The Regular Grunbaum Polyhedron of Genus 5

TL;DR

This paper analyzes a polyhedral embedding of a genus 5 regular map, confirming its combinatorial regularity, and introduces a new genus 11 vertex-transitive polyhedron, contributing to the understanding of high-genus regular polyhedra.

Contribution

It establishes the isomorphism and regularity of the Grunbaum polyhedron, and introduces a new high-genus vertex-transitive polyhedron, expanding the known examples.

Findings

The Grunbaum polyhedron is isomorphic to the Fricke-Klein map.

It is among the few known geometrically vertex-transitive polyhedra of genus > 2.

Only finitely many vertex-transitive polyhedra exist for genus > 2.

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.