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

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. .

Keywords:

Polyhedrons, Metric geometry, Isometry group, Octahedral group, Icosahedral group Rectified Archimedean solids,

PDF

___

[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

Mathematical Sciences and Applications E-Notes-Cover

ISSN: 2147-6268
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 2013
Yayıncı: -

Arşiv