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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

1.2The Paradigm Shift to the “Golden” Elementary Number Theory

1.3Fibonacci Microprocessors as a Prerequisite for the “Golden” Paradigm Shift in Computer Science

1.4Sergey Abachiev: Mathematics of Harmony Through the Eyes of the Historian and Expert of Methodology of Science

Chapter 2.The “Golden” Hyperbolic Functions as the “Golden” Paradigm Shift to the “Golden” Non-Euclidean Geometry

2.1The Concept of “Elementary Functions”

2.2Conic Sections and Hyperbola

2.3Hyperbolic Rotation

2.4Trigonometric Functions

2.5Geometric Analogies Between Trigonometric and Hyperbolic Functions and Basic Identities for Hyperbolic Functions

2.6Millennium Problems in Mathematics and Physics

2.7A New Look at the Binet Formulas

2.8Hyperbolic Fibonacci and Lucas Functions

2.9Recurrent Properties of the Hyperbolic Fibonacci and Lucas Functions

2.10Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions

2.11Formulas for Differentiation and Integration

Chapter 3.Applications of the Symmetric Hyperbolic Fibonacci and Lucas Functions

3.1New Geometric Theory of Phyllotaxis (“Bodnar Geometry”)

3.2The Golden Shofar

3.3The Shofar-Like Model of the Universe

Chapter 4.Theory of Fibonacci and Lucas λ-numbers and its Applications

4.1Definition of Fibonacci and Lucas λ-numbers

4.2Representation of the Fibonacci λ-numbers Through Binomial Coefficients

4.3Cassini Formula for the Fibonacci λ-numbers

4.4Metallic Proportions by Vera Spinadel

4.5Representation of the “Metallic Proportions” in Radicals

4.6Representation of the “Metallic Proportions” in the Form of Chain Fraction

4.7Self-similarity Principle and Gazale Formulas

4.8Hyperbolic Fibonacci and Lucas λ-functions

4.9Special Cases of Hyperbolic Fibonacci and Lucas λ-functions

4.10The Most Important Formulas and Identities for the Hyperbolic Fibonacci and Lucas λ-functions

Chapter 5.Hilbert Problems: General Information

5.1A History of the Hilbert Problems

5.2Original Solution of Hilbert’s Fourth Problem Based on the Hyperbolic Fibonacci and Lucas λ-Functions

5.3The “Golden” Non-Euclidean Geometry

5.4Complete Solution of Hilbert’s Fourth Problem, and New Challenges for the Theoretical Natural Sciences

5.5New 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

6.1Mathematics: A Loss of Certainty and Authority of Nature

6.2Strategic Mistakes in the Development of Mathematics: The View from the Outside

6.3Beauty and Aesthetics of Harmony Mathematics

6.4Mathematics of Harmony from an Aesthetic Point of View

Chapter 7.Epilogue

7.1A Brief History of the Concept of Universe Harmony

7.2More on the Doctrine of Pythagoreanism, Pythagorean MATHEMs, and Pythagorean Mathematical and Scientific Knowledge

7.3Mathematization of Harmony and Harmonization of Mathematics

7.4The Structure of Scientific Revolutions by Thomas Kuhn

7.5Main Conclusions and New Challenges

Bibliography