# Visualizing fractals with the Chaos Game

Many fractals may be described as the fixed set of an iterated function set (IFS). Perhaps most famously, the Sierpiński Triangle is such a fractal. Formally, an IFS is a set of maps on a metric space, eg. \mathbb{R}^n, which map points closer to each other.

Hutchinson proved in 1981 that an IFS has a unique compact fixed set S – a set where all points are mapped back into the set. Now, for some choices of IFS on the plane, the set S is very interesting and shows fractal properties. The Sierpiński Triangle is for example the fixed set of the following IFS:

(x,y) \mapsto \frac{1}{2}(x,y),\\ (x,y) \mapsto \frac{1}{2}(x-2,y), \\ (x,y) \mapsto \frac{1}{2}(x – 1, y – \sqrt{3})

A common way to visualise the fixed set of an IFS is by using the so-called Chaos game. Here, a point in the plane is picked at random. Then we apply one of the functions of the IFS, chosen at random, to the point. The result is another point in the plane which we again apply one of the function chosen at random on. At each step we plot the point, and we may continue for as long as we like and with as many initial points as we want.

Another possible fractal which may be constructed as the fixed set of an IFS is the Barnsley Fern. Here the functions are (with points written as column vectors):

\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0 & 0 \\ 0 & 0.16 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0.85 & 0.04 \\ -0.04 & 0.85 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0.20 & -0.26 \\ 0.23 & 0.22 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} -0.15 & 0.28 \\ 0.26 & 0.24 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}.

Here, the the probability to pick the first map should be 1%, the second should be 85% and the remaining two should be 7% each. This will yield the picture below: