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Welcome to Neal´s Mandelbrot Set and Julia Set Page!

This modest page is devoted to explaining two wonderful, related mathematical notions: the Mandelbrot Set and Julia sets.  If you have the Java 2 Plug-in, you can run a really fancy applet that allows you to explore these sets interactively.

Values in the Mandelbrot Set

The computations that go into determining membership in the Mandelbrot Set can be used to draw beautiful pictures.  How can this work?
  1. First, select an area of the complex plane.  Usually, you start with an area from about -2-1i to about 1+1i.
  2. Divide up this area into individual pixels, based on the height and width of the image you'd like to produce.  Each pixel gets a different complex value C.
  3. Select a palette of colors, and a maximum number of iterations you are willing to perform.
  4. For each pixel in the image, apply the Mandelbrot formula to the value C. If the point C appears to be in the set, color it black. Otherwise, select a color from your palette of colors based on how many applications of the formula were needed to pass the magnitude value 2.0 (for C1 above, that would 16).
Surprisingly, the colors of the points that are not in the set form fascinating swirls and spirals and branchings, in many respects far more interesting than the Set itself.
full Mandelbrot Set in color

The infinitely fractal nature of the Mandelbrot Set is evident in the patterns we observe around its boundaries.

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This page written by Neal Ziring , last modified 1/7/03.