Some Properties Of F-points In The Plane

Let F be the tiling (3~2.6~2;3.6.3.6) (which is not an Archimedean tiling) formed byregular triangles and regular hexagons, and let F denote the set of vertices of F. Apoint of F is called an F-point. In this thesis we apply some methods used to discussthe properties of the lattice points to investigate the properties of the vertices of non-Archimedean tiling, and obtain the following results related to F-points.Firstly we discuss the number of F-points lying on any given line in the plane, andprove that all the lines can be classified into three categories according to the numbersof F-points lying on them, namely, no F-point, one and only one F-point and infinitelymany F-points. Furthermore, those three types of lines are characterized by some nec-essary and sufficient conditions. Moreover, we consider the broadest paths that containno F-points in their interiors in any given directionθ∈[0,π). Secondly, we discuss thenumber of F-points N(n) lying inside or on the boundary of a circle C(√n) centeredat an F-point and with radius r =√n (n∈Z~+), and show that Finally we generalize two fundamental theorems in The Geometry of Numbers, thatis, Minkowski's Theorem and Blichfeldt's Theorem to the set of F-points, and prove aMinkowski-type Theorem and a Blichfeldt-type Theorem for F-points.