  A Platonic surface is a Riemann surface that underlies a regular map and so
we can consider its vertices, edge-centres and face-centres. A symmetry
(anticonformal involution) of the surface will fix a number of simple closed
curves which we call mirrors. These mirrors must pass through the vertices,
edge-centres and face-centres in some sequence which we call the pattern of the
mirror. Here we investigate these patterns for various well-known families of
Platonic surfaces, including genus 1 regular maps, and regular maps on Hurwitz
surfaces and Fermat curves. The genesis of this paper is classical. Klein in
Section 13 of his famous 1878 paper [11], worked out the pattern of the mirrors
on the Klein quartic and Coxeter in his book on regular polytopes worked out
the patterns for mirrors on the regular solids. We believe that this topic has
not been pursued since then.

Combinatorics