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 SO(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.
Two dimensions
The discrete point groups in 2 dimensional space consist of two infinite families
- cyclic groups C1, C2, C3, ... where Cn consists of all rotations about a fixed point by multiples of the angle 2
π /n - dihedral groups D1, D2, D3, ... where Dn consists of the rotations in Cn 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 C1 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 Cs (or C1h), Ci, 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
The last of these is the first of the uniaxial groups Cn, which are generated by a single rotation of angle 2
If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through
There is one more group in this family to mention, called Sn. This group is generated by an improper rotation of angle 2
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 2
- 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.
- 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. Td is isomorphic to S4, the symmetric group on 4 letters.
- 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, group has order 24. Th 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.
- Oh. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group has order 48, is isomorphic to S4 × C2, and is the symmetry group of the cube and octahedron.
- I, Ih (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 Ih having order 120. The group I is isomorphic to A5, the alternating group on 5 letters, and Ih 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.