FtG@rochester.UUCP (09/25/84)
From: FtG alice!td asks if there exists fair odd-sided polyhedral dice for all odd m=2n+1>=5. The following is an existence proof. Consider a 2n sided pyramid with all sides identical and consider varying the size of the base. If the base is extremely large, then that side will be favored. If the base is extremely tiny, then the sides will be favored. Clearly this is a continous deformation with a resulting continous function of the probability of landing on the base vs one of the sides. Ergo there exists a point where the probabilities are equal, QED. Note this is not constructive, but "I do and I do and I do for you kids and this is the thanks I get?" FtGone
wpt@fisher.UUCP (Bill Thurston) (09/28/84)
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