Our aim in the present article is to introduce and study new types of retractions of Lobachevsky space. Types of the deformation retracts of Lobachevsky space are presented. The relations between the folding and the d...Our aim in the present article is to introduce and study new types of retractions of Lobachevsky space. Types of the deformation retracts of Lobachevsky space are presented. The relations between the folding and the deformation retract of Lobachevsky space are deduced. Types of minimal retractions of Lobachevsky space are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced. Theorems governing this connection are achieved.展开更多
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 Geom...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 Phyl-lotaxis (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 scien-tific 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.展开更多
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 ...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 λ-Goniometry (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 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.展开更多
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—New...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—New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry ( 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 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.展开更多
Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural nu...Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural number), were introduced in mathematics. The principal distinction of the new classes of hyperbolic functions from the classic hyperbolic functions consists in the fact that they have recursive properties like the Fibonacci numbers (or Fibonacci l-numbers), which are “discrete” analogs of these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer numerical sequences does not exist. This unique property of the new hyperbolic functions has been confirmed recently by the new geometric theory of phyllotaxis, created by the Ukrainian researcherOleg Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These fundamental scientific results are overturning our views on hyperbolic geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”) and putting forward the challenge for theoretical natural sciences to search harmonic hyperbolic worlds of Nature. The goal of the present article is to show the uniqueness of these scientific results and their vital importance for theoretical natural sciences and extend the circle of readers. Another objective is to show a deep connection of the new results in hyperbolic geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.展开更多
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 Fibonacc...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.展开更多
文摘Our aim in the present article is to introduce and study new types of retractions of Lobachevsky space. Types of the deformation retracts of Lobachevsky space are presented. The relations between the folding and the deformation retract of Lobachevsky space are deduced. Types of minimal retractions of Lobachevsky space are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced. Theorems governing this connection are achieved.
文摘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 Phyl-lotaxis (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 scien-tific 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.
文摘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 λ-Goniometry (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 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.
文摘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—New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry ( 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 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.
文摘Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural number), were introduced in mathematics. The principal distinction of the new classes of hyperbolic functions from the classic hyperbolic functions consists in the fact that they have recursive properties like the Fibonacci numbers (or Fibonacci l-numbers), which are “discrete” analogs of these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer numerical sequences does not exist. This unique property of the new hyperbolic functions has been confirmed recently by the new geometric theory of phyllotaxis, created by the Ukrainian researcherOleg Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These fundamental scientific results are overturning our views on hyperbolic geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”) and putting forward the challenge for theoretical natural sciences to search harmonic hyperbolic worlds of Nature. The goal of the present article is to show the uniqueness of these scientific results and their vital importance for theoretical natural sciences and extend the circle of readers. Another objective is to show a deep connection of the new results in hyperbolic geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.
文摘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.
