Classification of convex polyhedra by their rotational orbit Euler characteristic

Jurij Kovič

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 voR(P), eoR(P),  foR(P) be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group G = GR(P) of all the rotations of the Euclidean space E3 preserving P. We define the rotational orbit Euler characteristic of P as the number EoR(P) = voR(P) − eoR(P) + foR(P).

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


Keywords


Polyhedron, rotational orbit, Euler characteristic

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ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications