On Symmetrical Equivelar Polyhedra of Type 3, 7 and Embeddings of Regular Maps

A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type p, q with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk. On the other hand, we have articles with polyhedral embeddings of regular maps by J. Bokowski and M. Cuntz, A. Boole Stott, U. Brehm, H. S. M. Coxeter, B. Grünbaum, E. Schulte, and J. M. Wills. The main concern of this partial survey article is to emphasize that all these articles should be seen as contributing to the common body of knowledge in the area of regular map embeddings. This article additionally provides a method for finding symmetrical equivelar polyhedral embeddings of type 3, 7 based on symmetrical graph embeddings on convex surfaces.