  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.
