Fractals, Chaos, Power Laws

Preface xiii
@=18
1 Introduction 1
...Einstein, Pythagoras, and Simple Similarity 3
...A Self-Similar Array of Self-Preserving Queens 4
...A Self-Similar Snowflake 7
...A New Dimension for Fractals 9
...A Self-Similar Tiling and a \l"Non-Euclidean\r" Paradox 13
...At the Gates of Cantor's Paradise 15
...The Sierpinski Gasket 17
...Sir Pinski's Game and Deterministic Chaos 20
...Three Bodies Cause Chaos 25
...Strange Attractors, Their Basins, and a Chaos Game 27
...Percolating Random Fractals 30
...Power Laws: from Alvarez to Zip! 33
...Newton's Iteration and How to Abolish Two-Nation Boundaries 38
...Could Minkowski Hear the Shape of a Drum? 40
...Discrete Self-Similarity: Creases and Center Folds 45
...Golden and Silver Means and Hyperbolic Chaos 49
...Winning at Fibonacci Nim 53
...Self-Similar Sequences from Square Lattices 55
...John Horton Conway's \l"Death Bet\r" 57
2 Similarity and Dissimilarity 61
...More Than One Scale 61
...To Scale or Not to Scale: A Bit of Biology and Astrophysics 63
...Similarity in Physics: Some Astounding Consequences 66
...Similarity in Concert Halls, Microwaves, and Hydrodynamics 68
...Scaling in Psychology 70
...Acousticians, Alchemy, and Concert Halls 72
...Preference and Dissimilarity: Concert Halls Revisited 74
3 Self-Similarity \em Discrete, Continuous, Strict, and Otherwise 81
...The Logarithmic Spiral, Cutting Knives, and Wideband Antennas 89
...Some Simple Cases of Self-Similarity 93
...Weierstrass Functions and a Musical Paradox 96
...More Self-Similarity in Music: The Tempered Scales of Bach 99
...The Excellent Relations between the Primes 3, 5, and 7 102
4 Power Laws: Endless Sources of Self-Similarity 103
...The Sizes of Cities and Meteorites 103
...A Fifth Force of Attraction 105
...Free of Natural Scales 107
...Bach Composing on All Scales 107
...Birkhoff's Aesthetic Theory 109
...Heisenberg's Hyperbolic Uncertainty Principle 112
...Fractional Exponents 115
...The Peculiar Distribution of the First Digit 116
...The Diameter Exponents of Trees, Rivers, Arteries, and Lungs 117
5 Noises: White, Pink, Brown, and Black 121
...Pink Noise 122
...Self-Similar Trends on the Stock Market 126
...Black Noises and Nile Floods 129
...Warning: World Warming 131
...Fractional Integration: A Modem Tool 131
...Brownian Mountains 133
...Radon Transform and Computer Tomography 134
...Fresh and Tired Mountains 135
6 Brownian Motion, Gambling Losses, and Intergalactic Voids: Random Fractals Par Excellence 139
...The Brownian Beast Tamed 140
...Brownian Motion as a Fractal 141
...How Many Molecules? 143
...The Spectrum of Brownian Motion 144
...The Gambler's Ruin, Random Walks, and Information Theory 145
...Counterintuition Runs Rampant in Random Runs 146
...More Food for Fair Thought 147
...The St. Petersburg Paradox 148
...Shannon's Outguessing Machine 149
...The Classical Mechanics of Roulette and Shannon's Channel Capacity 150
...The Clustering of Poverty and Galaxies 152
...Levy Flights through the Universe 155
...Paradoxes &om Probabilistic Power Laws 155
...Invariant Distributions: Gauss, Cauchy, and Beyond 157
7 Cantor Sets: Self-Similarity and Arithmetic Dust 161
...A Comer of Cantor's Paradise 161
...Cantor Sets as Invariant Sets 165
...Symbolic Dynamics and Deterministic Chaos 166
...Devil's Staircases and a Pinball Machine 167
...Mode Locking in Swings and Clocks 171
...The Frustrated Manhattan Pedestrian 172
...Arnold Tongues 174
8 Fractals in Higher Dimensions and a Digital Sundial 177
...Cartesian Products of Cantor Sets 177
...A Leaky Gasket, Soft Sponges, and Swiss Cheeses 178
...A Cantor-Set Sundial 181
...Fat Fractals 183
9 Multifractals: Intimately Intertwined Fractals 187
...The Distributions of People and Ore 187
...Self-Affine Fractals without Holes 190
...The Multi&actal Spectrum: Turbulence and Diffusion-Limited Aggregation 193
...Viscous Fingering 199
...Multifractals on Fractals 200
...Fractal Dimensions from Generalized Entropies 203
...The Relation between the Multifractal Spectrum f(\alpha) and the Mass Exponents \tau(q) 205
...Strange Attractors as Multi&actals 206
@+8
...A Greedy Algorithm for Unfavorable Odds 207
10 Some Practical Fractals and Their Measurement 211
...Dimensions &om Box Counting 213
...The Mass Dimension 215
...The Correlation Dimension 220
...Infinitely Many Dimensions 220
...The Determination of Fractal Dimensions &om Time Series 223
...Abstract Concrete 224
...Fractal Interfaces Enforce Fractional Frequency Exponents 225
...The Fractal Dimensions of Fracture Surfaces 230
...The Fractal Shapes of Clouds and Rain Areas 231
...Cluster Agglomeration 232
...Diffraction &om Fractals 233
11 Iteration, Strange Mappings, and a Billion Digits for Pi 237
...Looking for Zeros and Encountering Chaos 239
...The Strange Sets of Julia 243
...A Multifractal Julia Set 245
...The Beauty of Broken Linear Relationships 249
...The Baker's Transformation and Digital Musical Chairs 251
...Amol'd's Cat Map 253
...A Billion Digits for n 257
...Bushes and Flowers from Iterations 259
12 A Self-Similar Sequence, the Logistic Parabola, and Symbolic Dynamics 263
...Self-Similarity from the Integers 264
...The Logistic Parabola and Period Doubling 268
...Self-Similarity in the Logistic Parabola 272
...The Scaling of the Growth Parameter 274
...Self-Similar Symbolic Dynamics 277
...Periodic Windows Embedded in Chaos 279
...The Parenting of New Orbits 282
...The Calculation of the Growth Parameters for Different Orbits 286
...Tangent Bifurcations, Intermittency, and l/f Noise 289
...A Case of Complete Chaos 291
...The Mandelbrot Set 295
...The Julia Sets of the Complex Quadratic Map 297
13 A Forbidden Symmetry, Fibonacci's Rabbits, and a New State of Matter 301
...The Forbidden Fivefold Symmetry 301
...Long-Range Order &om Neighborly Interactions 304
...Generation of the Rabbit Sequence &om the Fibonacci Number System 307
...The Self-Similar Spectrum of the Rabbit Sequence 308
...Self-Similarity in the Rabbit Sequence 310
...A One-Dimensional Quasiperiodic Lattice 310
...Self-Similarity &om Projections 311
...More Forbidden Symmetries 315
14 Periodic and Quasiperiodic Structures in Space \em the Route to Spatial Chaos 319
...Periodicity and Quasiperiodicity in Space 320
...The Devil's Staircase for Ising Spins 321
...Quasiperiodic Spatial Distributions 322
...Beatty Sequence Spins 325
...The Scaling Laws for Quasiperiodic Spins 329
...Self-Similar Winding Numbers 330
...Circle Maps and Arnold Tongues 331
...Mediants, Farey Sequences, and the Farey Tree 334
...The Golden-Mean Route to Chaos 340
15 Percolation: from Forest Fires to Epidemics 345
...Critical Conflagration on a Square Lattice 346
...Universality 350
...The Critical Density 353
...The Fractal Perimeters of Percolation 353
...Finite-Size Scaling 354
16 Phase Transitions and Renormalization 357
...A First-Order Markov Process 357
...Self-Similar and Non-Self-Similar Markov Processes 358
...The Scaling of Markov Outputs 360
...Renormalization and Hierarchical Lattices 362
...The Percolation Threshold of the Bethe Lattice 363
...A Simple Renormalization 367
17 Cellular Automata 371
...The Game of Life 373
...Cellular Growth and Decay 375
...Biological Pattern Formation 382
...Self-Similarity &om a Cellular Automaton 383
...A Catalytic Converter as a Cellular Automaton 386
...Pascal' s Triangle Modulo N 387
...Bak's Self-Organized Critical Sandpiles 389
Appendix 391
References 395
Author Index 411
Subject Index 417