Modeling of Kepler-Poinsot Solid Using Isomorphic Polyhedral Graph

This paper presents an interactive modeling system of uniform polyhedra using the isomorphic graphs. Especially, Kepler-Poinsot solids are formed by modifications of dodecahedron and icosahedron. Keywords—Kepler-Poinsot solid, Shape modeling, Polyhedral graph, Graph drawing. I. INTRUDUCTION EPLER-POINSOT solids are subset of uniform polyhedra, which include 5 regular polyhedra (Platonic soilids), 13 semi-regular polyhedra (Archimedean solids), and 4 regular intersected polyhedra (Kepler-Poinst solids). Traditionally, Kepler-Poinsot solids are formed by stellating or faceting the ordinary dodecahedron and icosahedron, which are regular polyhedra. This paper presents another way to form and to model them using the isomorphic graphs. The system consists of three subsystems: graph input subsystem, wire frame subsystem, and polygon subsystem. II. KEPLER-POINSOT SOLIDS Four Kepler-Poinsot solids are listed in Table I-II, and illustrated in Fig. 1. The symbols {m, n} in the tables stand for Schlölfli’s symbols, which mean all the faces are congruent m-regular polygons (regular m-gons) and all the vertex figures are congruent n-regular polygons. Vertex figure of each vertex is the polygon formed by its adjacent vertices. It is similar to the segments joining the mid-point of the edges incident on the vertex, which is the conventional definition of vertex figure [1]. We define n-regular polygon by 2 / n π θ = with its regularity, where θ is the exterior angle of each vertex. The exterior angle of pentagram is 4 /5 π , then pentagram can be regarded as 5/2-regular polygon. Great dodecahedron {5, 5/2} and Great icosahedron {3, 5/2} are regular concave polyhedra with intersecting faces. They were introduced by L. Poinsot in 1809. Great stellated dodecahedron {5/2, 3} and Small stellated dodecahedron {5/2, 5} are regular concave polyhedra with pentagram (5/2) as faces. They were introduced by J. Kepler in 1619. It depends on the definition of face of pentagram that their faces are intersecting or not. If the face of pentagram is H. Nonaka is with Hokkaido University, Sapporo 060 0814, Japan (e-mail: [email protected]). defined by winding rule, faces are intersecting each other. On the other hand, if it is defined by even-odd rule, faces are not intersecting. Fig. 2 depicts the comparison of these two definitions. TABLE I THE LIST OF KEPLER-POINSOT SOLIDS (1) TABLE II THE LIST OF KEPLER-POINSOT SOLIDS (2) (a) Great dodecahedron {5,5/2} (b) Great icosahedron {3,5/2} (c) Great stellated dodecahedron (d) small stellated dodecahedron {5/2, 3} {5/2,5} Triangles(3), pentagons(5), and pentagrams(5/2) are colored with yellow, blue, and gray, respectively. Fig. 1 Four Kepler-Poinsot solids Modeling of Kepler-Poinsot Solid Using Isomorphic Polyhedral Graph