ON THE FAMILY OF METRICS FOR SOME PLATONIC AND ARCHIMEDEAN POLYHEDRA

Yıl 2016, Cilt: 4 Sayı: 2, 25 - 33, 15.10.2016

Özcan Gelişgen Zeynep Can

Öz

Convexity is an important property in mathematics and geometry. In geometry convexity is simply defined as; if every points of a line segment that connects any two points of the set are in the set then this set is convex. A polyhedra, when it is convex, is an extremely important solid in 3-dimensional analytical space. 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 many relationships between metrics and polyhedra. Some of them are given in previous studies. For example, in [7] the authors have shown that the unit sphere of Chinese Checkers 3-space is the deltoidal icositetrahedron. In this study, we introduce a family of metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are some well-known polyhedra.

Anahtar Kelimeler

Platonic solids, Archimedean solids, metric, Truncated cube, Cuboctahedron, Truncated octahedron

Kaynakça

• [1] Bulgarean V., The group Isodp (Rn) with p 6= 2, Automation Computers Applied Mathematics. Scientic Journal, 22(1)(2013),69-74.
• [2] Colakoglu H. B., Gelişgen O. and Kaya R., Area formulas for a triangle in the alpha plane, Mathematical Communications, 18(1)(2013),123-132.
• [3] Cromwell, P., Polyhedra, Cambridge University Press, 1999
• [4] Ermis T., Gelisgen O. and Kaya R., On Taxicab Incircle and Circumcircle of a Triangle, Scientic and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), 16 (2012), 3-12.
• [5] Ermis T. and Kaya R., Isometries the of 3- Dimensi3onal Maximum Space, Konuralp Journal of Mathematics, 3(1)(2015), 103-114.
• [6] Field, J.V., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50(3-4) (1997), 241-289.
• [7] Gelişgen O., Kaya R. and Ozcan M., Distance Formulae in The Chinese Checker Space, Int. J. Pure Appl. Math., 26(1)(2006),35-44.
• [8] Gelişgen O. and Kaya R., The Taxicab Space Group, Acta Mathematica Hungarica, DOI:10.1007/s10474-008-8006-9, 122(1-2) (2009), 187-200.
• [9] Gelişgen O. and Kaya R., Alpha(i) Distance in n-dimensional Space, Applied Sciences, 10 (2008), 88-93.
• [10] Gelişgen O. and Kaya R., Generalization of Alpha -distance to n-Dimensional Space, Scientic and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), 10 (2006), 33-35.
• [11] Kaya R., Gelisgen O., Ekmekci S. and Bayar A., On The Group of Isometries of The Plane with Generalized Absolute Value Metric, Rocky Mountain Journal of Mathematics, 39(2) (2009), 591-603.
• [12] Krause E. F., Taxicab Geometry, Addison-Wesley Publishing Company, Menlo Park, CA, 88p., 1975.
• [13] Millmann R. S. and Parker G. D., Geometry a Metric Approach with Models, Springer, 370p., 1991.
• [14] Thompson A. C., Minkowski Geometry, Cambridge University Press, Cambridge, 1996.
• [15] https://en.wikipedia.org/wiki/Cuboctahedron
• [16] http://en.wikipedia.org/wiki/Truncated cube
• [17] http://en.wikipedia.org/wiki/Truncated octahedron