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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

91.0. "Semi Regular Polyhedra" by HARE::STAN () Thu Jul 05 1984 09:23

Newsgroups: net.math
Path: decwrl!decvax!mcnc!philabs!cmcl2!seismo!rlgvax!cvl!umcp-cs!randy
Subject: Semi-regular polyhedra
Posted: Mon Jul  2 11:49:35 1984


The recent mail on geometric duality brings to mind a question I've
had for some time.  Maybe someone out there can help me.  I'm wondering
what the list of all semi-regular polyhedra is.  (Recall that a semi-
regular polyhedron has all faces regular but not necessarily congruent,
but every vertex is congruent.  Thus the cuboctahedron is composed of
triangles and squares with exactly two triangles and two squares meeting
at every vertex.)

I tried to make a list and found that there seem to be *lots* more than
I expected.  For example, there is an infinite set of "cylinders" built
as follows:  For base and top choose a regular K sided polygon.  Then
for the walls of the cylinder use K squares.  At every vertex, you'll
have one K-gon and 2 squares meeting.  (You can also get a similar infinite
set using triangles as wall constructors providing K > 3.  Then you need 2*K
triangles.)

Anyway, I've found all sorts of unexpected (and rather pleasing) polygons 
with the help of a lisp program to generate legal possibilities.  I can
send details if anyone is interested.  But this *must* have been done
somewhere before.  Anybody got pointers?  Thanks.

		- Randy
-- 
Randy Trigg
...!seismo!umcp-cs!randy (Usenet)
randy%umcp-cs@CSNet-Relay (Arpanet)

T.R Title User Personal
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91.1 HARE::STAN Tue Jul 10 1984 23:55 88

T.R	Title	User	Personal Name	Date	Lines
91.1		HARE::STAN		`Tue Jul 10 1984 23:55`	88
	From: ROLL::USENET "USENET Newsgroup Distributor" 10-JUL-1984 22:19 To: HARE::STAN Subj: USENET net.math newsgroup articles Newsgroups: net.math Path: decwrl!flairvax!ellis Subject: Re: Semi-regular polyhedra Posted: Mon Jul 9 06:37:31 1984 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 TV A=AV B=BV C=CV 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

From:	ROLL::USENET       "USENET Newsgroup Distributor" 10-JUL-1984 22:19
To:	HARE::STAN
Subj:	USENET net.math newsgroup articles

Newsgroups: net.math
Path: decwrl!flairvax!ellis
Subject: Re: Semi-regular polyhedra
Posted: Mon Jul  9 06:37:31 1984

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