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).

'Applications of Fibonacci Numbers' (Volume 7)Stakhov's lecture 'The Golden Section and Modern Harmony Mathematics'

Analysis of the fundamental ideas underlying the classical mathematics fundamentals was used for the creation of the Harmony Mathematics fundamentals. We take for the basis the following mathematics definition given by the famous Russian mathematician academician Kolmogorov:

"Mathematics is the science of the quantitative relations and the space forms of the real world".

Andrew Kolmogorov
Andrew Kolmogorov

It follows from Kolmogorov's definition that just concepts of the "Number" and the "Value" are fundamental concepts of mathematics. Developing such approach one may select three important mathematical concepts, which were put historically to the mathematics foundation:

  1. The concept of the Natural Number aroused from the "count problem" and brought into the "Number Theory".
  2. The concept of the Irrational Number, which arose at studying the concept of the "Value" and is the result of the "Mathematical Measurement Theory" development.
  3. The concept of the "fundamental mathematical constants", the p-number and Euler's number of e, are the next important mathematical concepts. They express the most important quantitative relations of the real world and "generate" the most important class of the "Elementary Functions" used widely for modeling of the quantitative regularities of the real world.

Analysis of these fundamental ideas allows us to advance the following fundamental ideas, which can underlay the Fibonacci Mathematics or "Harmony Mathematics":

  1. (1) The first idea is the new measurement theory called the Algorithmic Measurement Theory. The foundations of the Algorithmic Measurement Theory are stated in Stakhov's book "Introduction into Algorithmic Measurement Theory" (1977) and Stakhov's brochure "Algorithmic Measurement Theory" (1979).
Stakhov's book 'Introduction into the Algorithmic Measurement Theory' (1977)Stakhov's brochure 'Algorithmic Measurement Theory' (1979)

Later Prof. Sytakhov published the article "The Golden Section in the Measurement Theory" in the highly prestigious International Journal "Computers & Mathematics with Applications". This article was devoted to the algorithmic measurement theory as the new direction in the mathematical measurement theory.

The International Journal 'Computers and Mathematics with Applications'Stakhov's article 'The Golden Section in the Measurement Theory'
  1. The second idea consists of the following. It is suggested to add the golden ratio to the p and e-numbers, the main mathematical constants, and together with the golden ratio to add the new class of the elementary functions, the hyperbolic Fibonacci and Lucas function, to the classical elementary functions.
  2. The next idea is to give the New Number Geometric Definition based on the golden p-ratios concept. This one generalizes the classical Euclidean number definition and "generates" the new positional number systems, the number systems with irrational bases.

Table 1 gives comparison between the foundations of the "Classical Mathematics" and "Harmony Mathematics".

Table 1.

Foundation of the "Classical Mathematics"Foundation of the "Harmony Mathematics"
Euclidean number definition, natural numbers, number theoryNew number definition based on generalized golden sections, generalized Fibonacci numbers, Fibonacci numbers theory
Classical measurement theory, irrational numbers Algorithmic measurement theory, new number series following from the algorithmic measurement theory
Fundamental mathematical constants, the p and e-numbers, elementary functionsGolden section, generalized golden sections, hyperbolic Fibonacci and Lucas functions

We can see from this comparison that the foundation of the "Harmony Mathematics" is similar to the foundation of the "Classical Mathematics" but we have the "new number definition based on generalized golden sections" instead of the "Euclidean number definition", we have the "algorithmic measurement theory" instead of the "classical measurement theory" and at least we have the "Golden Section" and the "hyperbolic Fibonacci and Lucas functions" instead of the "fundamental mathematical constants, p and e-numbers, and the "classical elementary functions".

The algorithmic measurement theory, ascending to the Fibonacci "weighing problem", is the heart of the Harmony Mathematics (Fig.1)

The algorithmic measurement theory "generates" an infinite number of the new number series, in particular the p-Fibonacci numbers, the binomial coefficients, the binary and natural numbers. These number series could become the topic of the New Number Theory. Such an approach results in unlimited extension of the number theory topic and will promote to the natural joining of the classical number theory with the Fibonacci number theory and the binomial coefficient theory.

The algorithmic measurement theory results in the new approach to the positional number systems ascending to the Babylonian sexagesimal number system. Due to this approach there arises an infinite extension of the number systems and this "oldest" part of mathematics turns into the new mathematical theory developing and adding the classical theoretical arithmetic.

Binet's formulas generate the new fundamental system of elementary functions. These are the Fibonacci and Lucas hyperbolic functions, which are nothing as the generalization of Binet's formulas for "continues" domain. Due to these functions the Fibonacci number theory turns into the "continues" theory because each mathematical identity for Fibonacci and Lucas functions has its discrete analogy in the form of the corresponding identity for Fibonacci and Lucas numbers.

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:

  1. Simulation of biological processes (Bodnar's geometry).
  2. New computer theory (Fibonacci and "golden" computers).

As is well known the classical mathematical analysis based on the p- 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 p 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 p-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.