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.
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.