The regular Grünbaum polyhedron of genus 5

Abstract

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

The Farey Maps Modulo N

“…See figure 5 in the appendix, noting that 3 5 = 5 3 , 4 7 = 4 1 and 7 5 = 1 3 . Note that these give the cyclic orderings of the stars around our two points 3 5 and 1 2 . This is because the star of 1 0 is 0 1 , 1 1 , 2 1 .…”

“…The underlying Riemann surface is the unique Riemann surface of genus 5 with 192 automorphisms and this is known as the Fricke-Klein surface of genus 5. This map is particularly important as it is the regular map underlying the Grunbaum polyhedron which is one of the few regular maps that are known to admit a polyhedral embedding into Euclidean 3-space with convex faces [3]. For the uniqueness of this map and Riemann surface we refer to [1].…”

Vertex-Transitive Polyhedra of Higher Genus, I

“…All combinatorial types of vertex-transitive polyhedra of genus zero are known; they comprise the Platonic solids, the Archimedean solids, and the infinite families of prisms and antiprisms. Remarkably, two two-parameter families exist for genus one (see [16] and [10] for descriptions). In this article, we focus our attention on g ≥ 2 as only a few examples of higher genus are known, and the completeness of the list has never been established.…”

“…However, note that for toroidal vertex-transitive polyhedra, we must necessarily have simple transitivity of the symmetry group as well. This follows since reflections are also not allowed for g = 1 (see [10]), the torus cannot have Platonic symmetry at all (Theorem 2.3), and since a polyhedron cannot live in a plane. Infinite two-parameter families for the remaining possible symmetry groups of dihedral type have already been constructed, first in [16] and, independently, by Brehm (private communication), and described in more detail in [10].…”

Regular Polygon Surfaces

“…While the five convex regular polyhedra known as Platonic solids were described in Euclid's Elements, there is no characterization of generalized polyhedra. New examples of polyhedra (in R 3 ) are still being discovered (see [GSW14] for examples of regular polyhedra and [GS09] for examples of toroidal polyhedra). Moreover, mathematicians have not reached a consensus on the definition of a generalized polyhedron.…”