It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with 75 uniform polyhedra that do not belong to these classes.
For regular convex polyhedra (i.e., Platonic solids), quasi- and semiregular convex polyhedra (i.e., Archimedean solids) and regular nonconvex polyhedra (i.e., Kepler-Poinsot polyhedra), there are proofs that no more than 5 resp. 13 resp 4 can exist.
For the remaining 53 uniform star polyhedra I only know of a proof that relies on a computer search. Is there any proof that there are no more than 53 uniform star polyhedra that is done without the help of computer (e.g. in an anlaogous way as the proof for Archimedean solids?)