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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.