Abstract

Приведен обзор новых математических моделей Природы, основанных на золотом сечении и использовании гиперболических функций Фибоначчи и Люка и функции «Золотой Шофар». Также рассматриваются обобщенные числа Фибоначчи, обобщенные золотые пропорции, обобщенный принцип золотого сечения, «золотые» алгебраические уравнения, обобщенные формулы Бине, непрерывные функции для обобщенных чисел Фибоначчи и Люка, матрицы Фибоначчи и «золотые» матрицы. Статья опубликована в международном электронном журнале «Visual Mathematics», 2006, Vol. 3, No. 3 http://www.mi.sanu.ac.yu/vismath/stakhov/index.html

A survey of new mathematical models of Nature is presented based on the golden section and using a class of hyperbolic Fibonacci and Lucas functions, and a surface referred to as the Golden Shofar. Also considered are generalized Fibonacci numbers, generalized golden proportions, a Generalized Principle of the Golden Section, golden algebraic equations, generalized Binet formulas, generalized Lucas numbers, continuous functions for the generalized Fibonacci and Lucas numbers, Fibonacci matrices, and golden matrices.

Key words. The golden section, Fibonacci and Lucas numbers, Binet formulas, the Golden Shofar, hyperbolic functions, Dichotomy principle, the Golden Section principle.

1. Introduction

One of the major achievements of modern science is an understanding that the world of Nature is hyperbolic. The creator of non-Euclidean geometry was the Russian mathematician Nikolay Lobachevsky who derived a new geometric system based on hyperbolic functions in 1827. The need for new geometrical ideas became apparent in physics at the beginning of the 20^th century as the result of Einstein’s Special Theory of Relativity (1905). In 1908, three years after the publication of this great work, the German mathematician Herman Minkowsky gave a geometrical interpretation of the Special Theory of Relativity based on hyperbolic ideas.

The theory of hyperbolic functions has developed in ways that, at first sight, does not appear to have any connection to hyperbolic functions. However, it does relate to the theory of Fibonacci numbers, an actively developing branch of modern mathematics [1-3]. In 1993, the Ukrainian mathematicians Alexey Stakhov and Ivan Tkachenko developed a new approach to the theory of hyperbolic functions [4]. Using the so-called Binet formulas, they developed a new class of hyperbolic functions called hyperbolic Fibonacci and Lucas functions [4, 5]. This idea was further developed in Stakhov and Rozin’s paper [6] where they defined a class of symmetric hyperbolic Fibonacci and Lucas functions. In Stakhov and Rozin’s article [7] a new surface of the second degree called the Golden Shofar was developed. The hyperbolic Fibonacci and Lucas functions and the Golden Shofar surface are the most important ingredients of the «golden» mathematical models applicable to the description of the «hyperbolic worlds» of Nature.

Also useful for the study of such models are classes of generalized Fibonacci numbers [8], generalized golden proportions [8], golden algebraic equations [9], generalized Binet formulas [10], generalized Lucas numbers [10], continuous functions for the generalized Fibonacci and Lucas functions [11], Fibonacci matrices [12] and golden matrices [13]. These mathematical entities were used in algorithmic measurement theory [8, 14, 15], a new theory of real numbers [16], harmony mathematics [17-20], new computer arithmetics [8, 21-23], a new coding theory [24-25], and the formulation of the Generalized Principle of the Golden Section [13].

The main purpose of the present article is to give a brief survey of golden mathematical models used for describing the «hyperbolic worlds» of Nature developed by the authors in references [4-25].


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