Table of Contents

Preface 1

Getting Ready: Script and Image Files 3

1 Introduction to J and Graphics 5

• Some Arithmetic with J 5
• Lists, Arrays and Trigonometric Functions 7
• Experiment: Plotting Polygons 10
• Constructing Arrays 13
• Experiment: Creating a Raster Image 15
• Object versus Raster Graphics 16
• Defining Functions 17
• On Language 22
• Fancier Nouns and Array Computation 24
• Exercises 27

2 Object Graphics and Fractals 35

• Function Composition and Change of Coordinates. 35
• Experiment: Plotting Time Series, Functions and Curves 39
• More Function Composition 44
• Experiment: The Koch Snowflake 49
• Transformations of the Plane and Homogeneous Coordinates 52
• Experiment: Transformations and Animations 57
• Gerunds and Multiplots 61
• Experiment: Collages of Transformations 64
• Simple Verbs 67
• Exercises 70

3 Fractal Time Series 77

• Statistics and Least Squares Fit 77
• Experiment: Plot Driver 81
• Random Walks 83
• Experiment: Observing Trends 86
• R/S Analysis, the Hurst Exponent, and Sunspots 90
• Autocorrelation Functions 96
• Experiment: Random Midpoint Displacement 99
• Experiment: Fractal Forecasting 102
• Exercises 107

4 Raster Graphics and Fractals 113

• The RGB Color Model 113
• Experiment: Inner Product Fractals 116
• Agenda and the Hailstone Function 122
• Experiment: Probabilistic Iterated Function Systems 124
• Remarks on Iterated Function Systems 129
• Weighted Selection of Random Transformations 131
• Experiment: The Chaos Game 133
• Fractal Dimension 139
• Fractal Dimension via Box Counting 145
• Exercises 147

5 Chaotic Attractors and Symmetry 155

• Experiment: The Logistic Function and Plotting Frequency of Visitation 155
• Adverbs and Conjunctions 161
• Chaotic Attractors in the Plane 165
• Cyclic and Dihedral Symmetry 177
• Iterated Function Systems with Hyperbolic Symmetry 183
• Frieze Patterns 192
• Experiment: Crystallographic Symmetry on a Square Lattice 198
• Crystallographic Symmetry on a Hexagonal Lattice 203
• Experiment: Attractors Near Forbidden Symmetry 210
• Exercises 213

6 Automata and Image Processing 221

• One Dimensional Automata 221
• Fuzzy Logic and Fuzzy Automata 226
• Experiment: The Game of Life 232
• Creating Animation Files 236
• Experiment: Full Color Images 238
• Experiment: Filters for Image Processing 242
• Fast Fourier Transforms and Diffraction Patterns 245
• Experiment: Removing Motion Blur 249
• Image Cross-Correlation 252
• Exercises 255

7 Complex Dynamics and Contour Plots 259

• Experiment: Contour Plots 259
• Plasma Clouds 262
• Inverse Iterated Function Systems 263
• Experiment: The Julia Set 267
• The Mandelbrot Set 269
• Newton's Method in the Complex Plane 271
• Exercises 276

8 Visualization in Three Dimensions 279

• Experiment: Transformations in Three Dimensions 279
• Orthogonal Projection 283
• Experiment: Painter's Algorithm and Surface Plotting 286
• Perspective Projections 288
• Iterated Function Systems in 3-Dimensions 291
• The Lorenz Attractor 293
• Exercises 295

9 Ray Tracing 299

• Experiment: Introduction to POV-Ray 299
• Experiment: The Menger Sponge 304
• Animation 308
• Time Evolution of the Game of Life 309
• Rendering Surfaces 314
• Experiment: A Fractal Mountain 316
• Experiment: Collages Revisited 319
• Experiment: High Dimensional Sierpinski Fractals 323
• Hyperbolic Symmetry Revisited 330
• Exercises 334

10 GUI Programming and Open GL 269

• Open GL Experiment 337
• GUI Programming 341
• A Glance at Open GL 346
• Experiment: The Menger Sponge 350
• Experiment: A Surface from Polygons 351
• Exercises 352

Color Plates 355

Bibliography and References 365

Index 371