Creating Fractals
by Stevens, RogerEdition: 1st
Format: Paperback
Pub. Date: 2005-08-09
Publisher(s): Charles River Media
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Summary
Everything You'll Need to Create Thousands of Fractals! Fractals are the name given to certain types of iterated equations that produce very strange results and are capable of creating unusual and beautiful patterns. Creating Fractals describes the characteristics and mathematical background of fractals and shows the reader how the accompanying fractal-generating program is used to produce thousands of different kinds of fractals, to enlarge them, to color them, and to save them-- without any knowledge of computers or programming. The program works with any computer using Windows. In addition to producing artistic effects, the reader can gain an understanding of how each type of fractal is created and how it might be used to treat natural phenomena, e.g., the turbulence of liquids, the behavior of the stock market, and the compression of graphic images. Mathematical terminology is explained in elementary terms.
Table of Contents
| Introduction | p. 1 |
| "Monster" Curves | p. 3 |
| Working with Fractals | p. 3 |
| The Lorenz and Other Strange Attractors | p. 4 |
| What You Can Do with L-Systems Fractals | p. 5 |
| The Snowflake and Other von Koch Curves | p. 6 |
| Peano Curves | p. 6 |
| Generators with Different Length Line Segments | p. 6 |
| The Hilbert Curve | p. 7 |
| FASS Curves | p. 7 |
| Trees | p. 7 |
| Creating Your Own L-Systems Fractals | p. 7 |
| Newton's Method | p. 8 |
| Fractals with the Logistic Equation | p. 8 |
| Mandelbrot and Julia Sets | p. 10 |
| Working with Colors | p. 11 |
| Curves from Trigonometric and Exponential Functions | p. 12 |
| Fractals Using Orthogonal Functions | p. 12 |
| Phoenix Curves | p. 12 |
| The Mandela Fractal | p. 13 |
| Pokorny Fractals | p. 13 |
| Fractals Using Circles | p. 13 |
| Barnsley Fractals | p. 14 |
| Iterated Function Systems | p. 14 |
| Midpoint Displacement Fractals | p. 14 |
| Continuing on Your Own | p. 15 |
| References | p. 15 |
| What Are Fractals? | p. 17 |
| Iterated Functions | p. 18 |
| How Are Fractals Used? | p. 20 |
| Basic Considerations | p. 20 |
| Fractal Dimensions | p. 20 |
| References | p. 22 |
| The Lorenz and Other Strange Attractors | p. 23 |
| Strange Attractors | p. 25 |
| The Lorenz Attractor | p. 25 |
| Runge-Kutta Integration | p. 27 |
| Viewing the Lorenz Attractor | p. 28 |
| Other Strange Attractors | p. 29 |
| References | p. 31 |
| What You Can Do with L-System Fractals | p. 33 |
| How L-Systems Works | p. 34 |
| The Geometric Basis for L-Systems | p. 35 |
| Symbols Used in the L-Systems Language | p. 35 |
| Overview of the L-Systems Program | p. 36 |
| More Complex Generator Schemes | p. 37 |
| Recurrence in L-Systems | p. 39 |
| Creating L-Systems Fractals | p. 39 |
| References | p. 40 |
| The Snowflake and Other von Koch Curves | p. 41 |
| Snowflake Curve | p. 42 |
| Gosper Curve | p. 46 |
| 3-Segment Quadric von Koch Curve | p. 53 |
| 8-Segment Quadric von Koch Curve | p. 54 |
| 18-Segment Quadric von Koch Curve | p. 61 |
| 32-Segment Quadric von Koch Curve | p. 64 |
| 50-Segment Quadric von Koch Curve | p. 67 |
| Hexagonal 8-Segment von Koch Curve | p. 70 |
| Sierpinski Triangle | p. 74 |
| Islands Curve | p. 78 |
| Peano Curves | p. 83 |
| Original Peano Curve | p. 84 |
| Modified Peano Curve | p. 87 |
| Cesaro Triangle | p. 89 |
| Modified Cesaro Triangle | p. 93 |
| Polya Triangle | p. 95 |
| Peano-Gosper Curve | p. 98 |
| Harter-Heighway Dragon Curve | p. 102 |
| References | p. 105 |
| Generators with Different Sized Line Segments | p. 107 |
| Peano 7-Segment Snowflake | p. 108 |
| Peano 13-Segment Snowflake | p. 112 |
| von Koch Curve Using Complex Generator | p. 115 |
| Fractal Dimensions | p. 119 |
| The Hilbert Curve | p. 121 |
| Hilbert II Curve | p. 126 |
| Using the Hilbert Curve for Display Data Storage | p. 129 |
| References | p. 129 |
| Fass Curves | p. 131 |
| FASS 1 Curve | p. 132 |
| FASS 2 Curve | p. 135 |
| FASS 3 Curve | p. 136 |
| FASS 4 Curve | p. 138 |
| FASS 5 Curve | p. 140 |
| FASS 6 Curve | p. 143 |
| FASS 7 Curve | p. 146 |
| Tress | p. 149 |
| Real Trees | p. 150 |
| Tree Drawing with L-Systems | p. 152 |
| Second Tree | p. 154 |
| Third Tree | p. 157 |
| Fourth Tree | p. 159 |
| Fifth Tree | p. 162 |
| Sixth Tree | p. 165 |
| Bush | p. 167 |
| Randomness in Trees | p. 169 |
| References | p. 172 |
| Creating Your Own L-System Fractals | p. 173 |
| Creating an Initiator and Generator | p. 174 |
| Using the Create L-Systems Curve Command | p. 174 |
| More Complicated Fractals with L-Systems | p. 175 |
| Newton's Method | p. 177 |
| Noninteger Exponents | p. 181 |
| What You Can Do with Mandelbrot-like and Julia-like Fractals | p. 183 |
| Julia Sets | p. 184 |
| The Mandelbrot Set as a Map of Julia Sets | p. 185 |
| Expansion of the Display | p. 185 |
| References | p. 186 |
| The Mandelbrot and Julia Sets | p. 187 |
| Expanding the Mandelbrot Set | p. 191 |
| Julia Sets | p. 193 |
| The Mandelbrot Set as a Map of Julia Sets | p. 195 |
| Number of Iterations | p. 195 |
| Specifying Julia Set Parameters | p. 196 |
| Hypnoteyes | p. 197 |
| Binary Decomposition | p. 197 |
| References | p. 199 |
| Working with Colors | p. 201 |
| Coloring L-Systems Fractals | p. 202 |
| Mandelbrot Colors | p. 203 |
| Julia Colors | p. 203 |
| Dragon Colors | p. 204 |
| Phoenix and Phoenix 2 Colors | p. 204 |
| Blue and Silver | p. 205 |
| Random Colors | p. 205 |
| Custom Colors | p. 206 |
| Complex Colors | p. 207 |
| Gradient Colors | p. 208 |
| Binary Decomposition | p. 210 |
| Newton Colors | p. 210 |
| Lyapunov Colors | p. 210 |
| References | p. 222 |
| Fractals with the Logistic Equation | p. 211 |
| Bifurcation Diagrams | p. 213 |
| Expansion of the Display | p. 214 |
| "Period Three Implies Chaos" | p. 216 |
| The Fiegenbaum Number | p. 216 |
| Self-Squared Dragons | p. 217 |
| Lyapunov Fractals | p. 219 |
| References | p. 222 |
| Fractals Using Transcendental Functions | p. 223 |
| Cosine Fractal | p. 225 |
| Sine Fractal | p. 227 |
| Hyperbolic Cosine Fractal | p. 228 |
| Hyperbolic Sine Fractal | p. 229 |
| Exponential Fractal | p. 230 |
| Fractals Using Orthogonal Polynomials | p. 233 |
| Creating Fractals with Orthogonal Polynomials | p. 235 |
| Bernoulli Fractals | p. 236 |
| Chebyshev Polynomials | p. 237 |
| Legendre Polynomials | p. 240 |
| Laguerre Polynomials | p. 241 |
| Hermite Polynomials | p. 242 |
| References | p. 244 |
| Creating Your Own Second-Order to Seventh-Order Equations | p. 245 |
| Phoenix Curves | p. 249 |
| Relationship of Mandelbrot-Like and Julia-Like Phoenix Curves | p. 250 |
| Phoenix Fractal Coloring | p. 251 |
| The Mandela and Pokorny Fractals | p. 253 |
| Coloring the Mandela Curve | p. 254 |
| Pokorny Fractals | p. 255 |
| Fractals Using Circles | p. 259 |
| Apollonian Packing of Circles | p. 260 |
| Soddy's Formula | p. 261 |
| Creating the Apollonian Circle Packing Fractal | p. 261 |
| Inversion | p. 265 |
| Pharaoh's Breastplate | p. 266 |
| Self-Homographic Fractals | p. 266 |
| References | p. 268 |
| Barnsley Fractals | p. 271 |
| The First Barnsley Fractal | p. 272 |
| The Second Barnsley Fractal | p. 273 |
| The Third Barnsley Fractal | p. 274 |
| The Barnsley Sierpinski Triangle | p. 275 |
| References | p. 276 |
| Iterated Function Systems | p. 277 |
| Affine Transformation | p. 278 |
| The Collage Theorem | p. 283 |
| Creating Your Own IFS Fractals | p. 284 |
| References | p. 285 |
| Midpoint Displacement Fractals | p. 287 |
| Midpoint Displacement | p. 288 |
| Triangles with a Common Line | p. 290 |
| Midpoint Displacement in the Fractal Program | p. 290 |
| Coloring Mountains | p. 291 |
| Random Number Considerations | p. 292 |
| Further Considerations | p. 293 |
| About the CD-ROM | p. 295 |
| Index | p. 299 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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