Symmetry group

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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 Cn and Dn. The former is generated by a rotation by 2π/n radians about a particular point, and the latter by such a rotation together with a reflection about a line that runs through that point.

Examples (text really limits my options):

    ***           ***            ***        *
      **          * *            *         ***
      *           * *          ***          *
Asymmetric     Bilaterally      C2          D4
                symmetric

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

Three dimensions

Will come later...