rich burge wrote: > US patent #6926275 describes a fair five sided dice. There is some > empirical evidence given to support the fairness claim. I suppose the > dice as described are essentially fair. Why did the inventor not give > a proof of fairness?
I suppose that if he knew enough math to give a proof, he would have noticed that you can just roll an icosahedron and take the result mod 5.
It's actually quite hard to formulate what it means for a die to be fair. Let's try this: suppose the die is held above a table, just touching it, in some uniformly random orientation, then allowed to fall with no bouncing. Then it should fall with equal probability on any of the faces. For a simple five-faced polyhedron (that's stable on all five faces), this appears to be equivalent to requiring that the faces cover equal solid angles with respect to the center of gravity. For a center of gravity in the obvious place, I get that the six triangle edges should be longer than the other three edges by a ratio of sqrt[(15 + 9 sqrt(5)) / 10] ~ 1.87. That seems a reasonable result, but it's far from clear that my physical model was realistic, and the diagram in the patent doesn't appear to have those ratios at all, even taking the extra (unstable?) faces into account.