Abstract
We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid’s “Elements”. According to “Proclus hypothesis”, the main goal of Euclid was to create a full geometric theory of Platonic solids, associated with the ancient conception of the “Universe Harmony”. We consider the main periods in the development of the “Mathematics of Harmony” and its main mathematical results: algorithmic measurement theory, number systems with irrational bases and their applications in computer science, the hyperbolic Fibonacci functions, following from Binet’s formulas, and the hyperbolic Fibonacci l-functions (l = 1, 2, 3, …), following from Gazale’s formulas, and their applications for hyperbolic geometry, in particular, for the solution of Hilbert’s Fourth Problem.
Highlights
The harmony of the spheres is the ancient and medieval doctrine about the musical-mathematical construction of Cosmos, which goes back to the Pythagorean and Platonic philosophical tradition. Another mention about “the Mathematics of Harmony”, as the ancient Greek great discovery, we find in the book by Vladimir Dimitrov “A new kind of social science
It is this view on the “phyllotaxis geometry” gives us a right to claim that the hyperbolic Fibonacci functions (18) and (19) [17,21] and “Bodnar’s geometry” [5] are the fundamental discoveries of modern science
It is important to emphasize that the very name of “hyperbolic geometry” contains another way for the solution of Hilbert’s Fourth Problem: searching for the new classes of “hyperbolic functions,” which can be the basis for other hyperbolic geometries
Summary
The American philosopher Professor Scott Olsen, author of the excellent book: “The Golden Section: Nature’s Greatest Secret” [2] made a huge assistance in writing the book [1] and its preparation for the publication The publication of these books (see Figure 1) is a reflection of one of the most important trends in the development of modern science. These include: “the law of structural harmony of systems” by Edward Soroko [4], based on the golden p-proportions, and the “law of spiral biosimmetry transformation” by Oleg Bodnar [5], based on the “golden” Fibonacci hyperbolic functions It includes a new theory of the genetic code, based on the “golden genomatrices” (author—Doctor of Physical and Mathematical Sciences Sergey Petoukhov, Moscow) [6]. The purpose of this article is to give a survey of the main stages, events and scientific findings, which led to the creation of “Mathematics of Harmony”, the new interdisciplinary direction of modern science
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