Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site fisher.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!allegra!princeton!astrovax!fisher!wpt
From: wpt@fisher.UUCP (Bill Thurston)
Newsgroups: net.math
Subject: Re: odd-sided polyhedral dice
Message-ID: <331@fisher.UUCP>
Date: Thu, 27-Sep-84 23:03:40 EDT
Article-I.D.: fisher.331
Posted: Thu Sep 27 23:03:40 1984
Date-Received: Fri, 28-Sep-84 07:30:24 EDT
References: <892@rochester.UUCP>
Organization: Princeton Univ. Statistics
Lines: 37


	In the discussion of fair die, people have been overlooking the fact
that the favored sides will depend on conditions such as the surface that it
lands on and how it is thrown.
	For example, think about the difference between a surface which is very 
resilient (so the die bounces like a superball) and a surface which absorbs
energy rapidly. For the absorbent surface, the likelihood of ending up on a
particular face will depend much more on shape of the particular potential well,
 while for the resilient surface, the likelihood of ending up on a particular
face will depend much more one the values of potential energy.  A die which
should illustrate this principle is something shaped like a cylinder with a
polygonal cross-section, perhaps ten time as long as it is wide.  How many sides
does the polygon need to make it fair?  I maintain it will never be fair for
both kinds of surface.
	This variable cannot be just hypothesized away, since on an ideal
surface which does not absorb energy, a die would never rest.
	Another consideration is how the die is thrown.  Some shapes have axes
about which rotation is quite stable. (How can a frisbee be modified to make it
a fair two-sided die?)  But even assuming that the method of throwing does not
permit taking advantage of the frisbee phenomenon, at least for most assymetric
shapes there are bound to be differences in the results depending on the amount
of spin, and the energy and angle with which it first hits the table.
	I believe that the only shapes which can meet the criterion
of having an even distribution of results under all reasonable conditions
are probably the shapes whose symmetry group acts transitively on the faces.
This would limit the solutions to shapes like those which have been
discussed.  There are infinite series with dihedral symmetry of order
2n, for example two pyramids stuck together, and there are others which
can be derived from the Platonic solids by slicing away or building up in some
systematic pattern.  Of these, the one with the most faces is the 120-sided die
having as symmetry group the icosahedral group.  This can be formed from the
dodecahedron by dividing each face into ten triangles, then pushing out
say until it is inscribed in a sphere.

		Bill Thurston, Mathematics Department, Princeton U
		Princeton, NJ 08540
		allegra!fisher!wpt