Isometry Group of Truncated Truncated Cube and Truncated Truncated Octahedron Space

Öz

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 $\mathbb{R}^3$.

Anahtar Kelimeler

• Isometry group
• truncated truncated cube
• truncated truncated octahedron
• octahedral symmetry

Kesik Kesik Küp ve Kesik Kesik Sekizyüzlü Uzaylarının İzometri Grupları

Öz

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.

Anahtar Kelimeler

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

Kaynakça

1. Cromwell, P. R. (1997). Polyhedra, Cambridge University Press.
2. Field, J. V. (1997). Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler. Archive for History of Exact Sciences, 50(3/4), 241–289.
3. Senechal, M. (2013). Shaping space. Springer, New York, Heidelberg Dordrecht, London.
4. Can, Z., Çolak, Z., & Gelişgen, Ö. (2015). A note on the metrics induced by triakis icosahedron and disdyakis triacontahedron. Eurasian Academy of Sciences Eurasian Life Sciences Journal, 1, 1–11.
5. Can, Z., Gelişgen, Ö., & Kaya, R. (2015). On the metrics induced by icosidodecahedron and rhombic triacontahedron. KoG, 19, 17–23.
6. Çolak, Z., & Gelişgen, Ö. (2015). New metrics for deltoidal hexacontahedron and pentakis dodecahedron. SAU Fen Bilimleri Enstitüsü Dergisi, 19(3), 353–360.
7. Ermiş, T., & Kaya, R. (2015). On the isometries the of 3- dimensional maximum space. Konuralp Journal of Mathematics, 3(1), 103–114.
8. Ermiş, T., Savcı, Ü. Z., & Gelişgen, Ö. (2019). A note about truncated rhombicuboctahedron and truncated rhombicicosidodecahedron space. Sci. Stud. Res. Ser. Math. Inform., 29(1), 73–88.

9. Ermiş, T. (2020) Geometric analysis of the some rectified Archimedean solids spaces via their isometry groups. Mathematical Sciences and Applications E-Notes, 8(2), 96–109.
10. Gelişgen, Ö., Kaya, R., & Özcan M. (2006). Distance formulae in the chinese checker space. Int. J. Pure Appl. Math., 26(1), 35–44.
11. Gelişgen, Ö., & Kaya, R. (2009). The taxicab space group. Acta Mathematica Hungarica, 122(1-2), 187–200.
12. Gelişgen, Ö., & Kaya, R. (2015). The isometry group of chinese checker space. International Electronic Journal Geometry, 8(2), 82–96.
13. Gelişgen, Ö., & Çolak, Z. (2016). A family of metrics for some polyhedra. Automation Computers Applied Mathematics Scientific Journal, 25(1), 35–48.
14. Gelişgen, Ö., Ermis, T., & Gunaltılı, I. (2017). A note about the metrics induced by truncated dodecahedron and truncated icosahedron. International Journal of Geometry, 6(2), 5–16.
15. Gelişgen, Ö. (2017). On the relations between truncated cuboctahedron truncated icosidodecahedron and metrics. Forum Geometricorum, 17, 273–285.
16. Gelişgen, Ö., & Can, Z. (2016). On the family of metrics for some Platonic and Archimedean polyhedra. Konuralp Journal of Mathematics, 4(2), 25–33.
17. Gelişgen, Ö., & Yavuz, S. (2019). A note about isometry groups of chamfered dodecahedron and chamfered icosahedron spaces. International Journal of Geometry, 8(2), 33–45.
18. Gelişgen, Ö., & Yavuz, S. (2019). Isometry groups of chamfered cube and chamfered octahedron spaces. Mathematical Sciences and Applications e-Notes, 7(2), 174–182.
19. Gelişgen, Ö., & Ermiş, T. (2020). The metrics for rhombicuboctahedron and rhombicosidodecahedron. Palestine Journal of Mathematics, 9(1), 15–25.
20. Gelişgen, Ö., & Çolak, Z. (2024). The isometry groups of $\mathbb{R}_{DH}^{3},$ $\mathbb{R}_{PD}^{3}$ and $\mathbb{R}_{TI}^{3}$.. Hagia Sophia Journal of Geometry, 6(1), 1–9.
21. Gelişgen, Ö., & Can, Z. (2024). The group of transformations which preserving distance on some polyhedral space. Hagia Sophia Journal of Geometry, 6(1), 23–32.
22. Savcı, Ü. Z. (2019). Truncated truncated dodecahedron and truncated truncated icosahedron spaces. Cumhuriyet Science Journal, 40(2), 457–470.
23. Thompson, A. C. (1996). Minkowski geometry. Cambridge University Press.
24. Sacred Geometry, http://www.sacred-geometry.es/?q=en/content/archimedean-solids. [Accessed 30 July 2025].
25. Horvath, A. G. (2017). Isometries of Minkowski geometries. Linear Algebra and Its Applications, 512, 172–190.