Concept of the Harmony Mathematics

The golden ratio along with the numbers of pi and e have to occupy the prominent place in mathematics.

Thus, the neglect of the golden section and the harmony idea is one more strategic mistake not only mathematics but also theoretical physics. This mistake originated a number of other strategic mistakes in the mathematics development.

Concept of the Harmony Mathematics

The humanity became aware of a long time that it is the participant and the witness of huge number of different "worlds" surrounding it. And their own laws act in every from the "worlds". First of all there are the "mechanical world" and the "astronomical world" where "Newton's gravitation laws" act, the "electromagnetic world" where "Maxwell' equations" act, the world of "living nature", the world of "information", the world of "business", the "social world", the world of the "Art" and so on.

And the human science created the corresponding mathematical theories adapted very well to modeling of processes flowing in that or another "world". And this was the answer of mathematics to the "social need". The calculus created for modeling the "movement processes" and "gravitation laws" of mechanical objects was the brightest example of this. The electromagnetic theory created by Maxwell for modeling of the electromagnetic processes is the other brightest example.

We have described in our Museum a big number of examples where Fibonacci numbers and golden section play an important role. And the botanic phenomenon of phyllotaxis is the brightest example of this. And there arises a question: possibly there exist some "Fibonacci's world" subjected to Fibonacci numbers and golden section? Most likely the world of plants, animals and the man as the biological object is "Fibonacci's one". Recently the Ukrainian architect Oleg Bodnar showed that the geometry of living nature is the hyperbolic one. At that the growth processes are subjected to the hyperbolic Fibonacci and Lucas functions (we will tell about Bodnar' discovery more in detail). But possibly is the "world of business" the "Fibonacci world" too? And the highly unusual investigations of the American scientist Ralf Elliott ("Elliott's Waves") confirms this.

And like to the fact that the investigation of the "movement problem" brought into creation of the calculus, the most important mathematical apparatus of modern mathematics, and the investigation of the electromagnetic phenomena brought into the "Maxwell's equations", the modern scientific discoveries based on Fibonacci numbers and golden section demand on development of the corresponding mathematical apparatus adequate to the studying physical phenomena.

We showed in our Museum that the Fibonacci numbers theory was supplemented recently by some new mathematical results (the generalized Fibonacci numbers following from Pascal Triangle, the generalized golden ratios, the hyperbolic Fibonacci and Lucas functions being the extension of Binet's formulas to continues domain and so on). The algorithmic measurement theory being the generalization of Fibonacci "weighing problem" and also Bergman's number system and its generalization, "Codes of Golden Proportion", which, in essence, is the new number definition, play a special role in Fibonacci's field. These new mathematical results extend the topic of Fibonacci numbers theory and demand on systematisation of these new Fibonacci directions in the framework of some general idea. And just such attempt to systematize the different "Fibonacci's theories" was made by Prof. Stakhov in his lecture "The Golden Section and Modern Harmony Mathematics" delivered by him at the Seventh International Conference on Fibonacci Numbers and Their Applications (Austria, Graz, July 1996).

The hyperbolic Fibonacci and Lucas functions are the heart of the new phyllotaxis geometry (Bodnar's geometry), which presents by itself the brilliant confirmation of the effectiveness of the Fibonacci and Lucas hyperbolic functions for simulation of biological processes.

Thus it follows from this consideration at least two important modern applications of the Fibonacci numbers and the golden ratio theory namely:

Simulation of biological processes (Bodnar's geometry).

New computer theory (Fibonacci and "golden" computers).

As is well known the classical mathematical analysis based on the pi- and e-numbers was developed as the mathematical theory for simulation of mechanical processes (Newton's theory of gravitation). From comparison of the classical mathematical analysis with the Harmony Mathematics it follows that the latter, based on the golden ratio, is the interesting complementary to the classical mathematical analysis, its extension for simulation of biological and informational processes. Thanks to this approach the golden ratio along with the numbers of pi and e have to occupy the prominent place in mathematics.

Applications of the golden section in the Art are widely known. The Harmony Mathematics generates the new geometric proportions (the golden pi-ratios), which will be quite applicable to the art works. One may assume that the progress of the Harmony Mathematics will be able to influence to the progress of modern art.

Table 1.

Foundation of the "Classical Mathematics"

1. Euclidean number definition, natural numbers, number theory

2. Classical measurement theory, irrational numbers

3. Fundamental mathematical constants, the pi and e-numbers, elementary functions

Foundation of the "Harmony Mathematics"

1. New number definition based on generalized golden sections, generalized Fibonacci numbers, Fibonacci numbers theory

2. Algorithmic measurement theory, new number series following from the algorithmic measurement theory

3. Golden section, generalized golden sections, hyperbolic Fibonacci and Lucas functions

The main mathematical ideas and theories that underly the Harmony Mathematics:

1. Investigating the diagonal sums of Pascal triangle, the author came to the generalized Fibonacci numbers or Fibonacci p-numbers (ð = 0, 1, 2, 3,...), generalized the well-known Golden Section problem and developed the concept of the Generalized Golden Sections or the Golden ð-Sections (ð = 0, 1, 2, 3, ...) [2]. The author formulated a new scientific principle, the Generalized Principle of the Golden Section. This one includes in itself as special cases the "Dichotomy Principle" (ð = 0) and the classical "Golden Section Principle" (ð = 1) that came to us from the ancient science [3]. The Generalized Principle of the Golden Section underlies the following mathematical theories, which form in total the "Harmony Mathematics":

2. Algorithmic Measurement Theory [4] is a new direction in the mathematical measurement theory. In its origin this theory goes back to the problem of the best weights system choice (Fibonacci, 13th century). Its basic result is an infinite number of new, unknown until now measurement algorithms and new positional methods of number representations. They have practical and theoretical interest for modern computer and measuring systems. Fibonacci measurement algorithms based on the Fibonacci p-numbers, Fibonacci codes and Fibonacci arithmetic are one of the unexpected scientific results of the algorithmic measurement theory.

3. Theory of number systems with irrational radices is stated in author's book "Codes of the Golden Proportion" (1984) [5]. These number systems are a principally new class of the positional number systems that changes a correlation between rational and irrational numbers and concern to foundation of number theory [3]. New theory of number systems has fundamental interest computer science and can be used for creation of new computer projects. The ternary mirror-symmetric arithmetics [6] that is a synthesis of the classical ternary notation and Bergman's notation is one of the new results in this field.

4. Hyperbolic Fibonacci and Lucas functions [7-9] are a new class of hyperbolic functions. The Golden Section is the base of these functions. These functions have a "strategic" interest for theoretical physics if we take into consideration a role of the hyperbolic functions in Lobachevsky's geometry and Minkovsky's geometry (hyperbolic interpretation of special theory of relativity).

5. Fibonacci Matrices [10] that are based on the generalized Fibonacci numbers and the "Golden" matrices that are based on the hyperbolic Fibonacci and Lucas functions are a new class of the square matrices that have theoretical interest for modern matrix theory.

The "Golden" Projects:

1. New coding theory [11] that are based on the Fibonacci and "Golden" matrices can become the basis of new information technologies.

2. New theory of computers [4, 5, 6, 12] that is based on the Fibonacci codes and Codes of the Golden Proportion.

3. New theory of metrology and measurement systems [4, 5] that are based on the "golden" resistor dividers.

4. A reform of mathematical education based on the Golden Section.

5. Museum of Harmony and the Golden Section [13] (http://www.goldenmuseum.com/) as unique collection of all Nature, Science and Art works based on the Golden Section.

Concepts of Harmony Mathematics

http://www.goldenmuseum.com/1108HarmMath_engl.html

"Harmony Mathematics and its Application in Modern Science"

http://www.goldenmuseum.com/2204McMaster_engl.html

Doctor of Sciences in Physics and Mathematics, Professor

Gennady Shypov (Moscow):

In 1996 the author delivered the lecture "The Golden Section and Modern Harmony Mathematics" [1] on the 7th International Conference "Fibonacci Numbers and Their Applications" (Austria, Graz, 1996). This lecture was repeated by the author at the meeting of the Ukrainian Mathematical Society (Kiev, 1998) and then at the seminar "Geometry and Physics" of the Theoretical Physics Department of the Moscow University (Moscow, 2003).

Prof. Stakhov's lecture is irreproachable both under the form, and under the contents, that is, satisfies completely to the "Golden Section Principle". The most impression is the fact that such serious mathematical research is executed by one person. Results of Prof. Stakhov's researches have fundamental importance for development of mathematics and a computer science. Prof. Stakhov work is executed at so high level, that quite deserves its promotion on the Nobel Prize.

Professor Alan Rogerson,

the scientific supervisor of the International Project

"Mathematical Education of the 21-th century"

Museum of Harmony and the Golden Section

MATHEMATICAL CONNECTIONS IN NATURE, SCIENCE, AND ART

THE CONCEPT OF HARMONY AND THE GOLDEN SECTION

http://www.fenkefeng.org/essaysm18004.html

Mathematical Models of the Hyperbolic Worlds

http://www.mi.sanu.ac.yu/vismath/stakhov/index.html

Knots and Everything

http://www.mi.sanu.ac.yu/vismath/sg/sta ... /index.htm

From Euclid to Contemporary Mathematics and Computer Science

http://www.worldscibooks.com/mathematics/6635.html