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In the following sequence of applets we will investigate what happens if different geometric transformations interact.
Translational Symmetry:
The applet above shows the result of the simplest situation, an iterated application of a translation in direction of the horizontal axis.
What happens if we allow for many transformations that interact with each other? Depending on the type and direction of the tranformations we obtain very different (and often beautiful) looking patterns.
Two translations:
Let's begin with the simplest example. A second translation that points to another independent direction:
The entire plane is filled with copies of Dr. Stickler. It looks a bit like "Attack of the Clones" (Indeed the corresponding animation in the Star-Wars movie was generated in a similar way, start with one "Clone" and produce many copies of it by translation).
Two reflections:
If we iterate two reflections the situation looks quite different. Let us first look at reflections in parallel mirrors:
Dr. Stickler sees himself in the mirror infront of him. But there he also sees the mirror image of the mirror behind him. And the reflected reflected image ... and so forth. As usual mirrors and points can be dragged by the mouse.