Newton's method is a technique for solving equations that have several "roots"--the solutions to the equation. Typically, the roots are fixed points, like +/-1 and +/-i, the four roots of z^{4} - 1 = 0. To animate a Newton fractal, I decided to use four roots and to have them move around during the animation. In this case, the four roots move around on four different rose curves. The coloring is the typical iteration band coloring, just using black and white. And, to keep the visual noise down, I used a small number of iterations.

(See a higher-quality version on YouTube. While you're there, check out a variation based on circles.)

The technique has promise and can be expanded to different root paths and different functional forms. Here, I used a quartic polynomial because it's quick and easy, but there's no reason why rational or transcendental functions couldn't be used as well.