Isometries are transformations the plane which don't distort
shapes. For this reason, another common name for an isometry is a
rigid motion. The most familiar isometry is probably just
translation -- shifting a shape in a straight line. However,
another very familiar kind of isometry is rotation:
Unlike translation, this isometry has an obvious fixed
point. The pivot point around which everything rotates doesn't
move at all. Using the analogy of a square drawn on a large
transparent sheet of plastic, you might do a rotation by holding down
the pivot point with your finger, and whirling the sheet around it.
Translations have no fixed points at all, while rotations have
exactly one -- namely, if you stay away from the pivot point,
everything else moves. This is worth pointing out, since it suggests
a way of classifying any kind of isometry, by considering their
fixed point behavior.