Published by Lulu Available Perfect bound: http://www.lulu.com/content/635966 ISBN 978-1-4303-1980-1 or Coil bound: http://www.lulu.com/content/635938 no ISBN Abstract: An introduction to mathematical visualization including many fractals and using the J programming language. Designed for classroom use or individual learning. J is freely available and no prior experience with J is required. Experiments are hands on explorations that readers can duplicate. Topics include fractals, time series, iterated function systems, chaos and symmetry, cellular automata, complex dynamics, image processing, ray tracing and Open GL.

Table of Contents

Preface to the 3rd edition vi
Preface to the 2nd edition, 2000 vi
Getting Ready: Addon Files viii

Chapter 1 Introduction to J and Graphics 1
1.1 Some Arithmetic with J 1
1.2 Lists, Arrays and Trigonometric Functions 2
1.3 Experiment: Plotting Polygons 4
1.4 Constructing Arrays 5
1.5 Experiment: Creating a Raster Image 7
1.6 Object versus Raster Graphics 8
1.7 Defining Functions 8
1.8 On Language 11
1.9 Fancier Nouns and Array Computation 12
1.10 Exercises 14

Chapter 2 Plots, Verbs and First Fractals 19
2.1 Function Composition and Plots 19
2.2 Experiment: Plotting Time Series, Functions and Curves 21
2.3 More Function Composition 22
2.4 Experiment: The Koch Snowflake 25
2.5 Transformations of the Plane and Homogeneous Coordinates 27
2.6 Experiment: Transformations and Animations 29
2.7 Gerunds and Multiplots 32
2.8 Experiment: Collages of Transformations 33
2.9 Simple Verbs 35
2.10 Exercises 36

Chapter 3 Fractal Time Series 41
3.1 Statistics and Least Squares Fit 41
3.2 Experiment: Plot Driver 43
3.3 Random Walks 43
3.4 Experiment: Observing Trends 46
3.5 R/S Analysis, the Hurst Exponent, and Sunspots 48
3.6 Autocorrelation Functions 51
3.7 Experiment: Random Midpoint Displacement 52
3.8 Experiment: Forecasting via Best Analogs 54
3.9 Exercises 58

Chapter 4 Color, Raster Graphics and Fractals 61
4.1 The RGB Color Model 61
4.2 Experiment: Palettes and Inner Product Fractals 63
4.3 Agenda and the 3x+1 Function 67
4.4 Experiment: Probabilistic Iterated Function Systems 69
4.5 Remarks on Iterated Function Systems 72
4.6 Weighted Selection of Random Transformations 73
4.7 Experiment: The Chaos Game 74
4.8 Fractal Dimension 77
4.9 Fractal Dimension via Raster Box Counting 80
4.10 Exercises 81

Chapter 5 Chaotic Attractors and Symmetry 85
5.1 Experiment: The Logistic Function and Plotting Frequency of Visitation. 85
5.2 Adverbs and Conjunctions 89
5.3 Chaotic Attractors in the Plane 92
5.4 Cyclic and Dihedral Symmetry 98
5.5 Iterated Function Systems with Hyperbolic Symmetry 104
5.6 Frieze Patterns 109
5.7 Experiment: Crystallographic Symmetry on a Square Lattice 114
5.8 Crystallographic Symmetry on a Hexagonal Lattice 119
5.9 Experiment: Attractors Near Forbidden Symmetry 123
5.10 Exercises 125

Chapter 6 Cellular Automata 131
6.1 One Dimensional Automata 131
6.2 Fuzzy Logic and Fuzzy Automata 134
6.3 Experiment: The Game of Life 139
6.4 Creating Animation Files 142
6.5 Majority Rule and Spot Formation 143
6.6 Experiment: The Hodgepodge Rule 145
6.7 Hexagonal Lattice and the Packard-Wolfram Snowflake 148
6.8 A Snowflake Model Using Intermediate Values 150
6.9 Exercises 151

Chapter 7 Color Contours and Complex Dynamics 153
7.1 Experiment: Color Contour Plots 153
7.2 Plasma Clouds 155
7.3 Inverse Iterated Function Systems 156
7.4 Experiment: Julia Sets 159
7.5 The Mandelbrot Set 161
7.6 The 3x+1 Function in the Complex Plane 162
7.7 Newton's Method in the Complex Plane 163
7.8 Exercises 167

Chapter 8 Image Processing 169
8.1 Experiment: Full Color Images 169
8.2 Experiment: Thresholds and First Filters 171
8.3 More Smoothing Filters 173
8.4 Experiment: Color Spaces 176
8.5 Rotation, Tilt and Barrel Distortion 178
8.6 Fake Image Details using Best Analogs 182
8.7 Fast Fourier Transforms and Diffraction Patterns 184
8.8 Experiment: Removing Motion Blur 187
8.9 Image Cross-Correlation 188
8.10 Exercises 190

Chapter 9 Visualization in Three Dimensions 193
9.1 Experiment: Transformations in Three Dimensions 193
9.2 Orthogonal Projection 195
9.3 Experiment: Painter's Algorithm and Surface Plotting 197
9.4 Perspective Projections 198
9.5 Iterated Function Systems in 3-Dimensions 199
9.6 The Lorenz Attractor 200
9.7 Exercises 202

Chapter 10 Ray Tracing 205
10.1 Experiment: Introduction to POV-Ray 205
10.2 Experiment: The Menger Sponge 208
10.3 Experiment: Animation of the Menger Sponge 210
10.4 Time Evolution of the Game of Life 211
10.5 Rendering Surfaces 214
10.6 Experiment: A Fractal Mountain 215
10.7 Experiment: Collages Revisited 217
10.8 Experiment: High Dimensional Sierpinski Fractals 220
10.9 Exercises 224

Chapter 11 Open GL and GUI Interaction 227
11.1 Experiment: Open GL Sample Scene 227
11.2 The Menger Sponge 229
11.3 Experiment: A Surface from Polygons 230
11.4 A Peek Inside 231
11.5 A GUI 232
11.6 Starting and Stopping 236
11.7 Exercises 237

Bibliography and References 239
Index 243