Analytical Calculation of Geodesic Lengths and （分析计算测地线的长度和）.pdf

Analytical Calculation of Geodesic Lengths and Angle Measures on Sphere Tiling of Platonic and Archimedean Solids Kyongil Yoon Dept. of Computer Studies, Notre Dame of Maryland University 4701 North Charles Street, Baltimore, MD 21210, USA Abstract There are five Platonic solids and thirteen Archimedean solids, and they have many interesting characteristics. One of them is that their faces can be projected outward to a circumscribing sphere, producing tilings of the sphere. In this paper we show how to use analytical methods to calculate the lengths of the geodesics and the measures of the angles for these tilings. Introduction In art such as modular origami and architecture, regular and semi-regular polyhedra have been popular sub- jects [2][3]. These polyhedra have regular polygons as their faces and edges with the same length. Five Pla- tonic solids and thirteen Archimedean solids in Figure 1 are convex regular and semi-regular polyhedra [1]. One of interesting properties of these solids is that all the vertices are on the sphere that circumscribes the solid. The shortest distance on the sphere’s surface between any two adjacent points is obtained by the arc of a great circle. By the radial projection of edges of a polyhedron onto the surface, we get arcs which are called geodesics. These geodesics define a uniform tiling for each solid as in Figure 2. In this paper we analytically compute the length of a geodesic and interior angle measures of spherical polygons on the tiling of all the Platonic solids and Archimedean solids.