Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
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].
• Alexey Stakhov, Samuil Aranson, Assisted by Scott Olsen, The “Golden” Non-Euclidean Geometry, World Scientific, 2016 [52].
• Alexey Stakhov, Numeral Systems with Irrational Bases for Mission-Critical Applications, World Scientific, 2017 [53].
These books are fundamental in the sense that they are the first books in modern science, devoted to the description of the theoretical foundations and applications of the following new trends in modern science: the history of the golden section [78], the Mathematics of Harmony [6], the “Golden” Non-Euclidean geometry [52], ascending to Euclid’s Elements, and also the Numeral Systems with Irrational bases, ascending to the Babylonian positional numeral system, the decimal and binary system and Bergman’s system [54].
These books discuss the problems, which in modern mathematics are considered long resolved and therefore are not included in the circle of the studies of mathematicians, namely the new mathematical theory of measurement called the Algorithmic Measurement Theory [16, 17], the Mathematics of Harmony [6] as a new kind of elementary mathematics that has a direct relationship to the foundations of the mathematics and mathematical education, the new class of the elementary functions called the hyperbolic Fibonacci and Lucas functions [64, 75, 82] and finally, the new ways of real numbers representation, and the new binary and ternary arithmetic’s [55, 72], which have the fundamental interest for computer science and digital metrology.
In 2010, the Odesa I.I. Mechnikov National University (Ukraine) hosted the International Congress on the Mathematics of Harmony. The main goal of the Congress was to consolidate the priority of Slavic science in the development of this important trend and acquaint the scientific community with the main trends of the development of the Mathematics of Harmony as the new interdisciplinary direction of modern science.
In the recent years, the new unique books on the problems of Harmony and the history of the golden section have been published:
• The Prince of Wales. Harmony. A New Way of Looking at our World (coauthors Tony Juniper and Ian Skelly). An Imprint of HarperCollins Publisher, 2010 [49].
• Hrant Arakelian, Mathematics and History of the Golden Section. Moscow, Publishing House “Logos”, 2014 [50].
For the last 30 years, Charles, The Prince of Wales, had been known around the world as one of the most forceful advocates for the environment. During that period, he focused on many different aspects of our lives, when we continually confront with the real life from new angles of view and search original approaches. Finally, in Harmony (2010) [49], The Prince of Wales and his coauthors laid out their thoughts on the planet, by offering an in-depth look into its future. Here, we see a dramatic call to the action and an inspirational guide on the relationship of mankind with Nature throughout history. The Prince of Wales’s Harmony (2010) [49] is an illuminating look on how we must reconnect with our past in order to take control of our future.
The 2014 book [50] by the Armenian philosopher and physicist Hrant Arakelian is devoted to the golden section and to the complexity of problems connected with it. The book consists of two parts. The first part is devoted to the mathematics of the golden section and the second part to the history of the golden section. Undoubtedly, Arakelian’s 2014 book is one of the best modern books devoted to mathematics and the history of the golden section.
The International Congress on Mathematics of Harmony (Odessa, 2010) and the above-mentioned books by The Prince of Wales and Armenian philosopher Hrant Arakelian are brilliant confirmation of the fact that in modern science, the interest in the mathematics of the golden section and its history increases and further development of the Mathematics of Harmony can lead to revolutionary transformations in modern mathematics and science on the whole.
Why did the author decide to write the three-volume book The Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science? It should be noted that the author and other famous authors in this field published many original books and articles in this scientific direction. However, all the new results and ideas, described in the above-mentioned publications of Alexey Stakhov, Samuil Aranson, Charles, The Prince of Wales, Hrant Arakelian and other authors are scattered in their numerous articles and books, which makes it difficult to understand their fundamental role in the development of the modern mathematics, computer science and theoretical natural sciences on the whole.
This role is most clearly reflected in the following citations taken from Harmony by the Prince of Wales (2010) [49]:
“This is a call to revolution. The Earth is under threat. It cannot cope with all that we demand of it. It is losing its balance and we humans are causing this to happen.”
The following quote, placed on the back cover of Prince of Wales’s Harmony [49], develops this thought:
“We stand at an historical moment; we face a future where there is a real prospect that if we fail the Earth, we fail humanity. To avoid such an outcome, which will comprehensively destroy our children’s future or even our own, we must make choices now that carry monumental implications.”
Thus, The Prince of Wales has considered his 2010 book, Harmony. A New Way of Looking at our World, as a call to the revolution in modern science, culture and education. The same point of view is expressed in the above-mentioned books by Stakhov and Aranson [6, 46, 51–53]. Comparing the books of Prince of Wales [49] and Hrant Arakelian [50] to the 2009, 2016 and 2017 books of Alexey Stakhov and Samuil Aranson [6, 51–53], one can only be surprised how deeply all these books, written in different countries and continents,