Most transformation that repeats itself after every two moves can either be a mirror image or a rotation by half. Every movement in a flat surface can be made by reflecting it at most three times. Every small group of transformations has at least one point that doesn't move. To be an isometry, a function needs to keep the distances between all points in both places the same. The distance from x to y is the same as the distance from f(x) to f(y).