Research Article
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Year 2019, , 174 - 182, 15.10.2019
https://doi.org/10.36753/mathenot.542272

Abstract

References

  • [1] Z. Can, Z. Çolak and Ö. Geli¸sgen, A Note On The Metrics Induced By Triakis Icosahedron And DisdyakisTriacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences Journal / Avrasya Fen Bilimleri Dergisi 1, 1–11(2015).
  • [2] Z. Can, Ö. Geli¸sgen and R. Kaya, 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 (2015).
  • [3] P. Cromwell, Polyhedra, Cambridge University Press (1999).
  • [4] Z. Çolak and Ö. Geli¸sgen, New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron, SAU FenBilimleri Enstitüsü Dergisi 19(3), 353-360 (2015).
  • [5] T. Ermis and R. Kaya, Isometries the of 3- Dimensional Maximum Space, Konuralp Journal of Mathematics 3(1),103–114 (2015).
  • [6] J. V. Field, 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(1997).
  • [7] Ö. Geli¸sgen, R. Kaya and M. Ozcan, Distance Formulae in The Chinese Checker Space, Int. J. Pure Appl. Math.26(1), 35–44 (2006).
  • [8] Ö. Geli¸sgen and R. Kaya, The Taxicab Space Group, Acta Mathematica Hungarica 122(1-2), 187–200 (2009).
  • [9] Ö. Gelisgen and R. Kaya, The Isometry Group of Chinese Checker Space, International Electronic Journal Geometry8(2), 82–96 (2015).
  • [10] Ö. Geli¸sgen and Z. Çolak, A Family of Metrics for Some Polyhedra, Automation Computers Applied MathematicsScientific Journal 24(1), 3–15 (2015).
  • [11] Ö. Geli¸sgen, T. Ermis, and I. Gunaltılı, A Note About The Metrics Induced by Truncated Dodecahedron AndTruncated Icosahedron, InternationalJournal of Geometry, 6(2), 5–16, (2017).
  • [12] Ö. Geli¸sgen, On The Relations Between Truncated Cuboctahedron Truncated Icosidodecahedron and Metrics,Forum Geometricorum, 17, 273–285, (2017).
  • [13] Ö. Geli¸sgen and Z. Can, On The Family of Metrics for Some Platonic and Archimedean Polyhedra, KonuralpJournal of Mathematics, 4(2), 25–33, (2016).
  • [14] A. G.Horvath, Isometries of Minkowski geometries, Lin. Algebra and Its Appl, 512, 172-190 (2017).
  • [15] M. Senechal, Shaping Space, Springer New York Heidelberg Dordrecht London (2013).
  • [16] A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge (1996).
  • [17] http://www.sacred-geometry.es/?q=en/content/archimedean-solids

Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

Year 2019, , 174 - 182, 15.10.2019
https://doi.org/10.36753/mathenot.542272

Abstract

Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists and
artists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. There are
only five regular convex polyhedra known as the platonic solids. There are many relationships between
metrics and polyhedra. Some of them are given in previous studies. In this study, we introduce two new
metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are
chamfered cube and chamfered octahedron. Also we give some properties about these metrics. We show
that the group of isometries of the 3-dimesional space covered by CC􀀀metric and CO􀀀metric are the
semi-direct product of Oh and T(3), where octahedral group Oh is the (Euclidean) symmetry group of the
octahedron and T(3) is the group of all translations of the 3-dimensional space.

References

  • [1] Z. Can, Z. Çolak and Ö. Geli¸sgen, A Note On The Metrics Induced By Triakis Icosahedron And DisdyakisTriacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences Journal / Avrasya Fen Bilimleri Dergisi 1, 1–11(2015).
  • [2] Z. Can, Ö. Geli¸sgen and R. Kaya, 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 (2015).
  • [3] P. Cromwell, Polyhedra, Cambridge University Press (1999).
  • [4] Z. Çolak and Ö. Geli¸sgen, New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron, SAU FenBilimleri Enstitüsü Dergisi 19(3), 353-360 (2015).
  • [5] T. Ermis and R. Kaya, Isometries the of 3- Dimensional Maximum Space, Konuralp Journal of Mathematics 3(1),103–114 (2015).
  • [6] J. V. Field, 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(1997).
  • [7] Ö. Geli¸sgen, R. Kaya and M. Ozcan, Distance Formulae in The Chinese Checker Space, Int. J. Pure Appl. Math.26(1), 35–44 (2006).
  • [8] Ö. Geli¸sgen and R. Kaya, The Taxicab Space Group, Acta Mathematica Hungarica 122(1-2), 187–200 (2009).
  • [9] Ö. Gelisgen and R. Kaya, The Isometry Group of Chinese Checker Space, International Electronic Journal Geometry8(2), 82–96 (2015).
  • [10] Ö. Geli¸sgen and Z. Çolak, A Family of Metrics for Some Polyhedra, Automation Computers Applied MathematicsScientific Journal 24(1), 3–15 (2015).
  • [11] Ö. Geli¸sgen, T. Ermis, and I. Gunaltılı, A Note About The Metrics Induced by Truncated Dodecahedron AndTruncated Icosahedron, InternationalJournal of Geometry, 6(2), 5–16, (2017).
  • [12] Ö. Geli¸sgen, On The Relations Between Truncated Cuboctahedron Truncated Icosidodecahedron and Metrics,Forum Geometricorum, 17, 273–285, (2017).
  • [13] Ö. Geli¸sgen and Z. Can, On The Family of Metrics for Some Platonic and Archimedean Polyhedra, KonuralpJournal of Mathematics, 4(2), 25–33, (2016).
  • [14] A. G.Horvath, Isometries of Minkowski geometries, Lin. Algebra and Its Appl, 512, 172-190 (2017).
  • [15] M. Senechal, Shaping Space, Springer New York Heidelberg Dordrecht London (2013).
  • [16] A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge (1996).
  • [17] http://www.sacred-geometry.es/?q=en/content/archimedean-solids
There are 17 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Özcan Gelişgen 0000-0001-7071-6758

Serhat Yavuz This is me 0000-0003-4544-4257

Publication Date October 15, 2019
Submission Date March 20, 2019
Acceptance Date July 27, 2019
Published in Issue Year 2019

Cite

APA Gelişgen, Ö., & Yavuz, S. (2019). Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Mathematical Sciences and Applications E-Notes, 7(2), 174-182. https://doi.org/10.36753/mathenot.542272
AMA Gelişgen Ö, Yavuz S. Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Math. Sci. Appl. E-Notes. October 2019;7(2):174-182. doi:10.36753/mathenot.542272
Chicago Gelişgen, Özcan, and Serhat Yavuz. “Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces”. Mathematical Sciences and Applications E-Notes 7, no. 2 (October 2019): 174-82. https://doi.org/10.36753/mathenot.542272.
EndNote Gelişgen Ö, Yavuz S (October 1, 2019) Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Mathematical Sciences and Applications E-Notes 7 2 174–182.
IEEE Ö. Gelişgen and S. Yavuz, “Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces”, Math. Sci. Appl. E-Notes, vol. 7, no. 2, pp. 174–182, 2019, doi: 10.36753/mathenot.542272.
ISNAD Gelişgen, Özcan - Yavuz, Serhat. “Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces”. Mathematical Sciences and Applications E-Notes 7/2 (October 2019), 174-182. https://doi.org/10.36753/mathenot.542272.
JAMA Gelişgen Ö, Yavuz S. Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Math. Sci. Appl. E-Notes. 2019;7:174–182.
MLA Gelişgen, Özcan and Serhat Yavuz. “Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 2, 2019, pp. 174-82, doi:10.36753/mathenot.542272.
Vancouver Gelişgen Ö, Yavuz S. Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces. Math. Sci. Appl. E-Notes. 2019;7(2):174-82.

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