The Regular Gr unbaum Polyhedron of Genus 5

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary 3-space. This polyhedron was originally discovered by Gr unbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Gr unbaum polyhedron with the Fricke-Klein map...

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Bibliographic Details
Published in:arXiv.org 2012-12
Main Authors: Gevay, Gabor, Schulte, Egon, Wills, Jorg M
Format: Article
Language:English
Subjects:
Combinatorial analysis
Isomorphism
Polyhedra
Online Access:Get full text
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Summary:We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary 3-space. This polyhedron was originally discovered by Gr unbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Gr unbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Gr unbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g > 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g > 2.
ISSN:2331-8422