On Some Related Properties Of Archimedean Tilings

Let E, Q, W respectively be an Archimedean tiling of the plane formed by regulartriangles and regular hexagons; regular triangles, squares and regular hexagons; squares,regular hexagons and regular dodecagons. And let E, Q, W respectively denote the setof vertices of the tiling E, Q, W. A point of E （resp. Q, W） is called an E-point （resp.Q-point, W-point）. For the sake of convenience, these points are all called quasi gridpoints. In the thesis we mainly apply some methods used to discuss the properties ofthe lattice points to investigate the properties related to the quasi grid points mentionedabove.In chapter1, we discuss the number N （n） of quasi grid points lying inside or onthe boundary of D(2^1/2), and determine the value of （？）, where D(2^1/2) is a circlecentered at a quasi grid point and with radius r=2^1/2（n∈Z+）. In chapter2, wegeneralize Pick Theorem in the Geometry of Numbers to the set E, and prove a Pick-type Theorem for E-points.