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Alexey Stakhov. Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Series on Knots and Everything — Vol. 69. Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Volume 3. The “Golden” Paradigm of Modern Science: Prerequisite for the “Golden” Revolution in Mathematics, Computer Science, and Theoretical Natural Sciences
Contents
Preface to the Three-Volume Book
Introduction
About the Author
Acknowledgments
Chapter 1. Mathematics of Harmony as a Prerequisite for the “Golden” Revolution in Mathematics and Computer Science. 1.1.“Proclus Hypothesis” as a Prerequisite for the “Golden” Revolution in the History of Mathematics. 1.1.1.The significance of the “Proclus hypothesis” for the development of mathematics
1.1.2.Proclus hypothesis and the “golden” revolution in the history of mathematics
1.2.The Paradigm Shift to the “Golden” Elementary Number Theory. 1.2.1.What is “elementary number theory”?
1.2.2.Bergman’s system
1.2.3.New classes of the numeral systems with irrational bases
1.2.4.Fibonacci p-codes
1.2.5.Codes of the “golden” p-proportions
1.2.6.The “golden” number theory
1.3.Fibonacci Microprocessors as a Prerequisite for the “Golden” Paradigm Shift in Computer Science. 1.3.1.“Trojan Horse” of the binary system
1.4.Sergey Abachiev: Mathematics of Harmony Through the Eyes of the Historian and Expert of Methodology of Science. 1.4.1.Publication of Stakhov’s article in the popular Soviet scientific journal Technology to Youth
1.4.2.Scientific work of Prof. Stakhov at the Dresden Technical University (April–May 1988)
1.4.3.Speech at the Karl-Marx-Stadt Technical University
1.4.4.Publication of the interview in the newspaper “Pravda”
Chapter 2. The “Golden” Hyperbolic Functions as the “Golden” Paradigm Shift to the “Golden” Non-Euclidean Geometry. 2.1.The Concept of “Elementary Functions”
2.2.Conic Sections and Hyperbola. 2.2.1.Apollonius’ “conic sections”
2.2.2.Hyperbola
2.3.Hyperbolic Rotation. 2.3.1.Compression to a point
2.3.2.Compression to straight line
2.3.3.Hyperbolic rotation
2.4.Trigonometric Functions. 2.4.1. Geometric definition
2.4.2.The simplest identities for the trigonometric functions
2.5.Geometric Analogies Between Trigonometric and Hyperbolic Functions and Basic Identities for Hyperbolic Functions. 2.5.1.The hyperbola equation, related to the axes
2.5.2.Geometric analogies between trigonometric and hyperbolic functions
2.5.3.Analytic expressions for hyperbolic functions
2.5.4.Applications of the hyperbolic functions in geometry
2.6.Millennium Problems in Mathematics and Physics
2.7.A New Look at the Binet Formulas. 2.7.1.Extended Fibonacci and Lucas numbers
2.7.2.Deduction of the Binet formulas
2.7.3.A new look on the Binet formulas for the Fibonacci and Lucas numbers
2.8.Hyperbolic Fibonacci and Lucas Functions. 2.8.1.A brief history
2.8.2.Symmetric hyperbolic Fibonacci and Lucas functions
2.8.3.Parity property
2.8.4.Graphs of the symmetric hyperbolic Fibonacci and Lucas functions
2.9.Recurrent Properties of the Hyperbolic Fibonacci and Lucas Functions. 2.9.1.Analogy to the recurrent relations for Fibonacci and Lucas numbers
2.9.2.Generalization of the Cassini formula
2.9.3.Table of the recurrent properties of the symmetric hyperbolic Fibonacci and Lucas functions
2.9.4.The theory of the symmetric hyperbolic Fibonacci and Lucas functions as the “golden” paradigm of the “Fibonacci numbers theory”
2.10.Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions. 2.10.1.Parity property
2.10.2.Fundamental identities for the symmetric hyperbolic Fibonacci and Lucas functions
2.10.3.The table of the identities for the symmetric hyperbolic Fibonacci and Lucas functions
2.11.Formulas for Differentiation and Integration
2.11.1.Aesthetics of the symmetric hyperbolic Fibonacci and Lucas functions
Chapter 3. Applications of the Symmetric Hyperbolic Fibonacci and Lucas Functions. 3.1.New Geometric Theory of Phyllotaxis (“Bodnar Geometry”) 3.1.1.The riddle of phyllotaxis
3.1.2.Structural and numerical analysis of the phyllotaxis lattices
3.1.3.Key principle of dynamic symmetry
3.1.4.The “golden” hyperbolic functions
3.1.5.Connection of the “golden” hyperbolic functions with the symmetric hyperbolic Fibonacci functions
3.1.6.A brief history
3.1.7.The main Jewish religious symbol is the Shofar
3.1.8.Quasi-sinusoidal Fibonacci and Lucas functions
3.1.9.Graphs of the quasi-sinusoidal Fibonacci and Lucas functions
3.1.10.Recurrent properties of the quasi-sinusoidal Fibonacci and Lucas functions
3.1.11.Fibonacci 3D spiral
3.2.The Golden Shofar
3.3.The Shofar-Like Model of the Universe. 3.3.1.Hyperbolic Universes with horned topology
Chapter 4. Theory of Fibonacci and Lucas λ-numbers and its Applications. 4.1.Definition of Fibonacci and Lucas λ-numbers. 4.1.1.A brief history
4.1.2.Recurrent relation for the Fibonacci λ-numbers
4.1.3.Extended Fibonacci λ-numbers
4.2.Representation of the Fibonacci λ-numbers Through Binomial Coefficients
4.3.Cassini Formula for the Fibonacci λ-numbers
4.4.Metallic Proportions by Vera Spinadel
4.5.Representation of the “Metallic Proportions” in Radicals
4.6.Representation of the “Metallic Proportions” in the Form of Chain Fraction
4.7.Self-similarity Principle and Gazale Formulas. 4.7.1.Gazale’s formulas for the Fibonacci λ-numbers
4.7.2.Gazale formula for the Lucas λ-numbers
4.8.Hyperbolic Fibonacci and Lucas λ-functions. 4.8.1.Properties of the extended Fibonacci and Lucas λ-numbers
4.8.2.Definition of the hyperbolic Fibonacci and Lucas λ-functions
4.8.3.Graphs of the hyperbolic Fibonacci and Lucas λ-functions
4.9.Special Cases of Hyperbolic Fibonacci and Lucas λ-functions. 4.9.1.The “golden”, “silver”, “bronze” and “copper” hyperbolic Fibonacci and Lucas λ-functions
4.9.2.Connection with the classical hyperbolic functions
4.9.3.Connection to Pell’s numbers
4.10.The Most Important Formulas and Identities for the Hyperbolic Fibonacci and Lucas λ-functions. 4.10.1.The relations connecting the “metallic proportions” with the “golden proportion”
4.10.2.Recurrent properties
4.10.3.Hyperbolic properties
Chapter 5. Hilbert Problems: General Information. 5.1.A History of the Hilbert Problems [146–149]
5.2.Original Solution of Hilbert’s Fourth Problem Based on the Hyperbolic Fibonacci and Lucas λ-Functions
5.3.The “Golden” Non-Euclidean Geometry
5.3.1.Slavic “Golden” Group, International Club of the Golden Section, and Institute of the Golden Section
5.3.2.The classical metric form of the Lobachevsky plane
5.3.3.The metric λ-form of the Lobachevsky plane
5.3.4.A summary of the dramatic history of the solution of Hilbert’s Fourth Problem in the 20th and 21st centuries
5.4.Complete Solution of Hilbert’s Fourth Problem, and New Challenges for the Theoretical Natural Sciences. 5.4.1.Insolvability of the Fourth Hilbert Problem for hyperbolic geometries
5.4.2.The “silver” proportion as the next challenge for theoretical natural sciences (Tatarenko’s proposal)
5.4.3.The mathematical constantsand, Pell numbers and Pythagoras constant
5.5.New Approach to the Creation of New Hyperbolic Geometries: From the “Game of Postulates” to the “Game of Functions”
Chapter 6. Beauty and Aesthetics of Harmony Mathematics. 6.1.Mathematics: A Loss of Certainty and Authority of Nature. 6.1.1.Morris Klein’s book
6.1.2.The Authority of Nature
6.1.3.Appeal to the origins of mathematics
6.2.Strategic Mistakes in the Development of Mathematics: The View from the Outside. 6.2.1.The moving away of mathematics from theoretical natural sciences
6.2.2.Neglect of “beginnings”
6.2.3.Neglect of the “idea of harmony” and the “golden section”
6.2.4.The golden ratio in natural science
6.2.5.Disrespect to the “Proclus hypothesis”
6.2.6.One-sided look at the origin of mathematics
6.2.7.The greatest mathematical mystification of the 19th century
6.2.8.Underestimation of Binet formulas
6.2.9.Underestimation of the “icosahedral” idea of Felix Klein
6.2.10.The underestimation of the mathematical discovery of George Bergman
6.3.Beauty and Aesthetics of Harmony Mathematics. 6.3.1.Hutcheson aesthetic principles
6.3.2.Dirac’s principle of mathematical beauty
6.4.Mathematics of Harmony from an Aesthetic Point of View
6.4.1.Aesthetics of the golden section
6.4.2.Aesthetics of Fibonacci and Lucas numbers
6.4.3.Aesthetics of Fibonacci and Lucas p-numbers (p = 0, 1, 2, 3, . . .)
6.4.4.Properties of Fibonacci and Lucas λ-numbers and “metallic proportions”
Chapter 7. Epilogue. 7.1.A Brief History of the Concept of Universe Harmony
7.2.More on the Doctrine of Pythagoreanism, Pythagorean MATHEMs, and Pythagorean Mathematical and Scientific Knowledge
7.2.1.The most famous numerical achievements of Pythagoreans
7.2.2.Geometric achievements of Pythagoreans
7.2.3.Pythagorean theory of music
7.2.4.Pythagorean numerical Harmony
7.3.Mathematization of Harmony and Harmonization of Mathematics. 7.3.1.A brief history
7.4.The Structure of Scientific Revolutions by Thomas Kuhn
7.4.1.Kuhn’s criteria to Theory Choice
7.5.Main Conclusions and New Challenges
Bibliography
