International Club of the Golden Section

Theses of the lecture

"Harmony Mathematics and its Application in Modern Science"

Alexey Stakhov
Doctor of Engineering Sciences, Professor

Algebra and Geometry have one and the same fate. The rather slow successes followed after the fast ones in the beginning. They left a science at such step where it is still far from perfect. It has taken place, probably, because the Mathematicians paid all attention on the higher parts of the Analytics. They neglected the beginnings and did not wish to work above processing of such field, which they went one time and left it behind themselves.
Nikolay Lobachevsky

Historical preconditions of the Harmony Mathematics origin:

  1. As is known, two longest periods in development of mathematics, the mathematics prehistory period (mathematics of ancient civilizations) and the period of elementary mathematics (from ancient Greeks to logarithms discovery) preceded creation of the "Higher Mathematics". During these periods the major mathematical discoveries had been made and the first mathematical theories, which determined mathematics development for many millenia, had been created. To them concern: positional principle of number representation (Babylon), the binary coding method (Ancient China, "The Book of Exchanges"), the "doubling method"of the Egyptian arithmetic (Ancient Egypt), trigonometry (Babylon and Ancient Egypt), geometry (Ancient Egypt and Greece), natural numbers and elementary number theory (Ancient Greece), "incommensurable line segments", irrational numbers and mathematical measurement theory (Ancient Greece), decimal number system (India, 5-th century), the major classes of the "elementary functions", logarithms, the beginnings of algebra and combinatorial analysis. It is essential to emphasize that these mathematical achievements form the basis of modern school mathematical education and underlay modern computers (decimal and binary notations, the theory of numbers). Unfortunately, it is necessary to ascertain that after the creation of differential and integral calculus (Newton, Leibnitz), according to Lobachevsky's opinion, "the mathematicians paid all attention on the higher parts of the Analytics. They neglected the beginnings and did not wish to work above processing of such field, which they went one time and left it behind themselves".
  2. Development of computer science became the reason of interest to new methods of number representation and new computer arithmetic's, a process of the "global mathematization of sciences" became the reason of interest to the measurement problem, and cosmological researches revivaled the interest to the "Pythagorean" doctrine about numerical Harmony of the Universe. Those are the major tendencies of modern science and mathematics development: they as though return the modern science and mathematics to their beginnings on the new stage of science development. Here there is the idea of revision and development of mathematical achievements and fundamental concepts of antique mathematics. In it the historical preconditions for creation of new kind of "Elementary Mathematics" called the "Harmony Mathematics" consist.
  3. 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).

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 (р = 0, 1, 2, 3,...), generalized the well-known Golden Section problem and developed the concept of the Generalized Golden Sections or the Golden р-Sections (р = 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" (р = 0) and the classical "Golden Section Principle" (р = 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] ( as unique collection of all Nature, Science and Art works based on the Golden Section.


  1. Stakhov A.P. The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers, Volume 7, 1998.
  2. Stakhov A.P. The Golden Section in the Measurement Theory. An International Journal "Computers & Mathematics with Applications", Volume 17, No 4-6, 1989.
  3. Stakhov A.P. Generalized Golden Sections and a New Approach to the Geometric Definition of a Number. Ukrainian Mathematical Journal, 2004, Volume 56, No 8 (in Russian).
  4. Stakhov A.P. Introduction into Algorithmic Measurement Theory. Moscow: Soviet Radio, 1977 (in Russian).
  5. Stakhov A.P. Codes of the Golden Proportion. Moscow: Radio & Communications, 1984 (in Russian).
  6. Stakhov A.P. Brousentsov's Ternary Principle, Bergman's Number System and Ternary Mirror-symmetrical Arithmetic, The Computer Journal (British Computer Society), Vol. 45, No. 2, 2002.
  7. Stakhov A.P., Tkachenko I.S. Hyperbolic Fibonacci Trigonometry. Journal "Reports of the Ukrainian Academy of Sciences", Vol. 208, No 7, 1993. (in Russian).
  8. Stakhov A.P. Hyperbolic Fibonacci and Lucas Functions: A New Mathematics for the Living Nature. Vinnitsa: ITI, 2003.
  9. Stakhov A., Rozin B. On a new class of hyperbolic function. - Chaos, Solitons & Fractals, 2004, V. 23, No.2.
  10. Stakhov A.P. A generalization of the Fibonacci Q-matrix. Доклады Национальной Академии наук Украины, 1999, № 9.
  11. Stakhov A.P., Sluchenkova A.A., Massingua V. Introduction into Fibonacci Coding and Cryptography, Kharkov: Osnova, 1999.
  12. Stakhov A.P. (editor). Noise-tolerant codes. Fibonacci computer. Moscow: Znanie, Series "Radio Electronics & Communications", No 6, 1989 (in Russian).
  13. Stakhov A.P., Sluchenkova A.A. Museum of Harmony and the Golden Section: Mathematical connections in Nature, Science and Art. Vinnitsa, ITI, 2003

Reviews of the well-known scientists to author's scientific direction

Doctor of Sciences in Physics and Mathematics, Professor
Gennady Shypov (Moscow):

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"

Dear Alexey,
I was fascinated to read your paper, which is full of interesting information, some of it new to me. Your ideas are far reaching and their implementation in schools is obviously the next step. Are there teachers in the Ukraine or elsewhere who have started to use your ideas and your programme? I would be most interested to hear about their experiences.
Best wishes Alan.

Information about the author

Alexey Stakhov is Doctor of Sciences in Computer Engineering (1972), Full Professor (1974), Academician of the Academy of Engineering Sciences of Ukraine (1992). He worked as Visiting-Professor of many World Universities (Vienna Technical University, Austria, 1976, Jena University, Germany, 1986, Dresden Technical University, Germany, 1989, University Al Fateh, Tripoli, Libya, 1995-1997, Eduardo Mondlane University, Maputo, Mozambique, 1998-2000). His area of scientific interests are mathematics, computer science and art. He published about 400 works in this area. In 2001 he created on the Internet web-site "Museum of Harmony and Golden Section" ( that became a sensation in the World science. In May 2003 he delivered the lecture "A New Kind of Elementary Mathematics and Computer Science based on the Golden Section" at the seminar "Geometry and Physics" of the Theoretical Physics Department of the Moscow University. In October 2003 the International Conference "Problems of Harmony, Symmetry and the Golden Section in Nature, Science and Art" was held in Vinnitsa (Ukraine) under his scientific supervision. The International Club of the Golden Section was established at the Conference and Prof. Stakhov was elected by its President. In September 2004 the Scientific Counsel of the Taganrog University of Radio Engineering (Russia) awarded Prof. Stakhov to the rank "Honorable Professor of the University".