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Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2021 / Cilt: 70 - Sayı: 2

Finite Blaschke products and the golden ratio

Sonlu Blaschke Çarpımları ve Altın Oran

Nihal ÖZGÜR
Sumeyra UCAR
Pages
664–677
Publication Date
2021-12-31
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Geometric properties of finite Blaschke products have been intensively studied by many different aspects.
📊
Key Findings & Data:In this paper, our aim is to study geometric properties of finite Blaschke products related to the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$.
💡
Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Sonlu Blaschke Çarpımı · altın oran · altın üçgen

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.

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Cite This Article

ÖZGÜR, N., UCAR, S. (2021). Finite Blaschke products and the golden ratio. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Cilt: 70 - Sayı: 2, 664–677. https://ajindex.com/articles/746692

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References (9)

  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 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 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 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 Frantz, M., How conics govern Möbius transformations, Amer. Math. Monthly 111(9) (2004), 779-790. DOI: 10.1080/00029890.2004.11920141 Fujimura, M., Inscribed ellipses and Blaschke products, Comput. Methods Funct. Theory 13(4) (2013), 557-573. DOI 10.1007/s40315-013-0037-8
  3. Gau, H. W., Wu, P. Y., Numerical range and Poncelet property, Taiwanese J. Math. 7(2) (2003), 173-193. DOI: 10.11650/twjm/1500575056 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 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 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
  4. Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2001.
  5. 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
  6. Özgür, C., Özgür, N. Y., Metallic shaped hypersurfaces in Lorentzian space forms, Rev. Un. Mat. Argentina 58(2) (2017), 215-226.
  7. Özgür, N. Y., Uçar, S., On some geometric properties of finite Blaschke products, Int. Electron. J. Geom., 8(2) (2015), 97-105.
  8. Özgür, N. Y., Finite Blaschke products and circles that pass through the origin, Bull. Math. Anal. Appl. 3(3) (2011), 64-72.
  9. Özgür, N. Y., Some geometric properties of finite Blaschke products, Proceedings of the Conference RIGA 2011, (2011),