Symmetry group

The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept.

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, which are Lie groups.

Two dimensions

The two simplest point groups in 2-D space are the trivial group C₁, where no symmetry operations leave the object unchanged, and the group containing only the identity and reflection about a particular line, D₁. The other point groups form two infinite series, called C_n and D_n: the cyclic groups and the dihedral groups. 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):

     ***           ***            ***        *
       **          * *            *         ***
       *           * *          ***          *
      C1            D1            C2          D4

Groups including translation in a single direction are called frieze groups. There are seventeen 2-D lattice groups including translation in multiple directions, called wallpaper groups.

Three dimensions

The situation in 3-D is more complicated, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group, and then there are three groups of order 2, called C_s, C_i, and C₂. These have the single symmetry operation of reflection about a plane, about a point, and about a line (equivalent to a rotation of π), respectively.

The last of these is the first of the uniaxial groups C_n, which are generated by a single rotation of angle 2π/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group C_nh, or a set of n mirror planes containing the axis, giving the group C_nv. If both are added, there intersections give n axes of rotation through π, so the group is no longer uniaxial.

This new group is called D_nh, and its subgroup of rotations is D_n. There is one more group in this family, called D_nd, which has mirror planes containing the main rotation axis but located halfway between the others, so the perpendicular plane is not there. D_nh and D_nd are the symmetry groups for regular prisms and antiprisms, respectively.

There is one more group in this family to mention, called S_n. This group is generated by an improper rotation of angle 2π/n - that is, a rotation followed by a reflection about a plane perpendicular to its axis. For n even, the rotation and reflection are generated, so this becomes the same as C_nh, but it remains distinct for n odd.

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using C_n to denote an axis of rotation through 2π/n and S_n to denote an axis of improper rotation through the same, the groups are:

T. There are four C₃ axes, directed through the corners of a cube, and three C₂ axes, directed through the centers of its faces. There are no other symmetry operations, giving the group an order of 12.

T_d. This group has the same rotation axes as T, but with six mirror planes, each containing a single C₂ axis and four C₃ axes. The C₂ axes are now actually S₄ axes. This group has order 24, and is the symmetry group for a regular tetrahedron.

T_h. This group has the same rotation axes as T, but with mirror planes, each containing two C₂ axes and no C₃ axes. The C₃ axes become S₆ axes, and a center of inversion appears. Again, this group has order 24.

O. This group is similar to T, but the C₂ axes are now C₄ axes, and a new set of 12 C₂ axes appear, directed towards the edges of the original cube. Another group of order 24...

O_h. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group has order 48, and is the symmetry group of the cube and octahedron.

I, I_h.