Joe,

Here is convex polytope that could be a fair die. Consider the rhombicuboctahedron which is one of the Archimedean solids. There are three rings of square faces, each of which divides the polyhedron in half. Take one of these rings and rotate the top half by 45 degrees. This is a famous "fake" Archimedean solid. The symmetry group of this polytope is not transitive on all its vertices. There are two orbits of vertices of 8 and 16 vertices each. Take the dual of this polytope. Each face is a deltoid/kite. All 24 faces are congruent and at the same distance from the centroid, but it is not isohedral. I think that all the second moments are equal as well. Is this a fair die?

Bob C.

Joe,

Here is convex polytope that could be a fair die. Consider the rhombicuboctahedron which is one of the Archimedean solids. There are three rings of square faces, each of which divides the polyhedron in half. Take one of these rings and rotate the top half by 45 degrees. This is a famous "fake" Archimedean solid. The symmetry group of this polytope is not transitive on all its vertices. There are two orbits of vertices of 8 and 16 vertices each. Take the dual of this polytope. Each face is a deltoid/kite. All 24 faces are congruent and at the same distance from the centroid, but it is not isohedral. I think that all the second moments are equal as well. Is this a fair die?

Bob C.