On Some Problems Related To H-points

Let H be an Archimedean tiling of the plane formed by regular hexagons, and let H denote the set of vertices of the tiling H. A point of H is called an H-point. In this thesis we mainly apply some methods used to discuss the properties of the lattice points to investigate the properties related to H-points.Firstly we discuss the number of H-points lying on any given line in the plane, and then consider the broadest paths that contain no H-points in their interiors in any directionθ∈[0,π). Let D(n) be a circle centered at an H-point and with radius r=n (n∈Z+). Secondly we discuss the number ND(n)(H) of H-points lying inside or on the boundary of D(n), and show that where S is the area of the regular hexagonal tiles. Thirdly we generalize two fundamental principles in The Geometry of Numbers, namely, Minkowski's Theorem and Blichfeldt's Theorem, to the set H, and prove a Minkowski-type Theorem and a Blichfeldt-type Theorem for H-points respectively.A simple polygon with vertices in H is called an H-polygon. Let K be a convex H-polygon in the plane, and we define G(v)=min{iH(K):vH(K)=v}, where vH(K),iH(K) denote the number of vertices and the number of interior H-points of K respectively. Finally we determine the values of the function G(v), and prove that G(7)=G(8)=2, G(9)=4, G(10)=6 and G(v)≥...