On Some Properties of Distance in TO-Space

Yıl 2020, Cilt: 4 Sayı: 2, 113 - 126, 30.12.2020

Zeynep Can

https://doi.org/10.29002/asujse.688279

Öz

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3-dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, d_TO, in R^2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in R_TO^3, 3-dimensional analytical space furnished with the truncated octahedron metric has been found.

Anahtar Kelimeler

Metric, Convex polyhedra, Truncated octahedron, Distance of a point to a line, Distance of a point to a plane, Distance between two lines

Kaynakça

• [1] Cromwell, P. R., Polyhedra, Cambridge University Press, 443p., 1997.
• [2] Thompson, A.C., Minkowski Geometry. Cambridge University Press, 346p., 1996.
• [3] Can, Z., Gelişgen, Ö., Kaya, R., 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).
• [4] Gelişgen, Ö., Can, Z., On The Family of Metrics for Some Platonic and Archimedean Polyhedra, Konuralp Journal of Mathematics, 4, 25-33, (2016).
• [5] Can, Z., Çolak, Z., Gelişgen, Ö., A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences Journal, 1, 1-11, (2015).
• [6] Erçınar G.Z., Investigation of The Metric of Which Unit Ball is a Rhombic Dodecahedron, Eskişehir Osmangazi University. Graduate School Of Natural And Applied Sciences, Master Of Science Dissertation, 69s, Eskişehir, 2015.
• [7] Çolak Z., Gelişgen Ö., New Metrics For Deltoidal Hexacontahedron and Pentakis Dodecahedron, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 19(3), 353-360, (2015).
• [8] Gelişgen Ö., Çolak Z., A Family of Metrics for Some Polyhedra, Automation Computers Applied Mathematics Scientific Journal, 24(1), 3-15, (2015).
• [9] Gelişgen, Ö., Ermiş, T., Günaltılı, İ., A Note About The Metrics Induced by Truncated Dodecahedron And Truncated Icosahedron, International Journal Of Geometry, 2(6), 5 – 16, (2017).
• [10] Milmann, R. S., Parker, G. D., Geometry; A Metric Approach with Models, Springer, 361p., 1991.
• [11] Divjak, B., Notes on taxicab geometry, Scientific and Professional Information Journal of Croatian Society for Constructive Geometry and Computer Graphics (KoG), 5, 5–9, (2000).
• [12] Çolak, Z., Can, Z., Some Distance Properties Of Two Spaces Induced By Dual Convex Polyhedra, International Journal Of Geometry, 7, 12-24, (2018).
• [13] Akça, Z., Kaya, R., On The Distance Formulae in Three Dimensional Taxicab Space, Hadronic Journal, 27(5), 521-532, (2004).
• [14] Gelişgen, Ö., Kaya, R., Özcan, M. 2006, Distance Formulae in the Chinese Checker Space, International Journal of Pure and Applied Mathematical (IJPAM), 26(1), 35-44, (2006).