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Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups

Year 2020, Volume: 8 Issue: 2, 96 - 109, 15.10.2020
https://doi.org/10.36753/mathenot.643969

Abstract

There are two main motivations in this article. First,we give the new metrics and the metric spaces whose unit spheres are Rectified Archimedean Solids. Then, using the general technique which is quite simple, we show that isometry group of $\mathbb{R}^{3}$ endowed with these new metrics are the semi direct product of the translation group $T(3)$ of $\mathbb{R}^{3}\ $ with the Euclidean symmetry groups of Rectified Archimedean Solids.

                                                                                                                                                                                                                                             .

Supporting Institution

ESKISEHIR OSMANGAZI UNIVERSTY

Project Number

201719A235

Thanks

This work was supported by the Scientific Research Projects Commission of Eskisehir Osmangazi University under Project Number 201719A235.

References

  • [1] Atiyah, M., Sutcliffe, P.: Polyhedra in Physics, Chemistry and Geometry. Milan Journal of Mathematics. 71, 33-58 (2003).
  • [2] Berger, M.: Geometry I. Springer-Verlag (2004).
  • [3] Berger, M.: Geometry II, Springer-Verlag (2009).
  • [4] Carrizales, J.M.M., Lopez, J.L.R., Pal, U., Yoshida M.M., Yacaman M.J.: The Completion of the Platonic Atomic Polyhedra: The Dodecahedron. Small. 3 (2), 351-355 (2006).
  • [5] Ermiş, T.: Düzgün Çokyüzlülerin Metrik Geometriler ˙Ile ˙Ili¸skileri Üzerine. Ph.D. Eski¸sehir Osmangazi University (2014).
  • [6] Ermiş, T., Kaya, R.: Isometries the of 3􀀀Dimensional Maximum Space. Konuralp Journal of Mathematics. 3 (1), 103-114 (2015). [7] Gelişgen, Ö., Kaya, R.: The Taxicab Space Group. Acta Mathematica Hungarica. 122 (1-2), 187-200 (2009).
  • [8] Gelişgen, Ö., Çolak, Z.: A Family of Metrics for Some Polyhedra. Automation Computers Applied Mathematics Scienti c Journal. 24 (1), 3-15 (2015).
  • [9] Gelişgen, Ö., Can Z.: On The Family of Metrics for Some Platonic and Archimedean Polyhedra. Konuralp Journal of Mathematics. 4 (2), 2533 (2016).
  • [10] Gelişgen, Ö., Yavuz, S.: Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Mathematical Sciences and Applications E-Notes. 7 (2), 174-182 (2019).
  • [11] Griffiths, D.: Introduction to Elementary Particles.Wiley - VCH (1987).
  • [12] Horvath, A. G.: Semi-indefinite inner product and generalized Minkowski spaces. Journal of Geometry and Physics. 60 (9), 1190-1208 (2010).
  • [13] Horvath A. G.: Isometries of Minkowski geometries. Lin. Algebra and Its Appl. 512, 172-190 (2017).
  • [14] Lopez, J.L.R, Carrizales, J.M.M., Yacaman, M.J.: Low Dimensional Non-Crystallographic Metallic Nanostructures: Hrtem Simulation, Models and Experimental Results. Modern Physics Letters B. 20 (13), 725-751 (2006).
  • [15] Saller, H.: Operational Quantum Theory I-Nonrelativistic Structures.Springer-Verlag (2006).
  • [16] Schattschneider, D. J.: Taxicab group. Amer.Math. Monthly. 91, 423-428 (1984).
  • [17] Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996).
  • [18] http://dmccooey.com/polyhedra/RectifiedArchimedean.html
Year 2020, Volume: 8 Issue: 2, 96 - 109, 15.10.2020
https://doi.org/10.36753/mathenot.643969

Abstract

Project Number

201719A235

References

  • [1] Atiyah, M., Sutcliffe, P.: Polyhedra in Physics, Chemistry and Geometry. Milan Journal of Mathematics. 71, 33-58 (2003).
  • [2] Berger, M.: Geometry I. Springer-Verlag (2004).
  • [3] Berger, M.: Geometry II, Springer-Verlag (2009).
  • [4] Carrizales, J.M.M., Lopez, J.L.R., Pal, U., Yoshida M.M., Yacaman M.J.: The Completion of the Platonic Atomic Polyhedra: The Dodecahedron. Small. 3 (2), 351-355 (2006).
  • [5] Ermiş, T.: Düzgün Çokyüzlülerin Metrik Geometriler ˙Ile ˙Ili¸skileri Üzerine. Ph.D. Eski¸sehir Osmangazi University (2014).
  • [6] Ermiş, T., Kaya, R.: Isometries the of 3􀀀Dimensional Maximum Space. Konuralp Journal of Mathematics. 3 (1), 103-114 (2015). [7] Gelişgen, Ö., Kaya, R.: The Taxicab Space Group. Acta Mathematica Hungarica. 122 (1-2), 187-200 (2009).
  • [8] Gelişgen, Ö., Çolak, Z.: A Family of Metrics for Some Polyhedra. Automation Computers Applied Mathematics Scienti c Journal. 24 (1), 3-15 (2015).
  • [9] Gelişgen, Ö., Can Z.: On The Family of Metrics for Some Platonic and Archimedean Polyhedra. Konuralp Journal of Mathematics. 4 (2), 2533 (2016).
  • [10] Gelişgen, Ö., Yavuz, S.: Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Mathematical Sciences and Applications E-Notes. 7 (2), 174-182 (2019).
  • [11] Griffiths, D.: Introduction to Elementary Particles.Wiley - VCH (1987).
  • [12] Horvath, A. G.: Semi-indefinite inner product and generalized Minkowski spaces. Journal of Geometry and Physics. 60 (9), 1190-1208 (2010).
  • [13] Horvath A. G.: Isometries of Minkowski geometries. Lin. Algebra and Its Appl. 512, 172-190 (2017).
  • [14] Lopez, J.L.R, Carrizales, J.M.M., Yacaman, M.J.: Low Dimensional Non-Crystallographic Metallic Nanostructures: Hrtem Simulation, Models and Experimental Results. Modern Physics Letters B. 20 (13), 725-751 (2006).
  • [15] Saller, H.: Operational Quantum Theory I-Nonrelativistic Structures.Springer-Verlag (2006).
  • [16] Schattschneider, D. J.: Taxicab group. Amer.Math. Monthly. 91, 423-428 (1984).
  • [17] Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996).
  • [18] http://dmccooey.com/polyhedra/RectifiedArchimedean.html
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Temel Ermiş 0000-0003-4430-5271

Project Number 201719A235
Publication Date October 15, 2020
Submission Date November 7, 2019
Acceptance Date October 7, 2020
Published in Issue Year 2020 Volume: 8 Issue: 2

Cite

APA 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. https://doi.org/10.36753/mathenot.643969
AMA Ermiş T. Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups. Math. Sci. Appl. E-Notes. October 2020;8(2):96-109. doi:10.36753/mathenot.643969
Chicago Ermiş, Temel. “Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups”. Mathematical Sciences and Applications E-Notes 8, no. 2 (October 2020): 96-109. https://doi.org/10.36753/mathenot.643969.
EndNote Ermiş T (October 1, 2020) Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups. Mathematical Sciences and Applications E-Notes 8 2 96–109.
IEEE T. Ermiş, “Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups”, Math. Sci. Appl. E-Notes, vol. 8, no. 2, pp. 96–109, 2020, doi: 10.36753/mathenot.643969.
ISNAD Ermiş, Temel. “Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups”. Mathematical Sciences and Applications E-Notes 8/2 (October 2020), 96-109. https://doi.org/10.36753/mathenot.643969.
JAMA Ermiş T. Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups. Math. Sci. Appl. E-Notes. 2020;8:96–109.
MLA Ermiş, Temel. “Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, 2020, pp. 96-109, doi:10.36753/mathenot.643969.
Vancouver Ermiş T. Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups. Math. Sci. Appl. E-Notes. 2020;8(2):96-109.

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