Graphics with Mathematica - 1st Edition - ISBN: 9780444517609, 9780080472874

Graphics with Mathematica

1st Edition

Fractals, Julia Sets, Patterns and Natural Forms

Authors: Chonat Getz Janet Helmstedt
eBook ISBN: 9780080472874
Imprint: Elsevier Science
Published Date: 29th September 2004
Page Count: 334
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Description

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.

Key Features

· 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

Readership

Computer Science, Mathematics, Applied Mathematics, Departments of Universities, Colleges, Technikons, Teacher Training Colleges, Mathematica Users.

Table of Contents

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
1.6 Lists
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
Point
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
Function Systems
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

Appendices
Appendix to 5.4.2
Appendix to 7.1.1
Appendix to 7.1.2
Appendix to 8.3.1

Bibliography.

Details

No. of pages:
334
Language:
English
Copyright:
© Elsevier Science 2004
Published:
Imprint:
Elsevier Science
eBook ISBN:
9780080472874

About the Author

Chonat Getz

Affiliations and Expertise

Johannesburg, South Africa

Janet Helmstedt

Affiliations and Expertise

Johannesburg, South Africa

Reviews

(...) 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.