Regular maps of high density
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 xn+yn+zn=0, excepting the one corresponding to the tetrahedron.
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