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Isometries: Closing the Ring
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When we do two reflections, one followed by another, we have seen that the next effect is the same as either a translation or rotation, depending on whether or not their mirror lines cross. This observation allows us to see what happens when we compose any two isometries, since the net effect translations, rotations, and glide reflections can be produced by a sequence of reflections.

  

The idea is that if, for example, we compose a translation with a rotation, we first think of two reflections that do the translation, and then two reflections that do the rotation. Then, we can accomplish the whole transformation by doing four reflections.

  

Now for the tricky part: if we choose our reflections carefully, we can arrange for the middle two reflections to cancel out, and the net effect looks like just doing two. But we have already seen that two reflections produces either a translation or a rotation! Thus composing a translation and a rotation again gives a translation or rotation:

  

Similar things happen in all other cases, and we have now arrived at the end of the road. We can conclude that our tentative list of isometries is complete. Any isometry of the plane is a translation, rotation, reflection or glide reflection, and no matter how you compose them, you will always end up with another isometry on this list.

Next: "The Holy Grail"



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