Variable Polyhedric Patterns on a Sphere Derived from Regular and Semi-Regular Polytopes in Four-Dimensional Space

Koji Miyazaki

Graduate School of Human and Environmental Studies, Kyoto University, Sakyo-ku, Kyoto 606-8301, Japan

(Received March 31, 1993; Accepted June 11, 1998)

Keywords: Four-Dimensional Space, Regular Polytope, Semi-Regular Polytope

Abstract. Variable polyhedric patterns on a sphere are shown by using of orthogonal projection into 3-space of regular and semi-regular 4-dimensional polytopes. They are composed of polygons as faces of polyhedral cells of a 4-polytope, and are smoothly transformed from regular to irregular shapes, or vice versa, according to rotation of the polytope in 4-space. Such variable polyhedric patterns on a sphere may be seen in nature, especially as organic and growing forms, and also useful for original creation of dynamic shapes in art and technology.