AUTHOR=He Yang-Hui , Read James 

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

JOURNAL=Frontiers in Physics

VOLUME=3

YEAR=2015

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2015.00091

DOI=10.3389/fphy.2015.00091

ISSN=2296-424X

ABSTRACT=<p>Grothendieck's <italic>dessins d'enfants</italic> arise with ever-increasing frequency in many areas of twenty-first century mathematical physics. In this paper, we review the connections between <italic>dessins</italic> and the theory of <italic>Hecke groups</italic>. Focussing on the restricted class of highly symmetric <italic>dessins</italic> corresponding to the so-called <italic>Archimedean solids</italic>, 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 <italic>dessins</italic> 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 <italic>dessins</italic> 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.</p>