These images were made by the Fracture screensaver.
These images are generated by different root-finding methods: Newton's Method, Halley's Method, the Secant Method, and a simplified version of Newton's Method (which has no name of its own that I know of). These are different methods for finding the "roots" to an equation, via an iterative method of successive approximation. Like the Mandelbrot Set, points in the image correspond to points on the complex plane. For each point, the given root-finding method is applied iteratively until the point converges on a root of the equation. The coloration depends upon which root is converged upon, how long it took to converge, or (as is most typical) both.
It turns out that the behavior of different "initial guess" points is not at all simple. They do not, for example, simply converge upon the root that is nearest to them. Rather, they form "basins" of points that all converge on a given root, and the boundaries between basins are infinitely complex (and fractal), composed of ever smaller intertwined basins. Points that do not converge on any root are colored with their own color, typically black.
The mathematics of these images can be thought of as being very similar to that of Julia Sets, really, as will be described further in the collection of Periodicity Analysis images.
All of these images are copyright © 2001 Ben Haller. Personal use of these images is allowed; all other use, including any kind of redistribution or reproduction of these images, is forbidden without the express, written consent of Ben Haller.