Why no higher-genus regular polyhedra?

Question

It seems to be a fact that there are only five bounded connected non-selfintersecting polyhedra with identical regular-polygon faces and congruent vertices (i.e., you can pick a neighborhood of every vertex so that all the neighborhoods are congruent), namely the platonic solids. (Drop any one word in the above and you get more: unbounded allows an infinite linear chain of octahedra glued face to face; disconnected allows the union of two disjoint Platonic solids; self-intersecting allows two of the Kepler-Poinsot solids; non-identical allows Archimedean solids; irregular faces allows e.g. the noble disphenoid, and of course dropping congruent vertices allows a myriad of possibilities.) Note I have not included convex among the hypotheses.However, I am hard-pressed to put together a proof of this. If you add the hypothesis that the polyhedron is genus 0, then Euler's formula shows you there must be an overall angle defect, so the same angle defect at each vertex since the vertices are congruent, and now you can do the usual three times the vertex angle must be less than a full circle, etc. (Even in the case of genus 1, Euler's formula just tells you that the angle sum must be 0 at each vertex, and then there are infinite solutions for both six triangles and four squares at each vertex -- basically infinitely long prisms, which are projectively speaking tori -- so you will really need to use the boundedness...)Nevertheless, the genus-zero hypothesis isn't necessary; there in fact are no higher-genus bounded non-selfintersecting polyhedra with regular-polygon faces and congruent vertices. Can anyone outline or point me to a proof of this fact?

Answer 1

It is not strict proof.

In the plane, the interior angle of a regular polygon can rise almost up to 180 degrees. In polyhedron, to have congruent faces, we have an upper bound of the number of sides for the face. For regular polygons, that upper bound is 120 degrees. If we connect three hexagons, we cover the plane. In 3D, we can have no more than 3 pentagons, 3 squares, 3, 4 or 5 triangles that will fold to make solid. Hence, the 5 platonic solids.