Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

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 O h and T (3), where octahedral group O h is the (Euclidean) symmetry group of the octahedron and T (3) is the group of all translations of the 3-dimensional space.