regular maps, tiling, tessellation, surface topology, mathematical visualization
Abstract:A regular map is a symmetric tiling of a closed surface, in the sense that all faces, vertices, and edges are topologically indistinguishable. Platonic solids are prime examples, but also for surfaces with higher genus such regular maps exist. We present a new method to visualize regular maps. Space models are produced by matching regular maps with target shapes in the hyperbolic plane. The approach is an extension of our earlier work. Here a wider variety of target shapes is considered, obtained by duplicating spherical and toroidal regular maps, merging triangles, punching holes, and gluing the edges. The method produces about 45 new examples, including the genus~7 Hurwitz surface.