Abstract

The «Dichotomy Principle» and the classical «Golden Section Principle» are two of the most important principles of Nature, Science and also Art. The Generalized Principle of the Golden Section that follows from studying the diagonal sums of the Pascal triangle is a sweeping generalization of these important principles. This underlies the foundation of «Harmony Mathematics», a new proposed mathematical direction. Harmony Mathematics includes a number of new mathematical theories: an algorithmic measurement theory, a new number theory, a new theory of hyperbolic functions based on Fibonacci and Lucas numbers, and a theory of the Fibonacci and «Golden» matrices. These mathematical theories are the source of many new ideas in mathematics, philosophy, botanic and biology, electrical and computer science and engineering, communication systems, mathematical education as well as theoretical physics and physics of high energy particles.
© 2005 Published by Elsevier Ltd.

Algebra and Geometry have one and the same fate. The rather slow successes followed after the fast ones at the beginning. They left science at such step where it was still far from perfect. It happened, probably, because Mathematicians paid attention to the higher parts of the Analysis. They neglected the beginnings and did not wish to work on such field, which they finished with one time and left it behind. Nikolay Lobachevsky

1. Introduction: «Fibonacci World» and «Harmony Mathematics»

To realize the real world as a whole, to see the reflection of one and the same essence, to see one reason in the set of all life forms, to see behind the natural laws one general law, from which all of them result, is the main tendency of human culture. The idea of Harmony that descended to us from ancient science is the «launching pad» for the consideration of all natural laws from the general point of view. In the Ancient Greek's philosophy, Harmony was opposite to Chaos and meant the organization of the Universe. The ingenious Russian philosopher Alexey Losev wrote: «From Plato's point of view and generally from the view of all antique cosmology, the Universe by itself is the certain proportional whole that subordinates to the law of harmonic division, the Golden Section».

There is presently a huge interest of modern science in the application of the Golden Section and Fibonacci numbers [1-16], which leads to supposition regarding the existence of what surrounds us, the so-called Fibonacci World, which is based upon the laws of the Golden Section and Fibonacci numbers. The «world» of animals and plants, the «world» of Man (including his morphological envelope and spiritual contents), and also the «world» of Music and Art, concerns the phenomena of the «Fibonacci world». Modern research in crystallography [17], in astronomy [18], theoretical physics [19-29], and physics of the high energy particles [30-32] all confirm the fact that the «Physical World» is based on the Golden Section too.

For a simulation and mathematical description of the different «worlds», which surround us, mathematicians have always created a corresponding mathematical theory. For example, for a simulation of mechanical and astronomical phenomena mathematicians created differential and integral calculus. For a description of the electromagnetic phenomena, Maxwell created the electromagnetism theory. The theory of probabilities was created for simulation of the «stochastic world». These are but a few examples. In this connection there exists the idea of the creation of a new mathematical direction, Mathematics of Harmony, that would be intended for the mathematical simulation of those phenomena and processes of the objective world, for which Fibonacci numbers and the Golden Section are their objective essence (pine cone and pineapple, Shopen's music, cordial activity of mammals, quasi-crystals, movement of the Solar system's planets, the structure of genetic code, physics of the high energy particles, etc.). But this new mathematics can influence on the development of other areas of human culture, such as the basis of new computer projects and the reform of mathematical education.

These specific ideas became the contents of the article in [12], and also of the lecture «The Golden Section and Modern Harmony Mathematics» that was delivered at the 7th International conference «Fibonacci Numbers and Their Applications» (Austria, Graz, July, 1996) [13] and at the meeting of the Ukrainian Mathematical Society (Ukraine, Kiev, 1998) [14].

The goal of the present paper is to develop a new scientific principle called the Generalized Principle of the Golden Section. This one originates from the study of diagonal sums of the Pascal triangle and is a sweeping generalization of the Dichotomy Principle and the classical Golden Section Principle that descended to us from ancient science (Pythagoras, Plato, and Euclid). These are the principles, which have inspired our Harmony Mathematics, which forms the mathematical basis of various new ideas in mathematics, philosophy, general science (including computer science), engineering, botanic and biology, mathematical education, theoretical physics and the physics of the high energy particles.