%0 Journal Article
%T Hyperbolic Fibonacci and Lucas Functions, ¡°Golden¡± Fibonacci Goniometry, Bodnar¡¯s Geometry, and Hilbert¡¯s¡ª¡ªPart I. Hyperbolic Fibonacci and Lucas Functions and ¡°Golden¡± Fibonacci Goniometry
%A Alexey Stakhov
%A Samuil Aranson
%J Applied Mathematics
%P 74-84
%@ 2152-7393
%D 2011
%I Scientific Research Publishing
%R 10.4236/am.2011.21009
%X This article refers to the ¡°Mathematics of Harmony¡± by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discove-ries¡ªNew Geometric Theory of Phyllotaxis (Bodnar¡¯s Geometry) and Hilbert¡¯s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and ¡°Golden¡± Fibonacci  ¦Ë-Goniometry ( ¦Ë > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas¡ªthe ¡°golden mean¡±, which had been introduced by Euclid in his Elements, and its generalization¡ªthe ¡°metallic means¡±, which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the ¡°Mathematics of Harmony¡±, which originates from Euclid¡¯s Elements.
%K Euclid¡¯s Fifth Postulate
%K Lobachevski¡¯s Geometry
%K Hyperbolic Geometry
%K Phyllotaxis
%K Bodnar¡¯s Geometry
%K Hilbert¡¯s Fourth Problem
%K The ¡°Golden¡± and ¡°Metallic¡± Means
%K Binet Formulas
%K Hyperbolic Fibonacci and Lucas Functions
%K Gazale Formulas
%K ¡°Golden¡± Fibonacci ¦Ë-Goniometry
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=3813