Harmony, which underlies the ancient Greek doctrine of Nature, was the main “paradigm” of the Greek science, starting from Pythagoras and ending with Euclid. This paradigm relates directly to the golden section and the Platonic solids, which are the most important Greek geometric discoveries for the expression of the Universal Harmony.
Johannes Kepler (1571–1630), the prominent astronomer and the author of “Kepler’s laws”, expressed his admiration with the golden ratio in the following words [6]:
“Geometry has the two great treasures: the first of them is the theorem of Pythagoras; the second one is the division of the line in the extreme and mean ratio. The first one we may compare to the measure of the gold; the second one we may name the precious stone.”
We should recall again that the ancient task of dividing line segment in extreme and mean ratio is Euclidean language for the golden section!
The enormous interest in this problem in modern science is confirmed by the rather impressive and far from the complete list of books and articles on this subject, published in the second half of the 20th century and the beginning of the 21st century [1–100].
Ancient Greeks Mathematical Doctrine of Nature
According to the outstanding American historian of mathematics, Morris Kline [101], the main contribution of the ancient Greeks is the one “which had the decisive influence on the entire subsequent culture, was that they took up the study of the laws of Nature”. The main conclusion, from Morris Kline’s book [101] is the fact that the ancient Greeks proposed the innovative concept of the Cosmos, in which everything was subordinated to the mathematical laws. Then the following question arises: during which time this concept was developed? The answer to this question is also addressed in Ref. [101].
According to Kline [101], the innovative concept of the Cosmos based on the mathematical laws, was developed by the ancient Greeks in the period from VI to III centuries BC. But according to the prominent Russian mathematician academician A.N. Kolmogorov [102], in the same period in ancient Greece, “the mathematics was created as the independent science with the clear understanding of the uniqueness of its method and with the need for the systematic presentation of its basic concepts and proposals in the fairly general form.” But then, the following important question, concerning the history of the original mathematics arises: was there any relationship between the process of creating the mathematical theory of Nature, which was considered as the goal and the main achievement of ancient Greek science [101], and the process of creating mathematics, which happened in ancient Greece in the same period [102]? It turns out that such connection, of course, existed. Furthermore, it can be argued that these processes actually coincided, that is, the processes of the creation of mathematics by the ancient Greeks [102], and their doctrine of Nature, based on the mathematical principles [101], were one and the same processes. And the most vivid embodiment of the process of the Mathematization of Harmony [68] happened in Euclid’s Elements, which was written in the third century BC.
Introduction of the Term Mathematics of Harmony
In the late 20th century, to denote the mathematical doctrine of Nature, created by the ancient Greeks, the term Mathematics of Harmony was introduced. It should be noted that this term was chosen very successfully because it reflected the main idea of the ancient Greek science, the Harmonization of Mathematics [68]. For the first time, this term was introduced in the small article “Harmony of spheres”, placed in The Oxford Dictionary of Philosophy [103]. In this article, the concept of Mathematics of Harmony was associated with the Harmony of spheres, which was, also called in Latin as “harmonica mundi” or “musica mundana” [10]. The Harmony of spheres is the ancient and medieval doctrine on the musical and mathematical structure of the Cosmos, which goes back to the Pythagorean and Platonic philosophical traditions.
Another mention about the Mathematics 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 ratio of the two neighboring Fibonacci numbers in the limit tends to be the golden ratio.
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.
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