TITLE:
A History, the Main Mathematical Results and Applications for the Mathematics of Harmony
AUTHORS:
A. P. Stakhov
KEYWORDS:
Golden Ratio; Pascal’s Triangle and Fibonacci Numbers; Binet’s Formulas; Gazale’s Formulas; Hyperbolic Fibonacci Functions; Number Systems with Irrational Bases; Fibonacci λ-Numbers; “Metallic Proportions”; Hilbert’s Fourth Problem
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.3,
February
11,
2014
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.