Two Dimensions
Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
- Cyclic groups Cn of n-fold rotation groups
- Dihedral groups Dn of n-fold rotation and reflection groups
Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.
| 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: cyclic with reflections. Abstract group Dihn, the dihedral group. |
The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.
| Reflective | Rotational | Related polygons | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Group | Coxeter group | Coxeter diagram | Order | Subgroup | Coxeter | Order | |||
| D1 | A1 | 2 | C1 | + | 1 | Digon | |||
| D2 | A12 | 4 | C2 | + | 2 | Rectangle | |||
| D3 | A2 | 6 | C3 | + | 3 | Equilateral triangle | |||
| D4 | BC2 | 8 | C4 | + | 4 | Square | |||
| D5 | H2 | 10 | C5 | + | 5 | Regular pentagon | |||
| D6 | G2 | 12 | C6 | + | 6 | Regular hexagon | |||
| Dn | I2(n) | 2n | Cn | + | n | Regular polygon | |||
| D2×2 | A12×2 | ] = | = | 8 | |||||
| D3×2 | A2×2 | ] = | = | 12 | |||||
| D4×2 | BC2×2 | ] = | = | 16 | |||||
| D5×2 | H2×2 | ] = | = | 20 | |||||
| D6×2 | G2×2 | ] = | = | 24 | |||||
| Dn×2 | I2(n)×2 | ] = | = | 4n | |||||
Read more about this topic: Point Group
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