A note on the number of vertices of the Archimedean tiling

There are 11 Archimedean tilings in R-2. Let E(n) denote the ellipse of short half axis length n(n is an element of Z(+)) centered at an arbitrary vertex of the Archimedean tiling by regular polygons of edge length 1, and let N(E(n)) denote the number of vertices of the Archimedean tiling that lie inside or on the boundary of E(n). In this paper, we present an algorithm to calculate the number N(E(n)), and get a unified formula lim(n ->infinity) = m center dot pi/s, where S is the area of the central polygon, and m is the ratio of long half axis length and short half axis length of the ellipse. Let C be a cube-tiling by cubes of edge length 1 in R-3, and the vertex of cube-tiling is called a C-point. Let S(n) denote the sphere of radius n(n is an element of Z(+)) centered at an arbitrary C-point, and let N-C(S(n)) denote the number of C-points that lie inside or on the surface of S(n). In this paper, we present an algorithm to calculate the number N-C(S(n)) and get a formula lim(n ->infinity) NC(S(n))/n(3) = 4 pi/3v, where V is the volume of the cube.