Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

ABSTRACT

This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( 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 scien-tific 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 [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( 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 scien-tific 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 Formulas, Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci -Goniometry

Euclid’s Fifth Postulate, Lobachevski’s Geometry, Hyperbolic Geometry, Phyllotaxis, Bodnar’s Geometry, Hilbert’s Fourth Problem, the “Golden” and “Metallic” Means, Binet Formulas, Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci -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 III. An Original Solution of Hilbert’s Fourth Problem,"*Applied Mathematics*, Vol. 2 No. 3, 2011, pp. 283-293. doi: 10.4236/am.2011.23033.

A. Stakhov and S. Aranson, "Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem,"

References

[1] A. P. Stakhov, “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science,” World Scientific, New Jersey, 2009.

[2] 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

[3] A. P. Stakhov, “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the ‘Golden’ Cryptography,” 2006. http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm

[4] V. W. de Spinadel, “From the Golden Mean to Chaos,” 2nd Edition, Nueva Libreria, Nobuko, 2004.

[5] S. Kh. Aranson, “Qualitative Properties of Foliations on Closed Surfaces,” Journal of Dynamical and Control Systems, Vol. 6, No. 1, 2000, pp. 127-157. doi:10.1023/A:1009525823422

[6] S. Kh. Aranson, V. Z. Grines and E. V. Zhuzhoma, “Using Lobachevsky Plane to Study Surface Flows, Foliations and 2-Webs,” Proceedings of the International Conference BGL-4 (Bolyai-Gauss-Lobachevsky). Non-Euclidean Geometry in Modern Physics and Mathematics, Nizhny Novgorod, 7-11 September 2004, pp. 8-24.

[7] S. Kh. Aranson and E. V. Zhuzoma, “Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces,” Proceedings of the Steklov Institute of Mathematics, Vol. 244, 2004, pp. 2-17.

[8] S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, “Introduction to the Qualitative Theory of Dynamical Systems on Surfaces,” American Mathematical Society, Providence, 1996.

[9] D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines and Yu. S. Il’yashenko, “Ordinary Differential Equations and Smooth Dynamical Systems,” Springer, Berlin, 1997.

[10] D. V. Anosov, S. Kh. Aranson, et al., “Dynamical Sys- tems IX. Dynamical Systems with Hyperbolic Behaviour,” Springer, Berlin, 1995.

[11] P. S. Aleksandrov, “Hilbert Problems,” Nauka, Moscow, 1969.

[12] S. Kh. Aranson, “Once Again on Hilbert’s Fourth Problem,” Academy of Trinitarism, Moskow, 1 December 2009. http://www.trinitas.ru/rus/doc/0232/009a/02321180.htm

[13] H. Busemann, “On Hilbert’s Fourth Problem,” Uspechi mathematicheskich Nauk, Vol. 21, No. 1, 1966, pp. 155-164.

[14] A. V. Pogorelov, “Hilbert’s Fourth Problem,” Nauka, Moscow, 1974.

[15] A. P. Stakhov and I. S. Tkachenko, “Hyperbolic Fibonacci Trigonometry,” Reports of the National Academy of Sciences of Ukraine, Vol. 208, No. 7, 1993, pp. 9-14.

[16] Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” Publishing House, Lvov, 1994.

[17] A. P. Stakhov and S. K. Aranson, “Golden Fibonacci Goniometry, Fibonacci-Lorentz Transformations, Hilbert’s Fourth Problem,” Congressus Numerantium, Vol. 193, 2008, pp. 119-156.

[18] B. A. Dubrovin, S. P. Novikov and A. T. Fomenko, “Modern Geometry. Methods and Applications,” Nauka, Moscow, 1979.

[1] A. P. Stakhov, “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science,” World Scientific, New Jersey, 2009.

[2] 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

[3] A. P. Stakhov, “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the ‘Golden’ Cryptography,” 2006. http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm

[4] V. W. de Spinadel, “From the Golden Mean to Chaos,” 2nd Edition, Nueva Libreria, Nobuko, 2004.

[5] S. Kh. Aranson, “Qualitative Properties of Foliations on Closed Surfaces,” Journal of Dynamical and Control Systems, Vol. 6, No. 1, 2000, pp. 127-157. doi:10.1023/A:1009525823422

[6] S. Kh. Aranson, V. Z. Grines and E. V. Zhuzhoma, “Using Lobachevsky Plane to Study Surface Flows, Foliations and 2-Webs,” Proceedings of the International Conference BGL-4 (Bolyai-Gauss-Lobachevsky). Non-Euclidean Geometry in Modern Physics and Mathematics, Nizhny Novgorod, 7-11 September 2004, pp. 8-24.

[7] S. Kh. Aranson and E. V. Zhuzoma, “Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces,” Proceedings of the Steklov Institute of Mathematics, Vol. 244, 2004, pp. 2-17.

[8] S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, “Introduction to the Qualitative Theory of Dynamical Systems on Surfaces,” American Mathematical Society, Providence, 1996.

[9] D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines and Yu. S. Il’yashenko, “Ordinary Differential Equations and Smooth Dynamical Systems,” Springer, Berlin, 1997.

[10] D. V. Anosov, S. Kh. Aranson, et al., “Dynamical Sys- tems IX. Dynamical Systems with Hyperbolic Behaviour,” Springer, Berlin, 1995.

[11] P. S. Aleksandrov, “Hilbert Problems,” Nauka, Moscow, 1969.

[12] S. Kh. Aranson, “Once Again on Hilbert’s Fourth Problem,” Academy of Trinitarism, Moskow, 1 December 2009. http://www.trinitas.ru/rus/doc/0232/009a/02321180.htm

[13] H. Busemann, “On Hilbert’s Fourth Problem,” Uspechi mathematicheskich Nauk, Vol. 21, No. 1, 1966, pp. 155-164.

[14] A. V. Pogorelov, “Hilbert’s Fourth Problem,” Nauka, Moscow, 1974.

[15] A. P. Stakhov and I. S. Tkachenko, “Hyperbolic Fibonacci Trigonometry,” Reports of the National Academy of Sciences of Ukraine, Vol. 208, No. 7, 1993, pp. 9-14.

[16] Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” Publishing House, Lvov, 1994.

[17] A. P. Stakhov and S. K. Aranson, “Golden Fibonacci Goniometry, Fibonacci-Lorentz Transformations, Hilbert’s Fourth Problem,” Congressus Numerantium, Vol. 193, 2008, pp. 119-156.

[18] B. A. Dubrovin, S. P. Novikov and A. T. Fomenko, “Modern Geometry. Methods and Applications,” Nauka, Moscow, 1979.