Iterated geometric transformations

If one applies a transformation to the result of the same transformation, and then again the same transformation and so on …, the result is an iterated transformation. In the following applet you can observe the behaviour of Dr. Stickler under iterated application of the same transformation.

For example, with a spiral similarity, the first application results in a slightly contracted and rotated copy. Applying the transformation to that image, it will be rotated and contracted yet again. Step by step the images will align along a logarithmic spiral (as in the case of the powers of a complex number, as a spiral similarity corresponds to a complex multiplication).

In the applet the function can again be chosen arbitrarily (with free parameters

and

), and the position of Dr. Stickler can be modified.

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

Enter function:

Here the basic symmetries:

Translation	Spiral similarity	Reflection	Inversion

And here again some additional interesting examples:

Rotation by 90°
Rotation by 72°
Expansion by a factor of 2

Generic spiral similarity
Spiral similarity with center d
Reflected inversion

Download file:

KomplexMaps7En.cdy

Geometric transformations as complex functions $\hookleftarrow$ Contents $\hookrightarrow$ Möbius transformations