ON THE FAMILY OF METRICS FOR SOME PLATONIC AND ARCHIMEDEAN POLYHEDRA

Abstract

Convexity is an important property in mathematics and geometry. In geometry convexity is simply defined as; if every points of a line segment that connects any two points of the set are in the set then this set is convex. A polyhedra, when it is convex, is an extremely important solid in 3-dimensional analytical space. Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists and artists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. There are many relationships between metrics and polyhedra. Some of them are given in previous studies. For example, in [7] the authors have shown that the unit sphere of Chinese Checkers 3-space is the deltoidal icositetrahedron. In this study, we introduce a family of metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are some well-known polyhedra.

Keywords

• Platonic solids,Archimedean solids,metric,Truncated cube,Cuboctahedron,Truncated octahedron

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