Hecke Groups, Dessins d'Enfants and the Archimedean Solids

Understand

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.