Encyclopedia Polyhedra

 

 

 

Robert W. Gray

 

 

 

Copyright 2004

 

 

 

 

 

 

 

 

Table of Contents

 

 

 

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)

 

 

 

 

 

 

Introduction

 

 

         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.   

 

 

 

 

 

Notations and Relations

 

 

      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.

 

 

      It is often mentioned that the Golden Ratio is related to the Fibonacci numbers by the equation

 

      However, this is not unique.

 

      Any number sequence which can be defined by the relation

 

 

where f (n) is an integer will also have the property that

 

 

 

 

      I have often found the following table of trigonometric relations to be useful.

 

sin(0°)	=	0	=	cos(90°)
sin(15°)	=		=	cos(75°)
sin(18°)	=		=	cos(72°)
sin(30°)	=		=	cos(60°)
sin(36°)	=		=	cos(54°)
sin(45°)	=		=	cos(45°)
sin(54°)	=		=	cos(36°)
sin(60°)	=		=	cos(30°)
sin(72°)	=		=	cos(18°)
sin(75°)	=		=	cos(15°)
sin(90°)	=	1	=	cos(0°)

 

 

      A system (polyhedron) is defined by its angles.  For example, a Tetrahedron is a Tetrahedron regardless of its edge lengths.  As you scale a system, its lengths, areas and volume change but its angles all remain the same.  A system is therefore characterized by its angles.  We should therefore pay particular attention to the polyhedra's angles.

 

      In the Quantum Mechanics theory of angular momentum, the following space quantization equation is used

 

 

where j is the angular momentum quantum number and m_j is the quantum number  giving the projection of the angular momentum onto a preferred axis.  The quantum numbers j are positive integers or half-integers.  The m_j quantum numbers are restricted to

 

m_j = -j, -j+1, ..., j-1, j

 

      This space quantization angle is the angle between the angular momentum vector and the axis defined by the preferred space direction (as can be defined, for example and in one case, by the direction of an externally applied magnetic field.)

 

      For example, if j = 1/2 then m_j = -1/2, 1/2.  For the m_j = 1/2 case the space quantization angle q is

 

 

This is the half-cone angle for a cone defined by a spinning Tetrahedron.  The half-cone angle is defined to be the angle between the surface of the cone and the symmetry (spin) axis of the cone. 

 

      For all convex polyhedra

Number of vertices ´ 360° - Sum of surface angles = 720°

Note that the sum of the surface angles of any Tetrahedron (regular or irregular) is 720°.  R. Buckminster Fuller interprets this as meaning that to form a closed structure from a plane, you must take out (in surface angle amount) the equivalent of one Tetrahedron.  Or, thinking of this in the reverse situation, to make a structure lay flat in a plane, you must add to the structure 720° in surface angle.

 

      For 3-dimensional structures (polyhedra) Euler's equation is

Number of Faces + Number of Vertices = Number of Edges + 2

F + V = E + 2