Isometry Group of Truncated Truncated Cube and Truncated Truncated Octahedron Space

Year 2025, Volume: 7 Issue: 2, 16 - 28, 29.12.2025

Özcan Gelişgen , Temel Ermiş

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

Isometry group , truncated truncated cube , truncated truncated octahedron , octahedral symmetry

Kesik Kesik Küp ve Kesik Kesik Sekizyüzlü Uzaylarının İzometri Grupları

Abstract

Çokyüzlüler, özellikle geometriciler ve matematikçiler tarafından, zengin simetri özellikleri nedeniyle uzun yıllar boyunca kapsamlı bir şekilde incelenmiş ve araştırılmıştır. Özellikle, belirli metrikler ile çokyüzlü şekiller arasında dikkat çekici bağlantılar bulunmaktadır. Örneğin, maksimum metrik, taksi metrik ve Çin Dama metriklerinin birim kürelerinin sırasıyla küp, oktahedron ve deltoidal ikositetrahedron ile temsil edildiği gösterilmiştir. Bu çalışmada, iki yeni metrik tanıtmakta ve bunlara karşılık gelen üç boyutlu analitik uzaylarda birim kürelerin sırasıyla kesilmiş kesilmiş küp xoctahedron) biçiminde olduğunu ispatlamaktayız. Ayrıca, bu metriklerin çeşitli temel özellikleri incelenmiştir. Bunun yanı sıra, $TTC$-metrik ve $TTO$-metrik ile donatılmış üç boyutlu uzayların izometri gruplarının, oktahedronun Öklidyen simetri grubu olan oktahedral grup $O_h$ ile $\mathbb{R}^3$’teki tüm ötelemelerden oluşan öteleme grubu $T(3)$’ün yarıdirekt çarpımının $O_h \ltimes T(3)$’e izomorf olduğu ispatlanmıştır.

Keywords

İzometri grup , Kesik kesik küp , Kesik kesik sekizyüzlü , Oktahedral simetri

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