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

Abstract

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 -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 -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.

Keywords

• discrete geometry
• Archimedean tiling
• ellipse
• central polygon
• convex hull