Regular Surfaces and Regular Maps

A regular surface is a closed genus g surface defined as the tubular neighbourhood of the edge graph of a regular map. A regular map is a family of disc type polygons glued together to form a 2-manifold which is flag transitive. The visualization of this highly symmetric surface is an intriguing and challenging problem. Unlike regular maps, regular surfaces can always be visualized as 3D embeddings. In this paper, we introduce an algorithm to visualize the regular surface formed around the tubular neighborhood of a regular map. Our algorithm takes as input the symmetry group of a regular map and outputs a 3D realization of its regular surface. This surface can be interactively modified and used as a target shape for other regular maps. As a result, we find new realizations of regular maps ranging from genus 9 to 85.