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

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Journal of Applied Mathematics and Physics*,

**1**, 60-66. doi: 10.4236/jamp.2013.13010.

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