Table of Contents
Foreword Steven Strogatz ix
Introduction xi
1 What is the shape of the world around us? 1
1.1 Types of symmetries 1
1.2 Symmetry under magnification 2
1.3 The Sierpinski gasket 4
1.4 Fractals with memory 5
1.5 Self-affine fractals 6
1.6 Nonlinear fractals 6
1.7 Random fractals 7
1.8 Sonic non-fractals 8
1.9 Physical world examples 10
2 Self-similarity in geometry 15
2.1 A simple way to grow fractals 15
2.2 Some classical fractals 21
2.3 Fractal trees and ferns 29
2.4 Fractal spirals 33
2.5 IFS with memory 34
2.6 Random rendering of fractal images 42
2.7 Circle inversion fractals 46
2.8 Random fractals 51
2.9 And flavors stranger still 53
3 Self-similarity in the wild 55
3.1 Math World vs. Physical World 56
3.2 A foreshadowing of fractal dimension 56
3.3 Coastlines, mountains, and rivers 58
3.4 How (and why) the lungs are fractal 64
3.5 Power laws 67
3.6 Forests and trees 70
3.7 Our complex hearts 71
3.8 Metabolic rates 73
3.9 Fractals and DNA 75
3.10 How planets grow 80
3.11 Reversals of the Earth's magnetic field 82
3.12 The distribution of galaxies 83
3.13 Is this all? 85
4 Manufactured fractals 89
4.1 Chemical mixers 90
4.2 Capacitors 91
4.3 Wide-band antennas 92
4.4 Invisibility cloaks 94
4.5 Fractal metamaterials 95
4.6 Internet infrastructure 97
4.7 Music 99
4.8 Literature 105
4.9 Visual art 119
4.10 Building fractals 128
5 The Mandelbrot set: infinite complexity from a simple formula 129
5.1 Some pictures 130
5.2 The algorithm 131
5.3 Julia sets 134
5.4 The Mandelbrot, set 138
5.5 Other Mandelbrot sets 147
5.6 The universality of the Mandelbrot set 149
5.7 The Mandelbrot set in four dimensions 153
5.8 Unanswered questions 156
6 Quantifying fractals: What is fractal dimension? 159
6.1 Similarity dimension 160
6.2 Box-counting dimension 167
6.3 Mass dimension 177
6.4 Random, with memory, and nonlinear 179
6.5 Dimension rules 189
6.6 Are unearthly dimensions of earthly use? 196
6.7 A speculation about dimensions 201
7 Further developments 205
7.1 Driven IFS 205
7.2 Driven IFS and synchronization 216
7.3 Multifractals from IFS 226
7.4 Applications of multifractals 233
7.5 Fractals and stories, again 242
8 Valediction 249
A A look under the hood: Some technical notes 253
B Solutions to the problems 407
References 439
Acknowledgments 491
Figure credits 493
Index 495
