Document Type

Article

Department

Mathematics (HMC)

Publication Date

4-14-2011

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 ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple
combinatorial derivations of the rational generating functions for the number
of vertices in each generation.

Comments

First published in the Electronic Journal of Combinatorics in 2011, published by the American Mathematical Society.

Rights Information

© 2011 American Mathematical Society