Fair die

A fair die is a convex polytope which, if made of a homogeneous material and randomly thrown onto a flat surface, has an equal probability of landing on each facet. Convex isotopic polytopes are fair dice, but depending on definition there may be fair dice that are not isotopic. Diaconis and Keller argued that a very flat regular-$n$-gonal prism has a low chance of landing on its rectangular faces and a very tall regular-$n$-gonal prism has a high chance, so there must be a prism between these extremes where the landing probability of a rectangular face is equal to that of the $n$-gon. However, the optimal edge ratio depends on the exact definition of fairness, which is not agreed upon. In sophisticated models, physical parameters such as the coefficient of friction between the surface and the die must be taken into account.

There are also curved convex shapes which might be informally called dice. A bicone has two different ways it can roll on a flat surface, and it can land on either one with equal probability. Bowers suggested from symmetry considerations that the convex regular hard polytwisters are fair dice.