We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid’s “ Ele ments ” . According to “Proclus hypothesis ”, the main goal of Euclid was to create a full geometric theory of Pla tonic solids, associated with the ancient conception of the “ Universe Harmony ” . We consider the main periods in the development of the “Mathematics of Harmony” and its main mathematical results: algorithmic measurement theory, number systems with irrational bases and their applications in computer science, the hyperbolic Fibo nacci functions, following from Binet’s formulas, and the hyperbolic Fibonacci l -functions ( l = 1, 2, 3, … ) , fol lowing from Gazale’s formulas, and their applications for hyperbolic geometry, in particular, for the solution of Hilbert’s Fourth Problem.
KeywordsGolden RatioPascal’s Triangle and Fibonacci NumbersBinet’s FormulasGazale’s FormulasHyperbolic Fibonacci FunctionsNumber Systems with Irrational BasesFibonacci &lambda-Numbers“Metallic Proportions”Hilbert’s Fourth Problem
A. P. Stakhov, “The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science,” World Scientific (International Publisher), Singapore City, 2009.
S. Olsen, “The Golden Section: Nature’s Greatest Secret,” Walker (Publishing Company), New York, 2006.
A. P. Stakhov, “The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics,” Congressus Numerantium, Vol. 193, Utilitas Mathematica, Winnipeg, 2008, pp. 5-48.
E. M. Soroko, “Structural Harmony of Systems,” Nauka i Tekhnika, Minsk, 1984.
O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Art,” Svit, Lvov, 1994.
S. V. Petoukhov, “Metaphysical Aspects of the Matrix Analysis of Genetic Code and the Golden Section,” Metaphysics: Century XXI, BINOM, Moscow, 2006, pp. 216-250.
“Academician Mitropolsky’s Commentary on the Scientific Research of the Ukrainian Scientist Doctor of Engineering Sciences Professor Alexey Stakhov,” In: A. Stakhov, Ed., The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, World Scientific, Singapore, 2009.
A. P. Stakhov, “The Golden Section in the Measurement Theory,” Computers & Mathematics with Applications, Vol. 17, No. 4-6, 1989, pp. 613-638. http://dx.doi.org/10.1016/0898-1221(89)90252-6
A. P. Stakhov, “Application of Natural Redundancy of Fibonacci’s Number Systems for Computer Check,” Automation and Computer Engineering, Vol. 6, 1975, pp. 80-87.
A. P. Stakhov, “The Golden Ratio in Digital Technology,” Automation and Computer Technology, Vol. 1, 1980, pp. 27-33.
A. P. Stakhov, “The Golden Section and Modern Harmony Mathematics,” Applications of Fibonacci Numbers, Vol. 7, Kluwer Academic Publishers, Dordrecht, 1998, pp. 393-399. http://dx.doi.org/10.1007/978-94-011-5020-0_43
A. P. Stakhov, “Introduction into Algorithmic Measurement Theory,” Soviet Radio, Moscow, 1977.
A. P. Stakhov, “Algorithmic Measurement Theory,” Nauka, Moscow, 1979.
A. P. Stakhov, “Codes of the Golden Proportion,” Radio and Communication, Moscow, 1984.
A. P. Stakhov, “Noise-Tolerant Codes. Fibonacci Computer,” Znanie, Moscow, 1989.
A. Stakhov, V. Massingue and A. Sluchenkova, “Introduction into Fibonacci Coding and Cryptography,” Osnova, Kharkov, 1999.
A. P. Stakhov and I. S. Tkachenko, “Hyperbolic Fibonacci Trigonometry,” Reports of the Ukrainian Academy of Sciences, Vol. 208, No. 7, 1993, pp. 9-14.
A. P. Stakhov, “A Generalization of the Fibonacci Q-Matrix,” Reports of the National Academy of Sciences of Ukraine, Vol. 9, 1999, pp. 46-49.
A. P. Stakhov, “Brousentsov’s Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical Arithmetic,” The Computer Journal, Vol. 45, No. 2, 2002, pp. 221-236. http://dx.doi.org/10.1093/comjnl/45.2.221
A. P. Stakhov, “Hyperbolic Fibonacci and Lucas Functions: A New Mathematics for the Living Nature,” ITI, Vinnitsa, 2003.
A. Stakhov and B. Rozin, “On a New Class of Hyperbolic Functions,” Chaos, Solitons & Fractals, Vol. 23, No. 2, 2004, pp. 379-389. http://dx.doi.org/10.1016/j.chaos.2004.04.022
A. Stakhov and B. Rozin, “The Golden Shofar,” Chaos, Solitons & Fractals, Vol. 26, No. 3, 2005, pp. 677-684. http://dx.doi.org/10.1016/j.chaos.2005.01.057
A. Stakhov, “The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science, and Engineering,” Chaos, Solitons & Fractals, Vol. 26, No. 2, 2005, pp. 263-289. http://dx.doi.org/10.1016/j.chaos.2005.01.038
A. P. Stakhov, “The Generalized Golden Proportions and a New Approach to Geometric Definition of a Number,” Ukrainian Mathematical Journal, Vol. 56, No. 8, 2004, pp. 1143-1150. http://dx.doi.org/10.1007/s11253-005-0064-3
A. Stakhov, “Fundamentals of a New Kind of Mathematics Based on the Golden Section,” Chaos, Solitons & Fractals, Vol. 27, No. 5, 2006, pp. 1124-1146. http://dx.doi.org/10.1016/j.chaos.2005.05.008
A. Stakhov and B. Rozin, “The ‘Golden’ Algebraic Equations,” Chaos, Solitons & Fractals, Vol. 27, No. 5, 2006, pp. 1415-1421. http://dx.doi.org/10.1016/j.chaos.2005.04.107
A. Stakhov, “The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science, and Engineering,” Chaos, Solitons & Fractals, Vol. 26, No. 2, 2005, pp. 263-289. http://dx.doi.org/10.1016/j.chaos.2005.01.038
A. Stakhov and B. Rozin, “Theory of Binet Formulas for Fibonacci and Lucas p-Numbers,” Chaos, Solitons & Fractals, Vol. 27, No. 5, 2006, pp. 1162-1177. http://dx.doi.org/10.1016/j.chaos.2005.04.106
A. Stakhov and B. Rozin, “The Continuous Functions for the Fibonacci and Lucas p-Numbers,” Chaos, Solitons & Fractals, Vol. 28, No. 4, 2006, pp. 1014-1025. http://dx.doi.org/10.1016/j.chaos.2005.08.158
A. Stakhov, “Fibonacci Matrices, a Generalization of the ‘Cassini Formula’, and a New Coding Theory,” Chaos, Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 56-66. http://dx.doi.org/10.1016/j.chaos.2005.12.054
A. P. Stakhov, “The Generalized Golden Proportions, a New Theory of Real Numbers, and Ternary Mirror-Symmetrical Arithmetic,” Chaos, Solitons & Fractals, Vol. 33, No. 2, 2007, pp. 315-334. http://dx.doi.org/10.1016/j.chaos.2006.01.028
A. Stakhov and B. Rozin, “The ‘Golden’ Hyperbolic Models of Universe,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 159-171. http://dx.doi.org/10.1016/j.chaos.2006.04.046
A. Stakhov, “The ‘Golden’ Matrices and a New Kind of Cryptography,” Chaos, Solitons & Fractals, Vol. 32, No. 3, 2007, pp. 1138-1146. http://dx.doi.org/10.1016/j.chaos.2006.03.069
A. P. Stakhov, “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the ‘Golden’ Cryptography,” Academy of Trinitarizam, Moscow, 2006. http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm
A. P. Stakhov and B. N. Rozin, “The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature,” Visual Mathematics, Vol. 8, No. 3, 2006. https://eudml.org/doc/256745
A. Stakhov, “A Generalization of the Cassini Formula,” Visual Mathematics, Vol. 14, No. 2, 2012. http://www.mi.sanu.ac.rs/vismath/stakhovsept2012/cassini.pdf
A. Stakhov, “On the General Theory of Hyperbolic Functions Based on the Hyperbolic Fibonacci and Lucas Functions and on Hilbert’s Fourth Problem,” Visual Mathematics, Vol. 15, No. 1, 2013. http://www.mi.sanu.ac.rs/vismath/pap.htm
A. Stakhov and S. Aranson, “Hyperbolic Fibonacci and Lucas Functions, ‘Golden’ Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem, Part I. Hyperbolic Fibonacci and Lucas Functions and ‘Golden’ Fibonacci Goniometry,” Applied Mathematics, Vol. 2, No. 1, 2011, pp. 74-84. http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.21009
A. Stakhov and S. Aranson, “Hyperbolic Fibonacci and Lucas Functions, ‘Golden’ Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem, Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry),” Applied Mathematics, Vol. 2, No. 2, 2011, pp. 181-188. http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.22020
A. Stakhov and S. Aranson, “Hyperbolic Fibonacci and Lucas Functions, ‘Golden’ Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem, Part III. An Original Solution of Hilbert’s Fourth Problem,” Applied Mathematics, Vol. 2, No. 3, 2011, pp. 283-293. http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.23033
A. P. Stakhov, “Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature,” Applied Mathematics and Physics, Vol. 1, No. 3, 2013, pp. 60-66. http://www.scirp.org/journal/jamp/.
S. Blackburn, “Harmony of Spheres: The Oxford Dictionary of Philosophy,” Oxford University Press, Oxford, 1994, 1996, 2005.
V. Dimitrov, “A New Kind of Social Science. Study of Self-Organization of Human Dynamics,” Lulu Press, Morrisville, 2005.
N. N. Vorobyov, “Fibonacci Numbers,” Nauka, Moscow, 1961.
V. E. Hoggat, “Fibonacci and Lucas Numbers,” Houghton-Mifflin, Palo Alto, 1969.
S. Vajda, “Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications,” Ellis Horwood limited, Chichester, 1989.
G. D. Grimm, “Proportionality in Architecture,” ONTI, Leningrad-Moscow, 1935.
V. P. Shenyagin, “Appeal to the International Scientific Community: On the Appropriateness of Nomination of Professor Stakhov AP to Award the Abel Prize in Mathematics in 2014,” Academy of Trinitarizm, Moscow, 2013. http://www.trinitas.ru/rus/doc/0001/005a/00011304.htm
S. Olsen, “Professor Alexey Stakhov Is an Absolute Genius of Modern Science (in Honor of Alexey Stakhov’s 70th Birthday),” Academy of Trinitarizm, Moscow, 2009. http://www.trinitas.ru/rus/doc/0232/012a/02322061.htm
E. M. Soroko, “A Review on the Book ‘The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science’,” In: A. Stakhov, Ed., Academy of Trinitarizm, Moscow, 2010. http://www.trinitas.ru/rus/doc/0232/100a/02320065.htm
S. K. Abachiev, “The Mathematics of Harmony through the Eyes of Specialist in History and Methodology of Science,” Academy of Trinitarizm, Moscow, 2007. http://www.trinitas.ru/rus/doc/0232/009a/02321185.htm
A. A. Zenkin, “The Mistake by Georg Cantor,” Problems of Philosophy, Vol. 2, 2000, pp. 165-168.
D. Polya, “Mathematical Discovery,” Nauka, Moscow, 1970.
G. Bergman, “A Number System with an Irrational Base,” Mathematics Magazine, Vol. 31, 1957, pp. 98-119. http://dx.doi.org/10.2307/3029218
A. P. Stakhov, “Codes of the Golden Proportion, or the Number Systems for Future Computers?” Technology for Young People, Vol. 7, 1985, pp. 40-44.
A. N. Kolmogorov, “Mathematics in Its Historical Development,” Nauka, Moscow, 1991.
V. de Spinadel, “The Family of Metallic Means,” Visual Mathematics, Vol. 1, No. 3, 1999. http://members.tripod.com/vismath/
M. J. Gazale, “Gnomon. From Pharaohs to Fractals,” Princeton University Press, Princeton, 1999.
J. Kappraff, “Connections. The Geometric Bridge between Art and Science,” 2nd Edition, World Scientific, Singapore, 2001. http://dx.doi.org/10.1142/4668
A. A. Tatarenko, “The Golden Tm-Harmonies’ and Dm–Fractals,” Academy of Trinitarism, Moscow, 2005. http://www.trinitas.ru/rus/doc/0232/009a/02320010.htm
H. Arakelyan, “The Numbers and Magnitudes in Modern Physics,” Armenian Academy of Sciences, Yerevan, 1989.
V. P. Shenyagin, “Pythagoras, or How Everyone Creates His Own Myth,” The Fourteen Years after the First Publication of the Quadratic Mantissa’s Proportions. Academy of Trinitarism, Moscow, 2011. http://www.trinitas.ru/rus/doc/0232/013a/02322050.htm
S. Falcon, “Angel Plaza, On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624.
E. Lucas, “The Theory of Simply Periodic Numerical Functions,” American Journal of Mathematics, Vol. 1, No. 2, 1878, pp. 184-240, 289-321. http://www.fq.math.ca/Books/Complete/simply-periodic.pdf http://dx.doi.org/10.2307/2369308
“Hilbert’s Problems,” from Wikipedia, the free Encyclopedia. http://en.wikipedia.org/wiki/Hilbert's_problems
“Hilbert’s Fourth Problem,” From Wikipedia, the free Encyclopedia. http://en.wikipedia.org/wiki/Hilbert's_fourth_problem
“Lecture ‘Mathematical Problems’ by Professor David Hilbert,” 1900. http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob4
A. V. Pogorelov, “Hilbert’s Fourth Problem,” Nauka, Moscow, 1974.
B. H. Yandell, “The Honors Class-Hilbert’s problems and Their Solvers,” A. K. Peters/CRC Press, Boston, 2003.
M. Kline, “The Loss of Certainty,” Oxford University Press, New York, 1980.