Graphics with Mathematica: Fractals, Julia Sets, Patterns and Natural Forms

Chonat Getz, Janet Margaret Helmstedt044451760X, 9780444517609

In this book graphic images are generated using the software Mathematica. The programmes used 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 function systems as well as the construction of many different types of Julia sets and parameter sets such as the Mandelbrot set. The beginning Mathematica user will find this a very interesting way to learn the basic commands and programming techniques of Mathematica, such as pure functions, the use of tables etc., since the results of their efforts will be exciting graphic images.

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Table of contents :
Preface……Page 6
Acknowledgements……Page 8
Contents……Page 10
Basics……Page 14
Introduction……Page 15
Error Messages……Page 16
Using ‘The Mathematica Book’ Section of Help……Page 17
Using the Master Index……Page 19
Built-in Functions……Page 20
Getting Started……Page 21
Using Previous Results……Page 22
Entering 2D Expressions……Page 23
Entering Special Characters……Page 24
Naming Expressions……Page 25
Making Lists of Objects……Page 26
Constructing Lists using the Command Table……Page 27
Some Operations on Lists……Page 28
Standard Built-in Functions……Page 29
User-defined Functions……Page 31
Pure Functions……Page 33
Compiling Functions……Page 34
Functions as Procedures……Page 36
Logical Operators and Conditionals……Page 37
Options……Page 39
Plotting a Sequence of Points Using the Command ListPlot……Page 40
2D Graphics Elements……Page 41
Constructing a Sequence of Graphics Primitives……Page 44
Graphs of Equations of the Form y = f [x]……Page 48
Constructing 2D Parametric Plots……Page 55
Add-ons, ComplexMap……Page 59
Polar and Implicit Plots……Page 61
3D Graphics Elements……Page 62
Plotting Surfaces Using the Command Plot3D……Page 65
3D Parametric Curve Plots……Page 73
3D Parametric Surface Plots……Page 74
Constructing Surfaces from a 2D Parametric Plot……Page 77
Density Plots……Page 81
Contour Plots……Page 85
Exact Solutions of Algebraic Equations of Degree at most Four……Page 90
Approximate Solutions of Algebraic Equations……Page 91
Transcendental Equations……Page 92
Finding Co-ordinates of a Point on a 2D Plot……Page 94
Introduction……Page 95
Using Color Charts……Page 96
Making Color Palettes by Coloring a Sequence of Rectangles……Page 97
Patterns made with Sequences of Graphics Primitives……Page 99
Coloring Sequences of 2D Curves Using the Command Plot……Page 101
Coloring Sequences of 2D Parametric Curves……Page 102
Coloring Sequences of Similar 3D Parametric Curves……Page 107
Sequences of Similar 2D Curves in Parallel Planes……Page 111
3D Graphics Constructed by Rotating Plane Curves……Page 112
Plane Patterns Constructed from Curves with Parametric Equations of the Form: { 0, f[t], g[t] }……Page 119
Coloring 3D Parametric Surface Plots……Page 122
Making Palettes for the Use of ColorFunction in Density, Contour and 3D Plots……Page 124
Contour Plots……Page 125
Density Plots……Page 138
Coloring 3D Surface Plots……Page 141
First Method of Construction……Page 146
Second Method of Construction……Page 150
Assigning Multiple Colors to the Designs……Page 153
Orbits of Points Under a Nsub->Nsub Mapping……Page 155
Limits, Continuity, Differentiability of Complex Functions……Page 156
Calculating the Orbit of a Point……Page 157
Plotting the Orbit of a Point……Page 158
Bounded and Unbounded Orbits……Page 159
Convergent Orbits……Page 160
Boundary of a Subset of Nsub……Page 162
The Contraction Mapping Theorem for Nsub……Page 163
Attracting and Repelling Fixed Points……Page 164
Attracting and Repelling Cycles of Prime Period Greater than One……Page 165
Basin of Attraction of a Fixed Point……Page 168
Basin of Attraction of an Attracting Cycle of Period p > 1……Page 170
The Basin of Attraction of Infinity……Page 172
The ‘Symmetric Mappings’ of Michael Field and Martin Golubitsky……Page 173
Using Roman Maeder’s Packages AffineMaps, Iterated Function Systems and Chaos Game to Construct Affine Fractals……Page 174
Introduction……Page 175
Definitions……Page 176
Sheared Affine Transformations……Page 177
Definition of the Sierpinski Triangle……Page 178
Definition of an IFS……Page 179
Constructing the Sierpinski Triangle Using an Affine IFS……Page 180
H[R2]……Page 181
The Contraction Mapping Theorem for H [R2]……Page 182
Relatives of the Sierpinski Triangle……Page 183
Iterated Function Systems which Include the Identity Map……Page 185
The Collage Theorem……Page 186
Constructing Your Own Fractals……Page 188
Constructing Fractals with Initial Set a Collection of Graphics Primitives……Page 191
Constructing Tree-Like Fractals……Page 192
Fractals Constructed Using Regular Polygons……Page 196
Constructing Affine Fractals Using Parametric Plots……Page 200
Constructing Fractals from Polygonal Arcs……Page 203
Roman Maeder’s Package: The ChaosGame……Page 206
Introduction……Page 210
The Quadratic Family Qc……Page 211
Construction of Julia Sets Using the Deterministic Algorithm……Page 212
Construction of Julia Sets Using the Random Algorithm……Page 215
Attractors of Iterated Function Systems whose Constituent Maps are not Injective……Page 217
Attractors of 3D Affine Iterated Function Systems Using Cuboids……Page 218
Construction of Cylinders……Page 223
Scaling, Rotating and Translating Cylinders……Page 224
Constructing the Initial Branches of a Tree……Page 227
The Routine for Generating the Tree……Page 228
Constructing 3D Analogues of Relatives of the Sierpinski Triangle……Page 231
Constructing other 3D Fractals with Spheres……Page 233
Construction of Affine Fractals Using 3D Parametric Curves……Page 234
Constructing Affine Fractals Using 3D Parametric Surfaces……Page 237
Introduction……Page 242
Julia Sets and Filled Julia Sets of Polynomials……Page 243
Notes on Julia Sets of Rational Functions……Page 251
Julia Sets of Rational Functions with Numerator not of Higher Degree than Denominator……Page 255
Julia Sets of Rational Functions with Numerator of Higher Degree than Denominator……Page 265
Critical and Asymptotic Values of Entire Transcendental Functions……Page 273
Exponential Functions……Page 274
Trigonometric Functions……Page 277
The Mandelbrot Set……Page 280
Parameter Sets for Entire Transcendental Functions……Page 283
Classifying Starting Points for Newton’s Method……Page 284
Choosing a Starting Point for Using Newton’s Method to Solve Transcendental Equations……Page 286
Sierpinski Relatives as Julia Sets……Page 289
Patterns Formed from Randomly Selected Circular Arcs……Page 292
Shell Anatomy……Page 295
Shell Construction……Page 298
Coloring Methods……Page 300
Constructing Shell Images as 3D Surface Plots……Page 302
Appendix to 5.4.2……Page 306
Conjugate Mappings……Page 307
Appendix to 7.1.2……Page 308
Appendix to 8.3.1……Page 310
Bibliography……Page 312
Index……Page 314