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 CCmetric and COmetric 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.
Keywords
Polyhedron,Metric,Isometry Group,Octahedral Symmetry,Chamfered Cube,Chamfered Octahedron
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