Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s——Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci Goniometry

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 discove-ries—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 discove-ries—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 Formulas, 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 Formulas, Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci λ-Goniometry

Cite this paper

nullA. Stakhov and S. Aranson, "Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s——Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci Goniometry,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 74-84. doi: 10.4236/am.2011.21009.

nullA. Stakhov and S. Aranson, "Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s——Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci Goniometry,"

References

[1] N. N. Vorobyov, “Fibonacci Numbers,” In Russian, Nauka, Moscow, 1978.

[2] V. E. Hoggat, “Fibonacci and Lucas Numbers,” Houghton-Mifflin, Boston, 1969.

[3] A. P. Stakhov, “Codes of the Golden Proportion,” In Russian, Radio and Communications Publishers, Moscow, 1984.

[4] O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” In Russian, Publishing House “Svit”, Lvov, 1994.

[5] A. P. Stakhov, “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science,” World Scientific, Singapore, 2009. doi:10.1142/ 9789812775832

[6] 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.

[7] 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

[8] A. P. Stakhov and B. N. Rozin, “The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature,” Visual Mathematics, Vol. 8, No. 3, 2006. http://www.mi. sanu.ac.yu/vismath/stakhov/index.html

[9] A. P. Stakhov, “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the ‘Golden’ Cryptography,” Academy of Trinitarism, No. 77-6567, 2006, pp. 1-32. http://www. trinitas.ru/rus/doc/0232/004a/02321063.htm

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

[11] A. A. Andronov, A. A. Vitt and S. E. Khaikin, “Theory of Oscillations,” In Russian, Fizmatgiz, Moscow, 1959.

[12] N. N. Bautin and E. A. Leontovich, “Methods and Ways of Qualitative Study of Dynamic Systems on a Plane,” In Russian, Nauka, Moscow, 1976.

[13] J. I. Neimark, “Methods of Point Mappings in Theory of Non-Linear Oscillations,” In Russian, Nauka, Moscow, 1972.

[14] V. W. de Spinadel, “From the Golden Mean to Chaos,” Nueva Libreria, Buenos Aires, 1998.

[15] M. J. Gazale, “Gnomon. From Pharaohs to Fractals,” Princeton University Press, Princeton, 1999.

[16] A. P. Stakhov, “The ‘Golden’ Matrices and a New Kind of Cryptography,” Chaos, Solitons & Fractals, Vol. 32, No. 3, 2007, pp. 1138-1146. doi:10.1016/j.chaos.2006. 03.069

[1] N. N. Vorobyov, “Fibonacci Numbers,” In Russian, Nauka, Moscow, 1978.

[2] V. E. Hoggat, “Fibonacci and Lucas Numbers,” Houghton-Mifflin, Boston, 1969.

[3] A. P. Stakhov, “Codes of the Golden Proportion,” In Russian, Radio and Communications Publishers, Moscow, 1984.

[4] O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” In Russian, Publishing House “Svit”, Lvov, 1994.

[5] A. P. Stakhov, “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science,” World Scientific, Singapore, 2009. doi:10.1142/ 9789812775832

[6] 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.

[7] 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

[8] A. P. Stakhov and B. N. Rozin, “The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature,” Visual Mathematics, Vol. 8, No. 3, 2006. http://www.mi. sanu.ac.yu/vismath/stakhov/index.html

[9] A. P. Stakhov, “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the ‘Golden’ Cryptography,” Academy of Trinitarism, No. 77-6567, 2006, pp. 1-32. http://www. trinitas.ru/rus/doc/0232/004a/02321063.htm

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

[11] A. A. Andronov, A. A. Vitt and S. E. Khaikin, “Theory of Oscillations,” In Russian, Fizmatgiz, Moscow, 1959.

[12] N. N. Bautin and E. A. Leontovich, “Methods and Ways of Qualitative Study of Dynamic Systems on a Plane,” In Russian, Nauka, Moscow, 1976.

[13] J. I. Neimark, “Methods of Point Mappings in Theory of Non-Linear Oscillations,” In Russian, Nauka, Moscow, 1972.

[14] V. W. de Spinadel, “From the Golden Mean to Chaos,” Nueva Libreria, Buenos Aires, 1998.

[15] M. J. Gazale, “Gnomon. From Pharaohs to Fractals,” Princeton University Press, Princeton, 1999.

[16] A. P. Stakhov, “The ‘Golden’ Matrices and a New Kind of Cryptography,” Chaos, Solitons & Fractals, Vol. 32, No. 3, 2007, pp. 1138-1146. doi:10.1016/j.chaos.2006. 03.069