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Journals / Hagia Sophia Journal of Geometry / 2026 / Cilt: 6 Sayı: 1

The Group of Transformations which Preserving Distance on Some Polyhedral Space

Özcan Gelişgen(Eskişehir Osmangazi Üniversitesi)0000-0001-7071-6758
Zeynep Can(AKSARAY UNIVERSITY)
Pages
23–32
Publication Date
2024-06-30
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In this article we show that the group of isometries of the $3-$dimensional space covered by $RT-metric$, $ID-metric$ and $DT-metric$ are the semi-direct product of $I_{h} $ and $T(3)$, where Icosahedral group $I_{h}$ is the (Euclidean) symmetry group of the icosahedron and $T(3)$ is the group of all translations of the $3-$ dimensional space.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:$3$-dimensional analytical space which is covered by a metric is called a Minkowski geometry.
📊
Key Findings & Data:In the Minkowski geometries, the unit balls are symmetric, convex closed sets.
💡
Conclusion & Impact:In this article we show that the group of isometries of the $3-$dimensional space covered by $RT-metric$, $ID-metric$ and $DT-metric$ are the semi-direct product of $I_{h} $ and $T(3)$, where Icosahedral group $I_{h}$ is the (Euclidean) symmetry group of the icosahedron and $T(3)$ is the group of all translations of the $3-$ dimensional space.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Catalan solids · Archimedean solids · isometry group

Abstract

$3$-dimensional analytical space which is covered by a metric is called a Minkowski geometry. In the Minkowski geometries, the unit balls are symmetric, convex closed sets. So there are Minkowski geometries which unit spheres are rhombic triacontahedron, icosidodecahedron and disdyakis triacontahedron. One of the fundamental problems in geometry for a space with a metric is to determine the group of isometries. In this article we show that the group of isometries of the $3-$dimensional space covered by $RT-metric$, $ID-metric$ and $DT-metric$ are the semi-direct product of $I_{h} $ and $T(3)$, where Icosahedral group $I_{h}$ is the (Euclidean) symmetry group of the icosahedron and $T(3)$ is the group of all translations of the $3-$ dimensional space.

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Cite This Article

Gelişgen, Ö., Can, Z. (2024). The Group of Transformations which Preserving Distance on Some Polyhedral Space. Hagia Sophia Journal of Geometry, Cilt: 6 Sayı: 1, 23–32. https://ajindex.com/articles/1490430

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References (30)

  1. Atiyah, M., & Sutcliffe, P. (2003). Polyhedra in physics chemistry and geometry. Milan Journal of Mathematics, 71, 33–58.
  2. Koca, M., Koca, N., & Koç, R. (2010). Catalan solids derived from three-dimensional-root systems and quarternions. Journal of Mathematical Physics, 51:043501, 1–15.
  3. Bloch, E. D. (2015). Polygons, polyhedra, patterns and beyond. Lecture Notes, Spring.
  4. Thompson, A. C. (1996). Minkowski geometry. Cambridge University Press.
  5. Gelişgen Ö., & Çolak, Z. (2016). A family of metrics for some polyhedra. Automation Computers Applied Mathematics Scientific Journal, 25(1), 35–48.
  6. Gelisgen, Ö., & Kaya, R. (2015). The isometry group of Chinese Checker space. International Electronic Journal Geometry, 8(2), 82–96.
  7. Kaya, R., Gelisgen, Ö., Ekmekci, S., & Bayar, A. (2009). On the group of isometries of the plane with generalized absolute value metric. Rocky Mountain Journal of Mathematics, 39(2), 591–603.
  8. Yüksel, S., & Özcan, M. (2015). On some regular polygons in the Taxicab 3-space. Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), 19, 32–41.
  9. 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 / Avrasya Fen Bilimleri Dergisi, 1, 1–11.
  10. Can, Z., Gelişgen, Ö., & Kaya, R. (2015). On the metrics induced by icosidodecahedron and rhombic triacontahedron. Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), 19, 17–23.
  11. Martin, G. E. (1982). Transformation Geometry: An introduction to symmetry. Springer-Verlag New York Inc.
  12. Coxeter, H. S. M. (1969). Introduction to Geometry. John Wiley& Sons Inst.
  13. Kaya, R., Gelişgen, Ö., Ekmekci, S., & Bayar, A. (2006). Group of isometries of CC-plane. Missouri J. of Math. Sci., 18(3), 221-233.
  14. Gelişgen, Ö., & Kaya, R. (2009). The Taxicab space group, Acta Mathematica Hungarica, 122(1-2), 187–200.
  15. Çolak, Z., & Gelişgen, Ö. (2015). New metrics for deltoidal hexacontahedron and pentakis dodecahedron, SAU Fen Bilimleri Enstitüsü Dergisi, 19(3), 353–360.
  16. Ermiş, T., & Kaya, R. (2015). On the isometries the of 3- dimensional maximum space, Konuralp Journal of Mathematics, 3(1), 103–114.
  17. Gelişgen Ö., & Can, Z. (2016). On the family of metrics for some platonic and Archimedean polyhedra. Konuralp Journal of Mathematics, 4(2), 25-33.
  18. 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.
  19. 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.
  20. Gelisgen Ö., & Yavuz, S. (2019). A note about isometry groups of chamfered dodecahedron and chamfered icosahedron spaces. International Journal of Geometry, 8(2), 33–45.
  21. Gelisgen Ö., & Yavuz, S. (2019). Isometry groups of chamfered cube and chamfered octahedron spaces. Mathematical Sciences and Applications e-Notes, 7(2), 174–182.
  22. Savcı, Ü. Z. (2019). Truncated truncated dodecahedron and truncated truncated icosahedron spaces. Cumhuriyet Science Journal, 40(2), 457-470.
  23. 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.
  24. Gelisgen Ö., & Ermiş, T. (2020). The metris for rhombicuboctahedron and rhombicosidodecahedron. Palestine Journal of Mathematics, 9(1), 15–25.
  25. Wikipedia, https://en.wikipedia.org/wiki/Archimedean solid. [Accessed 30 July 2023].
  26. Wikipedia, https://en.wikipedia.org/wiki/Catalan solid. [Accessed 30 July 2023].
  27. Deza, M. M., & Deza, E. (2009). Encyclopedia of distances. Springer, Berlin.
  28. Horvath, A. G. (2017). Isometries of Minkowski geometries. Lin. Algebra and Its Appl, 512, 172-190.
  29. Can. Z. (2016). On the metrics of some convex polyhedra and the geometries of these metrics. PhD Thesis, Eskişehir Osmangazi University, Eskişehir.
  30. Can, Z., Gelişgen, Ö., & Kaya, R. (2015). On the metrics induced by icosidodecahedron and rhombic triacontahedron. Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), 19, 17–23.