Table of Contents

Prologue xi

Software and solutions xix

1 IFS Labs 1

1.1 Finding IFS for fractal images 5

1.2 Spiral fractals from IFS 13

1.3 Finding IFS rules from images of points 22

1.4 A fractal leaf by IFS 30

1.5 Fractal wallpaper 35

1.6 Cumulative gasket pictures 39

1.7 IFS and addresses 46

1.8 Decimals as addresses 49

1.9 IFS with memory 56

1.10 IFS with more memory 67

1.11 Data analysis by driven IFS 74

2 Dimension and Measurement Labs 91

2.1 Dimension by box-counting 93

2.2 Paper ball and bean bag dimensions 109

2.3 Calculating similarity dimension 118

2.4 Sierpinski tetrahedron 128

2.5 Koch tetrahedron 134

2.6 Sierpinski hypertetrahedron 141

2.7 Basic multifractals: f(α) curves 152

3 Iteration Labs 163

3.1 Visualizing iteration patterns 164

3.2 Synchronized chaos 174

3.3 Domains of compositions 183

3.4 Fractals and Pascal's triangles 187

3.5 Fractals and Pascal's triangle relatives 197

3.6 Mandelbrot sets and Julia sets 207

3.7 Circle inversion fractals 216

3.8 Fractal tiles 224

4 Labs in the Studio and in the Kitchen 235

4.1 Fractal painting: decalcomania 1 236

4.2 Fractal painting: decalcomania 2 245

4.3 Fractal painting: bleeds 250

4.4 Fractal painting: mixing 257

4.5 Fractal painting; dripping 263

4.6 Fractal paper folds 270

4.7 A closer look at leaves 278

4.8 Structures of vegetables 284

4.9 Cooking fractals 289

5 Labs in the Lab 295

5.1 Magnetic pendulum 296

5.2 Optical gasket 304

5.3 Video feedback fractals 308

5.4 Electrodeposition 318

5.5 Viscous fingering 325

5.6 Crumpled paper patterns 329

5.7 Fractal networks of resistors 335

5.8 Fractal networks of magnets 342

5.9 Synchronization in fractal networks of oscillators 346

6 What Else? 361

6.1 Building block fractals 361

6.2 Non-Euclidean tilings 363

6.3 Fractal perimeters 365

6.4 Multifractal finance 367

6.5 Fractal music 369

6.6 Other ideas 370

7 Why labs matter 375

A Specific Physical Supplies 377

B Technical Notes 381

B.1 Notes for finding IFS, Lab 1.1 381

B.2 Notes for spiral fractals, Lab 1.2 383

B.3 Notes for cumulative gasket pictures, Lab 1.6 385

B.4 Notes for IFS with more memory, Lab 1.10 387

B.5 Notes on entropy and partitions, Lab 1.11 388

B.6 Notes on linear regression. Lab 2.1 388

B.7 Notes on the algebra of dimensions, Labs 2.1 and 2.2 393

B.8 Notes on eigenvalues and the Moran equation, Lab 2.3 394

B.9 Notes on multifractal analysis, Lab 2.7 396

B.10 Notes on the Mandelbrot set and Julia sets. Lab 3.6 401

B.11 Notes on circle inversion fractals, Lab 3.7 407

B.12 Notes on fractal painting: dripping, Lab 4.5 413

B.13 Notes on power law measurements, Lab 4.9 414

B.14 Notes on magnetic pendulum differential equations, Lab 5.1 416

B.15 Notes on molarity calculations, Lab 5.4 417

B.16 Notes on fractal resistor networks Lab 5.7 418

B.17 Notes on synchronization in fractal networks of oscillators, Lab 5.9 419

Bibliography 421

Figure Credits 437

Acknowledgements 439

Index 441