Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces (original) (raw)

Abstract

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

only five regular convex polyhedra known as the platonic solids. There are many relationships between

metrics and polyhedra. Some of them are given in previous studies. In this study, we introduce two new

metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are

chamfered cube and chamfered octahedron. Also we give some properties about these metrics. We show

that the group of isometries of the 3-dimesional space covered by CC􀀀metric and CO􀀀metric are the

semi-direct product of Oh and T(3), where octahedral group Oh is the (Euclidean) symmetry group of the

octahedron and T(3) is the group of all translations of the 3-dimensional space.