I've been looking at polyhedra, incuding Platonic solids as well as Archimedean and Catalan solids. Catalan solids are face-transitive, which I believe implies that they are "fair dice", in the sense that each face is equally likely to land on top if the solid is "rolled" in some suitably non-biased fashion.
There is a solid called a pseudo-deltoidal icositetrahedron, and it is not a Catalan solid: https://en.wikipedia.org/wiki/Pseudo-deltoidal_icositetrahedron. It has the interesting property of being "monohedral" but not face-transitive. In other words, each face is the same polygon, but the symmetry group induces more than one orbit on the faces. In fact, there are two types of faces, symmetry-wise: polar faces (8) and equatorial faces (16). I'm wondering whether this shape is, nevertheless, a fair die.
There are two Catalan solids having 24 faces (the deltoidal icosidetrahedron and the tetrakis hexahedron), so it is possible to make a fair D-24 from either of those. I'm just curious whether this is a third option.
Note: I'm aware that this question comes up in the answers of a question on SO: https://mathoverflow.net/questions/46684/fair-but-irregular-polyhedral-dice. It is conjectured that this shape is a fair die, and someone agrees that they think it is, but nobody proves anything.
I'm very interested in any thoughts anyone can offer on this question. :)