Iterated Function System (IFS) by similarity transformations

We briefly explain how a randomized algorithm can produce a fractal pattern.

Start with two similarity transformations

z -> f1(z) and z-> f2(z).

The algorithm proceeds as follows: Start with an arbitrary point z0 in the plane. Chose one of the transformations f1(z) or f2(z) with a 50:50 probability and apply it to the point. Draw the resulting point and proceed with this new point itertively many times.

The whole process can also be applied with many transformations. If you write a computer program for it, it looks approximately as follows:

       z=startpoint;
       n=number of iterations;
       repeat n-times: (
            f=a random transformation from f_1, f_2,... f_k;
            z=f(z);
            draw(z);
       )

It is a good practise not to draw the first 100 points of this process since they may not be too near to the desired fractal.


In the applet below you can watch the process. The numer of iterations is adjustable by the slider. The fractal pattern emerges with a huge number of iterations.

Bitte schalten Sie Java ein, um eine Cinderella-Konstruktion zu sehen.


Even simple transformations create interesting patterns. We consider the transformation

fp(z):=(z+p)/2

which sets z to the average of z and p The following example shows the iterated function system generated by three of these mappings. fa(z), fb(z), fc(z). The so called Sierpinski triangle emerges.

Bitte schalten Sie Java ein, um eine Cinderella-Konstruktion zu sehen.