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 Message-ID: <2994@alice.UUCP> Date: Thu, 20-Sep-84 20:05:05 EDT Article-I.D.: alice.2994 Posted: Thu Sep 20 20:05:05 1984 Date-Received: Tue, 25-Sep-84 20:22:27 EDT References: <328@elecvax.OZ> Organization: AT&T Bell Laboratories, Murray Hill Lines: 12 In fact, there is a fairly large (infinite!) set of these fair polyhedral dice. For example, the duals of the Archemedian (semi-regular) polyhedra are all fair dice. The dual d(P) of a polyhedron P has a vertex at the center of each face of P, a face centered on to each vertex of P, and edges `perpendicular' to the edges of P (i.e. there is an edge joining two vertices of d(P) if the corresponding two faces of P share an edge.) It is not hard to prove that if P is Archemedian, then there is a rotation mapping d(P) onto itself which maps any face onto any other face. I.e., you can't tell one face from another by looking at it. Write numbers on the faces and you have a fair die. I believe that the rhombic dodecahedron mentioned in elecvax.328 is one of these Archemedian duals, although offhand I can't think which.