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 S 2 ). Let v o R ( P ), e o R ( P ), f o 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 E 3 preserving P . We define the rotational orbit Euler characteristic of P as the number E o R ( P ) = v o R ( P ) − e o R ( P ) + f o R ( P ) . Using the Burnside lemma we obtain the lower and the upper bound for E o R ( P ) in terms of the genus of the surface S ( P ) . We prove that E o R ∈ {2, 1, 0, − 1} for any convex polyhedron P . In the non-convex case E o R may be arbitrarily large or small.