Fractals, Julia Sets, Patterns and Natural Forms To order this title, and for more information, click here
By Chonat Getz, Johannesburg, South Africa Janet Helmstedt, Johannesburg, South Africa
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.
Audience
Computer Science, Mathematics, Applied Mathematics, Departments of Universities, Colleges, Technikons, Teacher Training Colleges, Mathematica Users.
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.
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