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.
© 2011 American Mathematical Society
Paul A., and Pippenger N. “A Census of Vertices by Generations in Regular Tessellations of the Plane”, Electronic Journal of Combinatorics, 18, 87 (2011).