Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

Abstract

Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists andartists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. There areonly five regular convex polyhedra known as the platonic solids. There are many relationships betweenmetrics and polyhedra. Some of them are given in previous studies. In this study, we introduce two newmetrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics arechamfered cube and chamfered octahedron. Also we give some properties about these metrics. We showthat the group of isometries of the 3-dimesional space covered by CC−metric and CO−metric are thesemi-direct product of Oh and T(3), where octahedral group Oh is the (Euclidean) symmetry group of theoctahedron and T(3) is the group of all translations of the 3-dimensional space.