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Two Notes on Maps and Surface Symmetry

  • Thomas W. TuckerEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

The first note of this paper determines for which g the orientable surface of genus g can be embedded in euclidean 3-space so as to have prismatic, cubical/octahedral, tetrahedral, or icosahedral/dodecahedral symmetry. The second note proves, through entirely elementary methods, that the clique number of the graph underlying a regular map is m = 2, 3, 4, 6; for m = 6 the map must be non-orientable and for m = 4, 6 the graph has a K m factorization. Here a regular map is one having maximal symmetry: reflections in all edges and full rotational symmetry about every vertex, edge and face.

Keywords

Riemann-Hurwitz equation Regular map Clique 

Subject Classifications

05C10 57M15 57M60 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsColgate UniversityHamiltonUSA

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