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.
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 Cn and Dn: 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.
*** *** *** * ** * * * *** * * * *** * C1 D1 C2 D4Groups 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.
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 Cs, Ci, 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 Cn, 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 Cnh, or a set of n mirror planes containing the axis, giving the group Cnv. 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 Dnh, and its subgroup of rotations is Dn. There is one more group in this family, called Dnd, which has mirror planes containing the main rotation axis but located halfway between the others, so the perpendicular plane is not there. Dnh and Dnd are the symmetry groups for regular prisms and antiprisms, respectively.
There is one more group in this family to mention, called Sn. 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 Cnh, 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 Cn to denote an axis of rotation through 2π/n and Sn 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.
Td. 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.
Th. 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...
I, Ih are the groups of symmetries and the group of rotations of the icosahedron (also of the dodecahedron and icosidodecahedron). The group of rotations is a normal subgroup of index 2 in the group of symmetries, with I having order 60 and Ih having order 120. The group of rotations is isomorphic to A5, the alternating group of order 5.