![[Mandelbrot Set thumbnail]](images/mset2.png "Mandelbrot Set Thumbnail")
Fractals are probably the most beautiful shapes in all of mathematics. They are infinitely detailed, often have a property called self-similarity - they contain similar copies of themselves - and most importantly, are generated by incredibly simple formulas that feed back to themselves, a process called iteration.
See the 640 X 480 version
See the 1280 X 1024 version
This image (created with Fractint, THE best fractal program ever), is an "overview" of the Mandelbrot set, named after the mathematician who made fractal geometry famous, Benoit B. Mandelbrot. It is generated by the formula:
z 
z2 + c
where c is a given location on the plane, expressed as a complex
number, and z is initially set to 0.
What this notation means is that z starts with some initial value (0 in this case), and at each iteration, the function on the right is evaluated, and the result is fed back to z. If, after a certain number of iterations (the higher, the better), z is within a circle of radius 2 centered at the origin (center of plane), it is determined to be inside the Mandelbrot set, and is colored accordingly (black in this case). If, however, it escapes this circle, the usual method is to color the point according to how many iterations it took. This provides a gorgeous series of contours around the set.
As I said before, fractals are infinitely detailed! Mandelbrot sets (this is just one of an infinite number) are not truely self-similar like other types of fractals. Therefore the details as one zooms in closer and closer gradually look different.
The "inside" region is the mathematically defined Mandelbrot set. However, it is not the most interesting part. The most interesting part is the set of extremely elaborate countours surrounding it everywhere. Thanks to the increasingly asymmetrical and branched structures, they form a wonderful structure with a beauty unmatched (IMHO) in mathematics.
The surrounding countours are "almost" within the set, meaning that if the maximum iteration value (see above) is set lower, the inner contours are found by the generating software to fall within the set, even though they don't. This is because, to see a completely accurate representation of the set, one would have to use the value infinity for maximum iteration, which is of course impossible; lower values make the set cover regions that for higher values are merely contours around it. This is why, as stated above, higher values for the maximum iteration are better. But they tend to lengthen the time required for computation.
Last updated: Tue May 27, 2008
E-mail (anti-spam mangled): ddawson at icehouse dot net
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