Classification of convex polyhedra by their rotational orbit Euler characteristic

Abstract

Let P be a polyhedron whose boundary consists of flat polygonal faces on some compact surface S(P) (not necessarily homeomorphic to the sphere S²). Let vo_R(P), eo_R(P),  fo_R(P) be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group G = G_R(P) of all the rotations of the Euclidean space E³ preserving P. We define the rotational orbit Euler characteristic of P as the number Eo_R(P) = vo_R(P) − eo_R(P) + fo_R(P).

Using the Burnside lemma we obtain the lower and the upper bound for Eo_R(P) in terms of the genus of the surface S(P). We prove that Eo_R ∈ {2, 1, 0,  − 1} for any convex polyhedron P. In the non-convex case Eo_R may be arbitrarily large or small.