%0 Journal Article
%T The Mathematics of Harmony, Hilbert¡¯s Fourth Problem and Lobachevski¡¯s New Geometries for Physical World
%A Alexey Stakhov
%A Samuil Aranson
%J Journal of Applied Mathematics and Physics
%P 457-494
%@ 2327-4379
%D 2014
%I Scientific Research Publishing
%R 10.4236/jamp.2014.27056
%X <p style=\"text-align:justify;\">
	<span style=\"font-size:10pt;font-family:'} ';\">We suggest an original approach to Lobachevski¡¯s geometry and Hilbert¡¯s
Fourth Problem, based on the use of the ¡°mathematics of harmony¡± and special
class of hyperbolic functions, the so-called hyperbolic Fibonacci </span><i style=\"line-height:1.5;\"><span style=\"font-size:10.0pt;font-family:Symbol;\">l</span></i><span style=\"font-size:10pt;font-family:'} ';\">-functions, which are based on the ancient ¡°golden proportion¡± and its generalization,
Spinadel¡¯s ¡°metallic proportions.¡± The uniqueness of these functions consists
in the fact that they are inseparably connected with the Fibonacci numbers and
their generalization</span><span style=\"font-size:10pt;font-family:'} ';\">¨D</span><span style=\"font-size:10pt;font-family:'} ';\"> Fibonacci </span><i style=\"line-height:1.5;\"><span style=\"font-size:10.0pt;font-family:Symbol;\">l</span></i><span style=\"font-size:10pt;font-family:'} ';\">-numbers (</span><i style=\"line-height:1.5;\"><span style=\"font-size:10.0pt;font-family:Symbol;\">l</span></i><i style=\"line-height:1.5;\"><span style=\"font-size:10.0pt;font-family:;\" \"=\"\"> </span></i><span style=\"font-size:10pt;font-family:'} ';\">&gt; 0 is a given real number) and have recursive
properties. Each of these new classes of hyperbolic functions, the number of
which is theoretically infinite, generates Lobachevski¡¯s new geometries, which
are close to Lobachevski¡¯s classical geometry and have new geometric and
recursive properties. The ¡°golden¡± hyperbolic geometry with the base <img src=\"Edit_b5b95889-ca6b-4bb0-b4aa-3fb21b9af773.bmp\" alt=\"\" /></span>
</p>
<p style=\"text-align:justify;\">
	<span></span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\"> </span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">(¡°Bodnar¡¯s
geometry) underlies the botanic phenomenon of phyllotaxis. The ¡°silver¡±
hyperbolic geometry with the base <img src=\"Edit_a1f1e288-41be-4b56-9abc-bd9b80f52a4b.bmp\" alt=\"\" /></span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">&nbsp;has the least distance to Lobachevski¡¯s
classical geometry. Lobachevski¡¯s new geometries, which are an original
solution of Hilbert¡¯s Fourth Problem, are new hyperbolic geometries for
physical world.</span> 
</p>
<p>
	<br />
</p>
%K Euclid¡¯s Elements
%K ¡°Golden¡± and ¡°Metallic¡± Proportions
%K Mathematics of Harmony
%K Hyperbolic Fibonacci Functions
%K Lobachevski¡¯s Geometry
%K Hilbert¡¯s Fourth Problem
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=46803