Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site alice.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!allegra!alice!td
From: td@alice.UUCP (Tom Duff)
Newsgroups: net.math
Subject: Re: Polyhedral dice (duals of Archemedians)
Message-ID: <3003@alice.UUCP>
Date: Sun, 23-Sep-84 11:55:36 EDT
Article-I.D.: alice.3003
Posted: Sun Sep 23 11:55:36 1984
Date-Received: Wed, 26-Sep-84 04:39:10 EDT
References: <3710@decwrl.UUCP>
Organization: AT&T Bell Laboratories, Murray Hill
Lines: 49

I've been receiving letters telling me that the dual of an Archemedian polyhedron
is another Archemedian polyhedron, and now I see the same blather in net.math.

In reply:  NO!

The dual of a *regular* (Platonic, NOT Archemedian) polyhedron is a regular
polyhedron.  The dual of an Archemedian (SEMI-regular -- don't you people *read*
things before running off at the mouth?) polyhedron generally doesn't have
regular faces, although all it's faces are all identical.

Archemdedian polyhedra have regular faces (not necessarily all with the same
number of edges) and the same configuration of faces at each vertex
(e.g. triangle-square-triangle-square.)
They are characterized by having a symmetry group under which
any vertex may be transformed into any other vertex.  These are the Archemedian
polyhedra:

	Vertex		Name
	Configuration
	3 3 3		Tetrahedron
	4 4 4		Cube
	3 3 3 3		Octahedron
	5 5 5		Dodecahedron
	3 3 3 3 3	Icosahedron
	3 6 6		Truncated Tetrahedron
	3 8 8		Truncated Cube
	3 10 10		Truncated dodecahedron
	3 4 3 4		Cuboctahedron
	3 5 3 5		Icosidodecahedron
	5 6 6		Truncated Icosahedron
	4 6 6		Truncated Octahedron
	4 6 8		Truncated Cuboctahedron
	4 6 10		Truncated icosidodecahedron
	3 4 4 4		Rhombicuboctahedron
	3 4 5 4		Rhombicosidodecahedron (My favorite name)
	4 3 3 3 3	Snub Cuboctahedron (My favorite shape)
	5 3 3 3 3	Snub Icosidodecahedron
	4 4 n		N-gonal Prism, n>=3 
	3 3 3 n		N-gonal Antiprism, n>=3 
43333 and 53333 exist in left- and right-handed versions.  Also, cube==square prism,
and octahedron==triangular antiprism.

As I stated in my previous message, the dual of any of these is a fair die, because
the dual's symmetry group transforms any faces into any other face.

Of course, there are other fair dice than these.  For example, take two n-gonal
pyramids and stick them back to back.  Lo and behold, a fair die with 2n faces
appears.  I don't know a general construction for fair dice with odd numbers of
faces, or even if they must exist for all 2n+1>=5.