Alexey
Stakhov, Samuil Aranson/30.03.2011
Hyperbolic
Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry,
Bodnar’s Geometry,
and Hilbert’s Fouth Problem
Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden”
Fibonacci Goniometry Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)Part III. An Original Solution of Hilbert’s Fourth ProblemAbstract

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 discoveries–New
Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem
based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci
λGoniometry (λ > 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 ideasthe “golden
mean,” which had been introduced by Euclid in his Elements, and its
generalization—the “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.
 