Symmetry group

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 possibly glide reflections. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.

If a figure is bounded, then all elements of its symmetry group have at least one common fixed point. Thus the symmetry group of a bounded figure can be represented as a subgroup of O(n) (by choosing the origin to be a fixed point).

Two dimensions

The discrete point groups in 2 dimensional space consist of two infinite families

• cyclic groups C₁, C₂, C₃, ... where C_n consists of all rotations about a fixed point by multiples of the angle 2πぱい/n
• dihedral groups D₁, D₂, D₃, ... where D_n consists of the rotations in C_n together with reflections in n axes that pass through the fixed point.

In the degenerate case n=1 we find that C₁ is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, and D₁ is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry. The dihedral groups D₃, D₄, ... are the symmetry groups of the regular polygons.

Providing the figure is bounded and topologically closed (so that the group is a complete point group) the only other possibility is the group SO(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. The closure condition here is a natural one for subsets of the plane that can be considered "figures", as it excludes non-drawable sets such as the set of all points on the unit circle with rational coordinates. The symmetry group of this set includes some, but not all, arbitrarily small rotations.

For non-bounded figures, the symmetry group can include translations, so that the seventeen wallpaper groups and seven frieze groups are possibilities.

Examples

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

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 (or C_1h), C_i, and C₂. These have the single symmetry operation of reflection in a plane, in a point, and in a line (equivalent to a rotation of πぱい), respectively.

The last of these is the first of the uniaxial groups C_n, which are generated by a single rotation of angle 2πぱい/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 horizontal and vertical reflection planes are added, their intersections give n axes of rotation through πぱい, so the group is no longer uniaxial. This new group is called D_nh. Its subgroup of rotations called D_n still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. There is one more group in this family, called D_nd (or D_nv), which has vertical mirror planes containing the main rotation axis but located halfway between the other 2-fold axes, so the perpendicular plane is not there. D_nh and D_nd are the symmetry groups for regular prisms and antiprisms, respectively. D_n is the symmetry group of a partially rotated prism.

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

Of these 7 families of groups described above, it has been noted that there are some coincidences for small values of n. In particular

• C_s, C_1h and C_1v are all groups of order 2 with a single reflection
• C₂ and D₁ are groups of order 2 with a single 180° rotation
• C_i and S₂ are groups of order 2 with a single inversion
• D_1h and C_2v are groups of order 4 with a reflection in a plane πぱい and a 180° rotation through a line in πぱい
• D_1d and C_2h are groups of order 4 with a reflection in a plane πぱい and a 180° rotation through a line perpendicular to πぱい

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ぱい/n and S_n to denote an axis of improper rotation through the same, the groups are:

• T (tetrahedral). 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. This group is isomorphic to A₄, the alternating group on 4 letters.

• 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_d is isomorphic to S₄, the symmetric group on 4 letters.

• 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, group has order 24. T_h is isomorphic to A₄ × C₂.

• O (octahedral). 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. This group of order 24 is also isomorphic to S₄.

• 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, is isomorphic to S₄ × C₂, and is the symmetry group of the cube and octahedron.

• I, I_h (icosahedral) are the groups of symmetries of the icosahedron and the dodecahedron. The group of proper rotations, I, is a normal subgroup of index 2 in the full group of symmetries, with I having order 60 and I_h having order 120. The group I is isomorphic to A₅, the alternating group on 5 letters, and I_h to A₅ × C₂....

Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme.

Related topics

• symmetry
• group action
• crystallographic group
  • crystallographic point group
  • space group