thechord and lie one in each face of thedihedral angle.

Sphere Center(0,0,0)

βDefinitions• Partial Dihedral Angle (Xi ξ )– An angle formed by two planes
meeting in a common line with one
plane being a face of the polyhedron
Polyhedron Faceand the other being the great circle
plane of the chord.
β– The angle is measured the same as
Great circle planethe dihedral angle β.
– For each dihedral angle β there areSphere Center
two partial dihedral angles ξ1 and ξ2.
(0,0,0)

Polyhedron Face

ξ1
ξ2

Definitions• Partial Dihedral Angle (Xi ξ )COS ξ1 = (TAN δ1 / 2) * TAN (90 - α1)• where:– δ1 = the central angle of the common
Polyhedron Faceedge of the dihedral angle beingmeasured.
– α1 = the face angle opposite thecommon dihedral being measured andβcontained in the face of the dihedral Great circle planeplane• The dihedral angle β = the sum of thetwo partial dihedral angles ξ1 and ξ2
that share a common edge.

Sphere Center(0,0,0)

Polyhedron Face

ξ1
ξ2

Definitions• Central Angle (Delta δ)– An angle formed by two radii of thepolyhedron passing through the end
points of a chord of the polyhedron.
– The vertex of the central angle ischosen as the point common to both,(the center of the polyhedron).
COS δ =

X1X2 + Y1Y2 + Z1Z2
r2

δ
Sphere Center(0,0,0)

Definitions• Axial Angle (Omega Ω)– An angle formed by a chord andthe radius from the center of thepolyhedron and meeting in acommon point.
– The vertex of the axial angle ischosen as the point common to thepolyhedron’s chord and radius.

Ω = (180°° - δ ) / 2Ω = ACOS (cf/2)
Sphere Center(0,0,0)ΩDefinitions• Chord Factor (cf)– The chord length of thepolyhedron.
– Based on a radius of a nondimensional unit of one for thecircumscribed sphere.
– With the end points of the chord
coincident with surface of thesphere.

cf = 2*SIN δ/2– The length of the chord for anystructure may be found by:l = cf * r where: r = radius
Sphere Center(0,0,0)cf

Definitions• Tangential Angle (Zeta ζ )– An angle formed by a chord of thepolyhedron and a normal to theradius of the polyhedron at thevertex of the chord where themeasure of the angle is taken.

NormalChord

ζ = 90°° - ΩSphere Center(0,0,0)

ζRadiusDefinitions• Frequency (Nu ν )– The number of subdivisions along
a path on a PPT to get from one
vertex to an adjacent vertex.
– b and c are counters along thepath.

ν=b+c

Definitions• Quotient of LE and SE (Eta h)– Quotient of the longest andshortest edge of the polyhedron– Polyhedron approaching equaledges
Where:h ≤ 2 sin 36ºDefinitions• Duality– The vertices of a polyhedroncorrespond to the faces of its dual,and vice versa.
– The edges of the two polyhedra
correspond directly to one another.

Nicholas Shea

Definitions• Euler’s theorem– The number of vertices (v), edges(e), and faces (f) in a sphericalpolyhedron are related by:

v+f=e+2

Definitions• Altitude (h)– A line constructed from a vertex ofa triangle ⊥ to the opposite edge.
– The three altitudes of a triangle
meet at one point (Orthocenter).

OrthocenterDefinitions• Median (m)– A line constructed from a vertex ofa triangle to the midpoint of theopposite side.
– The three medians of a triangle
meet at one point (Centroid) whose
distance from each vertex is twothirds the median from its vertex.
– The Centroid of the triangle is its
center of mass.

CentroidDefinitions• Angle Bisectors– The three bisectors of a triangle
meet in one point which isequidistant from the three edges.
– This point is known as the Incenter

IncenterDefinitions• Perpendicular Bisectors– The three ⊥ bisectors of the edges
of a triangle meet in one point whichis equidistant from the threevertices.
– This point is known as theCircumcenter.
CircumcenterDefinitions• Uniformity• Even Distribution– Two terms that have manymeanings

Definitions• Isomer (Chemistry and Biological term)– One of two or more substances which havethe same elementary composition, but differ instructure, and hence in properties

Definitions• Voronoi Diagram– The partitioning of a plane with npoints into n convex polygons such thateach polygon contains exactly one point.
– Every point in a given polygon iscloser to its center point than any other• Delaunay triangulation– The nerve of the cells in a Voronoi
Diagram.
– The triangular of the Convex Hull of
the points in the diagram.
Nicholass Shea

Topology• Topology– The mathematical study of propertiesof spatial objects where shape ispreserved through deformations andsize is not a consideration.
• Goldberg’s polyhedra• Geodesic Classes• Triangular Tessellation• Diamond Tessellation• Hexagonal Tessellation

Topology• Goldberg’s polyhedra– “The arrangement of the hexagons in a triangular patch is the same as ina 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, thesquare of the distance from the center of the patch to the vertex is equal to
a2 + ab + b2 …the total number of faces bounding the ‘polyhedron’ is10(a2 + ab +b2)+2;”for the icosahedral system,
4(a2 + ab +b2)+2 for the octahedral system, and2(a2 + ab +b2)+2 for the tetrahedral system.
– The a,b counters are directedalong the hex/tri paths from theparent vertex to its center.

a=1, b=2

Topology• {p,q+} b,c - Geodesic Classes (Clinton, Coxeter, Wenninger)– Class I Subscript b is an integer and c is always zero.
– Class II Subscript b is any integer and is always equal to c.
– Class III Subscript b and c can be any integers so long as b ≠ c– The b,c counters are directed along the tri paths from the parent triangle
vertex to its adjacent vertex

Class IIIClass IClass IIClass IIIClass I

Topology• Triangular Tessellation– Icosahedron• F = 20T
• E = 30 T
• V = 10T + 2• Where: T = b2 +bc +c2

Class IIIClass IClass IIClass III
Class I

Topology• Diamond Tessellation– Icosahedron• F = 10T
• E = 20 T
• V = 30T + 2• Where: T = b2 +bc +c2• T must be divisible by 3

Class IIIClass IClass IIClass III
Class I

Topology• Hexagonal Tessellation– Icosahedron• F = 10T/3 + 2• E = 10T
• V = 20T/3• Where: T = b2 +bc +c2• b and c must be divisible by 3

Class IIIClass IClass IIClass IIIClass IGeometry• Geometry– The mathematical study of figures inSpace of a given number of dimensions
and of a given type.
• Geodesic methods of tessellating thesurface of a sphere