  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.
