Mandelbrot Set

These images were made by the Fracture screensaver.

These are images of the Mandelbrot Set, probably the most famous fractal in existence. It is named after Benoit Mandelbrot, the mathematician who gave birth to the now-booming field of fractal mathematics with his 1977 book The Fractal Geometry of Nature (among other works). The Mandelbrot Set has for many become symbolic of fractals in general. Images of the Mandelbrot Set abound, so these images are not the most unusual in this gallery, but they are still beautiful.

The Mandelbrot Set is defined by the iteration of a formula applied to all of the points on the complex plane. For any point (x, y) in a given image, that point is represented by a complex number c = a + bi. The formula z ← z² + c is iterated (with an initial value for z of 0) to see what happens to z. If z grows larger and larger without bound (converging on infinity), then it is not part of the Mandelbrot Set, and is colored according to how quickly it converges towards infinity. If z stays bounded over some given iteration limit, then it is considered to be part of the Mandelbrot Set, and is colored with a chosen "set" color (typically black). These images are composed almost entirely of points that are not part of the set itself, but that converge towards infinity at different speeds, yielding very colorful panoramas.

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.