Symmetric tiling of closed surfaces : visualization of regular maps

A regular map is a tiling of a closed surface into faces, bounded by edges that join pairs of vertices, such that these elements exhibit a maximal symmetry. For genus 0 and 1 (spheres and tori) it is well known how to generate and present regular maps, the Platonic solids are a familiar example. We present a method for the generation of space models of regular maps for genus 2 and higher. The method is based on a generalization of the method for tori. Shapes with the proper genus are derived from regular maps by tubification: edges are replaced by tubes. Tessellations are produced using group theory and hyperbolic geometry. The main results are a generic procedure to produce such tilings, and a collection of intriguing shapes and images. Furthermore, we show how to produce shapes of genus 2 and higher with a highly regular structure.