A fractal is a geometrical structure that has a fragmented shape at all scales of measurement, from the most marco scale to the most micro scale. All mathematical properties of the structure (i.e.: perimeter of curves) behave as if the dimensions of the structure (read: fractal dimensions) are greater then the spatial dimensions. I am sure Wikipedia has more specific information then that in one of its articles, but that is the basic idea of a fractal.
For the past couple years, I have played with manipulating/warping fractals into patterns. Here are a couple of my favorites.
Sadly, WordPress is unable to upload (and most browsers are unable to show) a raw fractal. Neave Interactive has a fantastic demo using the same “Apophysis” algorithm. The Fractal is a raw one, without effects or changes like the ones in the gallery above.
Fractal by Derek Caswell is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.