The role of probabilty

One major component of the randomized process for the generation of limit sets is the "random" choice of one transformation for each step. So far we haven't specified the frequencies with which the transformations should be chosen; but instead assumed a uniform distribution. If the algorithm were to run infinitely, this assumption wouldn't matter: sonner or later there would be a point drawn in the vicinity of any limit point.

In practice, however, things look different: The algorithm should in a short time - not an infinitely long time - generate as good an image of the limit set as possible. Depending on the distribution of frequencies, different parts of the resulting point cloud differ in prominence.

In the following applet this effect can be examined: An IFS is generated from two rotating contractions. The first one maps the black segment $\overline{AB}$ onto the blue segment $\overline{AC}$ . The second transformatuon maps $\overline{AB}$ onto the red segment $\overline{DE}$ . The slider influences the probability with which each transformation gets chosen. One can observe that only a special choice of probabolities leads to satisfying images. If the red transformation is used too often, the fractal becomes rather thin towards the center. Too much weight on the blue transformation will loose structure in the outskirts.

Please enable Java for an interactive construction (with Cinderella).

One should also try modifying the positions of the line segments in order to see the rich structure of generated fractals.

Download file:

IFSProbEn.cdy (This file requires Cinderella 2.1 which is currently available as a beta version at http://beta.cinderella.de/public )

How an IFS is created $\hookleftarrow$ Contents $\hookrightarrow$ IFS from two similarities