In the following sequence of applets we 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 many transformations to interact with each other? Depending on the type and direction of the tranformations we obtain various -- often beautiful -- patterns.
Two translations:
Let's begin with the simplest example. A second translation which 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 movie Star Wars was generated in a similar way, namely, starting with one "Clone" and producing many copies of it by translation).
Two reflections:
If we iterate two reflections the situation appears quite differently. Let us first look at reflections in parallel mirrors:
Dr. Stickler sees himself in the mirror infront of him. He also sees there the mirror image of the mirror behind him, its reflection, and so forth. As usual, mirrors and points can be dragged by the mouse.