Hecke Groups, Dessins d'Enfants, and the Archimedean Solids

Frontiers in Physics

Read, James¹  He, Yang-Hui² 

[1] Department of Mathematics, City University London, London, UK;Merton College, University of Oxford, Oxford, UK

关键词: Hecke groups;    Dessins d';    enfants;    Archimedean solids;    Platonic solids;    Belyi maps;   

DOI  :  10.3389/fphy.2015.00091

学科分类：物理（综合）

来源: Frontiers

【 摘 要 】

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.

【 授权许可】

CC BY