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arXiv · 1212.6588

The Regular Grunbaum Polyhedron of Genus 5

Gabor Gevay·Egon Schulte·Jorg M. Wills
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Abstract

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 Grunbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Grunbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grunbaum 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.

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Gabor Gevay, Egon Schulte, Jorg M. Wills. 2012-12-29. The Regular Grunbaum Polyhedron of Genus 5. https://arxiv.org/abs/1212.6588

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