Fractal programming in C

Chapter 1: Introduction / 15
“Monster” Curves / 16
Strange Attractors / 17
Population Curves and Bifurcation Diagrams / 18
Mandelbrot and Julia Sets / 19
Iterated Function Systems / 20
The State of Science / 20
Why this Book is Different / 21
Chapter 2: What Are Fractals / 23
The Beginning of Fractal Curves / 23
How are Fractals Used / 24
Basic Considerations / 25
Fractal Dimensions / 26
Background Material / 27
Chapter 3: Hardware Requirements / 29
Display Considerations / 31
Processor Speed / 31
Where do We Go From Here / 33
Chapter 4: Saving and Compressing Display Data / 35
Format for Saving a Screen File / 36
Function to Save a Screen / 42
Function to Restore an EGA Screen / 47
Function to Restore a VGA Screen / 50
Chapter 5: Tools for Graphics Display Generation / 53
Setting the Display Mode / 54
Setting the EGA Palettes / 55
Clearing the Screen / 57
Plotting a Point on the Screen / 58
Displaying a Byte on the Screen / 60
Reading a Pixel from the Screen / 60
Reading a Byte from a Color Plane / 61
Drawing a Line / 62
Filling a Triangle / 64
Filling an Oval / 67
Turtle Graphics / 68
Point / 69
Turn / 69
Step / 70
Function to Display and Move Cursor / 71
Bounds Program / 77
Selecting Colors / 78
Chapter 6: The Lorenz
and Other Strange Attractors / 83
Strange Attractors / 84
The Lorenz Attractor / 84
Runge Kutta Integration / 86
Programming the Lorenz Attractor / 87
Other Strange Attractors / 91
Chapter 7: The Population
Equation and Bifurcation Diagrams / 95
The Population Equation / 95
Bifurcation Diagrams / 96
“Period Three Implies Chaos” / 100
The Feigenbaum Number / 101
Chapter 8: The Snowflake
and Other von Koch Curves / 107
The von Koch Snowflake / 107
Generic Initiator/Generator Program / 110
The Gosper Curve / 113
Three-Segment Quadric von Koch Curve / 117
Eight-Segment Quadric von Koch Curve / 121
Eighteen-Segment Quadric von Koch Curve / 125
32-Segment Quadric von Koch Curve / 130
Fifty-Segment von Koch Curve / 135
Using Other Initiators / 143
Complicated Generators / 146
Chapter 9: Peano Curves / 155
The Original Peano Curve / 155
Modified Peano Curve / 159
Cesaro Triangle Curve / 166
Modified Cesaro Triangle Curve / 170
Variation on the Cesaro Curve / 173
Polya Triangle Curve / 177
The Peano-Gosper Curve / 181
Peano Seven-Segment Snowflake / 185
Peano Thirteen-Segment Curve / 189
Chapter 10: The Hubert Curve / 197
Generating the Hilbert Curve / 198
Fractal Dimension of the Hubert Curve / 204
Hilbert Curve in Three Dimensions / 205
Using the Hilbert Curve for Display Data Storage / 208
Chapter 11: The Sierpinski Curve / 211
Sierpinski Gasket / 215
Another Method of Generating the Sierpinski Triangle / 218
Strange Cousins of the Sierpinski Triangle / 221
Sierpinski Box / 222
Chapter 12: Trees / 227
Real Trees / 227
Mathematical Representation of Trees / 228
Tree-Drawing Program / 229
Chapter 13: Working with Circles / 239
Appolonian Packing of Circles / 239
Soddy’s Formula / 240
Program for Appolonian Circle Packing / 241
Inversion / 244
Pharaoh’s Breastplate / 247
Chapter 14: The Mandelbrot Set / 253
Simplified Mandelbrot Set Program / 255
Improved Mandelbrot Set Program / 258
Precision Considerations / 263
Using Other Color Combinations / 264
Other Mandelbrot-Like Sets / 267
Chapter 15: Julia Sets / 275
Drawing Julia Sets / 276
Number of Iterations / 277
Selecting Colors / 281
Julia Set Displays / 282
Binary Decomposition / 283
Chapter 16: Dragon Curves / 289
Harter-Heightway Dragon / 289
Twin Dragon / 291
Julia Dragon / 294
Self-Squared Dragons / 295
San Marcos Dragon / 300
Dragon Outlines / 301
Color Section / 305
Chapter 17: Phoenix Curves / 321
Maps of the Phoenix Curves / 326
Chapter 18: Three-Dimensional Dragons / 329
Method of Projection / 329
Programming the Three-Dimensional Dragon / 330
Chapter 19: Newton’s Method / 335
Programs for Plotting Newton’s Method Curves / 336
Mathematical Meaning of the Curves / 339
Chapter 20: Brownian Motion / 343
One-Dimensional Brownian Motion / 344
Two-Dimensional Brownian Motion / 348
Chapter 21: Fractal Landscapes / 353
Midpoint Displacement Technique / 354
Oak Creek Canyon / 357
Pike’s Peak at Sunrise / 366
Earth Viewed from the Moon / 371
Chapter 22: Iterated Function Systems / 385
Affine Transformations / 385
The Deterministic Algorithm / 386
Generating a Deterministic Fern / 392
Using the Chaos Algorithm / 394
The Collage Theorem / 399
The Chaos Algorithm in Three Dimensions / 400
Appendix A / 405
Appendix B / 429
About the Author / 567
Bibliography / 569
Index / 573