I've long wanted to incorporate fractals into a ReBoot fanfic because of their complexity, beauty, and general neato-ness. They may look chaotic, yet they are intrinsically orderly, being based on simple mathematical functions. In Before the Dawn three different forms are mentioned specifically:
The Koch Snowflake is one of the earliest fractals "invented." The basic formula is to take a line, divide it into equal thirds, make the middle third the base of an equilateral triangle, and then remove that base line, connecting the new triangle with the original figure. Then repeat with the new, smaller lines. If you start with a triangle, on the second step you have what looks like a Star of David, and on successive iterations it gets more complicated and snowflake-like. This is what happened to the Unnamed System's Principle Office. Interestingly, Cyriak's The Bungelbrot Set applies this principle to, of all things, a teddy bear's head.
A Julia set. There is a tremendous amount of variation in this kind of fractal. I imagined something like this - thorny and dangerous-looking - when I was writing.
Ye classic Mandelbrot set. It may look only mildly interesting, but the more you zoom in on the complex stuff around the edges, the more fascinating stuff you can see - and, over and over, miniature copies of the original shapes appear within. As of this writing, Wikipedia's Mandelbrot page contains a gallery of beautiful images from a zoom sequence.