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@ njp@ 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 .
