Graphics with Mathematica (eBook)

Fractals, Julia Sets, Patterns and Natural Forms

Chonat Getz, Janet Margaret Helmstedt (Autoren)

eBook Download: PDF

2004 | 1. Auflage
334 Seiten
Elsevier Science (Verlag)
978-0-08-047287-4 (ISBN)

Infos

"In this book we generate graphic images using the software Mathematica thus providing a gentle and enjoyable introduction to this rather technical software and its graphic capabilities.
The programs we use for generating these graphics are easily adaptable to many variations.

These graphic images are enhanced by introducing a variety of different coloring techniques.

Detailed instructions are given for the construction of some interesting 2D and 3D fractals using iterated functions systems as well as the construction of many different types of Julia sets and parameter sets such as the Mandelbrot set.

The mathematics underlying the theory of Iterated function systems and Julia sets is given an intuitive explanation, and references are provided for more detailed study.

?Brilliant Graphic images
?Interesting Introduction to Mathematica for Beginners
?Easy constructions
?A variety of Coloring Techniques
?Programs Easily Adaptable to Many Variations
?Constructions useful for Dynamics and Fractals Courses"

Inhalt

Cover 1
Preface 5
Acknowledgements 7
Contents 9
Chapter 1. Basics 13
Introduction 14
The Booklet: Getting Started with Mathematica 15
Getting Started with Mathematica 15
Your First Mathematica Calculations 15
Error Messages 15
Using Help in Mathematica 16
Using 'The Mathematica Book' Section of Help 16
Using the Master Index 18
Built-in Functions 19
Front End (Other Information in Version 4.2) 20
Getting Started 20
Using Previous Results 21
Some Type-Setting 22
Using Palettes 22
Using Keyboard Shortcuts 22
Entering 2D Expressions 22
Entering Special Characters 23
Naming Expressions 24
Lists 25
Making Lists of Objects 25
Constructing Lists using the Command Table 26
Applying Built-in Functions to Lists 27
Some Operations on Lists 27
Mathematical Functions 28
Standard Built-in Functions 28
User-defined Functions 30
Pure Functions 32
Compiling Functions 33
Functions as Procedures 35
Logical Operators and Conditionals 36
2D Graphics 38
Options 38
Plotting a Sequence of Points Using the Command ListPlot 39
2D Graphics Elements 40
Constructing a Sequence of Graphics Primitives 43
Graphs of Equations of the Form y = f [x] 47
Constructing 2D Parametric Plots 54
Add-ons, ComplexMap 58
Polar and Implicit Plots 60
3D Graphics 61
3D Plots 61
3D Graphics Elements 61
Plotting Surfaces Using the Command Plot3D 64
3D Parametric Curve Plots 72
3D Parametric Surface Plots 73
Constructing Surfaces from a 2D Parametric Plot 76
2D Graphics Derived from 3D Graphics 80
Density Plots 80
Contour Plots 84
Solving Equations in one Variable 89
The Symbols = and == 89
Exact Solutions of Algebraic Equations of Degree at most Four 89
Approximate Solutions of Algebraic Equations 90
Transcendental Equations 91
Finding Co-ordinates of a Point on a 2D Plot 93
Chapter 2. Using Color in Graphics 94
Introduction 94
Selecting Colors 95
Using Color Selector 95
Using Color Charts 95
Coloring 2D Graphics Primitives 96
Syntax for Coloring Graphics Primitives 96
Making Color Palettes by Coloring a Sequence of Rectangles 96
Patterns made with Sequences of Graphics Primitives 98
Coloring Sequences of 2D Curves Using the Command Plot 100
Coloring Sequences of 2D Parametric Curves 101
Coloring Sequences of 3D Parametric Curves 106
Coloring Sequences of Similar 3D Parametric Curves 106
Sequences of Similar 2D Curves in Parallel Planes 110
3D Graphics Constructed by Rotating Plane Curves 111
Plane Patterns Constructed from Curves with Parametric Equations of the Form: { 0, f[t], g[t] } 118
Coloring 3D Parametric Surface Plots 121
Coloring Density and Contour Plots 123
Making Palettes for the Use of ColorFunction in Density, Contour and 3D Plots 123
Contour Plots 124
Density Plots 137
Coloring 3D Surface Plots 140
Chapter 3. Patterns Constructed from Straight Lines 145
Introduction 145
First Method of Construction 145
Second Method of Construction 149
Assigning Multiple Colors to the Designs 152
Chapter 4. Orbits of Points Under a N-> N Mapping
Introduction 155
Limits, Continuity, Differentiability 155
Limits of Sequences in Nsub 155
Limits, Continuity, Differentiability of Complex Functions 155
Constructing and Plotting the Orbit of a Point 156
Iterating a Function 156
Calculating the Orbit of a Point 156
Plotting the Orbit of a Point 157
Types of Orbits 158
Bounded and Unbounded Orbits 158
Fixed Points and Periodic Orbits 159
Convergent Orbits 159
The Contraction Mapping Theorem for Nsub 161
Contraction Mappings on Nsub 161
Boundary of a Subset of Nsub 161
Closed Subsets of Nsub, Closure 162
Compact Subsets of Nsub 162
The Contraction Mapping Theorem for Nsub 162
Attracting and Repelling Cycles 163
Attracting and Repelling Fixed Points 163
Attracting and Repelling Cycles of Prime Period Greater than One 164
Basins of Attraction 167
Basin of Attraction of a Fixed Point 167
Basin of Attraction of an Attracting Cycle of Period p > 1
The Basin of Attraction of Infinity 171
The 'Symmetric Mappings' of Michael Field and Martin Golubitsky 172
Chapter 5. Using Roman Maeder's Packages AffineMaps, Iterated Function Systems and Chaos Game to Construct Affine Fractals 173
Introduction 174
Affine Maps from R2 to R2 175
Definitions 175
Affine Maps which are Similarities 176
Sheared Affine Transformations 176
Definition of the Sierpinski Triangle 177
Iterated Function Systems 178
Contraction Mappings on Subsets of R2 and Compact Subsets of R2 178
Definition of an IFS 178
Constructing the Sierpinski Triangle Using an Affine IFS 179
Introduction to the Contraction Mapping Theorem for H [R2] 180
H[R2] 180
The Contraction Mapping Theorem for H [R2] 181
Constructing Various Types of Fractals using Roman Maeder's Commands 182
Relatives of the Sierpinski Triangle 182
Iterated Function Systems which Include the Identity Map 184
The Collage Theorem 185
Constructing Your Own Fractals 187
Constructing Fractals with Initial Set a Collection of Graphics Primitives 190
Constructing Tree-Like Fractals 191
Fractals Constructed Using Regular Polygons 195
Constructing Affine Fractals Using Parametric Plots 199
Constructing Fractals from Polygonal Arcs 202
Construction of 2D Fractals Using the Random Algorithm 205
Introduction 205
Roman Maeder's Package: The ChaosGame 205
Chapter 6. Constructing Non-Affine and 3D Fractals Using the Deterministic and Random Algorithms 209
Introduction 209
Construction of Julia Sets of Quadratic Functions as Attractors of Non-Affine Iterated Function Systems 210
Julia Sets and Filled Julia Sets 210
The Quadratic Family Qc 210
Construction of Julia Sets Using the Deterministic Algorithm 211
Construction of Julia Sets Using the Random Algorithm 214
Attractors of Iterated Function Systems whose Constituent Maps are not Injective 216
Attractors of 3D Affine Iterated Function Systems Using Cuboids 217
Construction of Affine Fractals Using 3D Graphics Shapes 222
Construction of Cylinders 222
Scaling, Rotating and Translating Cylinders 223
Constructing the Initial Branches of a Tree 226
The Routine for Generating the Tree 227
Constructing 3D Analogues of Relatives of the Sierpinski Triangle 230
Constructing other 3D Fractals with Spheres 232
Construction of Affine Fractals Using 3D Parametric Curves 233
Constructing Affine Fractals Using 3D Parametric Surfaces 236
Chapter 7. Constructing Julia and Mandelbrot Sets with the Escape-Time Algorithm and Boundary Scanning Method 241
Introduction 241
Julia Sets and Filled Julia Sets 242
Julia Sets and Filled Julia Sets of Polynomials 242
Notes on Julia Sets of Rational Functions 250
Julia Sets of Rational Functions with Numerator not of Higher Degree than Denominator 254
Julia Sets of Rational Functions with Numerator of Higher Degree than Denominator 264
Julia Sets of Entire Transcendental Functions which are Critically Finite 272
Critical and Asymptotic Values of Entire Transcendental Functions 272
Exponential Functions 273
Trigonometric Functions 276
Parameter Sets 279
The Mandelbrot Set 279
Parameter Sets for Entire Transcendental Functions 282
Illustrations of Newton's Method 283
Classifying Starting Points for Newton's Method 283
Choosing a Starting Point for Using Newton's Method to Solve Transcendental Equations 285
Chapter 8. Miscellaneous Design Ideas 288
Introduction 288
Sierpinski Relatives as Julia Sets 288
Patterns Formed from Randomly Selected Circular Arcs 291
Constructing Images of Coiled Shells 294
Shell Anatomy 294
Shell Construction 297
Coloring Methods 299
Constructing Shell Images as 3D Surface Plots 301
Appendices 305
Appendix to 5.4.2 305
Appendix to 7.1.1 306
Conjugate Mappings 306
Appendix to 7.1.2 307
Appendix to 8.3.1 309
Bibliography 311
Index 313

Produkt-Details

• Erscheint lt. Verlag: 29.9.2004
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Informatik ► Software Entwicklung
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Graphentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-047287-7 / 0080472877
• ISBN-13: 978-0-08-047287-4 / 9780080472874