Sonlu Blaschke Çarpımları ve Altın Oran

Abstract

Keywords

• Sonlu Blaschke Çarpımı
• altın oran
• altın üçgen
• altın elips
• altın dikdörtgen

Finite Blaschke products and the golden ratio

Abstract

Geometric properties of finite Blaschke products have been intensively studied by many different aspects. In this paper, our aim is to study geometric properties of finite Blaschke products related to the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$. Mainly, we focus on the relationships between the zeros of canonical finite Blaschke products of lower degree and the golden ratio. We show that the geometric notions such as "golden triangle, "golden ellipse" and "golden rectangle" are closely related to the geometry of finite Blaschke products.

Keywords

• Finite Blaschke product
• golden ratio
• golden triangle
• golden ellipse
• golden rectangle

References

1. Coxeter H. S. M., Introduction to Geometry, Second edition. John Wiley & Sons, Inc., New York-London-Sydney, 1969.
2. Crasmareanu, M., Hre¸tcanu, C-E., Golden differential geometry, Chaos Solitons Fractals 38 (2008), 1229-1238. DOI: 10.1016/j.chaos.2008.04.007
3. Crasmareanu, M., Hre¸tcanu, C-E., Munteanu, M-I., Golden- and product-shaped hypersurfaces in real space forms, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1320006, 9 pp. DOI:10.1142/S0219887813200065
4. Daepp, U., Gorkin, P., Mortini, R., Ellipses and Finite Blaschke products, Amer. Math. Monthly 109(9) (2002), 785-795. DOI: 10.1080/00029890.2002.11919914
5. Daepp, U., Gorkin, P., Voss, K., Poncelet's theorem, Sendov's conjecture, and Blaschke products, J. Math. Anal. Appl. 365(1) (2010), 93-102. DOI: 10.1016/j.jmaa.2009.09.058
6. Frantz, M., How conics govern Möbius transformations, Amer. Math. Monthly 111(9) (2004), 779-790. DOI: 10.1080/00029890.2004.11920141
7. Fujimura, M., Inscribed ellipses and Blaschke products, Comput. Methods Funct. Theory 13(4) (2013), 557-573. DOI 10.1007/s40315-013-0037-8
8. Gau, H. W., Wu, P. Y., Numerical range and Poncelet property, Taiwanese J. Math. 7(2) (2003), 173-193. DOI: 10.11650/twjm/1500575056

9. Gorkin P., Skubak, E., Polynomials, ellipses, and matrices: two questions, one answer, Amer. Math. Monthly 118(6) (2011), 522-533. DOI:10.4169/amer.math.monthly.118.06.522
10. Hopkins, A. B., Stillinger, H. F., Torquato, S., Spherical codes, maximal local packing density, and the golden ratio, J. Math. Phys. 51(4) (2010), 043302, 6 pp. DOI:10.1063/1.3372627
11. Hretcanu, C-E., Crasmareanu, M., Applications of the golden ratio on Riemannian manifolds, Turkish J. Math. 33(2) (2009), 179-191. DOI: 10.3906/mat-0711-29
12. Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2001.
13. Özgür, C., Özgür, N. Y., Classification of metallic shaped hypersurfaces in real space forms, Turkish J. Math., 39(5) (2015), 784-794. DOI: 10.3906/mat-1408-17
14. Özgür, C., Özgür, N. Y., Metallic shaped hypersurfaces in Lorentzian space forms, Rev. Un. Mat. Argentina 58(2) (2017), 215-226.
15. Özgür, N. Y., Uçar, S., On some geometric properties of finite Blaschke products, Int. Electron. J. Geom., 8(2) (2015), 97-105.
16. Özgür, N. Y., Finite Blaschke products and circles that pass through the origin, Bull. Math. Anal. Appl. 3(3) (2011), 64-72.
17. Özgür, N. Y., Some geometric properties of finite Blaschke products, Proceedings of the Conference RIGA 2011, (2011), 239-246.
18. Uçar, S., Finite Blaschke Products and Some Geometric Properties, Ph.D. Thesis, Balıkesir University, 2015.
19. Tutte, W. T., On chromatic polynomials and the golden ratio, J. Emphasized Theory, 9 (1970), 289-296.