Discrete Groups
There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.
| Group | Intl | Orbifold | Coxeter | Order | Description |
|---|---|---|---|---|---|
| Cn | n | nn | + | n | Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n. |
| Dn | nm | *nn | 2n | Dihedral: n-fold reflections. Abstract group Dihn, the dihedral group. |
Intl refers to Hermann-Mauguin notation or international notation, often used in crystallography. In the infinite limit, these groups become the one-dimensional line groups.
If a group is a symmetry of a two-dimensional lattice or grid, then the crystallographic restriction theorem restricts the value of n to 1, 2, 3, 4, and 6 for both families. There are thus 10 two-dimensional crystallographic point groups:
C1, C2, C3, C4, C6, D1, D2, D3, D4, D6
The groups may be constructed as follows:
- Cn. Generated by an element also called Cn, which corresponds to a rotation by angle 2π/n. Its elements are E (the identity), Cn, Cn2, ..., Cnn-1, corresponding to rotation angles 0, 2π/n, 4π/n, ..., 2(n-1)π/n.
- Dn. Generated by element Cn and reflection σ. Its elements are the elements of group Cn, with elements σ, Cnσ, Cn2σ, ..., Cnn-1σ added. These additional ones correspond to reflections across lines with orientation angles 0, π/n, 2π/n, ..., (n-1)π/n. Dn is thus a semidirect product of Cn and the group (E,σ).
All of these groups have distinct abstract groups, except for C2 and D1, which share abstract group Z2. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are: D1 ~ Z2 and D2 ~ Z2×Z2. In fact, D3 is the smallest nonabelian group.
For n even, the Hermann-Mauguin symbol nm is an abbreviation for the full symbol nmm, as explained below. The n in the H-M symbol denotes n-fold rotations, while the m denotes reflection or mirror planes.
| Parity of n | Full Intl | Reflection lines for regular polygon |
|---|---|---|
| Even n | nmm | vertex to vertex, edge center to edge center (2 families, 2 m's) |
| Odd n | nm | vertex to edge center (1 family, 1 m) |
Read more about this topic: Point Groups In Two Dimensions
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