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
Famous quotes containing the word dimensions:
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)