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, coincide in their ideas and goals.
Such an amazing coincidence can only be explained by the fact that in modern science, there is an urgent need to return to the “harmonious ideas” of Pythagoras, Plato and Euclid that permeated across the ancient Greek science and culture. The Harmony idea, formulated in the works of the Greek scholars and reflected in Euclid’s Elements turned out to be immortal!
We can safely say that the above-mentioned books by Stakhov and Aranson (2009, 2016, 2017) [6, 51–53], the book by The Prince of Wales with the coauthors (2010) [49] and book by Arakelian (2014) [50] are the beginning of a revolution in modern science. The essence of this revolution consists, in turning to the fundamental ancient Greek idea of the Universal Harmony, which can save our Earth and humanity from the approaching threat of the destruction of all mankind.
It was this circumstance that led the author to the idea of writing the three-volume book Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science, in which the most significant and fundamental scientific results and ideas, formulated by the author and other authors (The Prince of Wales, Hrant Arakelian, Samuil Aranson and others) in the process of the development of this scientific direction, will be presented in a popular form, accessible to students of universities and colleges and teachers of mathematics, computer science, theoretical physics and other scientific disciplines.
Structure and the Main Goal of the Three-Volume Book
The book consists of three volumes:
• Volume I. The Golden Section, Fibonacci Numbers, Pascal Triangle and Platonic Solids.
• Volume II. Algorithmic Measurement Theory, Fibonacci and Golden Arithmetic and Ternary Mirror-Symmetrical Arithmetic.
• Volume III. The “Golden” Paradigm of Modern Science: Prerequisite for the “Golden” Revolution in the Mathematics, the Computer Science, and Theoretical Natural Sciences.
Because the Mathematics of Harmony goes back to the “harmonic ideas” of Pythagoras, Plato and Euclid, the publication of such a three-volume book will promote the introduction of these “harmonic ideas” into modern education, which is important for more in-depth understanding of the ancient conception of the Universal Harmony (as the main conception of ancient Greek science) and its effective applications in modern mathematics, science and education.
The main goal of the book is to draw the attention of the broad scientific community and pedagogical circles to the Mathematics of Harmony, which is a new kind of elementary mathematics and goes back to Euclid’s Elements. The book is of interest for the modern mathematical education and can be considered as the “golden” paradigm of modern science on the whole.
The book is written in a popular form and is intended for a wide range of readers, including schoolchildren, school teachers, students of colleges and universities and their teachers, and also scientists of various specializations, who are interested in the history of mathematics, Platonic solids, golden section, Fibonacci numbers and their applications in modern science.
Introduction
It is known that the amount of irrational (incommensurable) numbers is infinite. However, some of them occupy a special place in the history of mathematics, science and education. Their significance lies in the fact that they are expressing some fundamental relationships, which are universal by their nature and appear in the most unexpected places.
The first of them is the irrational number
The next two irrational (transcendental) numbers are as follows: the number of π, which is equal to the ratio of the length of circumference to its diameter (this number lies at the basis of the trigonometric functions) and the Naperian number of e (this number underlies the hyperbolic functions and is the basis of natural logarithms). Between π and e, that is, between the two irrational numbers that dominate over the analysis, there is the following elegant relation derived by Euler:
where
Another famous irrational number is the “golden proportion” Φ = (1 +
In the preface, we already mentioned about the brilliant German astronomer, Johannes Kepler, who named the golden ratio as one of the “treasures
