The Mandelbrot Set

Where does the color come from?

We know the points in the set are coded black. The Java applet you will see took the function out to 100 iterates. After n iterates, the point may leave the set proceeding on its way to infinity. If n is a small number, then the point goes out to infinity very quickly. If n is close to 100, then the point's orbit, or iterate path, stays inside the set for awhile before it leaves.

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Let's take our earlier example, z₀ = 0.5 + 0.75 i. This appears to have left the set after the 1^st or 2^nd iterate. We can check this by finding the distance from z₂ to the origin. How do we compute distances between two complex numbers? Look it up. If the result is greater than a radius of 3, then the point has left the set. Since n = 2, it would be assigned the color for 2, say it is purple.

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Go on to the Julia set.

To see the Mandelbrot and Julia sets, go into the java applet. Acknowledgements

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