Fractals - Advanced Topics

A Julia set is the maximal set of points
that gets mapped onto itself under the
function Z' = Z² + C. Julia Sets are
produced with the same formula as the
Mandelbrot set, but the starting values are
different. There is an infinite number of
Julia Sets; one for each value of C.

Robert Devaney at Boston University explains elementary algorithms and mathematics behind some fractals.

Although some of this material is rather elementary, there is a broad collection of fractal demonstrations and papers gathered together in this site.

An object with Brownian motion traces a path that is very similar to a fractal. This site describes how a path created by Brownian motion and a coastline are similar. A much deeper paper on Brownian motion and fractals is available here. Note that the dimension for the Brownian planar frontier is not an integer!

Considerably more challenging is The Mandelbrot and Julia Sets Anatomy.