Isometry Group of Truncated Truncated Cube and Truncated Truncated Octahedron Space

Authors : Özcan Gelişgen, Temel Ermiş

Pages : 16-28

View : 95 | Download : 307

Publication Date : 2025-12-29

Article Type : Research Paper

Abstract :Polyhedra have been extensively studied and examined by scientists, especially geometers and also mathematicians over the years, primarily due to their rich symmetry properties. In particular, there exist notable connections between certain metrics and polyhedral shapes. For instance, it has been demonstrated that the unit spheres corresponding to the maximum metric, taxicab metric, and Chinese Checkers metric are represented, respectively, by the cube, the octahedron, and the deltoidal icositetrahedron. In the present work, we give two novel metrics and prove that the unit spheres in the associated three-dimensional analytic spaces take the form of the \\\\emph{truncated truncated cube} and the \\\\emph{truncated truncated octahedron}, respectively. Furthermore, we examine several fundamental properties of these metrics. In addition, we prove that the isometry groups of the three-dimensional spaces equipped with the $TTC$-metric and the $TTO$-metric are isomorphic to the semidirect product $O_h \\\\ltimes T(3)$, where $O_h$ denotes the octahedral group—the Euclidean symmetry group of the octahedron—and $T(3)$ denotes the translation group consisting of all translations in $\\\\mathbb{R}^3$.
Keywords : İzometri grup, Kesik kesik küp, Kesik kesik sekizyüzlü, Oktahedral simetri

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