A 3-orbit polyhedron is one in which the automorphism group has exactly 3 orbits on the set of incident triples (vertex, edge, face). Examples of such polyhedra are the prisms over regular n-gons as well as 5 of the Archimedean Solids. There are three classes of 3-orbit polyhedra, depending of the local configuration of their flags. In this paper, for each of the three classes, we find necessary and sufficient conditions on a set of generators of a group Gamma so that there exists a polyhedron P(Gamma) such that Gamma acts by automorphisms with 3-flag orbits, in the given class. The techniques used to do this combine the ideas of coset geometries used to construct regular and chiral polytopes, together with voltage assignments used on the so-called symmetry type graph of the constructed polyhedra. These twists to the classical techniques shed light on a possible way to generalize our results to polytopes of any rank and any number of orbits on flags (in fact, to polytopes of any possible symmetry type graph). Finally, we use the results obtained in the paper to show that, for any given class, almost all symmetric groups are the automorphism group of a 3-orbit polyhedron in that class.
展开▼
