The Isometry Group of Tetrakis Hexahedron and Disdyakis Dodecahedron Spaces

Abstract

Polyhedra are used in several fields by mathematicians and other scientists. It is easy to think of examples from architecture. Polyhedra have been used to scientifically explain the world around us. In the early days of study, polyhedra included only convex polyhedra. Since the ancient Greeks, many thinkers have worked on convex polyhedra. There are only five regular convex polyhedra known as Platonic solids, thirteen semi-regular convex polyhedra known as Archimedean solids, and thirteen irregular convex polyhedra which are duals of the Archimedean solids and known as Catalan solids. In this study, we show that the isometry group of the threedimensional analytic space formed by the metrics of the Tetrakis hexahedron and the Disdyakis dodecahedron is the semi-direct product of Oh and T(3), where the octahedral group Oh is the (Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the three-dimensional space.