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From: ellis@flairvax.UUCP (Michael Ellis)
Newsgroups: net.math
Subject: Re: Semi-regular polyhedra
Message-ID: <633@flairvax.UUCP>
Date: Mon, 9-Jul-84 09:37:31 EDT
Article-I.D.: flairvax.633
Posted: Mon Jul  9 09:37:31 1984
Date-Received: Tue, 10-Jul-84 08:02:11 EDT
Organization: Fairchild AI Lab, Palo Alto, CA
Lines: 78

The Semi-Regular polyhedra, for those who are unfamiliar with the term,
have faces that are all regular polygons, and identical vertices. From
this definition, one may infer that all edges are likewise identical,
and that the vertices all lie on a sphere.

I believe that all possible 3-D semiregulars are listed in the table below,
divided into three categories, depending on whether the symmetry groups
(not counting reflections) are A4, S4 or A5:
					
SYMBOL	NAME				F	E	V

3-3-3	tetrahedron			4	6	4
3-6-6	truncated tetrahedron		8	18	12

4-4-4	cube (hexahedron)		6	12	8
3-8-8	truncated cube			14	36	24
3-3-3-3	octahedron			8	12	6
4-6-6	truncated octahedron		14	36	24
3-4-3-4	cuboctahedron			14	24	12
3-4-4-4	??				26	48	24
4-6-8	??				26	72	48
3-3-3-3-4 snub cube			38	60	24

5-5-5	dodecahedron			12	30	20
3-10-10	truncated dodecahedron		32	90	60
3-3-3-3-3 icosahedron			20	30	12
5-6-6	truncated icosahedron		32	90	60
3-5-3-5	icosidodecahedron		32	60	30
3-4-5-4	??				62	120	60
4-6-10	??				62	180	120	
3-3-3-3-5 snub dodecahedron		92	150	60

Besides these 18 polyhedra there are also two infinite classes, with
symmetry groups Dn (ignoring reflections):

4-4-n	prisms				2+n	3n	2n
3-3-3-n	antiprisms			2+2n	4n	2n

I believe these solids were all investigated by the ancient greek geometers.

Notes:

1. The symbol `3-4-5-4' means a polyhedron with a regular triangle, square,
   regular pentagon, and square at every vertex. This is NOT the Schlaefli
   symbolism, by which, for instance, a cube is represented {3,4}.
2. The terms An and Sn mean the Alternating and Symmetric groups of n
   elements, respectively. Dn is the Dihedral group of the regular n-gon.
3. The snub polyhedra 3-3-3-3-4 and 3-3-3-3-5 have left and right handed
   variants.

========================================================================

One can generate all the `interesting' polyhedra from the tetrahedron
with six `operators', crudely described below:

V - swap faces and vertices (duality)
T - each edge generates a pair vertices (truncation)
A - each edge generates one new vertex
B - each n-gonal face generates n new vertices; each original edge 
    resulting in a square face.
C - each edge generates four new vertices
D - each n-gonal face generates n new vertices, each original edge
    resulting in a pair of triangular faces.

		V	T	T*V	A=A*V	B=B*V	C=C*V	D=D*V
3-3-3   	3-3-3	3-6-6	3-6-6	3-3-3-3	3-4-3-4	4-6-6	3-3-3-3-3
3-3-3-3		4-4-4	4-6-6	3-8-8	3-4-3-4	3-4-4-4 4-6-8	3-3-3-3-4
3-3-3-3-3	5-5-5	5-6-6	3-10-10	3-5-3-5	3-4-5-4	4-6-10	3-3-3-3-5

These also operate on the infinite plane tessellations, which can be regarded
as `degenerate' polyhedra:

4-4-4-4		4-4-4-4	4-8-8	4-8-8	4-4-4-4	4-4-4-4	4-8-8	4-3-4-3-3
3-3-3-3-3-3	6-6-6	6-6-6	3-12-12	3-6-3-6	3-4-6-4	4-6-12	3-3-3-3-6

There is at least one tessellation I've omitted, namely the ugly 4-4-3-3-3.

-michael