This is really really far from my specialty (you want a low-dimensional topologist here), but I believe the situation is that you want a die where all the faces are the same polygon and where it is “transitive” in the sense that you can get from any face to any other by rotations or reflections. This puts you in the category of Catalan solids, which are the duals of Archimedean solids (dual means vertices become faces and vice versa). The reason there is a limited number of Archimedean solids is that in any such you can write down some relational conditions between various integers that arise in a natural way in the figure, eg number of faces meeting at a vertex, and then show that there are only finitely many combinations of integers that meet these conditions. I know this is horribly vague, but it is the best I can do without either doing some research or tracking down my recently-retired colleague who really knows this stuff.
4 Likes
