Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

Abstract

This article refers to the "Mathematics of Harmony" by Alexey Stakhov [1], 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  -Goniometry (    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-The "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.

Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature

“…For the first time, this way (the "game of functions") is used in the works [7,8]. The "game of functions" led in [7,8] to the original solution of Hilbert's Fourth Problem. The essence of this solution consists in the following.…”

“…In the articles [7,8] the original solution of Hilbert's Fourth Problem is described. Hilbert's Fourth Problem [14,15], which relates to the non-Euclidean geometry, was formulated by David Hilbert as follows [16]: "The more general question now arises: Whether from other suggestive standpoints geometries may not be devised which, with equal right, stand next to Euclidean geometry".…”

“…The essence of this solution consists in the following. Developing the idea of the metric form of Lobachevski's plane, known in hyperbolic geometry, the following formula for the metric form of Lobachevski's plane, based on the hyperbolic Fibonacci -functions (9), (10), has been derived in [7,8]: Lobachevski's plane [7,8].…”

“…Thus, the main result of the research, described in [7][8][9], is a proof of the existence of an infinite number of the hyperbolic functions (9), (10), based on the "metallic proportions" (13). In addition, each class of the hyperbolic functions, corresponding to (9), (10), "generates" for the given 1, 2, 3,…”

“…Recently a number of the important scientific results have been obtained in modern science [1][2][3][4][5][6][7][8][9]. They relate to hyperbolic geometry and have a direct relation to the theoretical natural sciences in the whole.…”