Two translations

So far we only examined what happens if an object (e.g. Dr. Stickler) is repeatedly transformed by one single motion or its inverse. his creates images like the following:

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

The applet demonstrates the situation of a translation in the direction of the real axis.

So what happens if, instead of one single transformation, we allow for two distinct transformations? Depending on our choice of transformations, this will generate very different (and often quite pretty) structures.

Let's start with the most simple example - two translations in two diferent directions:

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The whole plane is filled by copies of Dr. Stickler, and it looks a bit like "Attack of the Dr. Stickler Clones". (In fact the animations in the corresponding Star Wars movie, where huge numbers of identical clones march in rigid formation, have been generated in a very similar way.)

Iterating two reflections, the situation looks different. Let's first consider reflections in two parallel mirrors:

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

First of all, Dr. Stickler sees his own reflection in front of him. There he also sees the reflection of the mirror behind him, and the reflected reflected mirror ... and so on. (The mirrors and the feet of Dr. Stickler can be moved with the mouse as usual.)

Download files:

Contents $\hookrightarrow$ Dr. Stickler in the hall of mirrors