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© Copyright 1998, Jim Loy
Above is the famous Mandelbrot set, discovered by Benoit Mandelbrot (the discoverer of fractals). The set is a fractal. This is an image made by Fractint, an amazing program that is free on the Internet. The Mandelbrot set is the central dark part of the picture. The colored part is drawn to show the great detail of the black part. The different colors are produced by the program, to show how many iterations (repetitions) were needed to decide that this particular point was not in the set.
So, just what is the Mandelbrot set? It is a graph, of the simple function z=z2+c (or z=z^2+c), where c is the point in question, on the complex plane. The origin is in the right center of the image. z starts at zero. And the formula gives the value of the next z. For most points on the plane, z quickly blows up, goes toward infinity. Near the Mandelbrot set, z blows up more slowly. In the Mandelbrot set, z never blows up. To graph the Mandelbrot set, the program selects a point c and a z=0, and then it solves for z over and over again. If z gets large, the program assumes that the point is not in the set, colors the point, and goes on to the next point. If z stays small for a long time, the program assumes that the point is in the set, colors the point black, and goes on to the next point. The picture above shows approximately where the origin (0,0) is.
The above picture uses different colors (to emphasize different features), and is the upper right portion of the set, above and slightly to the left of the origin.
Above is an image from Aset, a program from Image Laboratories. The location of this image is to the left of the largest crack in the set.
Fractals are a good model for chaos, as they exhibit many of the attributes of chaos, including great complexity. But, in general, a fractal is not chaotic, but is very very predictable. That is why a fractal can be a good model for chaos. It is not just another example of chaos. It is predictable, and therefore less of a mystery.
The Mandelbrot set (as well as other fractals) is more complicated, and shows more detail, than the entire known physical universe. This should not surprise anyone, as the set of real numbers has more points, and more detail in its own way, than there are subatomic particles in the entire known physical universe.
Benoit Mandelbrot is often considered the inventor of fractals. He coined the term "fractal," and essentially originated the field of fractals as a useful branch of mathematics. But many people created fractals, long before Mandelbrot. In fact, Euclid's use of polygons with more and more sides, to approach a circle, as a lot in common with fractals. Most of the other fractals listed in my Mathematics pages are much older than the Mandelbrot set.