Some Properties On The Vertices Of A Dihedral Tiling

A [4.8.8] tiling is an Archimedean tiling generated by squares and regular octagons. Now let D denote the set of vertices of the tiling [4.8.8]. A point of D is called a D-point. In this thesis, we mainly apply some methods used to discuss the properties of lattice points in the Geometry of Numbers to investigate the properties of D-points.Firstly we discuss the number of D-points lying on any given line in the plane. It is proved that all the lines can be classified into five categories according to the numbers of D-points lying on them, namely, no D-point, exactly one D-point, exactly two D-points, exactly four D-points and an infinite number of D-points. We also give the whole characterizations of these five types of lines by some necessary and sufficient conditions.Secondly we investigate the number (?)D(r) of D-points lying inside or on the boundary of a circle centered at a D-point and with radius r (r∈R+), and shows that (?) as r goes to infinity, where S is the sum of the areas of a regular octagon tile and a square tile.Furthermore, keeping the structure of tiling [4.8.8], that is, without changing the centers of square tiles but properly change the side length of them, we get a new tiling Dαconsisting of squares with side length a and octagons, Denote the set of vertices of the tiling Dαby Dα, and a point of Dαis called a Dα-point. On the basis of the above results, we discuss the number (?)Dα(r) of Dα-points lying inside or on the boundary of a circle centered at a Dα-point and with radius r (r∈R+). Although (?)Dα(r) is different when a changes, but we always have (?) as r goes to infinity, where Sαis the sum of the area of an octagon tile and a square tile in Dα.