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 l-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― Fibonacci l-numbers (l > 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 (“Bodnar’s geometry) underlies the botanic phenomenon of phyllotaxis. The “silver” hyperbolic geometry with the base ?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.
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