Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces

Yıl 2019, Cilt: 40 Sayı: 2, 457 - 470, 30.06.2019

Ümit Ziya Savcı

https://doi.org/10.17776/csj.534616

Öz

The theory of convex sets is a vibrant and classical
field of modern mathematics with rich applications. The more geometric aspects
of convex sets are developed introducing some notions, but primarily polyhedra.
A polyhedra, when it is convex, is an extremely important special solid in



















. Some examples of convex subsets of Euclidean
3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean
Duals or Catalan Solids. There are some relations between metrics and
polyhedra. For example, it has been shown that cube, octahedron, deltoidal
icositetrahedron are maximum, taxicab, Chinese Checker’s unit sphere,
respectively. In this study, we give two new metrics to be their spheres
truncated truncated dodecahedron and truncated truncated icosahedron.

Anahtar Kelimeler

Polyhedron, Metric, Truncated Truncated Dodecahedron, Truncated Truncated Icosahedron

Truncated Truncated Dodecahedron ve Truncated Truncated Icosahedron Uzayları

Öz

Konveks kümeler teorisi, zengin uygulamalara sahip
modern matematiğin canlı ve klasik bir alanıdır. Konveks kümelerin geometrik
yönleri, bazı kavramlar, fakat öncelikle çokyüzlülerin tanıtılmasıyla
geliştirilmiştir. Konveks olduğunda bir çokyüzlü,



















 de çok önemli
bir özel cisimdir. Öklid 3 boyutlu uzayın konveks alt kümelerinin bazı
örnekleri Platonik cisimler, Arşimet cisimleri ve Arşimet dualleri veya Katalan
cisimleridir. Metriklerle çokyüzlüler arasında bazı ilişkiler vardır. Örneğin,
küp, sekizyüzlü, deltoidal icositetrahedron'un sırasıyla, maksimum, Taksi, Çin
dama uzaylarının birim küresi olduğu görülmektedir. Bu çalışmada, kürelerinin
truncated truncated dodecahedron ve truncated truncated icosahedron olan iki
yeni metrik tanıtıldı.

Anahtar Kelimeler

Çokyüzlü, Metrik, Truncated Truncated Dodecahedron, Truncated Truncated Icosahedron

Kaynakça

• Ö. Gelisgen and R. Kaya, The Isometry Group of Chinese Checker Space, International Electronic Journal Geometry8-2 (2015) 82–96.
• Z. Can, Ö. Gelişgen 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(2015) 17–23.
• P. Cromwell, Polyhedra, Cambridge University Press (1999).
• A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge 1996.
• Z. Can, Z. Çolak and Ö. Gelişgen, 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 (2015) 1–11.
• Z. Çolak and Ö. Gelişgen, New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron, SAU Fen Bilimleri Enstitüsü Dergisi19-3 (2015) 353-360.
• T. Ermis and R. Kaya, Isometries the of 3- Dimensional Maximum Space, Konuralp Journal of Mathematics3-1 (2015) 103–114.
• Ö. Gelişgen, R. Kaya and M. Ozcan, Distance Formulae in The Chinese Checker Space, Int. J. Pure Appl. Math.26-1(2006) 35–44.
• Ö. Gelişgen and R. Kaya, The Taxicab Space Group, Acta Mathematica Hungarica122 1-2 (2009) 187–200.
• Ö. Gelişgen and Z. Çolak, A Family of Metrics for Some Polyhedra, Automation Computers Applied Mathematics Scientific Journal24-1 (2015) 3–15.
• Ö. Gelişgen, On The Relations Between Truncated Cuboctahedron Truncated Icosidodecahedron and Metrics, Forum Geometricorum, 17(2017) 273–285.
• Ö. Gelişgen and Z. Can, On The Family of Metrics for Some Platonic and Archimedean Polyhedra, Konuralp Journal of Mathematics, 4-2(2016) 25–33.
• M. Senechal, Shaping Space, Springer New York Heidelberg Dordrecht London 2013.
• 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 Sciences50 3-4 (1997) 241–289.
• http://www.sacred-geometry.es/?q=en/content/archimedean-solids Retrieved January, 5, 2019.
• T. Ermis, On the relations between the metric geometries and regular polyhedra, PhD Thesis, Eskişehir Osmangazi University, Graduate School of Natural and Applied Sciences, (2014).
• Ö. Gelişgen, T. Ermis, and I. Gunaltılı, A Note About The Metrics Induced by Truncated Dodecahedron And Truncated Icosahedron, InternationalJournal of Geometry, 6-2 (2017) 5–16.