The golden section, Fibonacci numbers, mathematics of harmony and “Golden” scientific revolution.
Abstract
The Publishing House “World Scientific” recently published two fundamental books: Alexey Stakhov “The Mathematics of Harmony” (2009) and Alexey Stakhov and Samuil Aranson “The “Golden” Non-Euclidean Geometry” (2016). In the given article the author develops the Mathematics of Harmony and “Golden” Non-Euclidean Geometry as a new interdisciplinary direction of modern science based on the golden section, Fibonacci numbers and their generalizations. The newest discoveries in different fields of modern science based on the Mathematics of Harmony, namely, mathematics (a general theory of hyperbolic functions and a solution to Hilbert’s Fourth Problem, algorithmic measurement theory and “golden” number theory), computer science (the “golden” information technology), crystallography (quasi-crystals), chemistry (fullerenes), theoretical physics and cosmology (Fibonacci-Lorentz transformations, the “golden” interpretation of special theory of relativity and “golden” interpretation of the Universe evolution), botany (new geometric theory of phyllotaxis), genetics (“golden” genomatrices) and so on, creats a general picture of the “Golden” Scientific Revolution, which can influence fundamentally on the development of modern science and education.
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References
Stakhov A.P. Introduction into algorithmic measurement theory / A.P. Stakhov. – Moscow: Soviet Radio, 1977. – 288 p. (In Russian).
Stakhov A.P. Algorithmic measurement theory / A.P. Stakhov. – Moscow: Znanie, 1979. – 64 p. – (Series “Mathematics and Cybernetics”. – 1979. – Issue 6). (In Russian).
Stakhov A.P. Codes of the golden proportion/ A.P. Stakhov. – Moscow: Radio and communications, 1984. – 151 p. (In Russian).
Noise-tolerant codes: Fibonacci computers. Moscow: Znanie, 1989. - (Series “Radio Electronics and Communications”. - 1989. – Issue 6). (In Russian).
Stakhov A.P. Computer Arithmetic based on Fibonacci Numbers and Golden Section: New Information and Arithmetic Computer Foundations / A.P. Stakhov. – Toronto: SKILLSET-Training, 1997.
Stakhov A.P. Introduction into Fibonacci Coding and Cryptography / A.P. Stakhov, V. Massingua, A.A. Sluchenkova. – Kharkov: Osnova, 1999. (In Russian).
The Golden Section: Theory and Applications: [ed. by A.P. Stakhov] // Boletin de Informatica University Eduardo Mondlane, Mozambique. – 1999. – № 9/10.
Stakhov A.P. Museum of Harmony and Golden Section: mathematical connections in Nature, Science and Art / A.P. Stakhov, A.A. Sluchenkova. – Vinnitsa: ITI, 2003.
Stakhov A.P. Hyperbolic Fibonacci and Lucas Functions: a New Mathematics for Living Nature / A.P. Stakhov. – Vinnitsa: ITI, 2003.
Stakhov A.P. Da Vinci code and Fibonacci series / A.P. Stakhov, A.A. Sluchenkova, I.G. Tscherbakov. – Sanct-Peterburgh: Piter, 2006. – 320 p.
Stakhov A.P. The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science/ A.P. Stakhov; [Assisted by Scott Olsen]. – New Jersey; London; Singapore; Hong Kong: World Scientific, 2009. – 748 p.
Stakhov A.P. The Golden Section in the measurement theory / A.P. Stakhov // Computers & Mathematics with Applications. – 1989. – Volume 17. – № 4 – 6.
Stakhov A.P. Hyperbolic Fibonacci trigonometry / A.P. Stakhov, I.S. Tkachenko // Reports of the Ukrainian Academy of Sciences. – 1993. – Volume 208. – № 7. (In Russian).
Stakhov A.P. The Golden Section and Modern Harmony Mathematics / A.P. Stakhov // Applications of Fibonacci Numbers. – 1998. – Volume 7. – P.323 – 399.
Stakhov A.P. A generalization of the Fibonacci Q-matrix / A.P. Stakhov // Reports of the Ukrainian Academy of Sciences. - 1999. – № 9.
Stakhov A.P. Brousentsov’s ternary principle, Bergman’s number system and ternary mirror-symmetrical arithmetic / A.P. Stakhov // The Computer Journal. – 2002. – Vol. 45. – № 2. – P. 222 – 236.
Stakhov A.P. The generalized golden sections and a new approach to geometric definition of a number / A.P. Stakhov // Ukrainian Mathematical Journal. – 2004. – Vol. 56. (In Russian).
Stakhov A. On a new class of hyperbolic function / A. Stakhov, B. Rozin // Chaos, Solitons & Fractals. – 2004. – Vol. 23. – Issue 2. – P.379 – 389.
Stakhov A. The Golden Shofar / A. Stakhov, B. Rozin // Chaos, Solitons & Fractals. – 2005. – Vol. 26. – Issue 3. – P.677–684.
Stakhov A. The “golden” algebraic equations/ A. Stakhov, B. Rozin // Chaos, Solitons & Fractals. – 2005. – Vol. 27. – Issue 5. – P.1415 – 1421.
Stakhov A. Theory of Binet’s formulas for Fibonacci and Lucas p-numbers / A. Stakhov, B. Rozin // Chaos, Solitons & Fractals. – 2005. – Vol. 27. – Issue 5. – P.1162 – 1177.
Stakhov A. The continuous functions for the Fibonacci and Lucas p-numbers / A. Stakhov, B. Rozin // Chaos, Solitons & Frac-tals. – 2006. – Vol. 28. – Issue 4. – P. 1014 – 1025.
Stakhov A. The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering / A. Stakhov // Chaos, Solitons & Fractals. – 2005. – Vol. 26. – Issue 2. – P. 263 – 289.
Stakhov A. Fundamentals of a new kind of Mathematics based on the Golden Section / A. Stakhov // Chaos, Solitons & Fractals. – 2005. – Vol.27. – Issue 5. – P.1124 – 1146.
Stakhov A. Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory / A. Stakhov // Chaos, Solitons & Fractals. – 2006. – Vol.30. – Issue 1. – P. 56-66.
Stakhov A. The “golden” matrices and a new kind of cryptography / A. Stakhov // Chaos, Solitons & Fractals. – 2007. – Vol.32. – Issue 3. – P. 1138 – 1146.
Stakhov A. The Golden Section, Fibonacci series and new hyperbolic models of Nature / A. Stakhov, B. Rozin // Visual Mathematics. – 2006. – Vol. 8. – № 3. – (http://members.tripod.com/vismath/pap.htm).
Stakhov A.P. The golden section, sacred geometry, and harmony mathematics / A.P. Stakhov // Metaphysics: Century XXI / [compiler and editor Y.S. Vladimirov]. – Moscow: BINOM, 2006 . – P. 174 – 215. (In Russian).
Stakhov A. Three “key” problems of mathematics on the stage of its origin, the “Harmony Mathematics” and its applications in contemporary mathematics, theoretical physics and computer science / A. Stakhov // Visual Mathematics. – 2007. – Vol. 9. – № 3. – (http://members.tripod.com/vismath/pap.htm).
Stakhov A.P. Gazale formulas, a new class of the hyperbolic Fibonacci and Lucas functions and the improved method of the “golden” cryptography / A.P. Stakhov // Academy of Trinitarism, Moscow: № 77 – 6567, publication 14098, 21.12.2006. – (http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm). (In Russian).
Stakhov A.P. Hyperbolic Fibonacci and Lucas functions: a history and applications / A.P. Stakhov // Academy of Trinitarism, Moscow, № 77 – 6567, publication 14429, 31.05.2007. – (www.trinitas.ru/rus/doc/0232/009a/02321057.htm). (In Russian).
Stakhov A.P. “Strategic mistakes” in the mathematics development/ A.P. Stakhov // Academy of Trinitarism, Moscow, № 77 – 6567, publication 14555, 27.08.2007. – (www.trinitas.ru/rus/doc/0232/004a/02321070.htm). (In Russian).
Stakhov A.P. Three “Key” Problems of Mathematics on the Stage of its Origin and the “Harmony Mathematics” as Alternative Way of Mathematics Development / A.P. Stakhov // Mathematics & Design: Fifth International Mathematics & Design Conference – V M&D. – Blumenau: Nova Letra, 2008. – P.88 – 102.
Stakhov A.P. Design of a New Coding Theory and Cryptography Based on the Fibonacci and “Golden” Matrices / A.P. Stakhov, A.A. Sluchenkova // Mathematics & Design: Fifth International Mathematics & Design Conference – V M&D. - Blumenau: Nova Letra, 2007. – P. 154 – 165.
Stakhov A.P. The "Strategic Mistakes" in the Mathematics Development and the Role of the Harmony Mathematics for Their Overcoming / A.P. Stakhov // Visual Mathematics. – 2008. – Vol.10. – №3. – (http://www.mi.sanu.ac.yu/ vis-math/ stakhov2008a/index.html).
A.P. Stakhov. The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics // Congressus Numeran-tium. – 2008. – № 193.
Stakhov A.P. “Golden” Fibonacci Goniometry, Fibonacci-Lorentz Transformations, and Hilbert’s Fourth Problem / A.P. Stakhov, S.Ch. Aranson // Congressus Numerantium. – 2008. – № 193. – P.119 – 156.
Stakhov A. A generalization of the Cassini formula / Alexey Stakhov // Visual Mathematics. – 2012. – Vol. 14. – № 2. – (http://www.mi.sanu.ac.rs/vismath/stakhovsept2012/cassini.pdf).
Kline M. Mathematics: The Loss of Certainty / Morris Kline. – New York: Oxford University Press, 1980. – 366 p.
Arnold V. A complexity of the finite sequences of zeros and units and geometry of the finite functional spaces: Lecture at the session of the Moscow Mathematical Society, May 13, 2007. – (http://elementy.ru/lib/430178/430281).
Olsen Scott The Golden Section: Nature’s Greatest Secret / Scott Olsen. – New York: Walker Publishing Company, 2006. – 58 p.
Herz-Fischler R. A Mathematical History of the Golden Number / Roger Herz-Fischler. – New York: Dover Publications, Inc., 1998. – 195 p.
Khinchin A.Ja. Continued Fractions / A.Ja. Khinchin. - Mineola, N.Y.: Dover Publications, 1997. – [first published in Moscow, 1935].
Vorobyov N.N. Fibonacci numbers / N.N. Vorobyov. – Moscow: Nauka, 1978. (In Russian).
Bodnar O. Y. The Golden Section and Non-Euclidean Geometry in Nature and Art / O. Y. Bodnar. – Lvov: Svit, 1994. (In Russian).
Soroko E.M. Structural harmony of systems / E.M. Soroko. – Minsk: Nauka i technika, 1984. (In Russian).
Metaphysics: Century XXI / [compiler and editor Y.S. Vladimirov]. – Moscow: BINOM, 2006. (In Russian).
Kolmogorov A.N. Mathematics in its historical development / A.N. Kolmogorov. – Moscow: Nauka, 1961. (In Russian).
Stakhov A.P. Proclus Hypothesis / A.P. Stakhov // British Journal of Mathematics & Computer Science. – 2016. – № 13 (06). – P.1 – 22.
Spinadel de Vera W. From the Golden Mean to Chaos / Vera W. de Spinadel. – Nueva Libreria, 1998. – [second edition in Nobuko, 2004].
Gazale Midhat J. Gnomon. From Pharaohs to Fractals / Midhat J. Gazale. – Princeton; New Jersey: Princeton University Press, 1999.
Kappraff Jay Beyond Measure. A Guided Tour Through Nature, Myth, and Number / Jay Kappraff. – Singapore; New Jersey; London; Hong Kong: World Scientific, 2002.
Hoggat Jr. V.E. Fibonacci and Lucas Numbers / V.E. Hoggat Jr. – Boston, MA: Houghton Mifflin, 1969.
Hilbert D. Hilbert's Problems / D. Hilbert; general edition P.S. Alexandrov. – Moscow: Nauka, 1969. (In Russian).
Hilbert’s Fourth Problem. Wikipedia. The Free Encyclopedia. – http://en.wikipedia.org/wiki/Hilbert's_fourth_ problem. – Title from the screen.
Dynamical theory of biological populations. – Moscow: Nauka, 1974.
Spears C.P. Asymmetric cell division: binomial identities for age analysis of mortal vs. immortal trees / C.P. Spears, M. Bicknell-Johnson // Applications of Fibonacci Numbers. – 1998. – Vol. 7.
Markov A.A. On a logics of constructive mathematics / A.A. Markov. – Moscow: Znanie, 1972. (In Russian).
Bergman G. A number system with an irrational base / G. Bergman // Mathematics Magazine. – 1957. – № 31.
Monteiro P. Minimal and maximal Fibonacci representations: Boolean generation / P. Monteiro, R. Newcomb // The Fibonacci Quarterly. – 1976. – Vol.14. – № 1.
Ligomenides P. Equivalence of some binary, ternary, and quaternary Fibonacci Computers / P. Ligomenides, R. Newcomb // Proceeding of the Eleventh International Symposium on Multiple-Valued Logic, Norman, Oklahoma, May 1981.
Ligomenides P. Complement representations in the Fibonacci computer / P. Ligomenides, R. Newcomb // Proceedings of the Fifth Symposium on Computer Arithmetic, Ann Arbor, Michigan, May 1981.
Hoang V.D. A class of arithmetic burst-error-correcting codes for the Fibonacci Computer: PhD Dissertation / V.D. Hoang. – University Maryland, December 1979.
Chernov V.M. Fivonacci-Mersenne and Fibonacci-Fermat discrete transforms / V.M. Chernov, M.V. Pershina. // Boletin de Informatica of the Eduardo Mondlane University. – 1999. – Special Issue: The Golden Section: Theory and Applications. – № 9/10.
Stankovic R.S. Fibonacci Decision Diagram / R.S. Stankovic, M.R. Stankovic, J.T. Astola, K. Egizarian. – Tampere International Center for Signal Processing, 2000.
Pospelov D.A. Arithmetical foundations of computers / D.A. Pospelov. – Moscow: Vyshaja shkola, 1970. (In Russian).
Knuth D.E. The Art of Computer Programming. Vol.2. Semi-Numerical Algorithms / D.E. Knuth. – Addison-Wesley, 1969.
Tsvetkov V.D. Heart, the Golden Section, and Symmetry / V.D. Tsvetkov. – Puschino: ONTI PNZ RAU, 1997. (In Russian).
Tsvetkov V.D. The Golden Section and a Heart / V.D. Tsvetkov. Puschino: ONTI PNZ RAU, 2008. (In Russian).
Petoukhov S.V. Metaphysical aspects of the matrix analysis of genetic code and the golden section / S.V. Petoukhov // Metaphysics: Century XXI. – Moscow: BINOM, 2006 . (In Russian).
Gratia D. Quasi-crystals / D. Gratia // Uspechi physicheskich nauk. – 1988. – Vol. 156. (In Russian).
Klein F. Lectures on isosahedron and solutions to the 5th degree equation / Felix Klein. – Moscow: Nauka, 1989.
Shilo N.A. Fenotypical system of atoms for the development of Mendeleev’s ideas / N.A. Shilo, A.V. Dinkov // Moscow: Academy of Trinitarism, Electronic publication № 77 – 6567, 14630, 09.11.2007. – (http://www.trinitas.ru/rus/doc/0232/ 009a/02321073.htm).
El Nashie M.S. Is Quantum Space a Random Cantor Set with a Golden Mean Dimension at the Core? / M.S. El Nashie // Chaos, Solitons & Fractals. – 1994. - Vol. 4. – Issue 2.
El Nashie M.S. On a class of general theories for high energy particle physics / M.S. El Nashie // Chaos, Solitons & Fractals. – 2002. – Vol. 14.
El Nashie M.S. Hilbert space, Poincaré dodecahedron and golden mean transfiniteness / M.S. El Nashie // Chaos, Solitons & Fractals. – 2007. – Vol. 31. – Issue 4.
Petrunenko V.V. The Golden Section of quantum states and its astronomical and physical manifestations / V.V. Petrunenko. – Minsk: Pravo i economika, 2005. (In Russian).
Shulman M.K. Reflections on the code of a nature and on self-organizing matter. – (http://www.chronos.msu.ru/ RRE-PORTS/shulman_razmyshlenia.htm). – Title from the screen.
Stakhov A.P. The “Golden” Number Theory and New Properties of Natural Numbers / A.P. Stakhov // British Journal of Mathematics & Computer Science. – 2015. – Vol. 11. – Issue 6. – P. 1-15.
Stakhov A.P. Hilbert’s Fours Problem as a Possible Candidate on the MILLENNIUM PROBLEM in Geometry / A.P. Stakhov, S.Ch. Aranson // British Journal of Mathematics & Computer Science. – 2016. – Vol. 12. – Issue 4. – P.1 -25.
Stakhov A.P. The Fine-Structure Constant as the Physical-Mathematical MILLENNIUM PROBLEM / A.P. Stakhov, S.Ch. Aranson // Physical Science International Journal. – 2016. – Vol. 9. – Issue 1. – P.1 - 36.
Stakhov A.P. Fibonacci p-codes and Codes of the “Golden” p-proportions: New Informational and Arithmetical Foundations of Computer Science and Digital Metrology for Mission-Critical Applications / A.P. Stakhov // British Journal of Mathematics & Computer Science. – 2016.
Stakhov A.P. Assisted by Scott Olsen. The “Golden” Non-Euclidean Geometry. Hilbert’s Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant / A.P. Stakhov, S.Ch. Aranson. – London; New Jersey; Singapore; Beijing; Shanghai; Hong Kong; Taipei; Chennai: World Scientific , 2016.
Mitropolsky Y.A. Commentary on the Scientific Research of Ukrainian Scientist Professor, Doctor of Computer Science Alexey Stakhov / Y.A. Mitropolsky // Academy of Trinitarism, Мoscow: № 77 – 6567, publication 12452, 2005. – (http://www.trinitas.ru/rus/doc/0232/006a/02320005.htm). – Title from the screen.
Olsen S.A. Professor Alexey Stakhov is an absolute genius of modern science (in Honor of Alexey Stakhov’s 70th Birthday) / Scott A. Olsen // Academy of Trinitarism, Мoscow: № 77 – 6567, publication 15281, 2009. – (http://www.trinitas.ru/rus/doc/ 0232/012a/02322061.htm). – Title from the screen.
