I've been researching polyhedral dice, and everywhere I look it states that face-transitive (and therefore fair) dice include polyhedra of various families, like the platonic solids, catalan solids, and trapezohedrons.
Is there a complete classification of face-transitive polyhedra? In other words, is there some proof that a face-transitive polyhedron must belong to one of a number of (possibly infinite) families?
Technically, the trapezohedrons are Catalan solids, in that their duals are 'Archimedean' in the sense of being fully vertex-transitive. (Some classifications of the Archimedean (or equivalently, Catalan) solids exclude the prisms and antiprisms so that they can have a finite list of solids, but that's never made much sense to me.) In any case, the two infinite families (prisms and antiprisms) and the well-known finite list of 'exceptional' solids are all that there are; there are no other infinite families. I believe the proof of this shows up in The Symmetries of Things as a consequence of the classification of spherical symmetry groups.