\input zb-basic \input zb-ioport \iteman{io-port 05903017} \itemau{Paul, Alice; Pippenger, Nicholas} \itemti{A census of vertices by generations in regular tessellations of the plane.} \itemso{Electron. J. Comb. 18, No. 1, Research Paper P87, 13 p., electronic only (2011).} \itemab Summary: 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 $p \geq 3$ and $q \geq 3$ with $1/p + 1/q \leq 1/2$, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation. \itemrv{~} \itemcc{} \itemut{} \itemli{emis:journals/EJC/Volume\_18/Abstracts/v18i1p87.html} \end