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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
Computer Science, Mathematics, Applied Mathematics, Departments of Universities, Colleges, Technikons, Teacher Training Colleges, Mathematica Users.
Chapter 1: Basics
1.1 The Booklet: Getting Started with Mathematica
1.2 Using Help in Mathematica
1.3 Using Previous Results
1.4 Some Type-setting
1.5 Naming Expressions
1.7 Mathematical Functions
1.8 2D Graphics
1.9 3D Graphics
1.10 2D Graphics Derived from 3D Graphics
1.11 Solving Equations in one Variable
Chapter 2: Using Color in Graphics
2.1 Selecting Colors
2.2 Coloring 2D Graphics Primitives
2.3 Coloring Sequences of 2D Curves Using the Command Plot
2.4 Coloring Sequences of 2D Parametric Curves
2.5 Coloring Sequences of 3D Parametric Curves
2.6 Coloring 3D Parametric Surface Plots
2.7 Coloring Density and Contour Plots
2.8 Coloring 3D Surface Plots
Chapter 3: Patterns Constructed from Straight Lines
3.1 First Method of Construction
3.2 Second Method of Construction
3.3 Assigning Multiple Colors to the Designs
Chapter 4: Orbits of Points Under a C->C Mapping
4.1 Limits, Continuity, Differentiability
4.2 Constructing and Plotting the Orbit of a
4.3 Types of Orbits
4.4 The Contraction Mapping Theorem for C
4.5 Attracting and Repelling Cycles
4.6 Basins of Attraction
4.7 The 'Symmetric Mappings' of Michael Field and Martin Golubitsky
Chapter 5: Using Roman Maeder's Packages Affine Maps,Iterated Function Systems and Chaos Game to Construct Affine Fractals
5.1 Affine Maps from R2 to R 2
5.2 Iterated Function Systems
5.3 Introduction to the Contraction Mapping Theorem for H[R2] 5.4 Constructing Various Types of Fractals Using Roman Maeder's Commands
5.5 Construction of 2D Affine Fractals Using the Random Algorithm
Chapter 6: Constructing Non-affine and 3D Fractals Using the Deterministic and Random Algorithms
6.1 Construction of Julia Sets of Quadratic
Functions as Attractors of Non-affine Iterated
6.2 Attractors of 2D Iterated Function Systems
whose Constituent Maps are not Injective
6.3 Attractors of 3D Affine Iterated Function
Systems Using Cuboids
6.4 Construction of Affine Fractals Using 3D Graphics Shapes
6.5 Construction of Affine Fractals Using 3D Parametric Curves
6.6 Attractors of Affine Iterated Function Systems Using 3D Parametric Surfaces
Chapter 7: Julia and Mandelbrot Sets Constructed Using the Escape - Time Algorithm and Boundary Scanning Method
7.1 Julia Sets and Filled Julia Sets
7.2 Parameter Sets
7.3 Illustrations of Newton's Method
Chapter 8: Miscellaneous Design Ideas
8.1 Sierpinski Relatives as Julia Sets
8.2 Patterns Formed from Randomly Selected Circular Arcs
8.3 Constructing Images of Coiled Shell
Appendix to 5.4.2
Appendix to 7.1.1
Appendix to 7.1.2
Appendix to 8.3.1
- No. of pages:
- © Elsevier Science 2004
- 29th September 2004
- Elsevier Science
- eBook ISBN:
Johannesburg, South Africa
Johannesburg, South Africa
(...) Getz and Helmstedt have delivered not merely a phrase book for the passing tourist but an exhortation to bold imagination for the fledgling poet. (...) the authors strike a good balance between computer platform particular, elementary mathematics, and principles of design aesthetics. Summing Up: Recommended. All levels. -- Reviewed CHOICE, June 2005 by D. V. Feldman, University of New Hampshire.
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