Exploring Fractals on the Macintosh

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Editorial Reviews

Booknews
This book/disk package discusses chaos theory, the nature of fractals, and where they can be found in the real world, and demonstrates how to create fractal figures on the Mac. Explains calculating fractal dimensions, image compression, and creating fractal programs. Emphasizes mathematical concepts and creativity. The accompanying software disk requires Macintosh Plus or later, 1 MB RAM, and Systems Software 4.2 or later. Illustrated with numerous b&w diagrams. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780201626308
  • Publisher: Addison Wesley Professional
  • Publication date: 12/8/1994
  • Edition number: 1
  • Pages: 368
  • Product dimensions: 7.38 (w) x 9.12 (h) x 0.76 (d)

Table of Contents

Preface
Ch. 1 Fractals in Our World 1
Fractals in Nature 2
Fractals in Science 6
Fractals in the Arts 10
Fractals and Computers 16
Ch. 2 Basics of Fractals 19
What Is Fractal Geometry? 19
What Is a Linear Fractal? 21
What Is a Nonlinear Fractal? 22
What Is Fractal Dimension? 23
Where Are Fractals Being Used Today? 24
What Is Chaos Mathematics? 25
Ch. 3 Creating Classical Fractals on the Macintosh 27
FractaSketch: A Tool for Exploring Fractal Geometry 28
The Cantor Set: Mathematical Dust 30
The Peano Curve 35
The Hilbert Curve 42
The Koch Curve 45
The Pascal Triangle Passed to Sierpinski 51
The Dragon Curve 57
Finite Segments and Proportional Widths 63
Ch. 4 Calculating Fractal Dimensions 75
Mathematical Tools 76
The Hausdorff-Besicovitch Dimension 78
Calculating Traditional Dimensions 78
Calculating Dimensions of Self-Similar Fractals 80
The Richardson Method of Varied Measured Lengths 92
The Box-Counting Method 99
Ch. 5 Exploring Nonlinear Fractals with MandelMovie 109
The Mandelbrot Set 110
Using MandelMovie to Explore the Mandelbrot Set 119
Julia Sets 121
Exploring Julia Sets with MandelMovie 127
Ch. 6 Fractals That Fell through the Cracks 131
The Infinite Staircase 131
Newton's Method 134
Randomness and Its Staggering Results 139
Fractal Mountains and Fractal Clouds 145
Multifractals 149
Fractal Image Compression 149
Aggregate Growth: The Fractal of Crystals 151
Iterated Function Systems 155
Ch. 7 Recipes from the Fractal Cookbook 187
The Basic Algorithm 187
The Relationships with Lindenmayer Systems 190
Techniques to Increase Speed and Accuracy 192
The Technique of Faking It 198
Calculating the Fractal Dimension 200
Low-Level Information about FractaSketch 203
Ch. 8 Creating Programs to Explore Fractals and Chaos 207
PostScript Code for Generating High-Resolution Linear Fractals 207
Algorithms for Mandelbrot and Julia Sets 215
C Code for Plotting Newton's Method 225
C Code for IFS Transformations 238
Basic Programs for Drawing Fractals and Creating Chaos Models 241
Creating Fractal Models with Mathematica 278
Ch. 9 FractaSketch User Manual 283
Basic Features 283
Overview of the Menu Commands 287
Added Features Found in Extended Versions of FractaSketch 292
Tips and Techniques 292
Ch. 10 MandelMovie User Manual 297
Basic Features 297
Overview of the Menu Commands 302
Added Features Found in Extended Versions of MandelMovie 317
Appendix A The Fractal Gallery 323
Benoit Mandelbrot's World 323
Inkblots of Nature 325
Fractal Potpourri 328
Appendix B Graph Paper with Different Grid Sizes 331
Appendix C Video Feedback 333
Works Cited 335
List of Contributors 341
Index 345
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