examples of the successful usage of Platonic solids, the golden section, and Fibonacci numbers in modern theoretical natural sciences, considered in Vols. I and II, allow expression of the idea that the process of harmonization of natural sciences has been realized actively in modern theoretical natural sciences. Fullerenes and quasi-crystals, awarded by the Nobel Prizes, are the most prominent examples of such harmonization, and this process requires a corresponding response from mathematics.
The development of modern Fibonacci numbers theory [7–9, 11] is a convincing example of harmonization of mathematics. This process obtained further reflection in Stakhov’s book The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science [6] as a new interdisciplinary direction of modern science and mathematics.
Mathematization of Harmony and Harmonization of Mathematics
By considering the history of the development of mathematics since the ancient Greeks to the present time, we can distinguish the two processes that are closely related to each other, despite the more than 2000-year time distance between them. This connection is carried out through the “golden” paradigm of ancient Greeks as a fundamental conception that permeates the entire history of science. The first of these is the process of Mathematization of Harmony. This process began developing in ancient Greece in the sixth or fifth century BC (Pythagoras and Plato’s mathematics) and ended in the third century BC by creating the greatest mathematical work of the ancient era, the Euclidean Elements. All efforts of the ancient Greeks were aimed at creating the mathematical doctrine of Nature, in the center of which the ancient Greeks placed the Idea of Harmony, which, according to Proclus hypothesis, had been expressed in the Euclidean Elements through Platonic solids (XIIIth Book of the Elements) and the golden section (Book II, Proposition of II.11).
The process of Mathematization of Harmony in the ancient period ended with the creation of Euclidean Elements; the main purpose of this process was the creation of the complete geometric theory of Platonic solids (Book XIII of the Elements), which expressed the Universal Harmony in Plato’s cosmology. To create this theory, Euclid already in Book II introduced the task of dividing a segment in extreme and mean ratio (the Euclidian name for the golden section), which was used by Euclid by creating the geometric theory of the dodecahedron, based on the golden ratio.
Harmonization of Mathematics is a process opposite to Mathematization of Harmony [68]. This process began developing most rapidly in the second half of the 20th century in the works of the Canadian geometer Harold Coxeter [7], the Soviet mathematician Nikolay Vorobyov [8], the American mathematician Verner Hoggatt [9], the English mathematician Stefan Waida [11], and other famous Fibonacci mathematicians.
The creators of the modern Fibonacci number theory [7–9, 11] have acted very wisely and cautiously, not attracting attention to the fact that Fibonacci numbers are one of the most important numerical sequences, which together with the golden section actually express Harmony of Nature. They “euthanized” the vigilance of modern orthodox mathematicians, which allowed them to establish the Fibonacci Association, the mathematical journal The Fibonacci Quarterly and, starting from 1984, regularly (once every 2 years) holding the International Conference on Fibonacci Numbers and their Applications. Thanks to the active work of the Fibonacci Association, it was possible to combine the efforts of a huge number of researchers, 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
