A. Stakhov and S. Aranson, “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, Vol. 2, No. 1, 2011, pp. 74-84. http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.21009

has been cited by the following article:

• 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.