Representing the Sporadic Archimedean Polyhedra as Abstract Polytopes

Details Representing the Sporadic Archimedean Polyhedra as Abstract Polytopes Representing the Sporadic Archimedean Polyhedra as Abstract Polytopes

2009-10-13

arxiv.org/abs/0910.2445v1

Mathematics

Combinatorics

17 pages, 2 figures, to appear in the Bled'07 special issue of Discrete Mathematics

Scientific paper

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterise which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).

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