The structure of mirrors on Platonic surfaces

TL;DR

This paper studies the patterns of mirrors fixed by symmetries on Platonic surfaces, extending classical results to various well-known families like genus 1 maps, Hurwitz surfaces, and Fermat curves.

Contribution

It systematically analyzes mirror patterns on Platonic surfaces, providing new insights into their symmetry structures beyond classical cases.

Findings

Mirror patterns are characterized for genus 1 regular maps.

Patterns are identified for regular maps on Hurwitz surfaces.

Mirror structures are described for Fermat curves.

Abstract

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.