[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).
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.
[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).
Can, Z. (2020). On Some Properties of Distance in TO-Space. Aksaray University Journal of Science and Engineering, 4(2), 113-126. https://doi.org/10.29002/asujse.688279