On Some Counting Problems In Archimedean Tilings (3.6.3.6) And (3~4.6)

Let D and I be the Archimedean tilings(3.6.3.6)and(34.6)of side length 1,and let D and I denote the set of vertices of D and I,respectively.A point in D(resp.I)is called a D-point(resp.I-point).In this thesis we apply some methods,which are used to discuss the properties of the lattice points,to investigate some properties related to D-points and I-points.In Chapter 1 and 2,we discuss the number of vertices lying on any given line in D and I respectively,and prove that all the lines can be classified into three categories according to the numbers of vertices lying on them,and the numbers are zero,one and infinity.Meanwhile,we characterize all kinds of lines by some necessary and sufficient conditions.Furthermore,we determine the maximum width of a path containing no vertex in their interiors in any given direction ??[0,?).In Chapter 3,we generalize two fundamental principles in the geometry of numbers,namely,Blichfedlt' s theorem and Minkowski' s theorem to the tiling D and tiling I.At first,we generalize the Blichfedlt' s theorem to a point set generated by a lattice by removing one residue class with respect to a sublattice,and then,we obtain Blichfedlt-type theorems for both D-points and I-points.Finally,on the basis of the Blichfedlt-type theorems we obtain Minkowski-type theorems with optimal lower bound for both.D-points and I-points.