Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Symmetry

Egon Schulte¹ 

[1] ¹Department of Mathematics, Northeastern University, Boston, MA 02115, USA ²Department Mathematik, University of Siegen, Emmy-Noether-Campus, D-57068 Siegen, Germany ^†Supported by NSF-Grant DMS–0856675.

关键词: Platonic solids;    regular polyhedra;    regular maps;    Riemann surfaces;    polyhedral embeddings;    automorphism groups;   

DOI  :  10.3390/sym4010001

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.