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{\bf Alice Paul and Nicholas Pippenger}
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{\bf A Census of Vertices by Generations in Regular Tessellations of the Plane}
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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 ${1/p + 1/q\le1/2}$,
we give simple combinatorial derivations of the rational generating
functions for the number of vertices in each generation.
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