Platonic solids have been studied since antiquity and in a multiplicity of artistic and scientific contexts. More generally, “polyhedral” maps are ubiquitous in chemistry and crystallography. Their properties have been studied since Kepler. In the present book we are going to study classes of maps on the sphere or the torus and make a catalog of properties that would be helpful and useful to mathematicians and researchers in natural sciences.

In particular, we are studying here two new classes of maps, interesting for applications, especially in chemistry and crystallography (on the sphere or the torus) generalizing Platonic polyhedra. Polycycles are 2-connected plane graphs having prescribed combinatorial type of interior faces and the same degree q for interior vertices, while at the most q for boundary vertices. Two-faced maps are the maps having at most two types of faces and the same degree of vertices. Many examples and various generalizations are given throughout the text. Pictures are given for many of the obtained graphs, especially when a full classification is possible. A lot of the presentation is necessarily compact but we hope to have made it as explicit as possible.

We are interested mainly in enumeration, symmetry, extremal properties, faceregularity, metric embedding and related algorithmic problems. The graphs in this book come from broad areas of geometry, graph theory, chemistry, and crystallography. Many new interesting spheres and tori are presented.