Lorenz Attractor

The Lorenz Attractor is an excellent and common example of attractors. In fractals, an attractor is an area that pulls points towards itself, much like the gravity around a star. Those points will remain chaotic and change orbits unpredictably. However, regardless of how chaotic the points become, they will always remain in an orbit around the attractor.
Edward Lorenz discovered this attractor during his research into weather patterns. He did this during the 1960's, before chaos' mainstream acceptance, when the accurate prediction of weather was thought to be easy. While attempting to simulate weather trends in a three dimensional computer model, Lorenz accidentally created this fascinating attractor.
The Lorenz attractor is created by iterating equations, just like many other fractals. It is somewhat unique, though, in that it uses three equations, X, Y, and Z, to make a three dimensional image, not the two dimensional ones that are more common. The three equations used to create this fractal are :

x_n = x_n-1 + d * a * (y_n-1 - x_n-1)
y_n = y_n-1 + d * (x_n-1 * (c - z_n-1) - y_n-1)
z_n = z_n-1 + d * (x_n-1 * y_n-1 - b * z_n-1)

You may have noticed four constants in the above equations. A, B, C, and D all have specific values, as determined by Lorenz, to produce this attractor. The values are as follows.

A	B	C	D
10	8/3	28	0.003

Of course, these four constants may be altered to produce different results. The image to the right was created by altering each value a small amount. As you can see, the resultant image follows the same shape and form as the Lorenz attractor, but there are noticeable differences in the actual location of each point. The general form remains; the details change. This behavior is present in most fractals - attractors included.
The Turbo Pascal program used to generate these images is available for download. The ZIP file includes the source code, and a DOS compiled executable file. You'll need a PC with at least 1 MB of video RAM to run this program.