2010/06/22 by Alice Paul, Paul, Alice, Nicholas Pippenger +1
Mathematics · Materials Science · #Point processes and geometric inequalities #Quasicrystal Structures and Properties #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1006.4356
We consider regular tessellations of the plane as infinite graphs in which\nq edges and q faces meet at each vertex, and in which p edges and p\nvertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the\nEuclidean plane; for 1/p + 1/q < 1/2 , they are tilings of the hyperbolic\nplane. We choose a vertex as the origin, and classify vertices into generations\naccording to their distance (as measured by the number of edges in a shortest\npath) from the origin. For all p\≥ 3 and q \≥ 3 with 1/p + 1/q \≤ 1/2\n, we determine the rational generating function giving the number of vertices\nin each generation.\n