Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Abstract

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E 3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 g 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 g 6, admitting realizations as convex-faced polyhedra in E 3 . For all admissible maps in this range, save Gordan's map of genus 4, and its dual, we rule out realizability by a polyhedron in E 3 .

On polyhedral realizations of Hurwitz's regular map {3, 7}_18 of genus 7 with geometric symmetries

“…With their other end, bounded by octagons, three of the tubes (mutually penetrating each other) are connected together via two equilateral triangles, while leaving free a 18-gon which will serve as the boundary of a "hole" of this complex. This way of connection is shown in Figure 19(a): the octagons are (6,14,28,34,40,32,19,13), (17,53,34,42,55,19,38,30), (23,36,42,28,44,38,32,46), the connecting triangles are shaded, and the "free" 18-gon is The other three antiprismatic tubes are connected together in an analogous way (hence providing a second free 18-gonal hole). Finally, the two 18-gonal holes are connected by a 7th tube, which is topologically a triangulated annulus (it is composed of 36 triangles).…”

“…E-mail addresses: juergen@bokowski.de (Jürgen Bokowski ), gevay@math.u-szeged.hu (Gábor Gévay ) In analogy with the regular maps, we say that a polyhedron P is combinatorially regular if its combinatorial automorphism group is transitive on the flags of P . Thus, a combinatorially regular polyhedron is a polyhedral realization in E 3 of a regular map on an orientable surface of some genus g. A summary of regular maps whose geometric realizations are known is given in Table 1 (as an extension of Table 1 in [34]).…”

Hurwitz's regular map 37 of genus 7: A polyhedral realization

“…However, there are also results in which polyhedral realizations of regular maps have been studied, see e.g. the corresponding articles of Jörg M. Wills and of his co-authors or other colleagues in [2,3,4,5,6,7,8,16,17], and [18]. This article is devoted to such a question that was studied by Jörg M. Wills for some time.…”

Icosahedral skeletal polyhedra realizing Petrie relatives of Gordan’s regular map

“…These give four of just five regular polyhedra of index 2 which are orientable and have planar faces (see [18] for the enumeration and Richter [15] [12]); for figures see [3,Fig. 6.4c], [5, De 2 f 2 on Plate XI], and [16].…”