关于[4.8.8]铺砌中椭圆上D-点数的研究

Research about the number of D-points of [4.8.8]-tiling in given ellipse

阿基米德平面铺砌是指用一种或多种正多边形铺砌全平面,且要求铺砌的每个顶点的顶点特征相同。阿基米德平面铺砌共有11种,针对其中的[4.8.8]铺砌,即每个铺砌顶点连接边长相同的一个正方形,两个正八边形,研究[4.8.8]铺砌上的椭圆所包含铺砌顶点数的特性,通过对椭圆内半弦上顶点列的分析,采用数的几何及数论中同余的方法给出顶点数的取值算法,并获得顶点数与椭圆短半轴长平方的比值的极限公式,证明极限值与对应铺砌的中心多边形的面积有关。所得算法及极限公式对其他阿基米德铺砌中相关问题的研究有借鉴作用。

An Archimedean tiling is a tiling of the plane by one type of regular polygon or several types of regular polygons,and every vertex of the tiling has the same vertex characteristics.There are 11 Archimedean tiling,and this paper studies[4.8.8]-tiling,which is an Archimedean tiling generated by squares and regular octagons in the plane,and every vertex is associated with one square and two octagons.This paper studies the number of vertices contained in an ellipse in [4.8.8]-tiling.Through analysing the sequence of vertices lying on half chord in the ellipse,and using the method of the geometry of number and congruence in number theory,it presents an algorithm about the value of the number of vertices contained in the ellipse,and obtains a formula of limit about the number of vertices and the square of short semi-axis of the ellipse.It is proved that the value of limit is connected with the area of the corresponding central polygon.The algorithm and the formula of limit are very useful for the study of related problems in other Archimedean tilings.