Samuil Aranson’s acquaintance to the golden section and the Fibonacci numbers began in 2001 after the reading of a very rare book “Chain Fractions” [107] by the famous Russian mathematician, Aleksandr Khinchin. In this book, Samuil Aranson found results, related to the representation of the “golden ratio” in the form of a continued fraction.
In 2007, Prof. Aranson read a wonderful Internet publication, Museum of Harmony and Golden Section, posted in 2001 by Professor Alexey Stakhov and his daughter Anna Sluchenkova. This Internet Museum covers various areas of modern natural sciences and tells about the different and latest scientific discoveries, based on the golden ratio and Fibonacci numbers, including the Mathematics of Harmony and its applications in modern natural sciences. After reading this Internet Museum, Samuil Aranson contacted Alexey Stakhov in 2007 through e-mail and offered him joint scientific collaboration in further application of the Mathematics of Harmony in various areas of mathematics and modern natural sciences. Scientific collaboration between Alexey Stakhov and Samuil Aranson turned out to be very fruitful and continues up to the present time.
New ideas in the field of elementary mathematics and the history of mathematics, developed by Stakhov (Proclus’s hypothesis as a new look at Euclid’s Elements and history of mathematics, hyperbolic Fibonacci and Lucas functions [64, 75] as a new class of elementary functions and other mathematical results) attracted the special attention of Prof. Aranson. Scientific collaboration between Stakhov and Aranson began in 2007. From 2007, they published the following joint scientific works (in Russian and English), giving fundamental importance for the development of mathematics and modern theoretical natural sciences:
Stakhov and Aranson’s Mathematical Monographs in English
1. Stakhov A., Aranson S., The Mathematics of Harmony and Hilbert’s Fourth Problem. The Way to the Harmonic Hyperbolic and Spherical Worlds of Nature. Germany: Lambert Academic Publishing, 2014.
2. Stakhov A., Aranson S., Assisted by Scott Olsen, The “Golden” Non-Euclidean Geometry: Hilbert’s Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant, World Scientific, 2016.
Stakhov and Aranson’s Scientific Papers in English
3. Stakhov A.P., Aranson S.Kh., “Golden” Fibonacci goniometry, Fibonacci-Lorentz transformations, and Hilbert’s fourth problem. Congressus Numerantium 193, (2008).
4. Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part I. Hyperbolic Fibonacci and Lucas functions and “Golden” Fibonacci goniometry. Applied Mathematics 2(1), (2011).
5. Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part II. A new geometric theory of phyllotaxis (Bodnar’s Geometry). Applied Mathematics 2(2), (2011).
6. Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part III. An original solution of Hilbert’s fourth problem. Applied Mathematics 2(3), (2011).
7. Stakhov A.P., Aranson S.Kh., The mathematics of harmony, Hilbert’s fourth problem and Lobachevski’s new geometries for physical world. Journal of Applied Mathematics and Physics 2(7), (2014).
8. Stakhov A., Aranson S., The fine-structure constant as the physical-mathematical millennium problem. Physical Science International Journal 9(1), (2016).
9. Stakhov A., Aranson S., Hilbert’s fourth problem as a possible candidate on the millennium problem in geometry. British Journal of Mathematics & Computer Science 12(4), (2016).
Chapter 1
The Golden Section: History and Applications
1.1. The Idea of the Universal Harmony in Ancient Greek Science
1.1.1. What is Harmony?
V.P. Shestakov, the author of the book Harmony as an Aesthetic Category [10], notes the following:
“In the history of aesthetic teachings, various types of understanding of harmony were put forward. The very concept of “harmony” was used extremely broadly and multivalently. It denoted both the natural structure of Nature and the Cosmos, and the beauty of the physical and moral world of man and the principles of the structure of the work of art, and the laws of aesthetic perception.”
Shestakov singles out three basic understandings of Harmony that evolved in the process of development of science and aesthetics:
(1) Mathematical understanding of Harmony or mathematical Harmony. In this sense, harmony is understood as the equality or proportionality of the parts with each other and the part with the whole. In the Great Soviet Encyclopedia, we find the following definition of Harmony, which expresses the mathematical understanding of harmony:
“Harmony is the proportionality of parts and the whole, the fusion of various components of an object into a single organic whole. Harmony is the outer revealing of inner order and the measure of existence.”
(2) Aesthetic harmony. Unlike the mathematical understanding, the aesthetic understanding is no longer just quantitative, but qualitative, expressing the inner nature of things. The aesthetic Harmony is associated with aesthetic experiences, with aesthetic evaluation. This type of harmony is most clearly manifested in the perception of the beauty of Nature.
(3) Artistic harmony. This type of Harmony is associated with art. The artistic Harmony is the actualization of the principle of Harmony in the material of art itself.
The most important aspect that follows from the above reasonings is the following: Harmony is a universal concept that has relation not only to mathematics and science but also to fine arts.
1.1.2. Numerical harmony of Pythagoreans
Pythagoras and Heraclitus were philosophers and thinkers, whose names are usually associated with the beginning of the philosophical doctrine of Harmony. According to many authors, the key idea of Harmony as a proportional unity of opposites belongs to Pythagoras. Pythagoreans first put forward the idea of harmonious construction of the whole world, including not only Nature and Man but also the entire cosmos. According to Pythagoreans, “harmony is an inner connection of things, without which the cosmos could not exist” [10]. Finally, according to Pythagoras, harmony has a
