Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
who found the Fibonacci numbers and the golden ratio in their scientific areas. Starting from the last decade of the 20th century, the so-called Slavic Golden Group began playing an active role in the development of this direction. The Slavic Golden Group was established in Kiev (the capital of Ukraine) in 1992 during the First International Workshop Golden Proportion and Problems of Harmony Systems. This scientific group included leading scientists and lovers of the golden ratio and Fibonacci numbers from Ukraine, Russia, Belarus, Poland, Armenia and other countries.
In 2003, according to the initiative of the Slavic Golden Group, the International Conference on Problems of Harmony, Symmetry and the Golden Section in Nature, Science and Art was held at Vinnitsa Agrarian University by the initiative of Professor Alexey Stakhov. According to the decision of the conference, the Slavic Golden Group was transformed into the International Club of the Golden Section.
In 2005, the Golden Section Institute was organized at the Academy of Trinitarism (Russia). In 2010, according to the initiative of the International Club of the Golden Section, the First International Congress on Mathematics of Harmony was held on the basis of the Odessa Mechnikov National University (Ukraine). All these provide evidence of the fact that the International Club of the Golden Section plays in the Russian-speaking scientific community the same role as the American Fibonacci Association in the English-speaking scientific community. The publication of Stakhov’s book The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2009) [6] was an important event in harmonization of modern science and mathematics.
What is Harmonization of Mathematics?
This, first of all, refers to the wide use of fundamental concept of Mathematics of Harmony, such as the Platonic Solids, the golden proportion, the Fibonacci numbers and their generalizations (the Fibonacci p-numbers, the metallic proportions or the “golden” p-proportions, etc.), as well as new mathematical concepts (the Fibonacci matrices, the “golden” matrices, the hyperbolic Fibonacci and Lucas functions [64, 75], etc.) to solve certain mathematical problems and create new mathematical theories and models.
The brilliant examples are the solution of Hilbert’s 10th Problem (Yuri Matiyasevich, 1970), based on the use of new mathematical properties of the Fibonacci numbers, and the solution of Hilbert’s Fourth problem (Alexey Stakhov and Samuil Aranson), based on the use of Spinadel’s metallic proportions. The theory of numeral systems with irrational bases (Bergman’s system and the codes of the golden ratio) and the concept of the “golden” number theory, arising from them, are examples of the original and far from trivial mathematical results, obtained in the framework of Mathematics of Harmony [6].
The main merit of the modern mathematicians in the field of golden ratio and Fibonacci numbers consisted in the fact that their researches caused the spark, from which the flame had ignited. The process of Harmonization of Mathematics is confirmed by a rather impressive and far from complete list of modern books in this field, published in the second half of the 20th century and early 21st century [1–53].
Among them, the following three books, published in the 21st century, deserve special attention:
(1)Stakhov Alexey. Assisted by Scott Olsen. The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2009) [6]
(2)The Prince of Wales with co-authors. Harmony. A New Way of Looking at our World (New York: Harpert Collins Publishers, 2010) [51]
(3)Arakelyan Hrant. Mathematics and History of the Golden Section (Moscow: Logos, 2014) [50] (Russian).
What Place Does Mathematics of Harmony Occupy in the System of Modern Mathematical Theories?
To answer this question, it is appropriate to consider a quote from the review of the prominent Ukrainian mathematician, academician Yuri Mitropolskiy on the scientific research of the Ukrainian scientist Professor Alexey Stakhov. In this review, Yuri Mitropolskiy reports the following:
“I have followed the scientific career of Professor Stakhov for a long time — seemingly since the publication of his first 1977 book, “Introduction into Algorithmic Measurement Theory”, which was presented by Professor Stakhov in 1979 at the scientific seminar of the Mathematics Institute of the Ukrainian Academy of Sciences. I became especially interested in Stakhov’s scientific research after listening his brilliant speech at the 1989 session of the Presidium of the Ukrainian Academy of Sciences. In his speech, Professor Stakhov reported on scientific and engineering results in the field of ‘Fibonacci computers’ that were conducted under his scientific supervision at the Vinnitsa Technical University. . .
One may wonder what place does take this work in the general theory of mathematics. As it seems to me, that in the last few centuries, as Nikolay Lobachevsky said, “Mathematicians have turned all their attention to the Advanced parts of analytics, and by neglecting the origins of mathematics and did not wishing to work in that field, which they passed and left behind. As a result, it was created a gap between ‘Elementary Mathematics’, the basis of modern mathematical education, and ‘Advanced Mathematics.’ In my opinion, the Mathematics of Harmony, developed by Professor Stakhov, fills out that gap. The Mathematics of Harmony is a big theoretical contribution to the development of the ‘Elementary Mathematics’, and the Mathematics of Harmony should be considered as great contribution to mathematical education.”
Note that Alexey Stakhov used Mitropolsky’s review as the Preface to Stakhov’s 2009 book The Mathematics of Harmony [6].
Thus, academician Mitropolsky in his review focuses on the historical aspect. His point of view is that Stakhov’s Mathematics of Harmony is, first of all, a new kind of elementary mathematics, based on the unusual interpretation of the Euclidean Elements, as historically the first version of Mathematics of Harmony, connected with the Platonic solids and the golden section.
But besides this, there are other aspects of Mathematics of Harmony: applied and aesthetical. First of all, we should note the applied nature of Mathematics of Harmony, which is the true Mathematics of Nature. Mathematics of Harmony is found in many natural phenomena, such as the movement of Venus across the sky (“Pentacle of Venus”), the pentagonal symmetry in Nature, the botanical phenomenon of phyllotaxis, the fullerenes, the quasicrystals, etc.
On the other hand, Mathematics of Harmony [6] by the name and by the contents fully satisfies Hutcheson and Dirac principles of mathematics beauty [154]. According to Dirac, the main mathematical ideas should be expressed in terms of excellent mathematics. This means that Mathematics of Harmony, which was aroused in the ancient Greek mathematics, is a beautiful mathematics, which must be embodied in the structures of Nature and contemporary science. This conclusion is confirmed by the modern scientific achievements, described in Vols. I and II and will be discussed in detail in this volume.
The harmonious combination of the applied aspect of Mathematics of Harmony, as the true Mathematics of Nature, with its aesthetic perfection (Hutcheson and Dirac principles [154]), gives us reason to suggest that it is Mathematics of Harmony that can become the “golden” paradigm of modern science, which will help overcome the crisis in modern mathematics [101]. Mathematics of Harmony, described in this three-volume book, is a very young mathematical theory, although in its origins it goes back to the Euclidean Elements.
The term Mathematics of Harmony was used first by Alexey Stakhov in the speech The Golden Section and Modern Harmony Mathematics [66], made in 1996 at the Seventh International Conference on Fibonacci Numbers and Its Applications (Austria, Graz, 1996).
The speech was perceived with great interest by the Fibonacci mathematicians, as evidenced