AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2014.53039AM-42742ArticlesPhysics&Mathematics A History, the Main Mathematical Results and Applications for the Mathematics of Harmony .P. Stakhov1*International Higher Education Academy of Sciences, Moscow, Russia;Institute of the Golden Section, Academy of Trinitarism, Moscow, Russia* E-mail:goldenmuseum@rogers.com070220140503363386November 15, 2013December 15, 2013 December 23, 2013© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

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