### The Chaos Game

The chaos game uses a random number generator to draw a fractal. We start with an IFS that describes a fractal (some IFS are listed in the drop down menu - or if you want to input your own parameters, choose user in the drop down menu and then type them in the spaces provided). If the IFS has k transformations w1, ..., wk, then at each iteration one of the numbers 1, ..., k is chosen randomly. Starting with some point pi, if the number l is chosen, then the transformation wl is applied to p0 to produce the next point p1. Then another number is chosen randomly, call it m, and the transformation wm is applied to p1 to obtain the next point p2, etc. In the end (after several thousand points are drawn) the fractal appears.

The columns a,b,c,d,e,f are the numbers that define the parameters for each transformation iin the IFS, the column p is the probability assigned to each transformation. That is, if pi is the probability assigned to the ith transformation wi, then the number i is picked with probability pi at each iteration (note that this means that the sum of the probabilities must add to 1). At first one may just assign equal probabilites to each transformation, i.e., if there are k transformations in the IFS, then each of the k transformations get assigned the probability 1/k (so pi = 1/k for each i = 1, ..., k). This works well for some fractals like the Sierpinski triangle and the von Koch curve, but not for other fractals like the Barnsley fern. The latter IFS has transformations with greatly differing rates of contractions. It remains an open mathematical problem as to what are the optimal probabilities (so that the fractal is drawn most efficiently) for a general IFS with transformations with differing rates of contraction. However, a general rule of thumb is to assign a higher probability to those transformations that have the least rate of contraction.

At the bottom of the applet is the fractal dimension of the object. This is computed using the box counting method with 3 grid sizes (a least squares fit).