Abstract
Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 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 E3. 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 E3.
Highlights
A polyhedron P, for the purpose of this paper, is a closed compact surface in Euclidean 3-spaceE made up from finitely many convex polygons, called the faces of P, such that any two polygons intersect, if at all, in a common vertex or a common edge
Since P is embedded in E3, the underlying surface is free of self-intersections and is necessarily orientable; and since the faces of P are convex, the underlying abstract polyhedron is necessarily a lattice, meaning here that any two distinct faces meet, if at all, in a common vertex or a common edge
A polyhedron P is said to be combinatorially regular if its combinatorial automorphism group Γ (P ) is transitive on the flags of P
Summary
A polyhedron P , for the purpose of this paper, is a closed compact surface in Euclidean 3-space. A combinatorially regular polyhedron is a polyhedral realization in E3 of a regular map on an orientable surface of some genus g (see [5]). Each such polyhedron or map has a (Schläfli) type {p, q} for some p, q > 3, describing the fact that the faces are p-gons, q meeting at each vertex. There are only finitely many regular maps in this genus range, and we wish to determine those which admit realizations as (convex-faced, combinatorially regular) polyhedra. For a discussion of regular toroidal polyhedra see Schwörbel [21]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
