Foreword by 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 Some 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 IPS 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 MathWorld vs. PhysicalWorld ............................ 56
3.2 A foreshadowing of fractal dimension ................... 56
3.3 Coastlines, mountains, and rivers ...................... 58
3.4 How (and why) the hmgs are fractal ..................... 64
3.5 Power laws ............................................. 67
3.6 Forests and trees ...................................... 70
3.7 Our complex hearts ..................................... 71
3.8 Metabolie 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
В Solutions to the problems .................................. 407
References .................................................... 439
Acknowledgments ............................................... 491
Figure credits ................................................ 493
Index ......................................................... 495
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