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Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

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DOI: 10.4236/am.2011.22020    5,115 Downloads   10,137 Views   Citations

ABSTRACT

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

A. Stakhov and S. Aranson, "Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)," Applied Mathematics, Vol. 2 No. 2, 2011, pp. 181-188. doi: 10.4236/am.2011.22020.

References

[1] O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” In Russian, Svit, Lvov, 1994.
[2] V. G. Shervatov, “Hyperbolic Functions,” In Russian Fizmatgiz, Moscow, 1958.
[3] A. P. Stakhov and B. N. Rozin, “On a New Class of Hyperbolic Function,” Chaos, Solitons & Fractals, Vol. 23, No. 2, 2004, pp. 379-389. doi:10.1016/j.chaos.2004.04. 022
[4] A. P. Stakhov and I. S. Tkachenko, “Hyperbolic Fibonacci Trigonometry,” Reports of the National Academy of Sciences of Ukraine, In Russian, Vol. 208, No. 7, 1993, pp. 9-14.

  
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