The mathematics of harmony
Subtitle: from Euclid to contemporary mathematics and computer science
By: by Alexey Stakhov ; assisted by Scott Olsen.
Contributors: Olsen, Scott Anthony
Series: Series on knots and everything -- v. 22
Publisher: World Scientific; World Scientific Publishing Co Pte Ltd
Published: 2009
Place of publication: [Hackensack], N.J
Language: eng
Number of pages: 694
ISBN-13: 9789812775825
Subjects: Fibonacci numbers; Golden section; Mathematics -- History; Computer science
Notes: Includes bibliographical references and index.
Table of contents: title: Introduction: Three "Key" Problems of Mathematics on the Stage of its Origin
pagenum: xix; title: Acknowledgements
pagenum: xxxix; title: Credits and Sources
pagenum: xliii; label: Part I
title: Classical Golden Mean, Fibonacci Numbers, and Platonic Solids; label: Chapter 1
title: The Golden Section; label: 1.1
title: Geometric Definition of the Golden Section
pagenum: 2; label: 1.2
title: Algebraic Properties of the Golden Mean
pagenum: 7; label: 1.3
title: The Algebraic Equation of the Golden Mean
pagenum: 11; label: 1.4
title: The Golden Rectangles and the Golden Brick
pagenum: 20; label: 1.5
title: Decagon: Connection of the Golden Mean to the Number π
pagenum: 24; label: 1.6
title: The Golden Right Triangle and the Golden Ellipse
pagenum: 25; label: 1.7
title: The Golden Isosceles Triangles and Pentagon
pagenum: 27; label: 1.8
title: The Golden Section and the Mysteries of Egyptian Culture
pagenum: 32; label: 1.9
title: The Golden Section in Greek Culture
pagenum: 37; label: 1.10
title: The Golden Section in Renaissance Art
pagenum: 41; label: 1.11
title: De Divina Proportione by Luca Pacioli
pagenum: 46; label: 1.12
title: A Proportional Scheme of the Golden Section in Architecture
pagenum: 51; label: 1.13
title: The Golden Section in the Art of 19th and 20th Centuries
pagenum: 53; label: 1.14
title: A Formula of Beauty
pagenum: 56; label: 1.15
title: Conclusion
pagenum: 59; label: Chapter 2
title: Fibonacci and Lucas Numbers; label: 2.1
title: Who was Fibonacci?
pagenum: 60; label: 2.2
title: Fibonacci's Rabbits
pagenum: 62; label: 2.3
title: Numerology and Fibonacci Numbers
pagenum: 67; label: 2.4
title: Variations on Fibonacci Theme
pagenum: 72; label: 2.5
title: Lucas Numbers
pagenum: 77; label: 2.6
title: Cassini Formula
pagenum: 83; label: 2.7
title: Pythagorean Triangles and Fibonacci Numbers
pagenum: 85; label: 2.8
title: Binet Formulas
pagenum: 90; label: 2.9
title: Fibonacci Rectangle and Fibonacci Spiral
pagenum: 95; label: 2.10
title: Chemistry by Fibonacci
pagenum: 97; label: 2.11
title: Symmetry of Nature and the Nature of Symmetry
pagenum: 100; label: 2.12
title: Omnipresent Phyllotaxis
pagenum: 105; label: 2.13
title: "Fibonacci Resonances" of the Genetic Code
pagenum: 110; label: 2.14
title: The Golden Section and Fibonacci Numbers in Music and Cinema
pagenum: 112; label: 2.15
title: The Music of Poetry
pagenum: 116; label: 2.16
title: The Problem of Choice: Will Buridan's Donkey Die?
pagenum: 120; label: 2.17
title: Elliott Waves
pagenum: 125; label: 2.18
title: The Outstanding Fibonacci Mathematicians of the 20th Century
pagenum: 129; label: 2.19
title: Slavic "Golden" Group
pagenum: 132; label: 2.20
title: Conclusion
pagenum: 136; label: Chapter 3
title: Regular Polyhedrons; label: 3.1
title: Platonic Solids
pagenum: 137; label: 3.2
title: Archimedean Solids and Star-shaped Regular Polyhedra
pagenum: 144; label: 3.3
title: A Mystery of the Egyptian Calendar
pagenum: 148; label: 3.4
title: A Dodecahedron-Icosahedron Doctrine
pagenum: 152; label: 3.5
title: Johannes Kepler: from "Mysterium" to "Harmony"
pagenum: 154; label: 3.6
title: A Regular Icosahedron as the Main Geometrical Object of Mathematics
pagenum: 160; label: 3.7
title: Regular Polyhedra in Nature and Science
pagenum: 163; label: 3.8
title: Applications of Regular Polyhedrons in Art
pagenum: 172; label: 3.9
title: Application of the Golden Mean in Contemporary Art
pagenum: 179; label: 3.10
title: Conclusion
pagenum: 182; label: Part II
title: Mathematics of Harmony; label: Chapter 4
title: Generalizations of Fibonacci Numbers and the Golden Mean; label: 4.1
title: A Combinatorial Approach to the Harmony of Mathematics
pagenum: 186; label: 4.2
title: Binomial Coefficients and Pascal Triangle
pagenum: 189; label: 4.3
title: The Generalized Fibonacci p-Numbers
pagenum: 192; label: 4.4
title: The Generalized Golden p-Sections
pagenum: 199; label: 4.5
title: The Generalized Principle of the Golden Section
pagenum: 202; label: 4.6
title: A Generalization of Euclid's Theorem II.11
pagenum: 204; label: 4.7
title: The Roots of the Generalized Golden Algebraic Equations
pagenum: 206; label: 4.8
title: The Generalized Golden Algebraic Equations of Higher Degrees
pagenum: 212; label: 4.9
title: The Generalized Binet Formula for the Fibonacci p-Numbers
pagenum: 214; label: 4.10
title: The Generalized Lucas p-Numbers
pagenum: 221; label: 4.11
title: The "Metallic Means Family" by Vera W. de Spinadel
pagenum: 227; label: 4.12
title: Gazale Formulas
pagenum: 232; label: 4.13
title: Fibonacci and Lucas m-Numbers
pagenum: 237; label: 4.14
title: On the m-Extension of the Fibonacci and Lucas p-Numbers
pagenum: 241; label: 4.15
title: Structural Harmony of Systems
pagenum: 249; label: 4.16
title: Conclusion
pagenum: 253; label: Chapter 5
title: Hyperbolic Fibonacci and Lucas Functions; label: 5.1
title: The Simplest Elementary Functions
pagenum: 255; label: 5.2
title: Hyperbolic Functions
pagenum: 259; label: 5.3
title: Hyperbolic Fibonacci and Lucas Functions (Stakhov-Tkachenko's Definition)
pagenum: 264; label: 5.4
title: Integration and Differentiation of the Hyperbolic Fibonacci and Lucas Functions and their Main Identities
pagenum: 268; label: 5.5
title: Symmetric Hyperbolic Fibonacci and Lucas Functions (Stakhov-Rozin Definition)
pagenum: 277; label: 5.6
title: Recursive Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions
pagenum: 280; label: 5.7
title: Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions and Formulas for Their Differentiation and Integration
pagenum: 283; label: 5.8
title: The Golden Shofar
pagenum: 286; label: 5.9
title: A General Theory of the Hyperbolic Functions
pagenum: 291; label: 5.10
title: A Puzzle of Phyllotaxis
pagenum: 299; label: 5.11
title: A Geometric Theory of the Hyperbolic Functions
pagenum: 301; label: 5.12
title: Bodnar's Geometry
pagenum: 307; label: 5.13
title: Conclusion
pagenum: 313; label: Chapter 6
title: Fibonacci and Golden Matrices; label: 6.1
title: Introduction into Matrix Theory
pagenum: 317; label: 6.2
title: Fibonacci \(Q\) -Matrix
pagenum: 322; label: 6.3
title: Generalized Fibonacci \(Q_p\) -Matrices
pagenum: 326; label: 6.4
title: Determinants of the \(Q_p\) -Matrices and their Powers
pagenum: 330; label: 6.5
title: The "Direct" and "Inverse" Fibonacci Matrices
pagenum: 333; label: 6.6
title: Fibonacci \(G_m\) -Matrices
pagenum: 334; label: 6.7
title: Fibonacci \(Q_{p,m}\) -Matrices
pagenum: 340; label: 6.8
title: Determinants of the \(Q_{p,m}\) -Matrices and their Powers
pagenum: 343; label: 6.9
title: The Golden \(Q\) -Matrices
pagenum: 345; label: 6.10
title: The Golden \(G_m\) -Matrices
pagenum: 348; label: 6.11
title: The Golden Genomatrices by Sergey Petoukhov
pagenum: 350; label: 6.12
title: Conclusion
pagenum: 357; label: Part III
title: Application in Computer Science; label: Chapter 7
title: Algorithmic Measurement Theory; label: 7.1
title: The Role of Measurement in the History of Science
pagenum: 360; label: 7.2
title: Mathematical Measurement Theory
pagenum: 364; label: 7.3
title: Evolution of the Infinity Concept
pagenum: 370; label: 7.4
title: A Constructive Approach to Measurement Theory
pagenum: 377; label: 7.5
title: Mathematical Model of Measurement
pagenum: 382; label: 7.6
title: Classical Measurement Algorithms
pagenum: 385; label: 7.7
title: The Optimal Measurement Algorithms Originating Classical Positional Number Systems
pagenum: 389; label: 7.8
title: Optimal Measurement Algorithms Based on the Arithmetical Square
pagenum: 392; label: 7.9
title: Fibonacci Measurement Algorithms
pagenum: 396; label: 7.10
title: The Main Result of Algorithmic Measurement Theory
pagenum: 401; label: 7.11
title: Mathematical Theories Isomorphic to Algorithmic Measurement Theory
pagenum: 408; label: 7.12
title: Conclusion
pagenum: 413; label: Chapter 8
title: Fibonacci Computers; label: 8.1
title: A History of Computers
pagenum: 416; label: 8.2
title: Basic Stages in the History of Numeral Systems
pagenum: 424; label: 8.3
title: Fibonacci p-Codes
pagenum: 429; label: 8.4
title: Minimal Form and Redundancy of the Fibonacci p-Code
pagenum: 434; label: 8.5
title: Fibonacci Arithmetic: The Classical Approach
pagenum: 443; label: 8.6
title: Fibonacci Arithmetic: An Original Approach
pagenum: 449; label: 8.7
title: Fibonacci Multiplication and Division
pagenum: 455; label: 8.8
title: Hardware Realization of the Fibonacci Processor
pagenum: 460; label: 8.9
title: Fibonacci Processor for Noise-tolerant Computations
pagenum: 465; label: 8.10
title: The Dramatic History of the Fibonacci Computer Project
pagenum: 470; label: 8.11
title: Conclusion
pagenum: 475; label: Chapter 9
title: Codes of the Golden Proportion; label: 9.1
title: Numeral Systems with Irrational Bases
pagenum: 476; label: 9.2
title: Some Mathematical Properties of the Golden p-Proportion Codes
pagenum: 480; label: 9.3
title: Conversion of Numbers from Traditional Numeral Systems into the Golden p-Proportion Codes
pagenum: 484; label: 9.4
title: Golden Arithmetic
pagenum: 488; label: 9.5
title: A New Approach to the Geometric Definition of a Number
pagenum: 492; label: 9.6
title: New Mathematical Properties of Natural Numbers (Z- and D-properties)
pagenum: 497; label: 9.7
title: The F- and L-Codes
pagenum: 500; label: 9.8
title: Number-theoretical Properties of the Golden p-Proportion Codes
pagenum: 505; label: 9.9
title: The Golden Resistor Dividers
pagenum: 511; label: 9.10
title: Application of the Fibonacci and Golden Proportion Codes to Digital-to-Analog and Analog-to-Digital Conversion
pagenum: 515; label: 9.11
title: Conclusion
pagenum: 520; label: Chapter 10
title: Ternary Mirror-Symmetrical Arithmetic; label: 10.1
title: A Ternary Computer "Setun"
pagenum: 523; label: 10.2
title: Ternary-Symmetrical Numeral System
pagenum: 527; label: 10.3
title: Ternary-Symmetrical Arithmetic
pagenum: 530; label: 10.4
title: Ternary Logic
pagenum: 533; label: 10.5
title: Ternary Mirror-Symmetrical Representation
pagenum: 538; label: 10.6
title: The Range of Number Representation and Redundancy of the Ternary Mirror-Symmetrical Numeral System
pagenum: 544; label: 10.7
title: Mirror-Symmetrical Summation and Subtraction
pagenum: 546; label: 10.8
title: Mirror-Symmetrical Multiplication and Division
pagenum: 553; label: 10.9
title: Typical Devices of Ternary Mirror-Symmetrical Processors
pagenum: 557; label: 10.10
title: Matrix and Pipeline Mirror-Symmetrical Summators
pagenum: 561; label: 10.11
title: Ternary Mirror-Symmetrical Digit-to-Analog Converter
pagenum: 565; label: 10.12
title: Conclusion
pagenum: 567; label: Chapter 11
title: A New Coding Theory Based on a Matrix Approach; label: 11.1
title: A History of Coding Theory
pagenum: 569; label: 11.2
title: Non-singular Matrices
pagenum: 579; label: 11.3
title: Fibonacci Encoding/Decoding Method Based upon Matrix Multiplication
pagenum: 581; label: 11.4
title: The Main Checking Relations of the Fibonacci Encoding/Decoding Method
pagenum: 584; label: 11.5
title: Error Detection and Correction
pagenum: 588; label: 11.6
title: Redundancy, Correcting Ability, and the Advantages of the Fibonacci Encoding/Decoding Method
pagenum: 597; label: 11.7
title: Matrix Cryptography
pagenum: 601; label: 11.8
title: Conclusion
pagenum: 613; label: Epilogue
title: Dirac's Principle of Mathematical Beauty and the Mathematics of Harmony: Clarifying the Origins and Development of Mathematics; label: E.1
title: Introduction
pagenum: 615; label: E.2
title: The "Strategic Mistakes" in the Development of Mathematics
pagenum: 620; label: E.3
title: Three "Key" Problems of Mathematics and a New Approach to the Mathematics Origins
pagenum: 632; label: E.4
title: The Generalized Fibonacci Numbers and the Generalized Golden Proportions
pagenum: 633; label: E.5
title: A New Geometric Definition of Number
pagenum: 641; label: E.6
title: Fibonacci and "Golden" Matrices
pagenum: 644; label: E.7
title: Applications in Computer Science: the "Golden" Information Technology
pagenum: 646; label: E.8
title: Fundamental Discoveries of Modern Science Based Upon the Golden Section and "Platonic Solids"
pagenum: 649; label: E.9
title: Conclusion
pagenum: 657; title: References
pagenum: 661; title: Appendix: Museum of Harmony and Golden Section
pagenum: 675; title: Index
pagenum: 685