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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A Census of Vertices by Generations in Regular Tessellations of the Plane. | A Census of Vertices by Generations in Regular Tessellations of the Plane Alice Paul and Nicholas Pippenger Department of Mathematics Harvey Mudd College Claremont CA 91711 USA apaul@hmc.edu njp@math.hmc.edu Submitted Jul 20 2010 Accepted Mar 28 2011 Published Apr 14 2011 Mathematics Subject Classifications 05A15 05C63 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. 1. Introduction. A regular tessellation is a planar graph in which every vertex has degree q 3 and every face has degree p 3. Following Coxeter C1 we denote such a graph by p q . This notation will not be used to denote a set with two elements. When 1 p 1 q 1 2 the graph p q can be drawn on a sphere in a regular way that is so that all edges have the same spherical length and all faces the same spherical area . These tessellations correspond to the Platonic solids 3 3 is the tetrahedron 4 3 is the cube 3 4 is the octahedron 5 3 is the dodecahedron and 3 5 is the icosahedron. When 1 p 1 q 1 2 the graph p q can be drawn in the Euclidean plane in a regular way. These tessellations correspond to tilings of the Euclidean plane by regular polygons 4 4 6 3 and 3 6 are the tilings by squares regular hexagons and equilateral triangles respectively. When 1 p 1 q 1 2 the graph p q can be drawn in the hyperbolic plane in a regular way that is so that all edges have the same hyperbolic length and all faces have the same hyperbolic area . See .
