Encyclopedia Polyhedra

Robert W. Gray

Copyright 2004

Introduction i

Notations and Relations ii

Tetrahedron (Platonic) 1

Truncated Tetrahedron (Archimedean)

1/4-Tetrahedron

A Quantum Module

B Quantum Module

C Quantum Module

D Quantum Module

The Tetrahelix

Octahedron (Platonic)

Truncated Octahedron (Archimedean)

1/8-Octahedron

Iceberg

B Quantum Module

Mite

Sytes

Syte: Bite

Syte: Rite

Syte: Lite

Kites

Kite: Kate

Kite: Kat

Octet

Coupler

Cube (a.k.a. Hexahedron) (Platonic Polyhedra)

Truncated Cube (Archimedean)

Cubeoctahedron (Archimedean)

(a.k.a. Vector Equilibrium)

Truncated Cubeoctahedron

The Jitterbug

Rhombic Dodecahedron

Icosahedron (Platonic Polyhedra)

Truncated Icosahedron (Archimedean)

Icosidodecahedron (Archimedean)

(a.k.a. 30-Verti)

S Quantum Module

Dodecahedron (Platonic Polyhedra)

Truncated Dodecahedron (Archimedean)

Rhombic Triacontahedron

E Quantum Module

T Quantum Module

Tetrakaidecahedron (Lord Kelvin’s Solid)

Rhombicuboctahedron

Snub Cube

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron

120 Polyhedron (Type I: 5 Octahedra)

120 Polyhedron (Type II: Jitterbugs)

120 Polyhedron (Type III: Dennis)

120 Polyhedron (Type IV: Sphere)

This book tabulates the numerical data
for the polyhedra mentioned and used in R. Buckminster Fuller’s books __Synergetics__
and __Synergetics 2__ as well as other polyhedra which I have found
interesting. I have calculated and
tabulated this data because I have been unable to find all of this data in any
other reference. Other references seem
to have bits and pieces of data for some of these polyhedra, often without
exact expressions. I have often needed,
for example, a particular angle for a particular part of a polyhedron and I
have found it a big distraction to have to stop my current train of thought and
calculate the needed angle. It is very
convenient to have a table of all these polyhedrons’ numerical data.

In addition to the polyhedra data, I have collected some useful information about how the polyhedra fit within one another or otherwise are related one to another. This information is presented at the end of each section.

I have found that the so called (by Lynnclaire Dennis) "120 Polyhedron" can define, using its 62 vertices, 10 Tetrahedra, 5 Octahedra, 5 Cubes, 5 rhombic Dodecahedra, 1 Icosahedron, 1 regular Dodecahedron, 1 rhombic Triacontahedron, as well as several “Jitterbugs”. For this reason, the coordinates for many of the polyhedra vertices are given in terms of the 62 vertex coordinates of the 120 Polyhedron. The Cartesian (x, y, z) coordinates for these vertices is tabulated in the chapter on the 120 Polyhedron (Type III Dennis). The coordinates are expressed in terms of the Golden Ratio.

This is a tabulation of most of the abbreviations used in the book. Other abbreviations are explain in the context where they are used.

The symbol “@” is to be read as “is approximately equal to.”

The symbol “º” is to be read as “is defined to be.”

EL º Edge Length.

FA º Face Altitude.

DFV º Distance from the center of a face to a vertex.

DFE º Distance from the center of a face to a mid-edge point.

DVV º Distance from the center of volume to a vertex.

DVE º Distance from the center of volume to a mid-edge point.

DVF º Distance from the center of volume to a face center point.

See the data for the Tetrahedron for an illustrative use of these abbreviations

The notation DF(V1.V2.V3)E(V2.V3) indicates the distance from the center of the face formed by the vertices V1, V2, and V3, to the mid-edge point along the edge formed by V2 and V3.

j º Golden Mean, Golden Ratio. "j" is the Greek letter phi.

j = @ 1.618 033 989

j ^{2} = @ 2.618 033 989 = j + 1

j ^{3} = @ 4.236 067 977 = 2 j + 1

2j ^{2} = @ 5.236 067 977 = 2 j + 2
= j ^{3} + 1

Note that for *n* an integer, j ^{n+1} = j ^{n} + j ^{n-1} .

The
Fibonacci numbers *f *(*n*) may be defined by

with *n* an integer and *n* > 0 and with *f*(0)
= 1, *f*(1) = 1.