Foreword ........................................................ 1
Introduction: Causality Principle, Deterministic Laws and
Chaos ............................................. 9
1. The Backbone of Fractals: Feedback and the Iterator ....... 15
1.1. The Principle of Feedback ........................... 17
1.2. The Multiple Reduction Copy Machine ................. 23
1.3. Basic Types of Feedback Processes ................... 27
1.4. The Parable of the Parabola — Or: Don't Trust Your
Computer ............................................ 37
1.5. Chaos Wipes Out Every Computer ...................... 49
2. Classical Fractals and Self-Similarity .................... 61
2.1. The Cantor Set ...................................... 65
2.2. The Sierpinski Gasket and Carpet .................... 76
2.3. The Pascal Triangle ................................. 80
2.4. The Koch Curve ...................................... 87
2.5. Space-Filling Curves ................................ 92
2.6. Fractals and the Problem of Dimension .............. 104
2.7. The Universality of the Sierpinski Carpet .......... 110
2.8. Julia Sets ......................................... 120
2.9. Pythagorean Trees .................................. 124
3. Limits and Self-Similarity ............................... 129
3.1. Similarity and Scaling ............................. 132
3.2. Geometric Series and the Koch Curve ................ 141
3.3. Corner the New from Several Sides: Pi and the
Square Root of Two ................................. 147
3.4. Fractals as Solutions of Equations ................. 162
4. Length, Area and Dimension: Measuring Complexity and
Scaling Properties ....................................... 173
4.1. Finite and Infinite Length of Spirals .............. 175
4.2. Measuring Fractal Curves and Power Laws ............ 182
4.3. Fractal Dimension .................................. 192
4.4. The Box-Counting Dimension ......................... 202
4.5. Borderline Fractals: Devil's Staircase and Peano
Curve .............................................. 210
5. Encoding Images by Simple Transformations ................ 215
5.1. The Multiple Reduction Copy Machine Metaphor ....... 217
5.2. Composing Simple Transformations ................... 220
5.3. Relatives of the Sierpinski Gasket ................. 230
5.4. Classical Fractals by IFSs ......................... 238
5.5. Image Encoding by IFSs ............................. 244
5.6. Foundation of IFS: The Contraction Mapping
Principle .......................................... 248
5.7. Choosing the Right Metric .......................... 258
5.8. Composing Self-Similar Images ...................... 262
5.9. Breaking Self-Similarity and Self-Affinity:
Networking with MRCMs .............................. 267
6. The Chaos Game: How Randomness Creates Deterministic
Shapes ................................................... 277
6.1. The Fortune Wheel Reduction Copy Machine ........... 280
6.2. Addresses: Analysis of the Chaos Game .............. 287
6.3. Tuning the Fortune Wheel ........................... 300
6.4. Random Number Generator Pitfall 311
6.5. Adaptive Cut Methods ............................... 319
7. Recursive Structures: Growing Fractals and Plants ........ 329
7.1. L-Systems: A Language for Modeling Growth .......... 333
7.2. Growing Classical Fractals with MRCMs .............. 340
7.3. Turtle Graphics: Graphical Interpretation of
L-Systems .......................................... 351
7.4. Growing Classical Fractals with L-Systems .......... 355
7.5. Growing Fractals with Networked MRCMs .............. 367
7.6. L-System Trees and Bushes .......................... 372
8. Pascal's Triangle: Cellular Automata and Attractors ...... 377
8.1. Cellular Automata .................................. 382
8.2. Binomial Coefficients and Divisibility ............. 393
8.3. IFS: From Local Divisibility to Global Geometry .... 404
8.4. HIFS and Divisibility by Prime Powers .............. 412
8.5. Catalytic Converters, or How Many Cells Are
Black? ............................................. 420
9. Irregular Shapes: Randomness in Fractal Constructions .... 423
9.1. Randomizing Deterministic Fractals ................. 425
9.2. Percolation: Fractals and Fires in Random
Forests ............................................ 429
9.3. Random Fractals in a Laboratory Experiment ......... 440
9.4. Simulation of Brownian Motion ...................... 446
9.5. Scaling Laws and Fractional Brownian Motion ........ 456
9.6. Fractal Landscapes ................................. 462
10. Deterministic Chaos: Sensitivity, Mixing, and Periodic
Points ................................................... 467
10.1. The Signs of Chaos: Sensitivity .................... 469
10.2. The Signs of Chaos: Mixing and Periodic Points ..... 480
10.3. Ergodic Orbits and Histograms ...................... 485
10.4. Metaphor of Chaos: The Kneading of Dough ........... 496
10.5. Analysis of Chaos: Sensitivity, Mixing, and
Periodic Points .................................... 509
10.6. Chaos for the Quadratic Iterator ................... 520
10.7. Mixing and Dense Periodic Points Imply
Sensitivity ........................................ 529
10.8. Numerics of Chaos: Worth the Trouble or Not? ....... 535
11. Order and Chaos: Period-Doubling and Its Chaotic
Mirror ................................................... 541
11.1. The First Step from Order to Chaos: Stable Fixed
Points ............................................. 548
11.2. The Next Step from Order to Chaos: The Period-
Doubling Scenario .................................. 559
11.3. The Feigenbaum Point: Entrance to Chaos ............ 575
11.4. From Chaos to Order: A Mirror Image ................ 583
11.5. Intermittency and Crises: The Backdoors to Chaos ... 595
12. Strange Attractors: The Locus of Chaos ................... 605
12.1. A Discrete Dynamical System in Two Dimensions:
Henon's Attractor .................................. 609
12.2. Continuous Dynamical Systems: Differential
Equations .......................................... 628
12.3. The Rossler Attractor .............................. 636
12.4. The Lorenz Attractor ............................... 647
12.5. Quantitative Characterization of Strange Chaotic
Attractors: Ljapunov Exponents ..................... 659
12.6. Quantitative Characterization of Strange Chaotic
Attractors: Dimensions ............................. 671
12.7. The Reconstruction of Strange Attractors ........... 694
12.8. Fractal Basin Boundaries ........................... 706
13. Julia Sets: Fractal Basin Boundaries ..................... 715
13.1. Julia Sets as Basin Boundaries ..................... 717
13.2. Complex Numbers — A Short Introduction ............. 722
13.3. Complex Square Roots and Quadratic Equations ....... 729
13.4. Prisoners versus Escapees .......................... 733
13.5. Equipotentials and Field Lines for Julia Sets ...... 744
13.6. Binary Decomposition, Field Lines and Dynamics ..... 756
13.7. Chaos Game and Self-Similarity for Julia Sets ...... 764
13.8. The Critical Point and Julia Sets as Cantor Sets ... 769
13.9. Quaternion Julia Sets .............................. 780
14. The Mandelbrot Set: Ordering the Julia Sets .............. 783
14.1. From the Structural Dichotomy to the Binary
Decomposition ...................................... 785
14.2. The Mandelbrot Set - A Road Map for Julia Sets ..... 797
14.3. The Mandelbrot Set as a Table of Content ........... 820
Bibliography .................................................. 839
Index ......................................................... 853
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