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Recently the new uniqueclasses of hyperbolic functions-hyperbolicFibonacci functions based on the -golden ratio-, and hyperbolic Fibonacci l-functions based on the -metallicproportions- l is a given natural number, were introduced in mathematics. The principaldistinction of the new classes of hyperbolic functions from the classic hyperbolicfunctions consists in the fact that they have recursive properties like theFibonacci numbers or Fibonacci l-numbers, which are -discrete- analogs ofthese hyperbolic functions. In the classic hyperbolic functions, such relationship with integernumerical sequences does not exist. This unique property of the new hyperbolicfunctions has been confirmed recently by the new geometric theory ofphyllotaxis, created by the Ukrainian researcherOleg Bodnar-Bodnar’s hyperbolic geometry. These new hyperbolic functions underlie the originalsolution of Hilbert’s Fourth Problem Alexey Stakhov and Samuil Aranson. Thesefundamental scientific results are overturning our views on hyperbolicgeometry, extending fields of its applications -Bodnar’s hyperbolic geometry-and putting forward the challenge for theoretical natural sciences to searchharmonic hyperbolic worlds of Nature. The goal of the present article is toshow the uniqueness of these scientific results and their vital importance fortheoretical natural sciences and extend the circle of readers. Anotherobjective is to show a deep connection of the new results in hyperbolicgeometry 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.





Author: A. P. Stakhov

Source: http://www.scirp.org/



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