On Some Counting Problems In Archimedean Tilings

In this thesis, we mainly study some counting problems in Archimedean tilings by using the theories and methods in the Geometry of Numbers.In chapter one, we study the number of vertices of the Archimedean tilings in given circles. In each of the 11 Archimedean tilings, let C(n) denote the circle of radius r=√n(n ∈Z+) centered at any vertex of the Archimedean tiling, and let N(n) denote the number of vertices of the Archimedean tiling lying inside or on the boundary of C(n).We introduce the definition “central polygon” for all the 11 Archimedean tilings and get a unified formula for the number of vertices of Archimedean tilings in given circles,i.e. limn�'∞N(n)n=πS, where S is the area of the central polygon.In chapter two, we firstly investigate a Pick-type theorem in the(3.3.4.3.4) tiling.We show that there are only four kinds of lattice segments which are paralleled to the edges of the tiling. Moreover, we define the symmetrical lattice segments. Then we prove that in the(3.3.4.3.4) tiling, if the boundary of a simple polygon P is made up of the lattice segments which are parallel to the edges of the tiling, or of the symmetrical lattice segments, or of the lattice segments which meet the asymmetric conditions, then the area of P is A(P)=18[(2+√3)b+(4+2√3)i+(2-√3)c+8√3-24], where b denotes the number of lattice points on the boundary of P, i denotes the number of lattice points in the interior of P, and c represents the boundary characteristic of P. Then by similar methods, we study the Pick-type theorems in the other Archimedean tilings formed by2 or 3 different kinds of regular polygons, and obtain a unified Pick-type formula for all of the 11 Archimedean tilings, that it, if a simple polygon P meets the condition of the Pick-type theorem, the boundary characteristic c of P is 2 times to the degree d of the vertex, and the adjacency characteristic e of P equals to 0, then A(P)=S·(b2+i-1),where b and i represent the number of lattice points on the boundary of P and in the interior of P respectively, S is the area of the central polygon.In chapter three, we obtain a Pick-type theorem in the non-Archimedean tiling(3.3.6.6; 3.6.3.6) for the first time and get the Pick-type formula A(P)=√324(8b + 16i + c- 24).