Abstract

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into «continuous» theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, («Lo-batchevski's hyperbolic geometry», «Four-dimensional Minkowski's world», etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631].

1. Introduction. A role of the Golden Section in modern science

There is well-known fact that two mathematical constants of Nature, the π- and e-numbers, play a great role in mathematics and physics. These functions reflect some deep relations and regularities of the «Physical World» surrounding us. Their importance consists in the fact that they «generate» the main classes of so-called «elementary functions»: sin; cosine (the π-number), exponential, logarithmic and hyperbolic functions (the e-number). It is impossible to imagine mathematics and physics without these functions. For example, there is the well-known greatest role of the classical hyperbolic functions in geometry (Hyperbolic Lobatchevski's geometry) and in cosmological researches (Four-dimensional Minkowki's world). However, there is the one more mathematical constant playing a great role in modeling of processes in living nature termed the Golden Section, Golden Proportion, Golden Ratio, Golden Mean. However, we should certify that a role of this mathematical constant is sometimes undeservedly humiliated in modern mathematics and mathematical education. What is a reason of this? Probably the reason consists in the wide usage of the Golden Section in so-called «esoteric sciences». There is the well-known fact that the basic symbols of esoteric (pentagram, pentagonal star, platonic solids etc.) are connected to the Golden Section closely. Moreover, the «materialistic» science together with its «materialistic» education had decided to «throw out» the Golden Section.

However, in modern science, an attitude towards the Golden Section and connected to its Fibonacci and Lucas numbers is changing very quickly. The outstanding discoveries of modern science (Shechtman's quasi-crystals [1], Bodnar's theory of phyllotaxis [2], Soroko's law of structural harmony of systems [3], Butusov's resonance theory of the Solar system [4], Stakhov's algorithmic measurement theory [5-7] and Staklhov's codes of the Golden Proportion [8]) based

on the Golden Section have a revolutionary importance for development of modern science. These are enough convincing confirmation of the fact that human science approaches to uncovering one of the most complicated scientific notions, namely, the notion of Harmony, which according to Pythagoras, underlies the Universe.

In this connection, it is necessary to point out a great interest of modern physics in the Golden Section. The papers [9-16] present a special interest. In the paper [9] written by Maudlin and Willams in 1986 «proved a theorem which at first sight may seem slightly paradoxical but we perceive as excitingly interesting. This theorem states that the Hausdorff dimension of a randomly constructed Cantor set is where is the Golden Mean» (quotation from

[13]). El Naschie's works [10-15] develop the Golden Mean applications into modern physics. In the paper [13] devoted to the role of the Golden Mean in quantum physics El Naschie conclude the following: «In our opinion it is very worthwhile enterprise to follow the idea of Cantorian space-time with all its mathematical and physical ramifications. The final version may well be a synthesis between the results of quantum topology, quantum geometry and maybe also Rossler's endorphysics, which like Nottale's latest work makes extensive use of the ideas of Nelson's stochastic mechanism».

In October 2003 in the Ukrainian town Vinnitsa, the International Conference «Problems of Harmony, Symmetry, and the Golden Section in Nature, Science, and Art» was hold. The conference had interdisciplinary character and attracted attention of philosophers, mathematicians, physicists, economists, engineers, and linguists from Russia, Ukraine, Byelorussia, Canada, the USA and other countries. Participation of physicists-theoreticians became the significant event of the Conference. Three lectures of physicists-theoreticians were given at the planar session of the Conference and then these were published in the Proceedings of the Conference [16-18]. The paper [16] written by the famous Moscow physicist-theoretician Prof. Vladimirov (Department of Theoretical Physics, Moscow University) and his recent book [19] present the special interest because these researches are related to the quark theory that is based on the Icosahedron and the Golden Section. In the conclusion of his book, Prof. Vladimirov writes [19]: «Thus, it is possible to assert, that in the theory of electroweak interactions there are the relations approximately coincident with the «Golden Section», playing an important role in various areas of Science and Art».

Thus, in the Shechtman's, Butusov's, Mauldin and Williams', El Naschie's, Vladimirov's works, the Golden Section occupied a firm place in modern physics and it is impossible to imagine the future progress in physical researches without the Golden Section.


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