Introduction: Three "Key" Problems of Mathematics on
the Stage of its Origin ....................................... xix
Acknowledgements ............................................ xxxix
Credits and Sources ......................................... xIiii
Part I Classical Golden Mean, Fibonacci Numbers, and
Platonic Solids
Chapter 1. The Golden Section
1.1 Geometric Definition of the Golden Section ................. 2
1.2 Algebraic Properties of the Golden Mean .................... 7
1.3 The Algebraic Equation of the Golden Mean ................. 11
1.4 The Golden Rectangles and the Golden Brick ................ 20
1.5 Decagon: Connection of the Golden Mean to the Number π .... 24
1.6 The Golden Right Triangle and the Golden Ellipse .......... 25
1.7 The Golden Isosceles Triangles and Pentagon ............... 27
1.8 The Golden Section and the Mysteries of Egyptian
Culture ................................................... 32
1.9 The Golden Section in Greek Culture ....................... 37
1.10 The Golden Section in Renaissance Art ..................... 41
1.11 De Divina Proportione by Luca Pacioli ..................... 46
1.12 A Proportional Scheme of the Golden Section in
Architecture .............................................. 51
1.13 The Golden Section in the Art of 19th and 20th
Centuries ................................................. 53
1.14 A Formula of Beauty ....................................... 56
1.15 Conclusion ................................................ 59
Chapter 2. Fibonacci and Lucas Numbers
2.1 Who was Fibonacci? ........................................ 60
2.2 Fibonacci's Rabbits ....................................... 62
2.3 Numerology and Fibonacci Numbers .......................... 67
2.4 Variations on Fibonacci Theme ............................. 72
2.5 Lucas Numbers ............................................. 77
2.6 Cassini Formula ........................................... 83
2.7 Pythagorean Triangles and Fibonacci Numbers ............... 85
2.8 Binet Formulas ............................................ 90
2.9 Fibonacci Rectangle and Fibonacci Spiral .................. 95
2.10 Chemistry by Fibonacci .................................... 97
2.11 Symmetry of Nature and the Nature of Symmetry ............ 100
2.12 Omnipresent Phyllotaxis .................................. 105
2.13 "Fibonacci Resonances" of the Genetic Code ............... 110
2.14 The Golden Section and Fibonacci Numbers in Music and
Cinema ................................................... 112
2.15 The Music of Poetry ...................................... 116
2.16 The Problem of Choice: Will Buridan's Donkey Die? ........ 120
2.17 Elliott Waves ............................................ 125
2.18 The Outstanding Fibonacci Mathematicians of the 20th
Century .................................................. 129
2.19 Slavic "Golden" Group .................................... 132
2.20 Conclusion ............................................... 136
Chapter 3. Regular Polyhedrons
3.1 Platonic Solids .......................................... 137
3.2 Archimedean Solids and Star-shaped Regular Polyhedra ..... 144
3.3 A Mystery of the Egyptian Calendar ....................... 148
3.4 A Dodecahedron-Icosahedron Doctrine ...................... 152
3.5 Johannes Kepler: from "Mysterium" to "Harmony" ........... 154
3.6 A Regular Icosahedron as the Main Geometrical Object
of Mathematics ........................................... 160
3.7 Regular Polyhedra in Nature and Science .................. 163
3.8 Applications of Regular Polyhedrons in Art ............... 172
3.9 Application of the Golden Mean in Contemporary Art ....... 179
3.10 Conclusion ............................................... 182
Part II. Mathematics of Harmony
Chapter 4. Generalizations of Fibonacci Numbers and the
Golden Mean
4.1 A Combinatorial Approach to the Harmony of Mathematics ... 186
4.2 Binomial Coefficients and Pascal Triangle ................ 189
4.3 The Generalized Fibonacci p-Numbers ...................... 192
4.4 The Generalized Golden p-Sections ........................ 199
4.5 The Generalized Principle of the Golden Section .......... 202
4.6 A Generalization of Euclid's Theorem 11.11 ............... 204
4.7 The Roots of the Generalized Golden Algebraic
Equations ................................................ 206
4.8 The Generalized Golden Algebraic Equations of Higher
Degrees .................................................. 212
4.9 The Generalized Binet Formula for the Fibonacci
p-Numbers ................................................ 214
4.10 The Generalized Lucas p-Numbers .......................... 221
4.11 The "Metallic Means Family" by Vera W. de Spinadel ....... 227
4.12 Gazale Formulas .......................................... 232
4.13 Fibonacci and Lucas m-Numbers ............................ 237
4.14 On the m-Extension of the Fibonacci and Lucas
p-Numbers ................................................ 241
4.15 Structural Harmony of Systems ............................ 249
4.16 Conclusion ............................................... 253
Chapter 5. Hyperbolic Fibonacci and Lucas Functions
5.1 The Simplest Elementary Functions ........................ 255
5.2 Hyperbolic Functions ..................................... 259
5.3 Hyperbolic Fibonacci and Lucas Functions (Stakhov-
Tkachenko's Definition) .................................. 264
5.4 Integration and Differentiation of the Hyperbolic
Fibonacci and Lucas Functions and their Main
Identities ............................................... 268
5.5 Symmetric Hyperbolic Fibonacci and Lucas Functions
(Stakhov-Rozin Definition) ............................... 277
5.6 Recursive Properties of the Symmetric Hyperbolic
Fibonacci and Lucas Functions ............................ 280
5.7 Hyperbolic Properties of the Symmetric Hyperbolic
Fibonacci and Lucas Functions and Formulas for Their
Differentiation and Integration .......................... 283
5.8 The Golden Shofar ........................................ 286
5.9 A General Theory of the Hyperbolic Functions ............. 291
5.10 A Puzzle of Phyllotaxis .................................. 299
5.11 A Geometric Theory of the Hyperbolic Functions ........... 301
5.12 Bodnar's Geometry ........................................ 307
5.13 Conclusion ............................................... 313
Chapter 6. Fibonacci and Golden Matrices
6.1 Introduction into Matrix Theory .......................... 317
6.2 Fibonacci Q-Matrix ....................................... 322
6.3 Generalized Fibonacci Qp-Matrices ........................ 326
XVI
6.4 Determinants of the Qp- Matrices and their Powers ........ 330
6.5 The "Direct" and "Inverse" Fibonacci Matrices ............ 333
6.6 Fibonacci Gm-Matrices .................................... 334
6.7 Fibonacci Qp,m-Matrices .................................. 340
6.8 Determinants of the Qp,m-Matrices and their Powers ....... 343
6.9 The Golden Q-Matrices .................................... 345
6.10 The Golden Gm-Matrices ................................... 348
6.11 The Golden Genomatrices by Sergey Petoukhov .............. 350
6.12 Conclusion ............................................... 357
Part III Application in Computer Science
Chapter 7. Algorithmic Measurement Theory
7.1 The Role of Measurement in the History of Science ........ 360
7.2 Mathematical Measurement Theory .......................... 364
7.3 Evolution of the Infinity Concept ........................ 370
7.4 A Constructive Approach to Measurement Theory ............ 377
7.5 Mathematical Model of Measurement ........................ 382
7.6 Classical Measurement Algorithms ......................... 385
7.7 The Optimal Measurement Algorithms Originating
Classical Positional Number Systems ...................... 389
7.8 Optimal Measurement Algorithms Based on the
Arithmetical Square ...................................... 392
7.9 Fibonacci Measurement Algorithms ......................... 396
7.10 The Main Result of Algorithmic Measurement Theory ........ 401
7.11 Mathematical Theories Isomorphic to Algorithmic
Measurement Theory ....................................... 408
7.12 Conclusion ............................................... 413
Chapter 8. Fibonacci Computers
8.1 A History of Computers ................................... 416
8.2 Basic Stages in the History of Numeral Systems ........... 424
8.3 Fibonacci p-Codes ........................................ 429
8.4 Minimal Form and Redundancy of the Fibonacci p-Code ...... 434
8.5 Fibonacci Arithmetic: The Classical Approach ............. 443
8.6 Fibonacci Arithmetic: An Original Approach ............... 449
8.7 Fibonacci Multiplication and Division .................... 455
8.8 Hardware Realization of the Fibonacci Processor .......... 460
8.9 Fibonacci Processor for Noise-tolerant Computations ...... 465
8.10 The Dramatic History of the Fibonacci Computer Project ... 470
8.11 Conclusion ............................................... 475
Chapter 9. Codes of the Golden Proportion
9.1 Numeral Systems with Irrational Bases .................... 476
9.2 Some Mathematical Properties of the Golden p-Proportion
Codes .................................................... 480
9.3 Conversion of Numbers from Traditional Numeral Systems
into the Golden p-Proportion Codes ....................... 484
9.4 Golden Arithmetic ........................................ 488
9.5 A New Approach to the Geometric Definition of a Number ... 492
9.6 New Mathematical Properties of Natural Numbers (Z- and
D-properties) ............................................ 497
9.7 The F- and I-Codes ....................................... 500
9.8 Number-theoretical Properties of the Golden
p-Proportion Codes ....................................... 505
9.9 The Golden Resistor Dividers ............................. 511
9.10 Application of the Fibonacci and Golden Proportion
Codes to Digital-to-Analog and Analog-to-Digital
Conversion ............................................... 515
9.11 Conclusion .............................................. 520
Chapter 10. Ternary Mirror-Symmetrical Arithmetic
10.1 A Ternary Computer "Setun" ............................... 523
10.2 Ternary-Symmetrical Numeral System ....................... 527
10.3 Ternary-Symmetrical Arithmetic ........................... 530
10.4 Ternary Logic ............................................ 533
10.5 Ternary Mirror-Symmetrical Representation ................ 538
10.6 The Range of Number Representation and Redundancy
of the Ternary Mirror-Symmetrical Numeral System ......... 544
10.7 Mirror-Symmetrical Summation and Subtraction ............. 546
10.8 Mirror-Symmetrical Multiplication and Division ........... 553
10.9 Typical Devices of Ternary Mirror-Symmetrical
Processors ............................................... 557
10.10 Matrix and Pipeline Mirror-Symmetrical Summators ........ 561
10.11 Ternary Mirror-Symmetrical Digit-to-Analog Converter .... 565
10.12 Conclusion .............................................. 567
Chapter 11. A New Coding Theory Based on a Matrix Approach
11.1 A History of Coding Theory ............................... 569
11.2 Non-singular Matrices .................................... 579
11.3 Fibonacci Encoding/Decoding Method Based upon Matrix
Multiplication ........................................... 581
11.4 The Main Checking Relations of the Fibonacci Encoding
Decoding Method .......................................... 584
11.5 Error Detection and Correction ........................... 588
11.6 Redundancy, Correcting Ability, and the Advantages of
the Fibonacci Encoding/Decoding Method ................... 597
11.7 Matrix Cryptography ...................................... 601
11.8 Conclusion ............................................... 613
Epilogue. Dirac's Principle of Mathematical Beauty and the
Mathematics of Harmony: Clarifying the Origins and
Development of Mathematics
E.l Introduction ............................................. 615
E.2 The "Strategic Mistakes" in the Development of
Mathematics .............................................. 620
E.3 Three "Key" Problems of Mathematics and a New Approach
to the Mathematics Origins ............................... 632
E.4 The Generalized Fibonacci Numbers and the Generalized
Golden Proportions ....................................... 633
E.5 A New Geometric Definition of Number ..................... 641
E.6 Fibonacci and "Golden" Matrices .......................... 644
E.7 Applications in Computer Science: the "Golden"
Information Technology ................................... 646
E.8 Fundamental Discoveries of Modern Science Based Upon
the Golden Section and" Platonic Solids" ................. 649
E.9 Conclusion ............................................... 657
References .................................................... 661
Appendix: Museum of Harmony and Golden Section ................ 675
Index ......................................................... 685
|
