There is an introduction to symmetry groups on the Mathematical group page. Basically, the symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. In Euclidean geometry, discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections, and infinite lattice groups, which also include translations and glide reflections. There are also continuous symmetry groups - see Lie groups, perhaps.

**Two dimensions**

The two simplest point groups in 2-D space are the trivial group, where no symmetry operations leave the object unchanged, and the group containing only the identity and reflection about a particular line. The other point groups form two infinite series, called *C _{n}* and

Examples (text really limits my options):

*** *** *** * ** * * * *** * * * *** *

Asymmetric Bilaterally C_{2}D_{4}symmetric

There are seventeen 2-D lattice groups, called wallpaper groups.

**Three dimensions**

Will come later...