A Census of Vertices by Generations in Regular Tessellations of the Plane

TL;DR

This paper analyzes regular tessellations of the plane, classifying vertices by generations based on distance from a chosen origin, and derives generating functions for counting vertices in each generation across Euclidean and hyperbolic tilings.

Contribution

It provides explicit rational generating functions for the number of vertices in each generation in all regular tessellations with given parameters, covering both Euclidean and hyperbolic cases.

Findings

Derived generating functions for Euclidean tessellations.

Extended analysis to hyperbolic tessellations.

Unified classification of vertices by generations.

Abstract

We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For $1/p + 1/q = 1/2$, these are tilings of the Euclidean plane; for $1/p + 1/q < 1/2 $, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all $p\ge 3$ and $q \ge 3$ with $1/p + 1/q \le 1/2 $, we determine the rational generating function giving the number of vertices in each generation.