Robert W. Gray
Notations and Relations ii
Tetrahedron (Platonic) 1
Truncated Tetrahedron (Archimedean)
A Quantum Module
B Quantum Module
C Quantum Module
D Quantum Module
Truncated Octahedron (Archimedean)
B Quantum Module
Cube (a.k.a. Hexahedron) (Platonic Polyhedra)
Truncated Cube (Archimedean)
(a.k.a. Vector Equilibrium)
Icosahedron (Platonic Polyhedra)
Truncated Icosahedron (Archimedean)
S Quantum Module
Dodecahedron (Platonic Polyhedra)
Truncated Dodecahedron (Archimedean)
E Quantum Module
T Quantum Module
Tetrakaidecahedron (Lord Kelvin’s Solid)
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.