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)

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.

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.

KEYWORDS

Euclid’s Fifth Postulate, Lobachevski’s Geometry, Hyperbolic Geometry, Phyllotaxis, Bodnar’s Geometry, Hilbert’s Fourth Problem, The “Golden” and “Metallic” Means, Binet Formukas, Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci λ-Goniometry

Euclid’s Fifth Postulate, Lobachevski’s Geometry, Hyperbolic Geometry, Phyllotaxis, Bodnar’s Geometry, Hilbert’s Fourth Problem, The “Golden” and “Metallic” Means, Binet Formukas, Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci λ-Goniometry

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.

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),"

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.

[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.