Enumeration of Vertices, Edges and Polygons in Tessellations of the Plane

摘要

In this work, we consider the tessellations (or tilings) of Euclidean and hyperbolic planes using copies of a regular polygon. We introduce the concept of k-type of vertices and edges, which allow a thorough control of these elements when the tessellation increases, and we obtain an enumeration for the vertices, edges, and polygons at a given distance.