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 of geometry” and compared it to the Pythagoras theorem. A prominent Soviet philosopher Alexey Losev, a researcher of the aesthetics of antiquity and the Renaissance, in his citation (see the preface) argues that “from Plato’s point of view, and in general in terms of the entire ancient cosmology, the Universe is determined as a certain proportional whole, which obeys to the law of harmonic division, the golden section.”
It is well known that the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . . ., introduced in the 13th century by the famous Italian mathematician Leonardo of Pisa (Fibonacci) for solving the “task of rabbits’ reproduction” are closely related to the golden ratio. The deep mathematical connection between the Fibonacci numbers and the golden ratio is that the ratio of the two neighboring Fibonacci numbers in the limit tends to be the golden ratio, which implies that this numerical sequence is also expressing Harmony.
The so-called Pascal’s triangle, the special table for the location of binomial coefficients, is one of the highly harmonious objects of mathematics. This table was proposed in the 17th century by the outstanding French mathematician and physicist Blaise Pascal (1623–1662). In the second half of the 20th century, the famous American mathematician George Polya (1887–1995) in his book [111] had described the connection of Fibonacci numbers to the so-called diagonal sums of Pascal’s triangle. The development of these ideas led to a generalization of the task of rabbit reproduction and the introduction of the so-called Fibonacci p-numbers [6].
Volume I of this three-volume book is devoted to the presentation of the foundations of the theory of these extremely beautiful mathematical objects (Fibonacci p-numbers), the interest in which will not fade for centuries or even millennia.
Volume I consists of four chapters. Chapter 1 “The Golden Section: History and Applications” begins with the analysis of a sensational hypothesis, the Proclus hypothesis, which overturns our ideas about Euclid’s Elements and the entire history of origin of the mathematics. According to this hypothesis, Euclid’s Elements were created under the powerful influence of the “Harmony idea”, which was the basic concept of ancient Greek science. The main goal of Elements was to create a complete geometric theory of Platonic solids. This theory was described by Euclid in the final (Book XIII) book. To give the completed theory of dodecahedron, Euclid, already in Book II, introduces and solves the task of “dividing the segment in the extreme and mean ratio” (Proposition II.11) [32], which in modern science is called the golden section.
Chapter 1 deals with the following: the geometric method of constructing the golden section, the algebraic equation of the golden section, the most famous algebraic identities for the golden ratio and also the geometric figures associated with the golden section