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A complex number c is a member of the Mandelbrot set if the sequence c, c2+c, (c2+c)2 + c, ((c2+c)2 + c)2 + c, ..., remains bounded. In fact, if it ever turns out that that sequence attains an absolute value of greater than 2, then the sequence becomes unbounded. The computer "pixelates" an area, picks the number at the center of each pixel, then calculates the above sequence for as long as you tell it to. The interesting thing is that you get quite different pictures, depending on how long you let the computer run. Below on this page we have various illustrations of approximately the same region of the complex plane. In the first picture, the program allowed the number to be in the Mandelbrot set if its 50th iteration had absolute value less than 2. In the second picture, the number had to still have its absolute value less than 2 after 100 iterations, and so on. The difficulty is that it may take many iterations before the sequence becomes large, but of course we are only willing to wait so long for the computer to produce the picture. I made these pictures with the Windows program Set Surfer, available here. In each of these pictures, the black part is the only part that is (allegedly) in the Mandelbrot set.
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Click on any image for a larger picture.
Better yet, here is an animation of the same thing
but with a different region.
Other microscopic parts of the Mandelbrot set