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Fractal Fun!

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Fractal Fun!

 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.   The custom procedures used throughout this Document are defined in its Startup code-region (from the Edit menu, select Startup Code).

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 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 .

This image can be easily enhanced using the ImageTools  package.

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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:

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 Mandelbrot Self-Similarity Misiurewicz Point The Misiurewicz Point is located at: -0.1011+ 0.9563i.   A threshold can be applied to the image in order to bring out the details of interest. 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.

 This image can be modified by shading according to a threshold value or by applying a custom coloring procedure. The coloring procedure can be found in this Document's Startup code-region (from the Edit menu, select Startup Code). An attractive three-layer color image can also be created, using one of the grayscale images above. This is accomplished by first creating an empty color image and then setting each of its red, green, and blue layers to modified versions of the grayscale image. The procedure Colorize is defined in this Document's Startup code-region (from the Edit menu, select Startup Code).

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.

The fractals below were all created by iterating this formula for different values of :

 Julia Set Fractals for different values of