Jan Poland  →  Math fun  →  Fractals

Fractals are beautiful...

It is not my purpose to provide anything comprehensive on fractals. I rather feature one particular, very basic type of fractal. It is related to both Iterated Function Systems and Lindenmayer Systems. The Koch curve is the most famous example in this class.

You can obtain such a fractal by the following procedure: Start with a base figure which consists of lines, e.g. a single line. Fix a "replacing figure" which consists of more than one line. Now replace each line in your base figure by the replacing figure. The result is the first step figure. In the second step, replace each line of the first step figure by the replacing figure. You may do arbitrary many steps of this construction, or even infinitely many. Since each step applies the same transformations to the previous result, we call this procedure recursive.

You can get a feeling for and play with this construction, there is a Java applet on the next page. It allows you to construct many fractals by choosing a base figure and a replacing figure.

This is not the only possibility to generate fractals of this type. You can always state an iterative method. The dragon curve you see here at the right has a particularly elegant iterative algorithm - can you figure it out? And can you find the replacing figure for the recursive construction? By the way: The dragon curve is space-filling and therefore has fractal dimension two. (Note: If there is no dragon curve, then probably the Java of your web browser is not working properly.)

Another way of obtaining the fractals is a Lindenmayer system or L-system. There, you do a similar recursive replacement with strings instead of lines. After computing a number of steps, the resulting string (which is probably quite long) may be graphically displayed. This link contains a very nice demonstration in Mathematica, which I found some time ago in the web. Unfortunately I lost the reference, and apparently the author does not maintain his page any more.

Yet another way to compute the fractals is an Iterated Function System (IFS). Here the replacement is represented as a mapping on the space of all compact (i.e. bounded and closed) subsets of the Euclidean space. If this mapping is a contraction (with respect to an arbitrary metric), repeated application to any starting figure converges. For fractals of the above type, the mapping is piecewise linear.

In his book Fractals Everywhere, Barnsley gives an elementary proof that an IFS always converges to an attractor. Using slightly more advanced math, it is possible to give a shorter proof.

Last update: Fri Mar 9 05:09:33 2007 GMT by Jan Poland, Page = "Fractals"