The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science (Series on Knots and Everything 22)

Contents......Page 14
1. The Main Stages of Mathematics Development......Page 20
2. A “Count Problem”......Page 21
3. A “Measurement Problem”......Page 22
4. Mathematics. The Loss of Certainty......Page 23
5. A “Harmony Problem”......Page 25
6. The Numerical Harmony of the Pythagoreans......Page 26
7. A “Harmony Problem” in Euclid’s Elements......Page 27
9. The First Book on the Golden Mean in the History of Science......Page 29
12. The Golden Section and Fibonacci Numbers in Science of the 20th and 21st Centuries......Page 30
13. The Lecture: “The Golden Section and Modern Harmony Mathematics”......Page 34
14. Two Historical Ways of Mathematics Development......Page 35
15. The Main Goal of the Present Book......Page 37
Acknowledgements......Page 40
Credits and Sources......Page 44
Part I. Classical Golden Mean, Fibonacci Numbers, and Platonic Solids......Page 53
1.1.1. A Problem of the Division in the Extreme and Mean Ratio (DEMR)......Page 54
1.1.2. The Origin of the Concept and Title of the Golden Section......Page 56
1.1.4. A “Double” Square......Page 58
1.2.1. Remarkable Identities of the Golden Mean......Page 59
1.2.2. The “Golden” Geometric Progression......Page 61
1.2.4. A Representation of the Golden Mean in “Radicals”......Page 62
1.3.1. Polynomials and Equations......Page 63
1.3.2. Quadratic Irrationals......Page 64
1.3.3. Poor “Studious” Niels Abel......Page 66
1.3.4. Mathematician and Revolutionist Evariste Galois......Page 68
1.3.5. The “Golden”Algebraic Equations of n th Degree......Page 69
1.4.1. A Golden Rectangle with a Side Ratio of τ......Page 72
1.4.2. A Golden Rectangle with a Side Ratio of τ 2......Page 73
1.4.3. The “Golden” Brick of Gothic Architecture......Page 74
1.5. Decagon: Connection of the Golden Mean to the Number π......Page 76
1.6.1. The Golden Right Triangle......Page 77
1.6.2. The “Golden” Ellipse......Page 78
1.7.1. Construction of the “Golden” Isosceles Triangle and Regular Pentagon in The Elements......Page 79
1.7.2. A Regular Pentagon......Page 80
1.7.3. The Pentagon......Page 81
1.7.4. The “Golden Cup” and the Golden Isosceles Triangle......Page 82
1.7.5. Pentagonal Symmetry in Nature......Page 83
1.8.1. Phenomenon of Ancient Egypt......Page 84
1.8.2. The Mysteries of the Egyptian Pyramids......Page 86
1.9.1. Pythagoras......Page 89
1.9.2. The Idea of Harmony in Greek Culture......Page 91
1.9.3. The Golden Section in Greek Sculpture......Page 92
1.10.1. The Idea of the “Divine Harmony” in the Renaissance Epoch......Page 93
1.10.3. “Mona Lisa” by Leonardo da Vinci......Page 95
1.11.1. Luca Pacioli......Page 98
1.11.2. About Pacioli’s Plagiarism......Page 101
1.11.3. Death and Oblivion of Luca Pacioli......Page 102
1.12. A Proportional Scheme of the Golden Section in Architecture......Page 103
1.13.1. Ivan Shishkin’s Picture “The Ship Grove”......Page 105
1.13.2. “Modulor” by Le Corbusier......Page 106
1.13.4. A Picture “Near to the Window” by Konstantin Vasiliev......Page 107
1.14.1. The Golden Ratio in the External Forms of a Person......Page 108
1.14.3. Nefertiti......Page 109
1.15. Conclusion......Page 111
2.1.1. Leonardo of Pisa Fibonacci......Page 112
2.1.2. Fibonacci and Abu Kamil......Page 113
2.2.1. The “Rabbit Reproduction Problem”......Page 114
2.2.2. About the Rabbits......Page 116
2.2.4. Honeybees and Family Trees......Page 117
2.3.1. Some Mathematical Properties of Numerological Values......Page 119
2.3.2. Fibonacci Numerological Series......Page 121
2.3.3. Another Periodic Properties of Fibonacci Numbers......Page 123
2.4. Variations on Fibonacci Theme......Page 124
2.4.1. Formulas for the Sums of Fibonacci Numbers......Page 125
2.4.2. Connection of Fibonacci Numbers to the Golden Mean......Page 126
2.4.3. “Iron Table” by Steinhaus......Page 127
2.5.1. Francois Edouard Anatole Lucas......Page 129
2.5.3. The “Extended” Fibonacci and Lucas Numbers......Page 131
2.5.5. Lucas Numbers and Numerology......Page 133
2.6.1. Great Astronomer Giovanni Domenico Cassini......Page 135
2.6.2. Cassini Formula for Fibonacci Numbers......Page 136
2.7.2. Pythagorean Triangles......Page 137
2.7.3. Fibonacci Pythagorean Triangles......Page 138
2.7.4. Lucas Pythagorean Triangles......Page 140
2.7.5. Fibonacci and Lucas Right Triangles......Page 141
2.8.2. Deducing Binet Formulas......Page 142
2.8.3. A Historical Analogy......Page 145
2.9.1. Fibonacci Rectangles......Page 147
2.9.3. Fibonacci Spirals in Nature......Page 148
2.10.1. Law of Multiple Ratios......Page 149
2.10.2. Research by the Ukrainian Scientist Nikolai Vasyutinsky......Page 150
2.11.1. Basic Concepts of Symmetry......Page 152
2.11.2. Symmetry of Crystals......Page 153
2.11.3. Symmetry Laws in Nature......Page 155
2.11.4. Symmetry Laws in Art......Page 156
2.12.1. A Helical Symmetry......Page 157
2.12.2. Densely Packed Phyllotaxis Structures......Page 159
2.12.3. Use of Phyllotaxis Lattices in Painting......Page 161
2.13.2. DNA SUPRA code (Jean Claude Perez’s Discovery)......Page 162
2.13.3. A Verification of Jean Claude Perez’s Law......Page 163
2.14.1. Pythagorean Theory of Musical Harmony......Page 164
2.14.2. Chopin’s Etudes in the Lighting of the Golden Section......Page 165
2.14.3. Rosenov’s Research......Page 166
2.14.5. The Golden Mean and Fibonacci Numbers in Cinema......Page 167
2.15.1. Pushkin’s Poetry......Page 168
2.15.2. Lermontov’s Poetry......Page 170
2.15.3. Shota Rustaveli’s Poem......Page 171
2.16.2. A Psychological Experiment......Page 172
2.16.3. What is an Implication?......Page 173
2.16.4. What is a Reflection?......Page 175
2.17.1. Ralph Nelson Elliott......Page 177
2.17.2. Rhythm in Nature......Page 178
2.17.3. Elliott’s Wave Principle......Page 179
2.18.1. Willem Abraham Wythoff......Page 181
2.18.3. Verner Emil Hoggatt......Page 182
2.18.5. Steven Vaida......Page 183
2.19. Slavic “Golden” Group......Page 184
2.20. Conclusion......Page 188
3.1.1. Regular Polygons......Page 189
3.1.2. Regular Polyhedra......Page 191
3.1.4. The Golden Section in the Dodecahedron and Icosahedron......Page 193
3.1.5. Plato’s Cosmology......Page 195
3.2.1. Archimedean Solids......Page 196
3.2.2. Star shaped Regular Polyhedra......Page 199
3.3.1. What is a Calendar?......Page 200
3.3.2. Structure of the Egyptian Calendar......Page 201
3.3.3. Connection of the Egyptian Calendar with the Numerical Characteristics of the Dodecahedron......Page 202
3.4.1. Sources of the Doctrine......Page 204
3.4.2. A Shape of the Earth......Page 205
3.5.1. “Mysterium Cosmographicum”......Page 206
3.5.2. Kepler’s Cosmic Cup......Page 207
3.5.3. Discovery of Kepler’s First Two Astronomical Laws......Page 209
3.5.4. “Harmonices Mundi”......Page 210
3.5.5. Life through the Centuries......Page 211
3.6.1. Felix Klein......Page 212
3.6.2. Elementary Mathematics from the Point of View of Higher Mathematics......Page 213
3.6.3. Role of the Icosahedron in Mathematical Progress......Page 214
3.7.1. Symmetry Groups of Regular Polyhedra......Page 215
3.7.2. Applications of Regular Polyhedra in the Living Nature......Page 217
3.7.3. “Parquet Problem” and Penrose Tiling......Page 219
3.7.4. Quasi crystals......Page 221
3.7.5. Fullerenes......Page 222
3.8.1. Leonardo da Vinci’s Methods of Regular Polyhedra Representation......Page 224
3.8.2. Pacioli’s Polyhedron......Page 225
3.8.3. Albrecht Durer......Page 227
3.8.4. Piero Della Francesca......Page 228
3.8.6. Salvador Dali’s Last Supper......Page 229
3.8.7. Escher’s Creative Work......Page 230
3.9.1 Abstract Art by Astrid Fitzgerald......Page 231
3.9.3. The Geometric Art of John Michell......Page 232
3.9.4. Quantum Connections by Marion Drennen......Page 233
3.10. Conclusion......Page 234
Part II. Mathematics of Harmony......Page 237
4.1.1. Mathematical, Aesthetic and Artistic Understanding of Harmony......Page 238
4.1.2. Concept of the Mathematical Theory of Harmony......Page 239
4.1.3. An Analogy between the Theory of Information and Mathematics of Harmony......Page 240
4.2.1. The Main Concepts of Combinatorial Analysis......Page 241
4.2.2. Binomial Formula......Page 242
4.2.3. Pascal Triangle......Page 243
4.3.1. Rectangular Pascal Triangle......Page 244
4.3.2. Pascal p Triangles and Fibonacci p Numbers......Page 245
4.3.4. A Representation of the Fibonacci p Numbers by the Binomial Coefficients......Page 247
4.3.5. The “Extended” Fibonacci р
Numbers......Page 248
4.3.6. Some Identities for the Sums of the Fibonacci р
Numbers......Page 249
4.3.7. The Ratio of Adjacent Fibonacci p Numbers......Page 250
4.4.1. A Generalization of the Division in Extreme and Mean Ratio (DEMR)......Page 251
4.4.2. Algebraic Properties of the Golden р
Proportions......Page 252
4.4.3. Geometric Progressions Based on the Golden p Proportions......Page 253
4.5.1. Dichotomy Principle......Page 254
4.5.3. The Generalized Principle of the Golden Section......Page 255
4.6.1. A Generalization of Euclid’s Theorem II.11 for the Case p=2......Page 256
4.6.2. Euclid’s Rectangular Parallelepiped......Page 257
4.7.1. Algebraic Equations......Page 258
4.7.2. Properties of the Roots of the Generalized Golden Algebraic Equations......Page 259
4.7.3. Some Corollaries of Theorems 4.1 and 4.2......Page 263
4.8.2. The Case of p=2......Page 264
4.8.4. The General Case......Page 265
4.9.1. A General Approach to the Synthesis of the Generalized Binet Formulas......Page 266
4.9.2. A Derivation of Binet Formulas for the Classical Fibonacci and Lucas Numbers......Page 267
4.9.3. Binet Formula for the Fibonacci 2 Numbers......Page 268
4.9.4. Binet Formula for the Fibonacci 3 Numbers......Page 269
4.9.5. Binet Formulas for the Fibonacci 4 Numbers......Page 270
4.9.6. A General Case of p......Page 271
4.10.1. Binet Formula for the Lucas p Numbers......Page 273
4.10.3. Binet Formula for the Lucas 3 Numbers......Page 274
4.10.4. Binet Formula for the Lucas 4 Numbers......Page 275
4.10.5. The “Extended” Lucas р
Numbers......Page 276
4.10.6. Identities for the Sums of the Lucas р
Numbers......Page 277
4.10.7. The Ratio of Adjacent Lucas p Numbers......Page 278
4.11.1. The Generalized Fibonacci Sequences......Page 279
4.11.2. The Metallic Means Family......Page 280
4.11.3. Other Types of the Metallic Means......Page 282
4.11.4. Pisot Vijayaraghavan Numbers and Metallic Means......Page 283
4.12.1. The Generalized Fibonacci m Numbers......Page 284
4.12.2. The Generalized Golden Mean of Order m......Page 285
4.12.3. Two Surprising Representations of the Golden m Proportion......Page 287
4.12.4. A Derivation of the Gazale Formula......Page 288
4.13.1. Fibonacci m Numbers......Page 289
4.13.3. Lucas m Numbers......Page 291
4.14.2. Some Properties of the Fibonacci (p,m) Numbers......Page 293
4.14.3. Characteristic Algebraic Equation for the Fibonacci (p, m)  Numbers......Page 294
4.14.4. The Golden (p,m) Proportions......Page 295
4.14.5. Properties of the Roots of the Characteristic Equation......Page 296
4.14.6. Generalized Binet Formulas for the Fibonacci (p,m) Numbers......Page 297
4.14.7. Generalized Binet Formulas for the Lucas (p,m) Numbers......Page 299
4.15.1. Soroko’s Law of the Structural Harmony of Systems......Page 301
4.15.2. Application of Soroko’s Law to Thermodynamic and Information Systems......Page 304
4.16. Conclusion......Page 305
5.1.1. Trigonometric Functions......Page 307
5.1.2. The Power Function......Page 308
5.1.3. An Exponential Function......Page 310
5.2.1. Definition of the Hyperbolic Functions......Page 311
5.2.2.2. Non Euclidean Geometry......Page 313
5.2.2.3. Minkowski’s Four dimensional World......Page 314
5.3.1. A Definition of Hyperbolic Fibonacci and Lucas Functions......Page 316
5.3.2. The Hyperbolic Fibonacci and Lucas Tangent and Cotangent......Page 318
5.3.3.3. Hyperbolic Lucas Sine......Page 319
5.4.1. Integration of the Function y=sF(x)......Page 320
5.4.4. Integration of the Function y=cL(x)......Page 321
5.4.6. Differentiation of the Function y=cF(x)......Page 322
5.4.8. Differentiation of the Function y=cLx......Page 323
5.4.9. The Main Identities for the Hyperbolic Fibonacci and Lucas Functions......Page 324
5.5. Symmetric Hyperbolic Fibonacci and Lucas Functions (Stakhov Rozin Definition)......Page 329
5.6. Recursive Properties of the Symmetric Hyperbolic Fibonacci and Lucas Functions......Page 332
5.7.1. Hyperbolic Properties......Page 335
5.7.2. Formulas for Differentiation and Integration......Page 336
5.8.1. The Quasi sine Fibonacci and Lucas Functions......Page 338
5.8.2. Recursive Properties of the Quasi sine Fibonacci and Lucas Functions......Page 339
5.8.3. Three dimensional Fibonacci Spiral......Page 340
5.8.4. The Golden Shofar......Page 341
5.9.1. A Definition of the Hyperbolic Fibonacci and Lucas m Functions......Page 343
5.9.2. General Properties of the Hyperbolic Fibonacci and Lucas m  Functions......Page 345
5.9.3. Partial Cases of the Hyperbolic Fibonacci and Lucas m Functions......Page 346
5.9.4. Comparison of the Classical Hyperbolic Functions with the Hyperbolic Lucas m Functions......Page 347
5.9.5. Recursive Properties of the Hyperbolic Fibonacci and Lucas m  Functions......Page 348
5.9.6. Hyperbolic Properties of the Hyperbolic Fibonacci and Lucas m  Functions......Page 349
5.10.1. The Phenomenon of Phyllotaxis......Page 351
5.10.2. Dynamic Symmetry......Page 352
5.11.1. Compression and Expansion......Page 353
5.11.2. Hyperbola......Page 354
5.11.3. Geometric Definition of the Hyperbolic Functions......Page 355
5.11.4. Hyperbolic Rotation......Page 358
5.12.1. Structural   numerical Analysis of Phyllotaxis Lattices......Page 359
5.12.3. The Key Idea of Bodnar’s Geometry......Page 361
5.12.4. The “Golden” Hyperbolic Functions......Page 362
5.12.5. The Connection of Bodnar’s “Golden” Hyperbolic Functions with the Hyperbolic Fibonacci Functions......Page 364
5.13. Conclusion......Page 365
6.1.1. A History of Matrices......Page 369
6.1.2. Definition of a Matrix......Page 370
6.1.3. Matrix Addition and Scalar Multiplication......Page 371
6.1.5.1. Definition of a Square Matrix......Page 372
6.1.5.3. Determinants......Page 373
6.2.2. Properties of the Q Matrix......Page 374
6.2.3. Binet Formulas for the Q Matrix......Page 375
6.3.2. The Main Theorems for the Qp Matrices......Page 378
6.4. Determinants of the Qp Matrices and their Powers......Page 382
6.5. The “Direct” and “Inverse” Fibonacci Matrices......Page 385
6.6.1. A Definition of the Fibonacci Gm Matrix......Page 386
6.6.4. Some Properties of the Gm Matrices......Page 388
6.6.5. The Inverse Matrices Gm n......Page 391
6.7.1. A Definition of the Qp,m Matrix......Page 392
6.7.2. The Main Theorems for the Powers of the Qp,m Matrices......Page 393
6.8.1. Determinant of the Qp,m Matrix......Page 395
6.8.2. Determinant of the Matrix Qp m n ,......Page 396
6.9.1. A Definition of the Golden Matrices......Page 397
6.9.2. The Inverse Golden Matrices......Page 398
6.9.3. Determinants of the Golden Matrices......Page 399
6.10.1. A Definition of the Golden Gm Matrices......Page 400
6.10.2. Inverse Golden Gm matrices......Page 401
6.11.1. The Genetic Code......Page 402
6.11.2. Symbolic Genomatrices......Page 403
6.12.3. Numerical Genomatrices......Page 404
6.12.4. The Golden Genomatrices......Page 405
6.12. Conclusion......Page 409
Part III. Application in Computer Science......Page 411
7.1.2. The “Differentiation Principle” of “Measurement Science”......Page 412
7.1.3. Applied and Fundamental Theories of Measurement......Page 413
7.1.4. What is the Fundamental Distinction between Physics and Mathematics?......Page 414
7.1.6. Purposes of the “Physical” Measurement Theory......Page 415
7.2.1. Evolution of the Measurement Concept in Mathematics......Page 416
7.2.2. Incommensurable Line Segments......Page 417
7.2.3. Eudoxus Archimedes and Cantor’s Axioms......Page 419
7.2.4. A Contradiction in the Continuity Axioms......Page 421
7.3.1. An Infinity Concept......Page 422
7.3.3. Potential and Actual Infinity......Page 423
7.3.4. An Origin and Application of the Infinity Idea in the Ancient Greek Mathematics......Page 424
7.3.5. Cantor’s Theory of Infinite Sets......Page 425
7.3.6. Antinomies of Cantor’s Theory of Infinite Sets......Page 426
7.3.7. Is Cantor’s Actual Infinity the Greatest Mathematical Mystification?......Page 427
7.4.1. The Rejection of Cantor’s Axiom from Mathematical Measurement Theory......Page 429
7.4.2. Bashet Mendeleev’s Problem......Page 430
7.4.3. Asymmetry Principle of Measurement......Page 432
7.4.4. A New Formulation of the Bachet Mendeleev Problem......Page 433
7.5.1. A Notion of the “Indicator Element”......Page 434
7.5.2. The (n, k, S) algorithms......Page 435
7.5.3. The Optimal (n, k, S) algorithms......Page 436
7.6.1. The “Binary” Algorithm......Page 437
7.6.2. The “Counting” Algorithm......Page 438
7.6.3. The “Ruler” Algorithm......Page 439
7.6.4. The Restriction S......Page 440
7.7.1. A General Method for the Synthesis of the Optimal (n,k,S)  algorithms......Page 441
7.7.2. The Optimal (n,k,0) algorithms......Page 442
7.7.3. Special Cases of the Optimal (n,k,0) algorithm......Page 443
7.8.1. The Optimal (n,k,1) algorithms......Page 444
7.8.2. Arithmetical Square......Page 446
7.8.4. The Example of the Optimal (n, k, 1) algorithm......Page 447
7.9.1. The Optimal Measurement Algorithms Based on the Fibonacci p numbers......Page 448
7.9.2. The Example of the Fibonacci Measurement Algorithm......Page 452
7.10.1. A Further Generalization of the Bachet Mendeleev Problem......Page 453
7.10.2. The Main Recursive Relation of the Algorithmic Measurement Theory......Page 454
7.10.3. The Unusual Results......Page 457
7.11.1. What is an Isomorphism?......Page 460
7.11.2. Isomorphism between the “Balance” and “Rabbits”......Page 461
7.11.3. The Generalized “Asymmetry Principle” of Organic Nature......Page 462
7.12. Conclusion......Page 465
8.1.1. Era of Information......Page 468
8.1.2. The Basic Milestones in Computer Progress......Page 469
8.1.2.2. Mechanical Calculation Machines......Page 470
8.1.2.3. Babbage’s Analytical Machine......Page 471
8.1.2.5. Electromechanical Computers by Zuse, Aiken and Stibitz......Page 472
8.1.2.6. ENIAC......Page 473
8.1.2.7. John von Neumann’s Principles......Page 474
8.1.2.8. The Phenomenon of Personal Calculators......Page 475
8.2.1. Babylonian Sexagecimal Numeral System......Page 476
8.2.3. Mayan System......Page 477
8.2.5. Binary System......Page 478
8.2.6. Exotic Number Systems......Page 480
8.3.1. Zeckendorf Sums......Page 481
8.3.2. Definition of the Fibonacci p Code......Page 482
8.3.3. The Range of Number Representation in the Fibonacci p Code......Page 483
8.3.4. Multiplicity of Number Representation......Page 485
8.4.1. Convolution and Devolution......Page 486
8.4.3. A Minimal Form of the Fibonacci p Code......Page 488
8.4.4. Comparison of Numbers in the Fibonacci p Codes......Page 491
8.4.5. Redundancy of the Fibonacci p Codes......Page 492
8.4.6. Surprising Analogies Between the Fibonacci and Genetic Codes......Page 494
8.5.1. Fibonacci Addition......Page 495
8.5.2. Direct Fibonacci Subtraction......Page 498
8.5.3. Fibonacci Inverse and Additional Codes......Page 499
8.6.1. Basic Micro Operations......Page 501
8.6.2. Logical Operations......Page 502
8.6.3. The Counting and Subtracting of Binary 1......Page 503
8.6.4. Fibonacci Summation......Page 505
8.6.5. Fibonacci Subtraction......Page 506
8.7.1. The Egyptian Method of Multiplication......Page 507
8.7.3. Fibonacci Multiplication......Page 509
8.7.5. Fibonacci Division......Page 510
8.7.5.2. The second stage......Page 511
8.8.1. A Device for Reduction of the Fibonacci Code to Minimal Form......Page 512
8.8.2. “Convolution” Register as a Self checking Device......Page 514
8.8.3. Combinative Logical “Convolution” Circuits......Page 515
8.8.4. “Devolution” devices......Page 516
8.9.1. Noise tolerant Computations......Page 517
8.9.2. Checking the Basic Micro Operations......Page 518
8.9.3. The Hardware Realization of a Noise tolerant Fibonacci Processor......Page 520
8.10.1. First Steps in the Development of Fibonacci Arithmetic......Page 522
8.10.2. Stakhov’s Lecture in Austria (1976)......Page 523
8.10.3. Patenting the Soviet Fibonacci Computer Inventions......Page 524
8.10.5. Fibonacci Digital Signal Processing......Page 526
8.11. Conclusion......Page 527
9.1.1. Bergman’s Numeral System......Page 528
9.1.2. A Definition of the Golden p Proportion Code......Page 529
9.1.3. Representation of the Golden p Proportion Powers......Page 530
9.2.1. A Minimal Form of the Golden p Proportion Code......Page 532
9.2.2. Comparison of Numbers......Page 533
9.2.3. Representation of Numbers with a Floating Comma......Page 535
9.3.2. The Conversion of Fractional Numbers......Page 536
9.3.2.3. The third multiplication:......Page 537
9.3.3.2. The second step of conversion......Page 538
9.3.3.6. The sixth step of conversion......Page 539
9.4.1. Golden Summation and Subtraction......Page 540
9.4.2. Golden Multiplication......Page 541
9.4.3. Golden Division......Page 542
9.5.1. The Geometric Definition of Number......Page 544
9.5.3. Numeral Systems with Irrational Radices as a New Definition of Real Numbers......Page 546
9.5.4. The Golden Representations of Natural Numbers......Page 547
9.6. New Mathematical Properties of Natural Numbers (Z  and D properties)......Page 549
9.7.1. Definition of the F  and L Codes......Page 552
9.7.2. A Numerical Example......Page 553
9.7.3. Some Properties of the F  and L Codes......Page 554
9.8.1. The Zp property of Natural Numbers......Page 557
9.8.2. Fp code......Page 559
9.8.3. Lр
code......Page 562
9.9.1. The Binary Resistor Divider......Page 563
9.9.2. The Golden Resistor Dividers......Page 565
9.10.1. The “Golden” Digital to Analog Converters......Page 567
9.10.3. The “Golden” ADC......Page 568
9.10.4. The Self correcting “Golden” ADC......Page 569
9.10.5. An Application of the Z property for Checking DAC......Page 570
9.11. Conclusion......Page 572
10.1.1. Brousentsov’s Ternary Principle......Page 575
10.1.2. The Dramatic History of the “Setun” Computer......Page 576
10.1.3. Ternary Technology......Page 578
10.2.1. Ternary Symmetrical Representation......Page 579
10.2.2. Ternary Inversion......Page 580
10.2.3. The Range of Number Representation......Page 581
10.3.1. Ternary Symmetrical Summation and Subtraction......Page 582
10.3.2. Ternary Symmetrical Multiplication......Page 583
10.3.3. Ternary Symmetrical Division......Page 584
10.4.1. Basic Functions of Ternary Logic......Page 585
10.4.2. Binary Realization of Ternary Logic Elements......Page 588
10.4.3. Flip flap flop......Page 589
10.5.1. A Conversion of the Binary “Golden” Representation to the Ternary “Golden” Representation......Page 590
10.5.3. The Representation of Negative Numbers......Page 593
10.5.5. The Radix of the Ternary Mirror Symmetrical Numeral System......Page 594
10.5.6. Comparison of Number in the Ternary Mirror Symmetrical Numeral System......Page 595
10.6.1. The Range of Number Representation......Page 596
10.6.2. The Redundancy of Ternary Mirror Symmetrical Representation......Page 597
10.7.1. Mirror Symmetrical Summation......Page 598
10.7.2. Ternary Mirror Symmetrical Multi Digit Summator......Page 600
10.7.3. Mirror Symmetrical Subtraction......Page 601
10.7.4. The “Swing” Phenomenon......Page 602
10.7.5. The “Doubling” Mirror Symmetrical Summator......Page 604
10.8.1. Mirror Symmetrical Multiplication......Page 605
10.8.2. Mirror Symmetrical Division......Page 606
10.8.3. The Main Arithmetical Advantages of Mirror Symmetrical Multiplication and Division......Page 608
10.9.1. A Converter for Binary “Golden” Code to Ternary Mirror  Symmetrical Representation......Page 609
10.9.2. Technical Realization of Mirror Symmetrical Checking......Page 610
10.9.4. Ternary Mirror Symmetrical Accumulator......Page 612
10.10.1. Matrix Mirror Symmetrical Summator......Page 613
10.10.3. Pipeline Mirror Symmetrical Multiplier......Page 616
10.11.1. The “Golden” Resistor Divider for Ternary Mirror Symmetrical Representation......Page 617
10.11.2. Ternary Mirror Symmetrical Digit to Analog Converter......Page 618
10.12. Conclusion......Page 619
11.1.1.1. Claude Elwood Shannon......Page 621
11.1.1.2. Mathematical Theory of Communication......Page 622
11.1.2. Error Correction Codes......Page 623
11.1.2.1. Codes Predating Hamming......Page 624
11.1.2.3. Hamming Code......Page 625
11.1.2.4. General Principles of Error Detection and Correction......Page 626
11.1.2.5. The Main Shortcomings of the Existing Error Correction Codes......Page 628
11.1.3.1. What is Cryptography?......Page 629
11.1.3.3. Public Key Cryptography......Page 630
11.2.1. A Definition of Non singular Matrices......Page 631
11.2.2. Non singular (2 ×2) matrices......Page 632
11.3.1. The Fibonacci Encoding/Decoding Method Based on the Qp,m Matrices......Page 633
11.3.2. Fibonacci Encoding/Decoding Method Based on the Gm and Q  Matrices......Page 634
11.3.3. An Example of the Fibonacci Encoding/Decoding Method Based on the Q Matrix......Page 635
11.4.1. Determinants of the Code Matrix......Page 636
11.4.2. Connections between the Elements of the Code Matrix......Page 637
11.5.2. Detection of “Errors”......Page 640
11.5.3. Correction of “Single” Errors......Page 642
11.5.4. A Numerical Example of “Single” Error Correction......Page 643
11.5.4.2. Situation (b)......Page 644
11.5.4.3. Situation (c)......Page 645
11.5.5. Correction of “Double” Errors......Page 647
11.6.1. Redundancy of the Fibonacci Encoding/Decoding Method......Page 649
11.6.2. Correction Ability of the Fibonacci Encoding/Decoding Method......Page 651
11.6.3. Advantages of the Fibonacci Encoding/Decoding Method......Page 652
11.7. Matrix Cryptography......Page 653
11.7.1. The Concept of Hybrid Cryptosystems......Page 654
11.7.2. General Principle of Matrix Cryptography......Page 655
11.7.3. Matrix Cryptography of Digital Sound Signals......Page 656
11.7.4. Matrix Cryptography of Digital Images......Page 659
11.7.5. The Problem of Checking Information in Cryptosystems......Page 660
11.7.6. “Golden” Cryptography......Page 661
11.7.7.1. The Main Checking Relations for the “Golden” Cryptography......Page 663
11.7.8. Project for a Cryptographic Mobile Phone......Page 664
11.8. Conclusion......Page 665
E.1.1. Dirac’s Principle of Mathematical Beauty......Page 667
E.1.3. Binomial Coefficients, the Binomial Formula, and Pascal’s Triangle......Page 668
E.1.4. Fibonacci and Lucas Numbers, the Golden Mean and Binet Formulas......Page 669
E.2.1. Mathematics: The Loss of Certainty......Page 672
E.2.2. The Neglect of the “Beginnings”......Page 674
E.2.3. The Neglect of the Golden Section......Page 675
E.2.4. The One sided Interpretation of Euclid’s Elements......Page 678
E.2.6. The Greatest Mathematical Mystification of the 19th Century......Page 679
E.2.7. The Underestimation of Binet Formulas......Page 681
E.2.9. The Underestimation of Bergman’s Number System......Page 682
E.3. Three “Key” Problems of Mathematics and a New Approach to the Mathematics Origins......Page 684
E.4.1. The Generalized Fibonacci P numbers, the Generalized P proportions, the Generalized Binet Formulas and the Generalized Lucas P numbers......Page 685
E.4.2. The generalized Fibonacci λ
numbers,“Metallic Means” by Vera

Spinadel, Gazale Formulas and General Theory of Hyperbolic Functions......Page 688
E.5.1. Euclidean and Newtonian Definition of Real Number......Page 693
E.5.2. Number Systems with Irrational Radices as a New Definition of Real Number......Page 694
E.6.1. Fibonacci Matrices......Page 696
E.6.2. The “Golden” Matrices......Page 697
E.7.1. Fibonacci Codes, Fibonacci Arithmetic and Fibonacci Computers......Page 698
E.7.3. A New Theory of Error Correcting Codes Based upon Fibonacci Matrices......Page 699
E.8.1. Shechtman’s Quasi Crystals......Page 701
E.8.3. El Naschie’s E infinity Theory......Page 702
E.8.5. Petoukhov’s “Golden” Genomatrices......Page 703
E.8.6. Fibonacci Lorenz Transformations and “Golden” Interpretation of the Universe Evolution......Page 704
E.8.7. Hilbert’s Fourth Problem......Page 706
E.9.1. The First Conclusion......Page 709
E.9.4. The Fourth Conclusion......Page 710
E.9.6. The Sixth Conclusion......Page 711
E.9.8. The Eighth Conclusion......Page 712
References......Page 713
Appendix: Museum of Harmony and Golden Section......Page 727
Index......Page 737