The chaos theory classic in complex number space.
Pick any point in the complex plane and call it c. Then, with z0 = 0, iterate so that zi+1 = zi2+c.
The first question is: is there a zi for which |zi| > 2; if so, all subsequent points will tend to infinity; if not, the points will be forever trapped close to 0 in the Mandelbrot Set.
If it tends to infinity, the second question is: how many iterations does it take to get there?
The Mandelbrot Set itself is colored black; all other colors represent different amounts of time to tend to infinity from that starting point.
Newton's method is a numerical technique for finding the roots of a polynomial by "guesswork". Starting with an arbitrary guess, figure both the function value and the differential. Use the value of the differential to come up with a better guess (where the root would be if the function were in fact a straight line). Then repeat.
Intuition suggests that there will be consistent areas where each root is found; but in fact the roots can go anywhere.
The colors are used to indicate which root was found from an initial guess.
The roots are in the complex plane.