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. Isotopic polytopes are fair dice, but there are fair dice that are not isotopic. For example, the dual of the elongated square gyrobicupola is a fair die. There are also fair dice where the facets are not congruent: 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.

Little is known about non-isotopic fair dice. Even the definition of fairness has not been formally worked out.

There are also curved convex shapes which might be informally called dice. If two congruent cones are joined at their circular faces, the resulting shape will roll on a flat surface but has two different ways it can do so with equal probability. Bowers suggested from symmetry considerations that the five convex regular hard polytwisters are fair dice.