Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
of Harmony in the connection to the ancient Greek mathematics is found in the book by Vladimir Dimitrov, A New Kind of Social Science, published in 2005 [44]. It is important to emphasize that in Ref. [44], the concept of Mathematics of Harmony is directly associated with the golden section, the most important mathematical discovery of the ancient science in the field of Harmony. This discovery at that time was called “dividing a segment into the extreme and mean ratio” [32].
From Refs. [44, 45], it is evident that prominent thinkers, scientists and mathematicians took part in the development of the Mathematics of Harmony for several millennia: Pythagoras, Plato, Euclid, Fibonacci, Pacioli, Kepler, Cassini, Binet, Lucas, Klein, and in the 20th century the well-known mathematicians Coxeter [7], Vorobyov [8], Hoggatt [9], Vaida [11], Knuth [123], and so on. And we cannot ignore this historical fact.
Fibonacci Numbers
The Fibonacci numbers, introduced into Western European mathematics in the 13th century by the Italian mathematician Leonardo of Pisa (known by the nickname Fibonacci), are closely related to the golden ratio. Fibonacci numbers from the numerical sequence, which starts with two units, and then each subsequent Fibonacci number is the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
The mathematical theory of Fibonacci numbers has been further developed in the works of the French mathematicians of the 19th century Binet (Binet formula) and Lucas (Lucas numbers). As mentioned above, in the second half of the 20th century, this theory was developed in the works of the Canadian geometer, Donald Coxeter [7], the Soviet mathematician, Nikolay Vorobyov [8], the American mathematician, Verner Hoggatt [9] and the English mathematician, Stefan Vajda [11], the outstanding American mathematician, Donald Knuth [123], and so on.
The development of this direction ultimately led to the emergence of the Mathematics of Harmony [6], a new interdisciplinary direction of modern science that relates to modern mathematics, computer science, economics, as well as to all theoretical natural sciences. The works of the well-known mathematicians, Coxeter [7], Vorobyov [8], Hoggatt [9], Vaida [11], Knuth [123], and others, as well as the study of Fibonacci mathematicians, members of the American Fibonacci Association, became the beginning of the process of Harmonization of Mathematics [68], which continues actively in the 21st century. And this process is confirmed by a huge number of books and articles in the field of the golden section and Fibonacci numbers published in the second half of the 20th century and the beginning of the 21st century [1–100].
Sources of the Present Three-Volume Book
The differentiation of modern science and its division into separate spheres do not allow us often to see the general picture of science and the main trends in its development. However, in science, there exist research objects that combine disparate scientific facts into a single whole. Platonic solids and the golden section are attributed to the category of such objects. The ancient Greeks elevated them to the level of “the main harmonic figures of the Universe”. For centuries or even millennia, starting from Pythagoras, Plato and Euclid, these geometric objects were the object of admiration and worship of the outstanding minds of mankind, during Renaissance, Leonardo da Vinci, Luca Pacoli, Johannes Kepler, in the 19th century, Zeising, Lucas, Binet and Klein. In the 20th century, the interest in these mathematical objects increased significantly, thanks to the research of the Canadian geometer, Harold Coxeter [7], the Soviet mathematician Nikolay Vorobyov [8] and the American mathematician Verner Hoggatt [9], whose works in the field of the Fibonacci numbers began the process of the “Harmonization of Mathematics”. The development of this direction led to the creation of the Mathematics of Harmony [6] as a new interdisciplinary trend of modern science.
The newest discoveries in the various fields of modern science, based on the Platonic solids, the golden section and the Fibonacci numbers, and new scientific discoveries and mathematical results, related to the Mathematics of Harmony (quasicrystals [115], fullerenes [116], the new geometric theory of phyllotaxis (Bodnar’s geometry) [28], the general theory of the hyperbolic functions [75, 82], the algorithmic measurement theory [16], the Fibonacci and golden ratio codes [6], the “golden” number theory [94], the “golden” interpretation of the special theory of relativity and the evolution of the Universe [87], and so on) create an overall picture of the movement of modern science towards the “golden” scientific revolution, which is one of the characteristic trends in the development of modern science. The sensational information about the experimental discovery of the golden section in the quantum world as a result of many years of research, carried out at the Helmholtz–Zentrum Berlin für Materialien und Energie (HZB) (Germany), the Oxford and Bristol Universities and the Rutherford Appleton Laboratory (UK), is yet another confirmation of the movement of the theoretical physics to the golden section and the Mathematics of Harmony [6].
For the first time, this direction was described in the book by Stakhov A.P., assisted by Scott Olsen, The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science, World Scientific, 2009 [6].
In 2006, the Russian Publishing House, “Piter” (St. Petersburg) published the book, Da Vinci Code and Fibonacci numbers [46] (Alexey Stakhov, Anna Sluchenkova and Igor Shcherbakov were the authors of the book). This book was one of the first Russian books in this field. Some aspects of this direction are reflected in the following authors’ books, published by Lambert Academic Publishing (Germany) and World Scientific (Singapore):
• Alexey Stakhov, Samuil Aranson, The Mathematics of Harmony and Hilbert’s Fourth Problem. The Way to Harmonic Hyperbolic and Spherical Worlds of Nature. Germany: Lambert Academic Publishing, 2014 [51].
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