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Alexey Stakhov: Fibonacci of XXI century. Mathematics of Harmony and Generalized Golden Section Principle

Alexey Stakhov: disharmony of harmonist

Dear Alexey Stakhov,

You mentioned in your letter (below) two questions, which I answer: 1. The GHA Honorary Title "World Harmony Creator" that you assessed as "silly joke" and 2. Your rejection now (2016) from this title (2011) under the pressure of Western technocratic Russophobic opinion.

1.You cooperated with the Global Harmony Association (GHA) during three years: 2008-2011. In this time your revealed o­ne-sidedness. On the o­ne hand, you have reached the undoubted success and great result in the mathematical understanding of binary mechanical Fibonaccis harmonic proportions, basing o­n which you created your math harmony as elementary arithmetic of harmony in disputes with its opponents and supporters (Sergienko, Soroko, Kharitonov and others), which are reflected o­n the GHA website "Peace from Harmony" (http://peacefromharmony.org/?cat=ru_c&key=488) and in The ABC of Harmony (GHA, 2012: http://peacefromharmony.org/?cat=en_c&key=478). You entered in the GHA Board, became an honorary member and coauthor of Harmonious Civilization (2009:http://peacefromharmony.org/?cat=en_c&key=404). Your achievements in its development, despite the resistance of traditional mathematics, have been observed up the GHA Highest Honorary Title "World Harmony Creator" (WHC) in 2011. This title was also assigned to the outstanding figures frommany areas of ​​the world harmony, for example, President of India, APJ Abdul Kalam; Prince of Wales; peacemakers, Johan Galtung and Noam Chomsky; politics, Vladimir Putin and Nursultan Nazarbayev and others (http://peacefromharmony.org/?cat=en_c&key=513).

On the other hand, you have been unable and powerless to go beyond the binary mechanical arithmetic of harmony and rise to the level of multidimensional social tetra-mathematics of harmony four-dimensional algebra and topology of spheral fractals in tetrasociology that noted in The ABC of Harmony (p. 52-53, etc.). An obstacle of that was your socio-cultural ignorance and caused it technocratic narrowness of your thinking, which ultimately led you to Russophobia and support of Ukrainian Nazism. Therefore, your evaluation of the Honorary Title WHC is ignorant and insignificant. It cannot be taken into account. Not you and your advocates but history will judge it.

2. Your claim to rewrite the history of GHA is ridiculous, ignorant joke. Let me remind you that any history, including the GHA history and your cooperation with it, cannot be rewritten o­n any request, including you. Manipulation of facts is the prerogative of dishonest people and doomed systems, including militaristic Western civilization, in the service of which you desperately want to put your math harmony and get in it the dividends. This math is organically unacceptable to this civilization, which is subject to violence and extinction. Prof. Tom Hastings succinctly expressed it in relation to its flagship - US: "We are waging war. We are the Nation of War. We destroy. We kill. Everyone fears us. Fewer and fewer admire us "(http://peacefromharmony.org/?cat=en_c&key=673).

In the same manipulation field is your advocate - unknown Belgian mathematician - reviewer, who proved himself as a liberal zombie of this civilization (The Zombie Doctrine by George Monbiot: http://peacefromharmony.org/?cat=en_c&key=684). I'm sad to see the scientists like you and your reviewer as disabled advocacy zombies.

After the Congress in Odessa in October 2010(as and before) you wrote enthusiastically to me and the GHA members: "Dear Lev! I admire your excellent report o­n the Odessa Congress and your wonderful photographs evoke pleasant memories of Odessa. And of course, I thank you for your appreciation of my scientific direction" (Http://peacefromharmony.org/?cat=ru_c&key=471). Now you renounce from harmonious memory in the name of Western militarism and Russophobia, awkwardly trying to dance to their tune. It is your right. But it does not honor you and dirties your math harmony. I am ashamed for you and your ridiculous claim to rewrite the GHA history because the historical facts are the scientific facts, and they cannot be transformed in favor of any person, including you. (Suddenly, tomorrow, you will again change your mind and again will require changes, restore you to the honorary title, regretting the refusal of it ...)

Therefore, any changes o­n the GHA website for your (or any) request do not expect. I will ignore all your requests of this nature as a funny joke of teenager, playing in "war games". Your ignorant hostility toward social harmony and its science - Tetrasociology and Global Peace Science from Harmony, marching around the continents (http://peacefromharmony.org) together with its high World Harmony Creators and Nobel Peace Laureatesis powerless to stop it. You cannot understand your actions and their ambiguous meaning. It is also a fact of history that exhausts our incident. On this adieu. Sincerely,
Dr. Leo Semashko, GHA Founder and Honorary President,

PS. My answer, along with your letter was published o­n your personal page: http://peacefromharmony.org/?cat=en_c&key=330

Dear Leo Semashko!

         I would like to inform you that my new book The Golden Non-Euclidean Geometry (World Scientific, 2016) has been reviewed o­n European Mathematical Society Newsletter. Please see: http://www.euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry The concluding citation of the review is the following:

The involvement of Stakhov with organizations such as the Global Harmony Association that awarded him the highest honory title of World Harmony Creator does not add to the credibility.

          That is, the reviewer Prof. Adhemar Bultheel accuses me that I collaborate with your organization Global Harmony Association.

You, Dear Leo, know that this award is your initiative and I cannot any relation to your Global Harmony Association and I did not collaborate with Global Harmony Association. This honorary title is your silly joke.

I do not need this honorary title, which is a shame for me in the eyes of the scientific community. 5 years ago I had asked you to remove the web site http://www.peacefromharmony.org/?cat=en_c&key=330 from Internet.

          I citate:

Alexey Stakhov asked to remove his from the GHA from May 11, 2011 in connection with a serious illness of his wife.

Also I would like to remove my name from all sites of Global Harmony Association, in particular, from the site: http://www.peacefromharmony.org/?cat=en_c&key=513

         Also I would like to ask Prof, Adhemar Bultheel to remove the citation about my participation in the Global Harmony Association from the review http://www.euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry

This honorary title is a silly joke of Leo Semashko.
Alexey Stakhov

Alexey Stakhov

Global Harmony Association Board Member

World Harmony/Peace Academy General Directorate Member

The Global Harmony Association highest Honorary Title:World Harmony Creator

Alexey Stakhov asked to remove his from the GHA since May 11, 2011 in connection with a serious illness of his wife.

Grand PhD in Computer Science, Professor

President of the International Club of the Golden Section

In Russian: http://peacefromharmony.org/?cat=ru_c&key=350 

Coathor of the GHA book Harmonious Civilization (2009):
http://peacefromharmony.org/?cat=en_c&key=404 (p. 251 and etc.)


Congress o­n Mathematics of Harmony, Odessa, October 8-10,2010
In Russian: http://peacefromharmony.org/?cat=ru_c&key=471
In English: http://peacefromharmony.org/?cat=en_c&key=441


Harmonious Civilization Spiritual Culture
Citizens of Earth!
Unite in harmony for love, peace, justice, fraternity and happiness!


Global Harmony Association (GHA)

Since 2005, GHA is an international NGO uniting more 460 members in 56 countries and more than o­ne million participants from the GHA collective members in 80 countries. GHA Founder and President is Dr. Leo Semashko, Address: 7/4-42 Ho-Shi-Min Street, St. Petersburg, 194356, Russia

Email: leo.semashko@gmail.com; GHA 1st Vice-president is Dr. Ernesto Kahan: Israel; GHA-USA President is Dr. Laj Utreja; GHA-Japan President is Kae Morii ;
GHA Web: www.peacefromharmony.org 
GHA 32 Projects List: http://www.peacefromharmony.org/?cat=en_c&key=472

GHA Mission from harmony to peace; or: Pave the scientific, based o­n harmony ABC, way for a harmonious civilization with the priority of spiritual culture and gradual rejection from material priority

Of Harmonious Era Calendar
June 21 - Global Harmony Day;

November 3, 2009 - the birthday of a harmonious civilization and the beginning of mankind Harmonious Era


GHA confersthe High Honor

World Harmony Creator


Alexey Stakhov, Professor

GHA Board Member

In connection with the 72-th anniversary, May 7, 2011

For his unprecedented contribution to:


Creation and development of the mathematics of harmony as the arithmetic of a harmonious civilization since 1996.

Publication of the fundamental book "Mathematics of Harmony" (2009, English), which became o­ne of the system signs of birth of a harmonious civilization in 2009, along with The ABC of harmony.

Organization and conduct of the First International Congress o­n Mathematics of harmony in Odessa University, October 2010.

As the harmony arithmetic was born in your theory, so ABC of social and individual harmony was born in the tetrasociological theory. Both are created in the GHA and constitute the core and foundation of the spiritual culture of harmonious civilization.

Harmonious civilization


Dear Alexey!

The GHA happy to congratulate you o­n your birthday; wish you the best health and new achievements in the development of your remarkable mathematics of harmony!

Your arithmetic of harmony with tetrasociological ABC of harmony are the core and foundation for the spiritual culture of harmonious civilization! GHA rightfully recognized you in the GHA Highest Honorary Title: WORLD HARMONY CREATOR that confirms the GHA appropriate certificate in the attachment! It is published o­n your personal page o­n the GHA website:

The GHA hopes for your active participation in the synthesis of arithmetic and alphabet of harmony that will be the greatest spiritual victory for humanity. It will win over all the wars and other pathologies of the industrial civilization. This will be the Harmonious Civilization Victory!

The GHA pleased to congratulate you and all GHA members with Victory Day o­n May 9 as a precursor for Harmonious Spiritual Victory of Humanity!

Dr. Leo Semashko,
GHA Founder and President,
May 7, 2011



A list of Stakhovs scientific publications:


1. Alexey Stakhov was born o­n May 7 at village Partisany of Kherson region (Ukraine)

2. Education. Since 1956 up to 1959 Alexey Stakhov was a student of Kiev Polytechnic Institute (now National Technical University Kiev Polytechnic Institute). He completed his engineering education in Kharkov Aviation Institute (now National Aero Cosmic University) during 1959-1961.

3. Scientific degrees.PhD in Technical Cybernetics (1966), Associate Professor (1968), Grand PhD in Computer Science (1972), Full Professor (1974), Academician of the Ukrainian Academy of Engineering Sciences (1992), Academician of the International Higher Education Academy of Sciences (2006)

4. Summary of Scientific Career.

O Assistant-professor, Associate Professor of Kharkov Institute for Radio Electronics (now National University for Radio Electronics), 1966 1971.

O Chairman of the Department of Information and Measurement System of Taganrog Radio Engineering Institute (now Taganrog State Radio Engineering University), 1971 1977

O Chairman of the Computer Engineering Department of the Vinnitsa Polytechnic Institute (now Vinnitsa National Technical University), 1977 1988.

O Director of Special Design Office Module at the Vinnitsa Technical University, 1986 1989.

O Chairman of the Applied Mathematics and Computer Systems Department of the Vinnitsa Polytechnic Institute (now Vinnitsa National Technical University), 1989 1995.

O Professor of the Mathematics Department of the Vinnitsa Pedagogical University, 2001-2002.

O Chairman of the Computer Science Department of the Vinnitsa State Agricultural University, 1997-2003.

5. Training PhDs. 30 PhDs and 4 Grand PhDs in Computer Science


6. Brief summary of scientific activity, basic scientific achievements:

6.1. He created a new direction in measurement theory Algorithmic Measurement theory. The foundations of this theory are stated in Stakhovs book Introduction into Algorithmic Theory (Moscow, Soviet Radio, 1977)

6.2. He created a new direction in computer engineering new number systems based o­n Fibonacci numbers and the Golden Mean -and put forward the Project of Fibonacci Computers. A World priority in this direction was protected by 65 patents of USA, Japan, England, Germany, France, Canada and other countries.The foundations of new computer theory are stated in Stakhovs books Introduction into Algorithmic Theory (Moscow, Soviet Radio, 1977) and Codes of the Golden Proportion (Moscow, Radio and Communications, 1984)

6.3. He developed a new computer arithmetic Ternary Mirror-symmetrical Arithmetic(see Alexey Stakhov Brousentsovs Ternary Principle, Bergmans Number System and Ternary Mirror-symmetrical Arithmetic, The Computer Journal, Vol. 45, No 2, 2002, 221-236)

6.4. He generalized the Golden Section Problem and discovered a new class of irrational numbers the Golden p-Proportions, which express some important features of Pascal Triangle (see Alexey Stakhov The generalized golden proportion, a new theory of real numbers , and ternary mirror-symmetrical arithmetic, Chaos, Solitons and Fractals, 33 (2007), 315-334)

6.4. He developed a theory of the golden algebraic equations (see Alexey Stakhov and Boris Rozin The golden algebraic equations, Chaos, Solitons and Fractals, 27 (2006), 1415-1421).

6.5. He developed a theory of the Fibonacci matrices and the golden matrices and created a new coding theory based o­n the Fibonacci and golden matrices. (see Alexey Stakhov Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos, Solitons and Fractals, 30 (2006), 56-66; Alexey Stakhov The golden matrices and a new kind of cryptography, Chaos, Solitons and Fractals, 32 (2007), 1138-1146).

6.6. He developed a new class of hyperbolic functions hyperbolic Fibonacci and Lucas functions (see Alexey Stakhov and Boris Rozin On a new class of hyperbolic functions, Chaos, Solitons and Fractals, 23 (2005), 379-389). From here it follows a continuous approach to the Fibonacci numbers theory and new theories of mathematics and theoretical physics (the golden Lobachevskis geometry and the golden Minkowskis geometry hyperbolic interpretation of Einsteins special theory of relativity).

6.7. He developed a new scientific principle the Generalized Golden Section Principle, which includes the Dichotomy principle and the classical Golden Section Principle as partial cases (see Alexey Stakhov The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering, Chaos, Solitons and Fractals, 26 (2005), 263-289).

6.8. He generalized Binet formulas and discovered a new class of recurrence numerical sequences the generalized Fibonacci and Lucas p-numbers (Alexey Stakhov and Boris Rozin Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos, Solitons and Fractals, 27 (2006), 1162-1177)

6.9. He developed Fibonacci-Lorentz transformations and new approach to Einsteins theory of relativity (see Alexey Stakhov and Samuil Aranson The golden Fibonacci goniometry, Fibonacci-Lorentz transformations, and the Fourth Hilbert Problem // Academy of Trinitarism, Moscow, 77-6567, publication 14816, 04.06.2008 (in Russian) http://www.trinitas.ru/rus/doc/0232/004a/02321087.htm

6.10. In 1996 he developed a concept of the Mathematics of Harmony as a new interdisciplinary direction, which concerns the foundations of mathematics, theoretical physics and computer science (see Alexey Stakhov The Golden Section and Modern Harmony Mathematics, Applications of Fibonacci Numbers, Vol. 7, 393-399). This theory is stated in Stakhovs book The Mathematics of Harmony, From Euclid to Contemporary Mathematics and Computer Science accepted for publication by the World Scientific (see http://www.worldscibooks.com/mathematics/6635.html,
. The
widened advertisement of the book was published in the International
electronic journal "Visual Mathematics"

6.11. In 2001 he created o­n the Internet the web site Museum of Harmony and Golden http://www.goldenmuseum.com/ (together with Anna Sluchenkova). The Museum is a unique collection of all creations of Nature, Science and Art based o­n the Golden Section.


6.12. The most important publications:


6.12.1 The books:


1. Introduction into Algorithmic Measurement Theory (Moscow: Soviet Radio, 1977) (in Russian)

2.Algorithmic Measurement Theory (Moscow: Znanie, 1979) (in Russian)

3.Codes of the Golden Proportion (Moscow: Radio and Communications, 1984) (in Russian)

4.Introduction into Fibonacci coding and cryptography by Alexey Stakhov, Vinancio Massimngua, Anna Sluchenkova (Kharkov: Osnova, 1999)

5.Sacred Geometry and the Mathematics of Harmony (Vinnitsa: ITI, 2003) (in Russian)

6.New Mathematics for Living Nature: Hyperbolic Fibonacci and Lucas Functions (Vinnitsa: ITI, 2003) (in Russian)

7.Da Vinci Code and Fibonacci Series by Stakhov A.P., Sluchenkova A.A.. Scherbakov I.G. (Sanct-Petersburg: Piter, 2006) (in Russian)

8.The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2008)


6.12.2. The Recent Publications in the Prestige Journals:


1.The Golden Section in the Measurement Theory, Computers & Mathematics with Applications, 1989, Vol. 17, No 4-6, 613-638.

2.The Golden Section and Science of System Harmony, Gazette of the Ukrainian Academy of Sciences, 1991, No 12

3.Fibonacci hyperbolic trigonometry by Alexey Stakhov and Ivan Tkachnko, 1993, Vol. 208, 7, 1993, 9-14 (in Russian). The article was recommended for publication by academician Mitropolski.

4.The Golden Section and Modern Harmony Mathematics, Applications of Fibonacci Numbers, Vol. 7, Kluwer Academic Publishers, 1998, 393-399. The article was written o­n the base of the lecture The Golden Section and Modern Harmony Mathematics presented by Alexey Stakhov at the 7th International Conference Fibonacci Numbers and Their Applications (Austria, Graz, July 1996)

5.A generalization of the Fibonacci Q-matrix, Reports of the Ukrainian Academy of Sciences, 1999, No 9, 46-49

6.Generalized Golden Sections and a new approach to geometric definition of a number. Ukrainian Mathematical Journal, 2004, Vol. 56, No. 8, 1143-1150 (inRussian).

7.Brousentsovs Ternary Principle, Bergmans Number System and Ternary Mirror-symmetrical Arithmetic, The Computer Journal (British Computer Society), 2002, Vol. 45, No. 2, 221-236. Prof. Donald Knut, the outstanding American scientist in Computer Science, Professor Emeritus of the Art of Computer Programming[2] at Stanford University, gave very positive review o­n this article.

8.O­n a new class of hyperbolic function by Stakhov A., Rozin B. Chaos, Solitons & Fractals, 2004, V. 23, No.2, 379-389.

9.The Generalized Principle of the Golden Section and its Applications in Mathematics, Science and Engineering.Chaos, Solitons & Fractals, 2005, 26, 263-289.

10. The Golden Shofar by Stakhov A., Rozin B. Chaos, Solitons & Fractals, 2005, 26, 677-684

11. Theory of Binet formulas for Fibonacci and Lucas p-numbers by Stakhov A., Rozin B. Chaos, Solitons & Fractals, 2006, Vol. 27, 1162-1177

12.The golden section, secrets of the Egyptian civilization and harmony mathematics. Chaos, Solitons & Fractals, 2006, Vol. 30, 490-505

13.The golden hyperbolic models of Universe by Stakhov A., Rozin B.Chaos, Solitons & Fractals, 2007, Volume 34, 159-171.

14.The Golden Section, Fibonacci series and new hyperbolic models of nature by Stakhov A., Rozin B.. Visual Mathematics, 2006, Vol. 8, No 3, http://www.mi.sanu.ac.yu/vismath/stakhov/index.html

15.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. Visual Mathematics, 2007, Vol. 9, No 3 http://www.mi.sanu.ac.yu/vismath/stakhov2007/stakhov1.htm

16. Gazale formulas, a new class of the hyperbolic Fibonacci and Lucas functions, and the improved method of the golden cryptography. Moscow: Academy of Trinitarism, 77-6567, publication 14098, 21.12.2006 http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm

17.The "Golden" Algebraic Equations by Stakhov A., Rozin B. Chaos, Solitons & Fractals 2006, Vol. 27, 1415-1421

18.Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos, Solitons and Fractals, 2006, Vol. 30, 56-66

19.The continuous functions for the Fibonacci and Lucas p-numbers by Stakhov A., Rozin B.Chaos, Solitons and Fractals, 2006,Vol.28, 1014-1025

20.Fundamentals of a new kind of mathematics based o­n the Golden Section, Chaos, Solitons and Fractals, 2006, Vol. 27, 1126-1146

21.The golden matrices and a new kind of cryptography, Chaos, Solitons and Fractals, 2007, Vol. 32, 1138-1146

22.The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic.Chaos, Solitons and Fractals, 2007, Vol.33, 315-334

23.The golden section, sacred geometry and harmony mathematics by Alexey Stakhov. The book Metaphysics: Century XXI. Moscow: Publishing House BINOM, 2006, 174-215 (in Russian).

24.The Mathematics of Harmony as a new interdisciplinary direction of modern science and its applications. Proceedings of the International Higher Education Academyof Sciences, 2006, No 2 (36), 52-64 (in Russian)

25.Three key problems of mathematics o­n the stage of its origin and the Harmony Mathematics as alternative way of mathematics development. Mathematics & Design. Fifth International Mathematics & Design Conference V M&D, 2007, 88-102

26.Design of a new coding theory and cryptography based o­n the Fibonacci and golden matrices. Mathematics & Design. Fifth International Mathematics & Design Conference V M&D, 2007, 154-162

27.Three key problems of mathematics o­n the stage of its origin and the Harmony Mathematics as alternative way of mathematics development. Totalogy-XXI, Kiev, National Academy of Sciences of Ukraine, 2007, No 17/18, 274-322

28. The golden Fibonacci goniometry, Fibonacci-Lorentz transformations, and the Fourth Hilbert Problem by Alexey Stakhov and Samuil Aranson // Academy of Trinitarism, Moscow, 77-6567, publication 14816, 04.06.2008 (in Russian) http://www.trinitas.ru/rus/doc/0232/004a/02321087.htm


7. The most important scientific lectures by Alexey Stakhov:


1. Lecture Algorithmic Measurement Theory and New Computer Aritmetics at the joint meeting of the Austrian Computer and Cybernetics Societies (Vienna, 1976). This lecture was a cause of the wide patenting of Stakhovs inventions in Fibonacci computers abroad (USA, Japan, England, Germany, France, Canada and other countries)

2. Lecture Algorithmic Measurement Theory at the plenary session of the International symposium Intellectual Measurements (Germany, Jean, 1986)

3. Lecture Fibonacci computers as new scientific direction at the meeting of the Ukrainian Academy of Sciences (Kiev, 1989)

4. Lecture The Golden Section and Modern Harmony Mathematics at the 7th International Conference Fibonacci Numbers and Their Applications (Austria, Graz, 1996)

5. Lecture The Mathematics of Harmony as a new direction in mathematics at the meeting of the Ukrainian Mathematical Society (Kiev, 1998)

6. 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 Moscow University (Moscow, 2003) http://www.goldenmuseum.com/20ReportPres_rus.html

7. Lecture o­n the plenary session of the International conference Problems of Harmony, Symmetry and the Golden Section in Nature, Science and Art (Vinnitsa, 2003)

8. Lecture at the meeting of the Alliance of Technology and Science Specialists of Toronto (Canada, Toronto, 2004)

9. Lecture The Mathematics of Harmony and New Innovational Projects for Modern Computer Science at the Innovation Forum -2008 (Canada, Toronto, 2008)

  1. Scientific awards, a work in foreign universities, international recognition of scientific direction:


8.1. Visiting-Professor:

Vienna Technical University(Austria, 1976),

Fridrich-Shiller University of Jena(Germany, 1986)

Dresden Technical University(Germany, 1988)

Al Fateh University(Tripoli, Libya, 1995-1997),

Eduardo Mondlane University(Maputo, Mozambique, 1998-2000),

Taganrog Radio Engineering University(Russia, 2001),

Kharkov National Aero Cosmic University(Ukraine, 2002),

Kharkov National University(Ukraine, 2003).


8.2. Prizes, awards, international recognition:

-Prize of the Ukrainian Ministry of Education for the best scientific publication (Brochure Algorithmic Measurement Theory), 1980.

-Honorary Medal by Henrich Barkhausen given by the Dresden Technical University as Visiting-Professor of the Henrich Barkhausen Department (1988)

-Prize for the best scientific paper of 1990 given by the Journal Gazette of the Ukrainian Academy of Sciences (1990)

-Biographical inclusion into the 7th Edition of the International Directory of Distinguished Leadership and into the 4th Edition of the International Whos Who of Contemporary Achievement (American Biographical Institute)

-The 2000 Millennium Medal of Honor given by the American Biographical Institute

-Honorary Diploma Man of the Year-2000 given by the Major of Vinnitsa Town (Ukraine)

-Scientific coordinator of the International conference Problems of Harmony, Symmetry and the Golden Section in Nature, Science and Art (Vinnitsa, 2003)

-President of the International Club of the Golden Section (2003)

-Honorary Professor of Taganrog Radio Engineering University (2004)

-A member of the Alliance of Technology and Science Specialists of Toronto (2004)

-A member of Shevchenko Scientific Society in Canada (2004)

-Director of the Institute of the Golden Section of Academy of Trinitarism (Russia, 2005)


Dear Leo! Dear colleagues!

I would like to express my point of view o­n the concept of "harmony" because Dr. Leo Semashko mentioned my name in his letter. Indeed, in October 2009, Publishing House "World Scientific" plans to publish my book "The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science" http://www.worldscibooks.com/mathematics/6635.html
. This large book (11 chapters, 700 pages), written by me in English, is the culmination of my 40-year-old researches in this area.

In order to exclude any misunderstanding about my "Mathematics of Harmony," I would like to limit clearly areas of applications of this mathematical theory. Here I would like to give an analogy with Shennons theory of information. After publication of Shennon's works o­n information theory, the boom associated with unduly broad application of this theory appeared. In a small note, "Bandvagon" Shannon clearly restrict the area of application of his theory for the solution of telecommunications tasks - and no more. Shennon's information has a clear quantitative definition; it is expressed through the entropy of the information source.

Speaking of "Mathematics of Harmony", we must give a clear quantitative definition of "harmony," which can be a subject of mathematical study. Is this possible? Yes, perhaps. The Russian philosopher Dr. Shestakov, o­ne of the classics of the Harmony Science ," asserts that there are, at least, three different understandings of harmony: mathematical, aesthetic and artistic.

The notion of the "mathematical harmony" goes back to the Pythagoreans, who led all things to numbers and relations between them. In mathematical sense, the harmony is defined as equality or proportionality of parts with o­ne another and with a part. This understanding was reflected in a certain numerical proportions. However, as Shestakov wrote in the book Harmony as an aesthetic category, 1973], the mathematical understanding of harmony fixed primarily quantitative definiteness of harmony, but it does not reflect an aesthetic harmony, its expressiveness, connection with beauty.

Thus, the Mathematics of Harmony is studying a harmony o­nly from quantitative, numerical point of view. The most striking examples of mathematical understanding of harmony are the "golden section", which has a strict geometric and algebraic definition, and Fibonacci numbers, which are expressed with a simple recurrence relations F (n) = F (n-1) + F (n-2). But in recent decades, several important generalizations of the "golden section" and Fibonaccis recurrence relation were obtained. First of all, this so-called p-Fibonacci numbers, which are generated by the recurrence relations F (n) = F (n-1) + F (n-p-1), where p = 0, 1, 2, 3, ....

When p = 0, this recurrence relation generates a classical binary sequence 1, 2, 4, 8, ..., and when p = 1, the classic Fibonacci numbers. A ratio of two p-Fibonacci numbers creates the golden p-proportions. This is a series of new mathematical constants: 2, 1, 618, 1,465, 1,380 ... The Belarusian philosopher Eduard Soroko used this idea (Structural Harmony of Systems, 1984) for his "Law of Structural Harmony of Systems", which greatly expanded a number of new "harmonic proportions", which is the golden p-proportions ."

Recently, a number of mathematicians (Shpinadel, Gazale, Kappraff, Tatarenko) came to another interesting generalization of the "golden proportion". Those are "metallic proportions, the roots of the simple algebraic equation x ^ 2-mx-1 = 0, where m> 0 is a given positive rational. The formula for the "metallic proportions " is expressed through the number m; while when m = 1, this formula sets the classical "golden proportion".

From these elementary mathematical reasoning, it is clear what is a object of the "Mathematics of Harmony." This is a theory of new recurrence relations, new "golden" algebraic equations and new mathematical constants, which express a more broad class of "harmonies" than the classical "golden proportion" or the classical Fibonacci numbers.

But it is not even the main thing. Applications of these results in modern mathematics, computer science and theoretical natural sciences are the most important. And here we obtained amazing results. Every scientist knows, or at least heard about hyperbolic functions, which were used by Nikolai Lobachevski in his "hyperbolic geometry". The number of e a base of natural logarithms - is their base.

A general theory of hyperbolic functions, called "golden" Fibonacci goniometry, was created in the "Mathematics of Harmony." The metallic proportions are bases of all these new hyperbolic functions. And since the number of "metallic proportions" is infinite, the number of new hyperbolic functions is infinite too.

This idea was used by Alexei Stakhov and Samuel Aranson for the solution of "Hilberts 4-th Problem, which for more than a century not to be solved in mathematics. Already this example is sufficient to prove the highest efficiency of the "Mathematics of Harmony." But there are other new results of fundamental character obtained in the "Mathematics of Harmony." This is - first and foremost, a new "golden" interpretation of Einstein's special theory of relativity and the "golden" interpretation of the Universe evolution, beginning with the "Big Bang".

But there is another unique application of the Mathematics of Harmony. This is the creation of a "Golden" Information Technology and the development of the" Golden "Scientific Revolution.

Those who want more detail to get acquainted with these results and the Mathematics of Harmony, I recommend to read the following articles:

1. A. Stakhov. Diracs Principle of Mathematical Beauty, Mathematics of Harmony and Golden Scientific Revolution http://atss.brinkster.net/Noosphere/En/Magazine/Default.asp?File=20090110_Stakhov.htm

2. A. Stakhov B. Rozin. The Golden Section, Fibonacci series and new hyperbolic models of Nature. Visual Mathematics, Volume 8, No. 3, 2006 (http://members.tripod.com/vismath/pap.htm)

3. A. Stakhov, S. Aranson. Golden Fibonacci Goniometry, Fibonacci-Lorentz Transformations, and Hilberts Fourth Problem. Congressus Numerantium, 193 (2006), pp.119-156 http://atss.brinkster.net/Noosphere/En/Magazine/Default.asp?File=20090512_Stakhov.htm

Therefore, despite the seemingly very "narrow" interpretation of the concept of "harmony" in the "Mathematics of Harmony", this "narrow" interpretation has led to the creation of a new mathematical theory, which is an alternative to the development of "classical mathematics".

I am sure that the Mathematics Harmony is just the first step in exploring such complex scientific concept as harmony. But I am sure that it is very important step in the creation of the Science of Harmony, which can transform all modern sciences and education.

Sincerely yours

Alexey Stakhov
Doctor of Sciences in Computer Science, Professor

President, International Club of the Golden Section

Director, Institute of the Golden Section, Academy of Trinitarism

Awarded by the Honored Diploma and Medal Knight of Science and Arts from the Russian Academy of Natural Sciences




Dear Friends

I enjoyed very much reading the letter of Alexey Stakhov o­n "Mathematics of Harmony" , It is great to be part of GHA, a group composed by intellectuals of such impressive level



Prof. Dr. Ernesto Kahan MD MPH

University Professor - Poet Physician

Tel Aviv University, Israel. Bar Ilan University , Israel

President- Israeli Association Writers. Spanish Branch

Vice President- World Academy of Arts and Culture USA

Former Vice President of IPPNW (Nobel Peace Prize) and the actual president of theIsraeli Branch

Honorary President of SIPEA International Society of Poets Writers and Artists A.C.

Vice President Intl Forum for Literature and Culture of Peace (IFLAC)


Vice Director of General Directorate Global Harmony Association (GHA)




Dear Professor Ernesto Kahan! Dear Friends!

It is very great honor for me to get high evaluation of my "Mathematics of Harmony" from o­ne of the outstanding researchers in this area. Thank you very much, Prof. Kahan, for your letter.

I would like to attach to my letter some recent materials concerning the Mathematics of Harmony. First of them is my article Proclus hypothesis: a new view o­n Euclids Elements and the mathematics of harmony submitted to o­ne of scientific journals. I am developing in this article a new approach to Euclid's Elements based o­n Proclus hypothesis. According to Proclus hypothesis, Euclid wrote his Elements with the o­nly purpose to givefull geometric theory of Platonic Solids (Book 13 of Euclid's Element), which reflect the "Universe harmony" in Plato's cosmology. In order to construct dodecahedron, Euclid introduced in Book 2 the division in extrem and mean ratio (the golden section). This means that Euclid's Elements can be considered from the "harmonic" point of view. The Mathematics of Harmony goes back to Euclid's Elements, wich can be considered as the first mathematical theory of Harmony, which had been reflected in ancient science with Platonic Solids. If we take into consideration Proclus hypothesis, we should conclude that the Mathematics of Harmony is a very important mathematical theory, which caninfluence o­n the development of modern mathematics, mathematical education, computer scienceand theoretical natural sciences. This means that the Mathematics of Harmony is a revolution in modern science!


The next material is the theses of my lecture The Mathematics of Harmony, Golden Information Technology and Golden Scientific Revolution. I am ready to present this lecture for scientists and professors of all World Universities in Russian and English if I can get invitation as Visiting-Professor.

One more material is the program of the mathematical course The Mathematics of Harmony and Golden Scientific Revolution. I am ready to lecture this course for mathematics and computer science students of all World Universities (in Russian and English).

I believe that my lecture and my course are so much interesting for modern mathematics, computer science and general science that it would be useful to organize a tour of my lectures in the World Universities.

Sincerely yours

Alexey Stakhov



Dear Professor Alexey Stakhov

Thank you very much for the important material you sent me in relation to mathematics of Harmony. Your lecture is like a key to open a door to understand a new understanding of the universe. I will show it to my colleagues.

AHHH! If I was a young man! I will try to search the two ways of the tendencies, towards harmony (Plato's theories) and toward entropy.

As a humanistic poet I am working with words in order to transform our society in a solid crystal with harmonic relationships

Ernesto Kahan




Dear Professor Ernesto Kahan!

My idea of Harmony relates to Plato and Euclid theories and my "Mathematics of Harmony" begins from Euclid's Elements. I think that the ideas of Harmony and Entropy can be united together if we use the approach developed by Belorussian philosopher Eduard Soroko. As is well known, the Golden Section reflects a numerical harmony between two parts of the whole. If the whole consists of three, four or N parts, we cannot use the Golden Section. For this case we can use a notion of entropy H, which is some real number joining all parts of the whole. Soroko considers the so-called "reduced" entropy, which is equal to H divided by log N. For this case, the "reduced" entropy is between 0 and 1. Then Soroko proves that the system with N parts is harmonious when its entropy is equal to o­ne of the golden p-proportions (p=1, 2, 3, ...): 0.618 (golden proportion); 0.6823 (golden 2-proportion); 0.7245 (golden 3-proportion); and so o­n. Such approach is useful when the system consists of N parts with entropy H=f(p1, p2, ..., pN). For example, the entropy of the dry air, which consists of N= 6 components, is equal to 0.683, what is very close to the golden 2-proportion 0.6823. This means that the system "dry air" is optimal or harmonious. You can study Soroko's entropy approach to harmony by looking my web site "Museum of Harmony and Golden SEction
Soroko's theory is described in Soroko's book "Structural Harmony of Systems" (1984). I consider Soroko's book as o­ne of the best World books in the Harmony field. Unfortunately, this book is not translated into English and many West scientists don't have any information o­n Soroko's Law of Structural Harmony of Systems.

Sincerely yours

Alexey Stakhov



Dear admired Dr Alexey


Im amazed to reading you here, I always thought that Universe is in perfect numericalorder , ageometry that Human beings are constantly disjointed in predjudice of evolution and it explode the anger , wars, violence in so many differents angles all over the planet,.so then the perfect machine of the armonious light that created us is broken and dont respect the numerical order of the creation. I think that Mathematic is in intimate relation with the cience of words andspirit like a perfect ecuation at the same time., As a poet Im I dont know the cience of numbers, I admired the nature of feelings and behaivor of the humankind, the proceses of cultures, I dont know ifnumberstake a visible place here, but I feel they are present all the time,in the air, as energy we are,wegenerate this precious piece of our being that fit properly in the same sytem of living in our relations during our lives o­n the planet.Since our birth our lives is an eternnal addition of calculations..


Your work is admirable!!!


All my respect and best wishes in this light of harmony

Susana Roberts




Dear Profs. Alexey Stakhov and Ernesto Kahan:

I read with interest and partial understanding of the article, "Mathematics of Harmony."

Harmony and Entropy o­n the other hand are exclusive of each other. The reason is simple-

Any system which performs action o­n its environment, or in reverse, the environment works o­n the system, there is production of heat and the entopy of the system increases (meaning disorder increases).Consequently, in a universe which is continuously expanding and expending work, the entropy of the universe is increasing continuously (e.g., formation and dying of stars, motion of the planets, etc.) This leads to o­nly a single state of harmony, before the big-bang, which was an action less state with constant entropy. And in that state, there was o­nly God in bliss (separated from His power, meaning in His unmanifested state).





Dear Dr. Laj Lutreja! Dear Prof. Ernesto Kahan! Dear Friends!

Harmony and Entropy are not different concepts, whichexclude o­ne another. The Law of Universe "Heat Death" is antiquated. Laureate of Nobel Prize Ilja Prigogin developed a concept of selforganization of system. According to this concept there is the following correlation between the information I and entropy H for every self-organizing system: I+H=logN, where N is a number of components of system. However, there is optimal, harmonious state of system for which the system strives in process of itsselforganization. The entropy canincrease or decrease in process of self-organization to its optimal stte. Soroko's Law of Structural Harmony of systems (1984) gives a proportion between I and H in the harmonies state. This proportion is the golden p-proportion, which is a root of the following algebraic equation: x^(p+1)+x-1=0, where p=1, 2, 3, .... For the case p=1 we have "golden" equation x^2+x=1=0. The golden p-proportions are a generalization of the classical golden proportion 0.618.

Sincerely yours

Alexey Stakhov



Dear Alexey and all:

After reading your outstanding, remarkable and genial article: "Harmony Mathematics and its Application in Modern Science" Alexey Stakhov, Doctor of Engineering Sciences, Professor

I must confess I dont agree with the concept of harmony as a perfect , mathematic equation

I have always thought that chaos is responsible of human violent behaviour. Also I believe in harmonization of society as a possibility, built over the shoulders of education.

I see Harmony as a balance, renewed every second, among free, floating energies that act and react o­ne against the other, continuously. Human beings are half rational, half irrational, so harmony is o­nly possible through nurturing human beings to become more rational. I cannot find any chance to find a naturalharmony among human beings. There is part of their souls that is severely oriented to Thanatos.

Harmony is present in Nature. Perhaps violence and not peaceis more nearnatural Human existence if we consider Humanity o­nly from the perspective of its body.

However,Human beings are also conformed by a soul, a rational, free spirit that generates in this human being a need of peace

Dont you think? I can explain my idea a little more if you want.

María Cristina Azcona



Dear Maria and all:

I agree with all your reasonings concerning harmony of human behaviour. However, like to you, I belive in Harmony of Nature. Here the Harmony is expressed in the form of different mathematical constants and numerical sequencessimilar to the Golden Mean and Fibonacci numbers. In geometry, this harmony is expressed in the form of different perfect geometric figures as Platonic Solids. A genius of Phythagoras, Plato and Euclid consists of the fact that they guessed right the harmonious proportions and perfect figures of Nature for millenia ahead! Phyllotaxis, genetic code, quasicrystals, fullerenes are based o­n these harmonious proportions and figures.

My Mathematics of Harmony http://www.worldscibooks.com/mathematics/6635.html not has any relation tohuman behaviour. However, this mathematics putsthe ancient idea ofharmony o­n the level of the most important problems of modern mathematics, mathematical education, computer science (the "golden" information technology), theoretical natural sciences and so o­n. My Mathematics of Harmony demands to return back to Euclid's Elements to begin new mathematics, based o­n the idea of Harmony. This new mathematics should become the source of new science, based o­n "harmonious ideas".My Mathematics of Harmony not has direct relation to art although its new numerical sequences and mathematical constants, whichare a generalization of Fibonacci numbers and golden ratio, can become a source for artistic inspiration.

Sincerely yours

Alexey Stakhov



 Dear Alexey, Maria, Laj, Susana and all

Although Mathematics of Harmony was developed to explain the Harmony of Nature and not the harmony of human behavior, by extrapolation we can learn from nature in order to understand the possible ways to harmony of society. Alexey's explanations are superb for a wonderful dialogue between mathematics and sociology.

One of the concepts we should discuss is if something is "Perfect" either in nature as the crystals, or in society, because such an acceptation implies the end of evolution, and that is a philosophical question. What is perfect will not change towards some evolution. When the chimpanzee was perfect I appeared as a very imperfect creature.

Entropy and Harmony are the same in terms of 1 and -1. Agree if the Golden Section reflects a numerical harmony between two parts of the whole, but in a constant interaction. I suppose Alexey pointed out (in my words) that Harmony can also be a stable state in which entropy is present. If that can be explained by a mathematical constant, then there is a hope for a million time maybe years of harmonic social stability under a spectrum of entropic situations given by the conflicts.

Well, a crystal diamond is enough adult and beautiful, and it is acceptable for any women. My question as a mortal person in his 75% of his life, is How long will take our society to build the ethical crystal that will be accepted for the main forces of humankind, or if entropy will prevail before that? With respect and friendship

Ernesto Kahan



Dear Prof. Ernesto Kahan:

You are absolutely right. We should study harmonious correlations of Nature and try to use them in order to understand the possible ways to harmony of society. The works of American psychologist Vladimir Lefewre are very interesting for explanation of the fact why a man uses the Golden Section and Fibonacci numbers for making different decisions (for example how to divide heaps of seeds into two parts). These Fibonacci's decisions can explain why Fibonacci numbers appear in market processes, which are a result of numerous human's decisions (Elliott'Waves). Fibonacci's model of Elliotte' Wave is similar to Fibonacci's resonances of genetic code and harmonious musical compositions. These are brilliant examples of the use mathematics (Fibonacci numbers and golden section) formodeling social, biological processes and perfect musical compositions.

The modern Mathematics of Harmony gives new harmonious models, which can be used in social processes. However, o­nly additional researches in specific areas can answer to the question: are this mathematical models adequate to this or that social or natural phenomena.

That is why, all researchers in social or artistic spheres should study carefully new mathematical models of the Harmony Mathematics and try to apply them in their specific areas. Of course, this demands of certain efforts and mathematical knowledge. o­nly the Mathematics of Nature (or Mathematics of Harmony) offers us a way for the creation of harmonious society. Of course, we should try to use new social models (for example, Leo Semashko's tetrasociology) together with the Mathematics of Harmony.

Sincerely yours




Dear Dr Alexey

I observedformulas in mathematic, they are so interesting but I dont understand so much, as a creator with words in this art called Literature I think in the evolution andthe natural entropy as evolution transforming with energy what was created in a natural way, it is like to paint a picture, we the creators pretend all the time make the nature to speak giving warm , in this interrelationshipthe ingredients in the social sphere where in a natural way, in this system we fit perfectly in the gear of the universe of the matter, may be we can be in a geometric or numerical way where anybody, nothing is occasional, the pace of life , the rhythm of life and the energy development of our conscience, our perfect machine our body and mindbuilding or destroying at the path. We feel and breathe as perfect the portion of air and light we need to live and grow. We are awake to calculate the mistakes in the human actions and share them looking for solutions, therefore the harmony is necessarily urgent to obtainthe law zero where all the sciences work together in o­ne perfect order of cooperation(does-it exist any numerical platform for this?) Here, your work in harmonious mathematics is super important and brilliant!!!

All my admiration to you and all.




Dear Alexey, Leo, Ernesto, Susana and all

It is proved that dancing helps to overwhelm any kind of stressing experience.

Dear Alexey, Dont you think Maths could possibly explain the relation between rhythmical movements, pleasure and peace of mind? Is this perhaps a clue to understand harmonious Mathematics?

María Cristina Azcona



That's right Dear Maria, I think any movement create energy. If people are generating good energy, warm thoughts and rhythmical dance contribute to eliminate stress, I think in the interrelations of different movements creating positive energy are embodied in numerical equations...Here Dr Alexey have explained the possible incidence in new social models. It is amazing how much o­ne discipline and the other , they could act in silence as a mutual contribution to the harmony. With Love and admiration




Dear Maria:

I think dancing andmelodic music is good examples for applications of harmonious Mathematics. There are many sites in Internet o­n applications of the Golden Section in music (Russian musicologist Sabaneev and others). By moving in this way, we can find connections between rhythmical movements, musicand Harmony Math.




Dear Maria and Susana

I was surprised by your amazing question and comment. I don't know is there is an answer for that

Ernesto Kahan



Hi Leo, Ernesto, Susana, Laj, Alexey

Alexey Stakhov says that we need to increase our research o­n the relationship between Maths and Social Sciences, under the perspective of Harmonization. Hesuggests that a new kind of education could be born from this research An education with the ability to develop individuals prepared for harmony as a way of life into social groups.

I thinkthat it would be great to create a new way to teach Mathematics, where the children will be able to learn how to solve a problem, but also how to use mathematicsto harmonize their lives. However, I think thatthe concept of Harmony in Maths needs to be used referred to particular aspects of human life In social life, harmony is found o­n the resolution of a conflict. I cannot find the relation to Mathematical equations, such as the Golden Ratio, in the resolution of a conflict between two different nations. Instead, I am aware of the presence of Mathematics , and specially the existence of preexistent laws as Golden Ratio, within theprocess of harmonization between body and soul, because I experience it all the timeasa pupil,during my Choreography lessons. In the past, I´vestudied classic dance but recently I added some classes of Choreography. At these new lessons I learned the importance ofstrictnessand respect to the infinitesimal second, the exact time when to move your feet at the same time when the others are moving their feet, and all at the exact moment when the musical compass is listened.

I´ve learned that harmony is not present in the find out of perfection but it is surely present when a human group, all together and synchronized, is searching for perfection seen as a common objective.

The search for this " Golden horizon" ( thesynchronizationof twenty dancers following a particular music piece,with their arms in perfect waves: four to the left, four to the right, four up, four down) creates a relationship of love, mutual respect and mutual encouragement, just as we all are experimenting while researching these important, crucial, meanings, all at o­nce.

As Laj says, the positive actions perhaps dont modify the entropy of the universe but probably will modify our own life, and our relatives o­nes, harmonizing our existence and through this harmonization, giving happiness to all our friends and beloved o­nes.

María Cristina Azcona




Dear Professor Leo Semashko!


I live in Canada since 2004. I work intensively in Canada andpublish my articles in the International Journals (Chaos, Solitons and Fractals, Visual Mathematics and so o­n), and also o­n the site Academy of Trinitarism http://www.trinitas.ru/rus/002/a0208001.htm where I am Director of the Institute of the Golden Section.


During four decades I am developing original scientific direction called the Mathematics of Harmony. My idea is very simple. After the careful study of the mathematics history, I concluded that starting from Euclids Elements, the two fundamental directions started to develop in mathematics: Classical Mathematics based o­n Euclidean axioms and the Mathematics of Harmony based o­n the Golden Section (Theorem 2.11 of Euclids Elements), Platonic Solids (the Book 13 of Euclids Elements), and Fibonacci numbers (13th century).These two directions were developing independently o­ne from another. In 20th century their rapprochement o­ne to another had started. This important tendency is confirmed by the great scientific discoveries of contemporary science (Shechtmans quasi-crystals, fullerenes (Nobel Prize-1996), the golden genomatrices by Sergey Petoukhov, a new geometric theory of phyllotaxis bt Oleg Bodnar).


My mission is to restore and develop in contemporary science the Mathematics of Harmony as alternative way of the mathematics development and a new interdisciplinary of modern science. My new book The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science is devoted to this important topic. The book has been accepted for publication by the International Publishing House World Scientific and will be published in the end of 2008. By the way, my book The da Vinci Code and Fibonacci series waspublished by the Russian Publisher Piter (Sanct-Peterburgh) in 2006 .


We can judge about the applications of the Mathematics of Harmony o­n the base of my recent article The golden Fibonacci goniometry, Fibonacci-Lorentz transformations, and the Fourth Hilbert Problem (Alexey Stakhov and Samuil Aranson) // Academy of Trinitarism, Moscow, 77-6567, publication 14816, 04.06.2008 (in Russian) http://www.trinitas.ru/rus/doc/0232/004a/02321087.htm. We are talking in this article about a new approach to Einsteins special relativity theory, Lobachevskis geometry and the Fourth Hilbert Problem. This new approach to fundamental theories of modern science is based o­n the so-called hyperbolic Fibonacci and Lucas functions. I consider these functions o­ne of my most important scientific results. This means that we are talking about the revision of the most important scientific theories of modern science by using the Golden Section and Fibonacci numbers.


The development of the Mathematics of Harmony is essence of my scientific project, which I can fulfill if this project will be financed. Around the International Club of the Golden Section headed by me since 2003, the golden fund of modern science in this field is concentrated. These high-qualified specialists are able to fulfill interesting and complex projects, which can open new ways in the development of modern science.


I would like to attract attention to o­ne more scientific projects Museum of Harmony and Golden Section http://www.goldenmuseum.com/. This bilingual (English-Russian) site was presented o­n the Internet in 2001. Now, it is considered o­ne of the best Internet manual o­n the Golden Section. There is an idea of architectural realization of this unique nature-science-history-art Museum.

Sincerely yours

Alexey Stakhov


Alexey Stakhov

The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics

Download paper in PDF format (940 KB).

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