The Mathematics of Harmony

The following sections are included:

• Preface

• Contents

• Introduction

• Acknowledgements

• Credits and Sources

The following sections are included:

• Geometric Definition of the Golden Section
  • A Problem of the Division in the Extreme and Mean Ratio (DEMR)
  • The Origin of the Concept and Title of the Golden Section
  • A Way of Geometrical Construction of the Golden Section
  • A “Double” Square

• Algebraic Properties of the Golden Mean
  • Remarkable Identities of the Golden Mean
  • The “Golden” Geometric Progression
  • A Representation of the Golden Mean in the Form of Continued Fraction
  • A Representation of the Golden Mean in “Radicals”

• The Algebraic Equation of the Golden Mean
  • Polynomials and Equations
  • Quadratic Irrationals
  • Poor “Studious” Niels Abel
  • Mathematician and Revolutionist Evariste Galois
  • The “Golden” Algebraic Equations of n-th Degree

• The Golden Rectangles and the Golden Brick
  • A Golden Rectangle with a Side Ratio of τ
  • A Golden Rectangle with a Side Ratio of τ²
  • The “Golden” Brick of Gothic Architecture

• Decagon: Connection of the Golden Mean to the Number π

• The Golden Right Triangle and the Golden Ellipse
  • The Golden Right Triangle
  • The “Golden” Ellipse

• The “Golden” Isosceles Triangles and Pentagon
  • Construction of the “Golden” Isosceles Triangle and Regular Pentagon in The Elements
  • A Regular Pentagon
  • The Pentagon
  • The “Golden Cup” and the Golden Isosceles Triangle
  • Pentagonal Symmetry in Nature

• The Golden Section and the Mysteries of Egyptian Culture
  • Phenomenon of Ancient Egypt
  • The Mysteries of the Egyptian Pyramids

• The Golden Section in Greek Culture
  • Pythagoras
  • The Idea of Harmony in Greek Culture
  • The Golden Section in Greek Sculpture

• The Golden Section in Renaissance Art
  • The Idea of the “Divine Harmony” in the Renaissance Epoch
  • “Vitruvian Man” by Leonardo da Vinci
  • “Mona Lisa” by Leonardo da Vinci

• De Divina Proportione by Luca Pacioli
  • Luca Pacioli
  • About Pacioli's Plagiarism
  • Death and Oblivion of Luca Pacioli

• A Proportional Scheme of the Golden Section in Architecture

• The Golden Section in the Art of the 19th and 20th Centuries
  • Ivan Shishkin's Picture “The Ship Grove”
  • “Modulor” by Le Corbusier
  • The Building of the United Nations Headquarters in New York
  • A Picture “Near to the Window” by Konstantin Vasiliev

• A Formula of Beauty
  • The Golden Ratio in the External Forms of a Person
  • The Beauty of a Woman's Face
  • Nefertiti

• Conclusion

The following sections are included:

• Who was Fibonacci?
  • Leonardo of Pisa Fibonacci
  • Fibonacci and Abu Kamil
  • Influence of Fibonacci's Works on the Development of the European Mathematics

• Fibonacci's Rabbits
  • The “Rabbit Reproduction Problem”
  • About the Rabbits
  • Dudeney's Cows
  • Honeybees and Family Trees

• Numerology and Fibonacci Numbers
  • Some Mathematical Properties of Numerological Values
  • Fibonacci Numerological Series
  • Another Periodic Properties of Fibonacci Numbers

• Variations on Fibonacci Theme
  • Formulas for the Sums of Fibonacci Numbers
  • Connection of Fibonacci Numbers to the Golden Mean
  • “Iron Table” by Steinhaus

• Lucas Numbers
  • Francois-Edouard-Anatole Lucas
  • Some Properties of Lucas Numbers
  • The “Extended” Fibonacci and Lucas Numbers
  • Other Remarkable Identities for Fibonacci and Lucas Numbers
  • Lucas Numbers and Numerology

• Cassini Formula
  • Great Astronomer Giovanni Domenico Cassini
  • Cassini Formula for Fibonacci Numbers

• Pythagorean Triangles and Fibonacci Numbers
  • Pythagorean Theorem
  • Pythagorean Triangles
  • Fibonacci-Pythagorean Triangles
  • Lucas-Pythagorean Triangles
  • Fibonacci and Lucas Right Triangles

• Binet Formulas
  • Jacques Philippe Marie Binet
  • Deducing Binet Formulas
  • A Historical Analogy

• Fibonacci Rectangle and Fibonacci Spiral
  • Fibonacci Rectangles
  • Fibonacci Spiral
  • Fibonacci Spirals in Nature

• Chemistry by Fibonacci
  • Law of Multiple Ratios
  • Research by the Ukrainian Scientist Nikolai Vasyutinsky

• Symmetry of Nature and the Nature of Symmetry
  • Basic Concepts of Symmetry
  • Symmetry of Crystals
  • Symmetry Laws in Nature
  • Symmetry Laws in Art

• Omnipresent Phyllotaxis
  • A Helical Symmetry
  • Densely Packed Phyllotaxis Structures
  • Use of Phyllotaxis Lattices in Painting

• “Fibonacci Resonances” of the Genetic Code
  • The Initial Data about the Genetic Code
  • DNA SUPRA-code (Jean-Claude Perez's Discovery)
  • A Verification of Jean-Claude Perez's Law

• The Golden Section and Fibonacci Numbers in Music and Cinema
  • Pythagorean Theory of Musical Harmony
  • Chopin's Etudes in the Lighting of the Golden Section
  • Rosenov's Research
  • Bella Bartok's Music
  • The Golden Mean and Fibonacci Numbers in Cinema

• The Music of Poetry
  • Pushkin's Poetry
  • Lermontov's Poetry
  • Shota Rustaveli's Poem

• The Problem of Choice: Will Buridan's Donkey Die?
  • A Parable about Buridan's Donkey
  • A Psychological Experiment
  • What is an Implication?
  • What is a Reflection?

• Elliott Waves
  • Ralph Nelson Elliott
  • Rhythm in Nature
  • Elliott's Wave Principle

• The Outstanding Fibonacci Mathematicians of the 20th Century
  • Willem Abraham Wythoff
  • Edouard Zeckendorf
  • Verner Emil Hoggatt
  • Alfred Brousseau (1907-1988)
  • Steven Vaida
  • Herta Taussig Freitag

• Slavic “Golden” Group

• Conclusion

The following sections are included:

• Platonic Solids
  • Regular Polygons
  • Regular Polyhedra
  • Numerical Characteristics of Platonic Solids
  • The Golden Section in the Dodecahedron and Icosahedron
  • Plato's Cosmology

• Archimedean Solids and Star-shaped Regular Polyhedra
  • Archimedean Solids
  • Star-shaped Regular Polyhedra

• A Mystery of the Egyptian Calendar
  • What is a Calendar?
  • Structure of the Egyptian Calendar
  • Connection of the Egyptian Calendar with the Numerical Characteristics of the Dodecahedron
  • The Mayan Calendar

• A Dodecahedron-Icosahedron Doctrine
  • Sources of the Doctrine
  • A Shape of the Earth

• Johannes Kepler: from “Mysterium” to “Harmony”
  • “Mysterium Cosmographicum”
  • Kepler's Cosmic Cup
  • Discovery of Kepler's First Two Astronomical Laws
  • “Harmonices Mundi”
  • Life through the Centuries

• The Regular Icosahedron as the Main Geometrical Object of Mathematics
  • Felix Klein
  • Elementary Mathematics from the Point of View of Higher Mathematics
  • Role of the Icosahedron in Mathematical Progress

• Regular Polyhedra in Nature and Science
  • Symmetry Groups of Regular Polyhedra
  • Applications of Regular Polyhedra in the Living Nature
  • “Parquet Problem” and Penrose Tiling
  • Quasi-crystals
  • Fullerenes

• Applications of Regular Polyhedra in Art
  • Leonardo da Vinci's Methods of Regular Polyhedra Representation
  • Pacioli's Polyhedron
  • Albrecht Durer
  • Piero Della Francesca
  • Art of Intarsia
  • Salvador Dali's Last Supper
  • Escher's Creative Work

• Application of the Golden Mean in Contemporary Art
  • Abstract Art by Astrid Fitzgerald
  • The World of Matjuska Teja Krasek
  • The Geometric Art of John Michell
  • Quantum Connections by Marion Drennen

• Conclusion

The following sections are included:

• A Combinatorial Approach to the Harmony of Mathematics
  • Mathematical, Aesthetic and Artistic Understanding of Harmony
  • Concept of the Mathematical Theory of Harmony
  • An Analogy between the Theory of Information and Mathematics of Harmony

• Binomial Coefficients and Pascal Triangle
  • The Main Concepts of Combinatorial Analysis
  • Binomial Formula
  • Pascal Triangle

• The Generalized Fibonacci p-Numbers
  • Rectangular Pascal Triangle
  • Pascal p-Triangles and Fibonacci p-Numbers
  • Partial Cases of the Fibonacci p-Numbers
  • A Representation of the Fibonacci p-Numbers by the Binomial Coefficients
  • The “Extended” Fibonacci p-Numbers
  • Some Identities for the Sums of the Fibonacci p-Numbers
  • The Ratio of Adjacent Fibonacci p-Numbers

• The Generalized Golden p-Sections
  • A Generalization of the Division in Extreme and Mean Ratio (DEMR)
  • Algebraic Properties of the Golden p-Proportions
  • Geometric Progressions Based on the Golden p-Proportions

• The Generalized Principle of the Golden Section
  • Dichotomy Principle
  • Classical Golden Section Principle
  • The Generalized Principle of the Golden Section

• A Generalization of Euclid's Theorem II. 11
  • A Generalization of Euclid's Theorem II.11 for the Case p=2
  • Euclid's Rectangular Parallelepiped
  • The General Case of p

• The Roots of the Generalized Golden Algebraic Equations
  • Algebraic Equations
  • Properties of the Roots of the Generalized Golden Algebraic Equations
  • Some Corollaries of Theorems 4.1 and 4.2

• The Generalized Golden Algebraic Equations of Higher Degrees
  • The Case of p=1
  • The Case of p=2
  • The Case of p=3
  • The General Case

• The Generalized Binet Formula for the Fibonacci p-Numbers
  • A General Approach to the Synthesis of the Generalized Binet Formulas
  • A Derivation of Binet Formulas for the Classical Fibonacci and Lucas Numbers
  • Binet Formula for the Fibonacci 2-Numbers
  • Binet Formula for the Fibonacci 3-Numbers
  • Binet Formulas for the Fibonacci 4-Numbers
  • A General Case of p

• The Generalized Lucas p-Numbers
  • Binet Formula for the Lucas p-Numbers
  • Binet Formula for the Lucas 2-Numbers
  • Binet Formula for the Lucas 3-Numbers
  • Binet Formula for the Lucas 4-Numbers
  • The “Extended” Lucas p-Numbers
  • Identities for the Sums of the Lucas p-Numbers
  • The Ratio of Adjacent Lucas p-Numbers

• The “Metallic Means Family” by Vera W. de Spinadel
  • The Generalized Fibonacci Sequences
  • The Metallic Means Family
  • Other Types of the Metallic Means
  • Pisot-Vijayaraghavan Numbers and Metallic Means
  • “Silver Means” by Jay Kappraff

• Gazale Formulas
  • The Generalized Fibonacci m-Numbers
  • The Generalized Golden Mean of Order m
  • Two Surprising Representations of the Golden m-Proportion
  • A Derivation of the Gazale Formula

• Fibonacci and Lucas m-Numbers
  • Fibonacci m-Numbers
  • The Generalized Cassini Formula
  • Lucas m-Numbers

• On the m-Extension of the Fibonacci and Lucas p-Numbers
  • A Recursive Relation for the Fibonacci (p,m)-Numbers
  • Some Properties of the Fibonacci (p,m)-Numbers
  • Characteristic Algebraic Equation for the Fibonacci (p, m)-Numbers
  • The Golden (p,m)-Proportions
  • Properties of the Roots of the Characteristic Equation
  • Generalized Binet Formulas for the Fibonacci (p,m)-Numbers
  • Generalized Binet Formulas for the Lucas (p,m)-Numbers

• Structural Harmony of Systems
  • Soroko's Law of the Structural Harmony of Systems
  • Application of Soroko's Law to Thermodynamic and Information Systems

• Conclusion

The following sections are included:

• The Simplest Elementary Functions
  • Trigonometric Functions
  • The Power Function
  • An Exponential Function
  • Logarithms

• Hyperbolic Functions
  • Definition of the Hyperbolic Functions
  • History and Applications of the Hyperbolic Functions
    • Lambert and Riccati
    • Non-Euclidean Geometry
    • Minkowski's Four-dimensional World
    • Vernadsky's Ideas

• Hyperbolic Fibonacci and Lucas Functions (Stakhov-Tkachenko's Definition)
  • A Definition of Hyperbolic Fibonacci and Lucas Functions
  • The Hyperbolic Fibonacci and Lucas Tangent and Cotangent

• Some Properties of the Hyperbolic Fibonacci and Lucas Functions
  • Hyperbolic Fibonacci Sine
  • Hyperbolic Fibonacci Cosine
  • Hyperbolic Lucas Sine
  • Hyperbolic Lucas Cosine

• Integration and Differentiation of the Hyperbolic Fibonacci and Lucas Functions and their Main Identities
  • Integration of the Function y=sF(x)
  • Integration of the Function y=cF(x)
  • Integration of the Function y=sL(x)
  • Integration of the Function y=cL(x)
  • Differentiation of the Function y=sF(x)
  • Differentiation of the Function y=cF(x)
  • Differentiation of the Function y=sLx
  • Differentiation of the Function y=cLx
  • The Main Identities for the Hyperbolic Fibonacci and Lucas Functions

• Symmetric Hyperbolic Fibonacci and Lucas Functions (Stakhov - Rozin Definition)

• Recursive Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions

• Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions and Formulas for Their Differentiation and Integration
  • Hyperbolic Properties
  • Formulas for Differentiation and Integration

• The Golden Shofar
  • The Quasi-sine Fibonacci and Lucas Functions
  • Recursive Properties of the Quasi-sine Fibonacci and Lucas Functions
  • Three-dimensional Fibonacci Spiral
  • The Golden Shofar
  • A General Model of the Hyperbolic Space with a “Shofarable” Topology

• A General Theory of the Hyperbolic Functions
  • A Definition of the Hyperbolic Fibonacci and Lucas m-Functions
  • General Properties of the Hyperbolic Fibonacci and Lucas m-Functions
  • Partial Cases of the Hyperbolic Fibonacci and Lucas m-Functions
  • Comparison of the Classical Hyperbolic Functions with the Hyperbolic Lucas m-Functions
  • Recursive Properties of the Hyperbolic Fibonacci and Lucas m-Functions
  • Hyperbolic Properties of the Hyperbolic Fibonacci and Lucas m-Functions

• A Puzzle of Phyllotaxis
  • The Phenomenon of Phyllotaxis
  • Dynamic Symmetry

• A Geometric Theory of the Hyperbolic Functions
  • Compression and Expansion
  • Hyperbola
  • Geometric Definition of the Hyperbolic Functions
  • Hyperbolic Rotation

• Bodnar's Geometry
  • Structural - numerical Analysis of Phyllotaxis Lattices
  • The Dynamic Symmetry of Phyllotaxis Objects
  • The Key Idea of Bodnar's Geometry
  • The “Golden” Hyperbolic Functions
  • The Connection of Bodnar's “Golden” Hyperbolic Functions with the Hyperbolic Fibonacci Functions

• Conclusion

The following sections are included:

• Introduction to Matrix Theory
  • A History of Matrices
  • Definition of a Matrix
  • Matrix Addition and Scalar Multiplication
  • Matrix Multiplication
  • Square Matrices
    • Definition of a Square Matrix
    • Invertible Matrices
    • Determinants

• Fibonacci Q-Matrix
  • A Definition of the Q-Matrix
  • Properties of the Q-Matrix
  • Binet Formulas for the Q-Matrix

• Generalized Fibonacci Q_p-Matrices
  • A Definition of the Q_p-Matrices
  • The Main Theorems for the Q_p-Matrices

• Determinants of the Q_p-Matrices and their Powers

• The “Direct” and “Inverse” Fibonacci Matrices

• Fibonacci G_m-Matrices
  • A Definition of the Fibonacci G_m-Matrix
  • Determinants of the G_m-Matrices
  • G_m-Matrix and Cassini Formula for the Fibonacci m-Numbers
  • Some Properties of the G_m-Matrices
  • The Inverse Matrices

• Fibonacci Q_p,m-Matrices
  • A Definition of the Q_p,m-Matrix
  • The Main Theorems for the Powers of the Q_p,m-Matrices

• Determinants of the Q_p,m-Matrices and their Powers
  • Determinant of the Q_p,m-Matrix
  • Determinant of the Matrix

• The Golden Q-Matrices
  • A Definition of the Golden Matrices
  • The Inverse Golden Matrices
  • Determinants of the Golden Matrices

• The Golden G_m-Matrices
  • A Definition of the Golden G_m-Matrices
  • Inverse Golden G_m-matrices
  • Determinants of the Golden G_m-Matrices

• The Golden Genomatrices by Sergey Petoukhov
  • The Genetic Code
  • Symbolic Genomatrices
  • Numerical Genomatrices
  • The Golden Genomatrices

• Conclusion

The following sections are included:

• The Role of Measurement in the History of Science
  • What is a Measurement?
  • The “Differentiation Principle” of “Measurement Science”
  • Applied and Fundamental Theories of Measurement
  • What is the Fundamental Distinction between Physics and Mathematics?
  • A Thesis about Inevitability of Measurement Errors
  • Purposes of the “Physical” Measurement Theory

• Mathematical Measurement Theory
  • Evolution of the Measurement Concept in Mathematics
  • Incommensurable Line Segments
  • Eudoxus-Archimedes and Cantor's Axioms
  • A Contradiction in the Continuity Axioms

• Evolution of the Infinity Concept
  • An Infinity Concept
  • The Origin of the Symbol∞
  • Potential and Actual Infinity
  • An Origin and Application of the Infinity Idea in the Ancient Greek Mathematics
  • Cantor's Theory of Infinite Sets
  • Antinomies of Cantor's Theory of Infinite Sets
  • Is Cantor's Actual Infinity the Greatest Mathematical Mystification?

• A Constructive Approach to Measurement Theory
  • The Rejection of Cantor's Axiom from Mathematical Measurement Theory
  • Bashet-Mendeleev's Problem
  • Asymmetry Principle of Measurement
  • A New Formulation of the Bachet-Mendeleev Problem

• Mathematical Model of Measurement
  • A Notion of the “Indicator Element”
  • The (n, k, S)-algorithms
  • The Optimal (n, k, S)-algorithms

• Classical Measurement Algorithms
  • The “Binary” Algorithm
  • The “Counting” Algorithm
  • The “Ruler” Algorithm
  • The Restriction S
  • Connection between Measurement Algorithms and Positional Number Systems

• The Optimal Measurement Algorithms Originating Classical Positional Number Systems
  • A General Method for the Synthesis of the Optimal (n,k,S)-algorithms
  • The Optimal (n,k,0)-algorithms
  • Special Cases of the Optimal (n,k,0)-algorithm

• Optimal Measurement Algorithms Based on the Arithmetical Square
  • The Optimal (n,k,1)-algorithms
  • Arithmetical Square
  • The Optimal Measurement Algorithms Based on Arithmetical Square
  • The Example of the Optimal (n, k, 1)-algorithm

• Fibonacci Measurement Algorithms
  • The Optimal Measurement Algorithms Based on the Fibonacci p-numbers
  • The Example of the Fibonacci Measurement Algorithm

• The Main Result of Algorithmic Measurement Theory
  • A Further Generalization of the Bachet-Mendeleev Problem
  • The Main Recursive Relation of the Algorithmic Measurement Theory
  • The Unusual Results

• Mathematical Theories Isomorphic to Algorithmic Measurement Theory
  • What is an Isomorphism?
  • Isomorphism between the “Balance” and “Rabbits”
  • The Generalized “Asymmetry Principle” of Organic Nature
  • Isomorphism between Measurement Algorithms and Positional Number Systems

• Conclusion

The following sections are included:

• A History of Computers
  • Era of Information
  • The Basic Milestones in Computer Progress
    • Abacus
    • Mechanical Calculation Machines
    • Babbage's Analytical Machine
    • Hollerith's Tabulators
    • Electromechanical Computers by Zuse, Aiken and Stibitz
    • ENIAC
    • John von Neumann's Principles
    • The Phenomenon of Personal Calculators

• Basic Stages in the History of Numeral Systems
  • Babylonian Sexagecimal Numeral System
  • Alphabetic and Roman Systems of Numeration
  • Mayan System
  • Decimal System
  • Binary System
  • Exotic Number Systems

• Fibonacci p-Codes
  • Zeckendorf Sums
  • Definition of the Fibonacci p-Code
  • The Range of Number Representation in the Fibonacci p-Code
  • Multiplicity of Number Representation

• Minimal Form and Redundancy of the Fibonacci p-Code
  • Convolution and Devolution
  • Radix of the Fibonacci p-Code
  • A Minimal Form of the Fibonacci p-Code
  • Comparison of Numbers in the Fibonacci p-Codes
  • Redundancy of the Fibonacci p-Codes
  • Surprising Analogies Between the Fibonacci and Genetic Codes

• Fibonacci Arithmetic: The Classical Approach
  • Fibonacci Addition
  • Direct Fibonacci Subtraction
  • Fibonacci Inverse and Additional Codes

• Fibonacci Arithmetic: An Original Approach
  • Basic Micro Operations
  • Logical Operations
  • The Counting and Subtracting of Binary 1
  • Fibonacci Summation
  • Fibonacci Subtraction

• Fibonacci Multiplication and Division
  • The Egyptian Method of Multiplication
  • The Egyptian Method of Division
  • Fibonacci Multiplication
  • Fibonacci Division
    • The first stage
    • The second stage

• Hardware Realization of the Fibonacci Processor
  • A Device for Reduction of the Fibonacci Code to Minimal Form
  • “Convolution” Register as a Self-checking Device
  • Combinative Logical “Convolution” Circuits
  • “Devolution” devices

• Fibonacci Processor for Noise-tolerant Computations
  • Noise-tolerant Computations
  • Checking the Basic Micro-Operations
  • The Hardware Realization of a Noise-tolerant Fibonacci Processor

• The Dramatic History of the Fibonacci Computer Project
  • First Steps in the Development of Fibonacci Arithmetic
  • Stakhov's Lecture in Austria (1976)
  • Patenting the Soviet Fibonacci Computer Inventions
  • The Fibonacci Computer Developments in the U.S.
  • Fibonacci Digital Signal Processing

• Conclusion

The following sections are included:

• Numeral Systems with Irrational Bases
  • Bergman's Numeral System
  • A Definition of the Golden p-Proportion Code
  • Representation of the Golden p-Proportion Powers

• Some Mathematical Properties of the Golden p-Proportion Codes
  • A Minimal Form of the Golden p-Proportion Code
  • Comparison of Numbers
  • Representation of Numbers with a Floating Comma
  • Conversion of Numbers from Traditional Numeral Systems into the Golden p-Proportion Codes
  • The Table Method
  • The Conversion of Fractional Numbers
    • The first multiplication:
    • The second multiplication:
    • The third multiplication:
    • A Conversion of Integers
      • The first step of conversion
      • The second step of conversion
      • The third step of conversion
      • The fourth step of conversion
      • The fifth step of conversion
      • The sixth step of conversion

• Golden Arithmetic
  • Golden Summation and Subtraction
  • Golden Multiplication
  • Golden Division

• A New Approach to the Geometric Definition of a Number
  • The Geometric Definition of Number
  • Newton's Definition of Real Numbers
  • Numeral Systems with Irrational Radices as a New Definition of Real Numbers
  • The Golden Representations of Natural Numbers

• New Mathematical Properties of Natural Numbers (Z- and D-properties)

• The F- and L-Codes
  • Definition of the F- and L-Codes
  • A Numerical Example
  • Some Properties of the F- and L-Codes
  • Algebraic Summation of Integers

• Number-heoretical Properties of the Golden p-Proportion Codes
  • The Z_p-property of Natural Numbers
  • F_p-code
  • L_p-code

• The Golden Resistor Dividers
  • The Binary Resistor Divider
  • The Golden Resistor Dividers

• Application of the Fibonacci and Golden Proportion Codes to Digital-to-Analog and Analog-to-Digital Conversion
  • The “Golden” Digital-to-Analog Converters
  • Checking the “Golden” DAC
  • The “Golden” ADC
    • The first step
    • The second step
    • The third step
    • The Self-orrecting “Golden” ADC
    • An Application of the Z-property for Checking DAC

• Conclusion

The following sections are included:

• A Ternary Computer “Setun”
  • Brousentsov's Ternary Principle
  • The Dramatic History of the “Setun” Computer
  • Ternary Technology

• Ternary Symmetrical Numeral System
  • Ternary Symmetrical Representation
  • Ternary Inversion
  • The Range of Number Representation

• Ternary-Symmetrical Arithmetic
  • Ternary-Symmetrical Summation and Subtraction
  • Ternary-Symmetrical Multiplication
  • Ternary-Symmetrical Division

• Ternary Logic
  • Basic Functions of Ternary Logic
  • Binary Realization of Ternary Logic Elements
  • Flip-flap-flop

• Ternary Mirror-Symmetrical Representation
  • A Conversion of the Binary “Golden” Representation to the Ternary “Golden” Representation
  • Ternary F- and L-Representations
  • The Representation of Negative Numbers
  • Mirror-Symmetrical Property of Integer Representation
  • The Radix of the Ternary Mirror-Symmetrical Numeral System
  • Comparison of Number in the Ternary Mirror-Symmetrical Numeral System

• The Range of Number Representation and Redundancy of the Ternary Mirror-Symmetrical Numeral System
  • The Range of Number Representation
  • The Redundancy of Ternary Mirror-Symmetrical Representation

• Mirror-Symmetrical Summation and Subtraction
  • Mirror-Symmetrical Summation
  • Ternary Mirror-Symmetrical Multi-Digit Summator
  • Mirror-Symmetrical Subtraction
  • The “Swing” Phenomenon
  • The “Doubling” Mirror-Symmetrical Summator

• Mirror-Symmetrical Multiplication and Division
  • Mirror-Symmetrical Multiplication
  • Mirror-Symmetrical Division
  • The Main Arithmetical Advantages of Mirror-Symmetrical Multiplication and Division

• Typical Devices of Ternary Mirror-Symmetrical Processors
  • A Converter for Binary “Golden” Code to Ternary Mirror-Symmetrical Representation
  • Technical Realization of Mirror-Symmetrical Checking
  • Technical Realization of Ternary Mirror-Symmetrical Arithmetic
  • Ternary Mirror-Symmetrical Accumulator

• Matrix and Pipeline Mirror-Symmetrical Summators
  • Matrix Mirror-Symmetrical Summator
    • The first stage
    • The second stage
  • Pipeline Mirror-Symmetrical Summator
  • Pipeline Mirror-Symmetrical Multiplier

• Ternary Mirror-Symmetrical Digit-to-Analog Converter
  • The “Golden” Resistor Divider for Ternary Mirror-Symmetrical Representation
  • Ternary Mirror-Symmetrical Digit-to-Analog Converter

• Conclusion

The following sections are included:

• A History of Coding Theory
  • Types of Coding
    • Claude Elwood Shannon
    • Mathematical Theory of Communication
  • Error-Correction Codes
    • Codes Predating Hamming
    • Richard Wesley Hamming
    • Hamming Code
    • General Principles of Error Detection and Correction
    • The Main Shortcomings of the Existing Error-Correction Codes

• Cryptographic Coding
  • What is Cryptography?
  • Symmetric-Key Cryptography
  • Public-Key Cryptography

• Non-singular Matrices
  • A Definition of Non-singular Matrices
  • Non-singular (2×2)-matrices

• Fibonacci Encoding/Decoding Method Based Upon Matrix Multiplication
  • The Fibonacci Encoding/Decoding Method Based on the Q_p,m-Matrices
  • Fibonacci Encoding/Decoding Method Based on the G_m and Q-Matrices
  • An Example of the Fibonacci Encoding/Decoding Method Based on the Q-Matrix

• The Main Checking Relations of the Fibonacci Encoding/Decoding Method
  • Determinants of the Code Matrix
  • Connections between the Elements of the Code Matrix

• Error Detection and Correction
  • A Notion of “Errors”
  • Detection of “Errors”
  • Correction of “Single” Errors
  • A Numerical Example of “Single” Error Correction
    • Situation (a)
    • Situation (b)
    • Situation (c)
    • Correction of “Double” Errors
    • Correction of “Triple” Errors

• Redundancy, Correction Ability, and the Advantages of the Fibonacci Encoding/Decoding Method
  • Redundancy of the Fibonacci Encoding/Decoding Method
  • Correction Ability of the Fibonacci Encoding/Decoding Method
  • Advantages of the Fibonacci Encoding/Decoding Method

• Matrix Cryptography
  • The Concept of Hybrid Cryptosystems
  • General Principle of Matrix Cryptography
  • Matrix Cryptography of Digital Sound Signals
  • Matrix Cryptography of Digital Images
  • The Problem of Checking Information in Cryptosystems
  • “Golden” Cryptography
  • Checking Information in the “Golden” Cryptosystem
    • The Main Checking Relations for the “Golden” Cryptography
    • Checking Encoder
    • Checking Channel and Decoder
  • Project for a Cryptographic Mobile Phone

• Conclusion

The following sections are included:

• Epilogue Dirac's Principle of Mathematical Beauty and the Mathematics of Harmony: Clarifying the Origins and Development of Mathematics

• Introduction
  • Dirac's Principle of Mathematical Beauty
  • Platonic Solids
  • Binomial Coefficients, the Binomial Formula, and Pascal's Triangle
  • Fibonacci and Lucas Numbers, the Golden Mean and Binet Formulas
• The “Strategic Mistakes” in the Development of Mathematics
  • Mathematics: The Loss of Certainty
  • The Neglect of the “Beginnings”
  • The Neglect of the Golden Section
  • The One-sided Interpretation of Euclid's Elements
  • The One-sided Approach to the Origin of Mathematics
  • The Greatest Mathematical Mystification of the 19th Century
  • The Underestimation of Binet Formulas
  • The Underestimation of Felix Klein's Idea Concerning the Regular Icosahedron
  • The Underestimation of Bergman's Number System
• Three “Key” Problems of Mathematics and a New Approach to the Mathematics Origins
• The Generalized Fibonacci Numbers and the Generalized Golden Proportions
  • The Generalized Fibonacci P-numbers, the Generalized P-proportions, the Generalized Binet Formulas and the Generalized Lucas P-numbers
  • The generalized Fibonacci λ-numbers,“Metallic Means” by Vera Spinadel, Gazale Formulas and General Theory of Hyperbolic Functions
• A New Geometric Definition of Number
  • Euclidean and Newtonian Definition of Real Number
  • Number Systems with Irrational Radices as a New Definition of Real Number
• Fibonacci and “Golden” Matrices
  • Fibonacci Matrices
  • The “Golden” Matrices
• Applications in Computer Science: the “Golden” Information Technology
  • Fibonacci Codes, Fibonacci Arithmetic and Fibonacci Computers
  • Ternary Mirror-Symmetrical Arithmetic
  • A New Theory of Error-Correcting Codes Based upon Fibonacci Matrices
  • Matrix and “Golden” Cryptography
• Fundamental Discoveries of Modern Science Based Upon the Golden Section and “Platonic Solids”
  • Shechtman's Quasi-Crystals
  • Fullerenes (Nobel Prize for Chemistry of 1996)
  • El-Naschie's E-infinity Theory
  • Bodnar's Geometry
  • Petoukhov's “Golden” Genomatrices
  • Fibonacci-Lorenz Transformations and “Golden” Interpretation of the Universe Evolution
  • Hilbert's Fourth Problem
• Conclusion
  • The First Conclusion
  • The Second Conclusion
  • The Third Conclusion
  • The Fourth Conclusion
  • The Fifth Conclusion
  • The Sixth Conclusion
  • The Seventh Conclusion
  • The Eighth Conclusion

• References

• Appendix I. Museum of Harmony and Golden Section

• Index