Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 1-37 (2001)

Platonic Hypermaps

Antonio J. Breda d'Azevedo; Gareth A. Jones

Departamento de Matematica, Universidade de Aveiro, 3800 Aveiro, Portugal; Department of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom

Abstract: We classify the regular hypermaps (orientable or non-orientable) whose full automorphism group is isomorphic to the symmetry group of a Platonic solid. 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), or their associates. We conclude by showing that any rotary Platonic hypermap is regular, so there are no chiral Platonic hypermaps.

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