Regular maps of high density

TL;DR

This paper classifies regular maps with reflection symmetry and high-density graphs, showing they mostly belong to a family related to Fermat curves, except for the tetrahedron case.

Contribution

It identifies and classifies all regular maps with reflection symmetry and density > 1/2, linking them to Fermat curves, except for the tetrahedron.

Findings

All such regular maps are associated with Fermat curves x^n + y^n + z^n = 0.

The tetrahedron is the only exception among these maps.

Regular maps with these properties have a specific geometric and algebraic structure.

Abstract

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and we conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x^n+y^n+z^n=0, excepting the one corresponding to the tetrahedron.