There are four kinds of distance preserving symmetries of the plane:

**Translations:**- Move all points the same distance in the same direction.
**Rotations:**- by an angle about a fixed center of rotation.
**Reflections:**- in a fixed line (
*the mirror*). **Glide Reflections:**- reflection in a mirror followed by a translation parallel to that mirror line.

Applying first one symmetry *s* and then another *t* is called *composition*
of the symmetries. We'll be interested in collections of symmetries
which form a *group*. This means the composition of two symmetries
in the collection is again in the collection. And the *inverse* of
a symmetry in the collection is again in it. (The inverse of a symmetry
involves reversing the correspondence of points in the symmetry. For
example, the inverse of rotation counterclockwise is rotation clockwise
by the same angle. Or the inverse of translation left is translation
right by the same distance.)