Fractal Fun! - Maple Help

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Fractal Fun!

1. Mandelbrot Set
1.1 The Original Mandelbrot Set
1.2 Mandelbrot Self-Similarity
1.3 Mandelbrot Variation

Introduction

A simple search in your favorite search engine will attest to the popularity of fractal art. That said, many people are often shocked to learn that these visually stunning images are created by iterating a simple complex formula to create a fractal object. A fractal object is any geometric object that posses the property of self-similarity. Self-similarity is a term attributed to Benoît Mandelbrot, to describe any object that appears roughly the same at any level of magnification. Fractal objects are readily prevalent in nature and can be easily seen by examining the intricate shape of sea shells, snowflakes and lightning bolts.

This application illustrates how Maple can be used to generate the two most famous fractal objects: the Mandelbrot Set and the Julia Set.

1. The Mandelbrot Set

The Mandelbrot Set is a mathematical set that is generated by iterating a simple formula where is any point on the complex plane, is the corresponding exponent, and . A point is within the set if is bounded, and outside the set if it’s unbounded; in practice, this is determined by noting whether exceeds a bailout value after a specified number of iterations.

The Original Mandelbrot Set

The image shown at right is the most famous Mandelbrot set. It was generated by iterating the equation: . The procedure used to generate this fractal is defined in the following code-edit region.

Mandelbrot Procedure

The first step in creating the Mandelbrot is to generate a list (or in this case, a Matrix) of complex numbers which will vary the point .

Zc := GenerateComplex(400, -2, 1, -1.35, 1.35); sol := LinearAlgebra[Map](MyMandel, Zc); img := Create(sol)

This image can be easily enhanced using the ImageTools package.

CheckImage := Checkerboard(20, 20, 20, foreground = .7, background = .2); imageCheck := Mask(Complement(img), CheckImage)

Mandelbrot Self-Similarity

As mentioned previously, a unique property common to all fractals is the property of self-similarity. The Mandelbrot Set shown in the previous section is self-similar in the neighborhood of the Misiurewicz point and the Feigenbaum point. These points can be found by magnifying into the area bounded by the white box below:

Mandelbrot Similarity Procedure

Mandelbrot Self-Similarity
Misiurewicz Point
The Misiurewicz Point is located at: -0.1011+ 0.9563i
Feigenbaum Point
The Feigenbaum Point is located at: -0.1528+ 1.0397i

Mandelbrot Variation

Another spellbinding Mandelbrot image can be obtained by magnifying into the area specified by the point -0.7454+ 0.1130i. The results of the magnification can be seen below.

Mandelbrot Variation Procedure

Zc1 := GenerateComplex(400, -.74542138, -.74543162, .11300388, .11301412); sol3 := LinearAlgebra[Map](MandelVariation, Zc1); imgVariation := Create(sol3)

This image can be rendered even more stunning by applying some custom-made colorizing algorithms. These algorithms can be found in the code-edit region defined above.

2. The Julia Set

Julia Set fractals are formed in a similar manner to Mandelbrot Set fractals. The only difference is that the Julia Set varies the complex number while keeping constant, while the Mandelbrot Set varies and the initial starting value of is . This difference, causes there to be an infinite number of Julia Set fractals for every distinct Mandelbrot Set.