Enumeration and Growth Functions for Uniform Tilings of the Hyperbolic Plane

The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence \(5\) and for general vertices with valence \(6\), with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.