Iterated similarity transformations

So far we only composed transformations that didn't change the sizes of objects. Spiral similarities and circle incersions explicitely aren't part of wallpaper groups. There is a good reason for this: if the transformations don't preserve the sizes of objects, it is hard to align the transformations in such a way that the resulting structures aren't chaotic.

If chaos can be prevented, through, some very aesthetic structures can emerge. The following applet shows the result of composing two rotating contractions in all conceivable ways. The transformations are defined by the positions of the three points

. The first transformation maps the segment $\overline{AB}$ onto the segment $\overline{AC}$ . The second transformation maps $\overline{AB}$ onto $\overline{CB}$ (both without changing orientation).

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

In the applet the white slider controls the number of chained operations. The positions of the points

and the position of Dr. Stickler can be adjusted as well, of course.

To better understand how the different images are generated, the following applet annotates each image of Dr. Stickler woth the sequence of transformations leading to that image. For example, iterating transformation

will cause the images to spiral towards point

. Iterating transformation

will spiral towards point

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

One can also observe that - depending on iteratoon depth - the copies of Dr. Stickler become smaller and smaller. In fact the position of deeply iterated copies is only insignificantly dependent on the position of the original Dr. Stickler (tr it!). The copies are drwan to a certain set of points, which only depends on the transformations used. This set is called the limit set. The study of limit sets for iterated Möbius transformations is one of the main topics of Indra's Pearls.

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Wallpaper groups $\hookleftarrow$ Contents