Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature

A. P.  Stakhov

doi:10.4236/jamp.2013.13010

A. P. Stakhov^*

Abstract: Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural number), were introduced in mathematics. The principal distinction of the new classes of hyperbolic functions from the classic hyperbolic functions consists in the fact that they have recursive properties like the Fibonacci numbers (or Fibonacci l-numbers), which are “discrete” analogs of these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer numerical sequences does not exist. This unique property of the new hyperbolic functions has been confirmed recently by the new geometric theory of phyllotaxis, created by the Ukrainian researcherOleg Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These fundamental scientific results are overturning our views on hyperbolic geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”) and putting forward the challenge for theoretical natural sciences to search harmonic hyperbolic worlds of Nature. The goal of the present article is to show the uniqueness of these scientific results and their vital importance for theoretical natural sciences and extend the circle of readers. Another objective is to show a deep connection of the new results in hyperbolic geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.

Keywords: hyperbolic Geometry of Lobachevski; Hyperbolic Fibonacci Functions; Bodnar’s Geometry of Phyllotaxis; Hilbert’s Fourth Problem; New Hyperbolic Worlds of Nature

Cite this paper: Stakhov, A. (2013) Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature. Journal of Applied Mathematics and Physics, 1, 60-66. doi: 10.4236/jamp.2013.13010.

References

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

[2] A. Stakhov and B. Rozin, “On a New Class of Hyperbolic Functions,” Chaos, Solitons & Fractals, Vol. 23, No. 2, 2004, pp. 379-389. http://dx.doi.org/10.1016/j.chaos.2004.04.022

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

[4] O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Arts (in Russian),” Svit, Lvov, 1994.

[5] O. Y. Bodnar, “Dynamic Symmetry in Nature and Architecture,” Visual Mathematics, Vol. 12, No. 4, 2010. http://www.mi.sanu.ac.rs/vismath/BOD2010/index.html

[6] A. P. Stakhov, “Gazale Formulas, a New Class of Hyperbolic Fibonacci and Lucas Functions and the Improved Method of the ‘Golden’ Cryptography,” Academy of Trinitarism, Moscow. http://www.trinitas.ru/rus/doc/0232/004a/023210 63.htm

[7] A. P. Stakhov, “On the General Theory of Hyperbolic Functions Based on the Hyperbolic Fibonacci and Lucas Functions and on Hilbert’s Fourth Problem,” Visual Mathematics, Vol. 15, No. 1, 2013. http://www.mi.sanu.ac.rs/vismath/2013stakhov/hyp.pdf

[8] A. Stakhov and S. Aranson, “Hyperbolic Fibonacci and Lucas Functions, ‘Golden’ Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem,” Applied Mathematics, Vol. 2, No. 3, 2011, pp. 283-293.

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

[10] V. G. Shervatov, “Hyperbolic Functions (in Russian),” Fizmatgiz, Moscow, 1958.

[11] V. de Spinadel, “The family of Metallic Means,” Visual Mathematics, Vol. 1, No. 3, 1999. http://members.tripod.com/vismath/

[12] Pell Numbers. http://en.wikipedia.org/wiki/Pell_number

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

[14] Hilbert’s Problems. http://en.wikipedia.org/wiki/Hilbert’s_problems

[15] Hilbert’s Fourth Problem. http://en.wikipedia.org/wiki/Hilbert’s_fourth_problem

[16] D. Hilbert, “Mathematical Problems”. http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob4

[17] A. V. Pogorelov, “Hilbert’s Fourth Problem (in Russian),” Nauka, Moscow, 1974.

[18] B. H. Yandell, “The Honours Class—Hilbert’s Problems and Their Solvers,” A K Peters/CRC Press, Natick, 2003.

[19] A. P. Stakhov, “Non-Euclidean Geometries. From the ‘Game of Postulates’ to the ‘Game of Function’ (in Russian),” Academy of Trinitarizm, Мoscow. http://www.trinitas.ru/rus/doc/0016/001d/00162125.htm

[20] Fullerene. http://en.wikipedia.org/wiki/Fullerene

[21] Quasicrystal. http://en.wikipedia.org/wiki/Quasicrystal

[22] S. Petoukhov, “Matrix Genetics, Algebra of the Genetic Code, Noise Immunity (in Russian),” Regular and Chaotic Dynamics, Moscow, 2008.

[23] A. P. Stakhov, “Introduction into Algorithmic Measurement Theory (in Russian),” Soviet Radio, Moscow, 1977.

[24] A. P. Stakhov, “Algorithmic Measurement Theory (News in Life, Science and Technology, Series “Mathematics and Cybernetics) (in Russian),” Knowledge, Moscow, 1979.

[25] A. P. Stakhov, “The Golden Section in the Measurement Theory,” Computers & Mathematics with Applications, 1989, Vol. 17, No. 4-6, pp. 613-638. http://dx.doi.org/10.1016/0898-1221(89)90252-6

[26] G. Bergman, “A Number System with an Irrational Base,” Mathematics Magazine, Vol. 31, No. 2, 1957, pp. 98-119. http://dx.doi.org/10.2307/3029218

[27] A. P. Stakhov, “Codes of Golden Proportion (in Russian),” Radio and Communications, Moscow, 1984.

[28] A. P. Stakhov, “Generalized Golden Sections and a New Approach to Geometric Definition of a Number,” Ukrainian Mathematical Journal, Vol. 56, No. 8, 2004, pp. 1143-1150. http://dx.doi.org/10.1007/s11253-005-0064-3

[29] A. P. Stakhov, “Brousentsov’s Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical Arithmetic,” The Computer Journal, Vol. 45, No. 2, 2002, pp. 221-236. http://dx.doi.org/10.1093/comjnl/45.2.221

[30] A. P. Stakhov, “A Generalization of the Fibonacci Q-matrix,” Reports of the National Academy of Sciences of Ukraine, Vol. 9, 1999, pp. 46-49.

[31] A. Stakhov “Fibonacci Matrices, a Generalization of the ‘Cassini formula,’ and a New Coding Theory,” Chaos, Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 56-66. http://dx.doi.org/10.1016/j.chaos.2005.12.054

[32] A. P. Stakhov, “The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics,” Congressus Numerantium, Vol. 193, 2008, pp. 5-48.

[33] A. P. Stakhov, “The ‘Golden’ Matrices and a New Kind of Cryptography,” Chaos, Solitons & Fractals, Vol. 32, No. 3, 2007, pp. 1138-1146. http://dx.doi.org/10.1016/j.chaos.2006.03.069

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

[35] A. P. Stakhov, “The Golden Section and Modern Harmony Mathematics,” Applications of Fibonacci Numbers, Vol. 7, 1998, pp. 393-399..