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Platonic hypermaps. (English) Zbl 0968.05022

Authors’ abstract (extended): A hypermap on any surface is called regular if its full automorphism group, including both orientation-preserving and orientation-reversing automorphisms, acts transitively on the set of blades (hyperedge-vertex-face incidence triples). A hypermap on an orientable surface is called rotary if its orientation-preserving automorphism group acts transitively on the set of darts (hyperedge-vertex incidence pairs). The associates of a hypermap are obtained by applying hyperedge-vertex-face duality as discussed in A. Machì [On the complexity of a hypermap, Discrete Math. 42, 221-226 (1982; Zbl 0503.05024)].
We classify the regular hypermaps (orientable or non-orientable) whose full automorphism group is isomorphic to the symmetry group of a Platonic solid: \(S_4\) for the tetrahedron, \(S_4\times C_2\) for the cube and octahedron and \(A_4\times C_2\) for the icosahedron and dodecahedron. There are 185 of them, of which 93 are maps. We also classify the regular hypermaps with automorphism group \(A_5\): there are 19 of these, all non-orientable, and 9 of them are maps. These hypermaps are constructed as combinatorial and topological objects, many of them arising as coverings of Platonic solids and Kepler-Poinsot polyhedra (viewed as hypermaps) [H. S. M. Coxeter, Regular polytopes (1st ed. 1948 (Zbl 0031.06502), 2nd ed. 1963 (Zbl 0118.35902), and 3rd ed. 1973)], or their associates. We conclude by showing that any rotary Platonic hypermap is regular, so there are no chiral Platonic hypermaps.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C65 Hypergraphs
52B10 Three-dimensional polytopes
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