the three-reflections theorem

Here is a striking economy: out of all the ways to move the plane rigidly, you never need more than three flips. The three-reflections theorem states that every isometry of the plane can be written as a composition of at most three reflections. Reflections are the single fundamental move, and translations, rotations, and glide reflections are just reflections done in sequence. The mirror is the atom of motion.

Watch how the other motions are built from flips. Reflect across one line, then reflect across a second line parallel to it: the net effect is a translation, perpendicular to both mirrors, by twice the distance between them. Reflect across two intersecting lines instead: the net effect is a rotation about their crossing point, by twice the angle between the mirrors. So every direct isometry (translation or rotation) is two reflections. The opposite isometries need an odd number: a single reflection is one, and a glide reflection is three (or, more economically, can always be reduced to exactly one reflection plus the leftover, never needing a fourth). The count of reflections matches the orientation — an even number preserves orientation, an odd number reverses it.

This theorem is the structural backbone behind the classification of plane isometries: once you know reflections generate everything and that two parallel mirrors make a translation while two crossing mirrors make a rotation, the four-type list almost writes itself. It also gives a practical recipe — to realize a desired rotation by angle theta about a point, just pick any two lines through that point meeting at angle theta/2 and reflect across each in turn. The bound 'at most three' is sharp: a glide reflection genuinely needs three and cannot be done with fewer, while two suffice for any direct motion.