Appendix 2

Modern state of the Golden Section theory and its Applications

In the second half of the 20th century the following scientific discoveries and achievements indicated fundamental role of the Golden Section in development of modern Science and Culture are obtained:


  1. Recently in this area the important scientific discovery overturning our ideas about the knowledge level of the ancient Egyptians is made. Investigating drawings represented on the wooden boards, extracted from the burial place of the Egyptian architect Khesi-Ra, the Russian architect Shmelev came to conclusion, that the graphics images represented on Khesi-Ra boards are "Harmony Theory" encoded by geometry language. This theory was the main science of the Egyptian culture.
  2. The Theory of Conjugate Measures of the "Double Square" system was developed (Shevelev). This theory allowed deciphering the working methods of determination of the architectural forms by the Egyptian architects and architects of the Antique Classics and Ancient Russia.
  3. A new hypothesis about the origin of the Egyptian calendar is put forward (Stakhov, Soroko). According to this hypothesis the Egyptians put in the basis of their calendar and time and angular values measurement system the numerical parameters of such "sacral" figure as Dodecahedron (12, 30, 60 and 360 = 12 ´ 30), which symbolized "Universe Harmony" since olden times.

Philosophy of Science and Theory of Symmetry

  1. "The Law of Structural System Harmony" based on the generalized Golden Sections and concerning all self-organizing systems of Nature is formulated (Soroko). "Soroko's Law" expands considerably our ideas on structural harmony of the Universe and can become a source of new ideas and discoveries in development of science and culture.
  2. The Meta-language of the Alive Nature is developed and it is executed the complex of the Golden Section structure research since point of view of principles, dominating in natural sciences, namely, dichotomy, invariance, symmetry (Shevelev). The fundamental researches in the field of application of the Golden Section for the substantiation of the Universe Harmony are executed by the Russian scientists Marutaev and Tatarenko
  3. In Kudrevatova and Pokrovski's works the changed laws of logic are offered and the "Wave of Symmetry", a new concept in symmetry theory, is developed; the latter is of interest for developers of computer and communication system logic structures. It is introduced the definition of the triad-information resonance in these systems that allows to make the equivalent electrical block diagrams for simulation of cyclic processes in complicated self-organizing systems.
  4. The Russian mathematician Kharitonov suggested the measures of Chaos and Ordering (instead of one measure used traditionally) and Law of Information Reservation based on the Golden Section.
  5. Russian scientist Balakshin obtained new outcomes on application of the Harmony, Symmetry and the Golden Section principles to the problem of self-development of dynamically similar systems.


  1. Solution of the 10th Hilbert's Problem. In the second half of the 20th century several famous mathematicians, in particular, the American mathematicians Julia Robinson and Martin Davis and also the young Russian mathematician Jury Matijasevich, approached. But the original solution of this problem was made by Jury Matijasevich in 1970; it is "based on the theorem proved by the Soviet mathematician Nikolay Vorob'ev in the 3-d extended edition of his popular book". That is, the 10th Hilbert problem was solved in modern mathematics extremely due to use of some number-theoretic properties of Fibonacci numbers stated by Vorob'ev in his famous brochure "Fibonacci Numbers"!
  2. Generalizations of the Golden Section Problem (Stakhov, Soroko, Tatarenko) are the most important achievements in mathematics area. These mathematical discoveries brought into formulation of the "Law of Structural System Harmony" (Soroko), new definition of a Number and following from it new number theory and new computer theory (Stakhov) and new concept of the "General Poly-Harmony of the Universe". It is necessary to note that the concept of Structural System Harmony based on the generalized Golden Sections attracted for wide attention of the contributors in the various science domains (Raduk: research of manifestations of the Second Golden Section 1,465 in Nature; Shaparenko and Guminski: applications to human morphology; Kaspler: preference of formats in the Fine Art).
  3. Algorithmic measurement theory based on the generalized Fibonacci numbers (Vitenko, Stakhov). This theory concerns to the mathematics foundations and is a constructive analog of the classical measurement theory (Eudox-Archimedus' and Cantor's axioms) and is also a wide generalization of the "weighing problem", formulated by Fibonacci in the 13th century. There is an isomorphism between measurement algorithms, methods of number representation, problem of "rabbit's multiplication" and division of biological cells (Rosin). These facts allow stating the supposition that the new measurement theory could become a source of new ideas in modern science.
  4. Hyperbolic Fibonacci and Lucas functions (Stakhov, Tkachenko). This mathematical discovery can play important role in development of modern science and mathematics if to take into consideration the role played by the hyperbolic functions in mathematics and physics ("hyperbolic Lobatchevski's geometry", "four-measured Minkovski's world", etc.). These functions, being the extension of Binet's formulas on continuous domain, result to overestimation of the Fibonacci numbers theory developed by Fibonacci mathematicians; using similar functions the Ukrainian architect Bodnar developed the original geometric theory of phyllotaxis.
  5. Generalized Fibonacci matrices (Stakhov, Petoukhov) based on the generalized Fibonacci numbers and being further development of the Q-matrix theory developed by Verner Hoggatt and other Fibonacci mathematicians. A new coding theory (Stakhov) is effective application of these matrices in computer science. The Russian scientist Petoukhov found interesting applications of the Fibonacci matrices in genetic coding.
  6. The theory of the "golden" algebraic equations of arbitrary degree with radicals of the Golden Ratio or the generalized Golden Ratios ("Golden p-ratios, = 0, 1, 2, 3, ...) is developed (Stakhov). As the equation of the Golden Ratio of the 4-th degree describes the butadiene power condition (Feinman), it means, that the theory of the "golden" algebraic equations can become the basis for search of the "Golden Section" in an extensive class of chemical compounds and also for the solution of many optimization problems reduced to the solution of the algebraic equations.
  7. The concept of the "Harmony Mathematics" stated by Stakhov for the first time on the 7th International Conference "Fibonacci Numbers and their Applications" (Austria, Graz, 1996). "Harmony Mathematics" based on the algorithmic measurement theory, new number theory, hyperbolic Fibonacci and Lucas functions and Fibonacci matrices is a new mathematical apparatus, which can be effectively used for simulation of processes in the alive nature and for development of Computer Science.

Computer Sciences and Electric Communication

  1. New methods of number representation based on Fibonacci numbers and the Golden Section (Zekendorf, Bergman, Stakhov), Fibonacci computers projects (Stakhov, Newcomb, Vishnjakov) and new "calculating logic" (Nikitin).
  2. Ternary mirror-symmetrical arithmetic (Stakhov) being a synthesis of the classical ternary arithmetic and "Bergman's number system" could bring to new class of fault-tolerant computers.
  3. New coding theory based on Fibonacci matrices (Stakhov). Its main feature consists of the fact that object of error correction are not bits but huge information files.
  4. New fast algorithms for digital signal processing based on Fibonacci's and "golden" representations (Chernov)
  5. The theory of electrical circuits based on the Golden Sections is developed (Semenjuta).
  6. Russian scientist Jasinski made complex of researches on application of the Golden Section in Electric Communication. New system of logarithms based on the Golden Proportion and new logarithmic measurement unit (1 Fidiy) is introduced.

Physics, chemistry, astronomy, crystallography, geology

  1. Dodecahedron-icosahedron doctrine, ascending to Plato's cosmology, runs through all history of science (Kepler's model of the Solar system based on the "Platonic Solids"; Felix Klein's idea about regular icosahedron as the main mathematics object connecting together all branches of mathematics; Bimon and Poincare's hypothesis about dodecahedron shape of the Earth and its development by the Russian geologist Kislizin)
  2. Discovery of quasi-crystals based on "pentagonal" or "icosahedron" symmetry (Israel physicist Dan Shechtman, 1982) is one of the revolutionary discoveries in modern crystallography. "A concept of quasi-crystals presents fundamental interest, because it generalizes and finishes definition of crystal Its significance in the world of minerals can be put in one rank with adding of irrationals to rational numbers in mathematics" (Gratia).
  3. Basing on the "resonance theory of the Solar system", the Russian astronomer Butusov found that the ratio of cycle times of the adjacent planets round the Sun is equal to the Golden Proportion 1,618 or its quadrate 2,618. "It seems rather surprising that Kepler who wrote about the Golden Section and studied the problem of the Universe harmony had not discovered this regularity!" (Butusov). In this connection Jasinski's researches on determination of the standard "golden" factors for the Earth have important significance.
  4. Modern physics gives many confirmations that the Golden Section is one of the Nature fundamental mathematical constants: Vladimirov's theory the "Quark Icosahedron", "Lorentz-Planck's Golden Section" (Maiboroda), the "phenomenon of deca-logarithmic periodicity" (Petrunenko).
  5. Researches by the Ukrainian chemist Vasutinski brought into conclusion that the chemical compounds based on Fibonacci numbers (in particular, Uranium oxides) ate the most stable ones!
  6. It is established the Golden Proportion is universal constant of the Solar system describing connection of the solar light to photosynthesis processes on our planet; it ensures a relative constancy of gas composition of atmosphere, and also harmonic ratio in gas exchange of the alive organisms (Jasinski).

Botany, biology, medicine

  1. Discovery of the "Fibonacci's resonances" in genetic code (SUPRA-code by Jean-Claude Perez, 1990). This experimental discovery, indicating an availability of the Golden Proportion between nitrous basis's in the structure of the genetic code and in any its part, can play important role in development of genetic engineering and gives "natural" explanation of the reasons of broad manifestation of the Golden Section in the Alive Nature and Art.
  2. In the 20th century the basic researches on disclosure of the "Phyllotaxis mystery" (Bodnar, Malygin, Veise, etc.). The law of the spiral symmetry transformation discovered by the Ukrainian architect Bodnar and based on the Fibonacci and Lucas hyperbolic functions is convincing confirmation of Vernadski's hypothesis on non-Euclidean character of the Alive Nature geometry.
  3. The "Golden Wurf" concept suggested by the Russian scientist Petoukhov became the important conceptual idea in development of modern biomedical researches (Shaparenko, Guminski).
  4. In the second half of the 20th century the researches on applications of the Golden Section in medicine were actuated. The researches of the Russian biologist Tsvetkov on search of "Fibonacci's regularities" in activity of mammal heart are well known in modern biology. The Russian scientist Subbota found many manifestations of the Golden Section in medicine, the Ukrainian scientists Shaparenko and Guminski found the generalized Golden Proportions in morphology of human body.

Economic Sciences

  1. "Elliott Waves". The stock market laws based on Fibonacci numbers and the Golden Section were discovered by the American accountant Ralf Elliott in the first half of the 20th century. The following Elliott's words were put in the base of his discovery: "All human operations have three features: model, time and relation; all of them are subjected to Fibonacci numbers". This discovery has got broad distribution in the American economic science (Prechter, Fisher).

Fine Art

  1. In the 20th century a huge number of Art works indicating broad manifestation and use of the Golden Section in all Fine Art spheres was executed: in music ("Shopin's etudes in lighting the Golden Section" by Sabaneev), in poetry ("The Golden Section in Shota Rustavely's poem" by academician Zereteli,), in cinema (Eisenstein), in architecture (Grimm, Shevelev), painting (Kovalev). In this connection the researches of the Russian philologist Grinbaum on revealing "Fibonacci's regularities" in Pushkin's poetry and the Russian philosopher Voloshinov on research of the mathematical beginnings of shape formation in music, architecture, painting and literature have a great interest.

Education and Pedagogics

  1. A concept of mathematical education based on the Golden Section is formulated and the programs of educational disciplines "Mathematics of Harmony ", "Harmony of systems ", "Applied "golden" mathematics" are developed (Stakhov, Jasinski, Mitrjasova)