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

A. P. Stakhov

International Higher Education Academy of Sciences, Moscow, Russia.

DOI: 10.4236/jamp.2013.13010                

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

Conflicts of Interest

The authors declare no conflicts of interest.

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