Strange

Mathematicians will (I hope) be surprised and delighted to find that sets thus far reputed exceptional should in a sense be the rule, that constructions deemed pathological should evolve naturally from very concrete problems, and that the study of Nature should help solve old problems and yield so many new ones. (Mandelbrot, 1977)

The screenshot below shows a gallery of Clifford Attractors. Each image is formed by beginning with an (x, y) point, and repeatedly applying the following transformation to it:

new x = sin(ay) + c cos(ax)
new y = sin(bx) + d cos(by)

Each set of coefficients {a, b, c, d} produces a different image. This program tries to automatically search for "interesting" attractors - those that generate images that are not too sparse - and displays them in a gallery. In the program, you can click on any image in the gallery to see a larger version of it.

As the transformation is applied over and over again, the point traces out a path, curving first in one direction and then another, returning over and over again to each region of the image. By plotting each point, an image of the attractor is generated. The points are colored based on the "velocity" of the point - the hue represents what direction the point moved in when the transformation was applied, and the saturation of each pixel represents the magnitude of the velocity. The brightness of each pixel is determined based on how many points were plotted in that pixel.

To see more examples of the surprising variety of images produced by this simple equation, see the screenshots page, or download the program and try it out for yourself.

If you are interested in fractals and chaos, I would suggest checking out Paul Bourke's website, which is what inspired me to write Strange. You may also enjoy browsing electricsheep.org.

Strange is free, and open source.