Platonic tessellations of Riemann surfaces as models in chemistry: non-zero genus analogues of regular polyhedra

摘要

The concept of regular polyhedra can be generalized to regular polygonal networks on surfaces of non-zero genus, i.e. platonic tessellations of Riemann surfaces. Examples of such figures include the genus 3 Klein {7,3}(8) tessellation of 24 heptagons and the Dyck (8,3)(6) tessellation of 12 octagons used to describe possible allotropes of carbon and boron nitride exhibiting low density negative curvature zeolite-like structures. The dual of a related genus 3 double cube {4,2.3} tessellation consisting of four hexagons meeting at each vertex is used by Sadoc and Charvolin in their topological theory of bilayers in liquid crystal and micellar structures. Similar platonic tessellations of non-zero genus provide geometric models for the double point groups used in molecular physics such as the genus 2 {3,4 + 4} tessellation for the double octahedral group and the genus 5 {3,2.5} tessellation for the double icosahedral group. (C) 2003 Elsevier B.V. All rights reserved. [References: 27]