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Isometries: Reflections on Reflections
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The four basic types of isometries of the plane (sometimes called rigid motions because they do not distort shapes) are translation, rotation, reflection and glide reflection. A little experimentation moving piece of paper around a table top should lead you to strongly suspect that these are the only kinds of isometries.

However, it frequently takes a minute to see how you can accomplish some sequence of transformations in one single operation from our list. In fact, it can easily be so obscure that we are left with a nagging doubt that there might some some other exotic isometries that only arise as a complicated sequence of transformations.

What happens when you do one isometry followed by another? Obviously the result is another isometry, since neither transformation introduces any distortion. The question is whether or not it is already on our list. To make things more concrete, consider reflections.

If we do one reflection followed by another, notice the result will have to be orientation-preserving; doing the first reflection will produce a mirror image, but the second reflection will do a mirror image of a mirror image -- that is, we will return to our original orientation. That suggests a product of reflections should be either a translation or a rotation.

In fact, this is exactly what happens. Which type you get depends on whether or not the mirror lines of the two relfections cross. You should be able to figure out which is which just by noticing that if the mirror lines cross, the intersection point will be a fixed point of the product, since neither reflection will move this point.

This leads us to a fascinating conclusion. Since we can emulate the effect of translation by doing two reflections, you should be able to see how we can also emulate the effect of a glide reflection by doing three. As a result, we can really just focus on the behavior of reflections, since the other three types of isometry can be built up from reflections.

Next: "Closing the Ring"

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