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.