The Seven Infinite Series of Axial Groups
The infinite series of axial or prismatic groups have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry, see cyclic symmetries, and three with additional axes of 2-fold symmetry, see dihedral symmetry. They can be understood as point groups in two dimensions extended with an axial coordinate and reflections in it.
They are related to the frieze groups; they can be interpreted as frieze-group patterns repeated n times around a cylinder. This table lists several notations for point groups: Hermann–Mauguin notation, Schönflies notation, orbifold notation, and Coxeter notation. The latter two are not only conveniently related to its properties, but also to the order of the group, see below. It is a unified notation, also applicable for wallpaper groups and frieze groups. The crystallographic groups have n restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer.
The series are:
| Hermann–Mauguin | Schönflies | Orbifold | Coxeter | Frieze | Order | Abstract group | Comments | |
|---|---|---|---|---|---|---|---|---|
| Even n | Odd n | |||||||
| n | Cn | nn | + | p1 | n | Zn | n-fold rotational symmetry | |
| 2n | n | S2n | nx | p11g | 2n | Z2n | Not to be confused with the symmetric groups | |
| n/m | 2n | Cnh | n* | p11m | 2n | Zn × Z2 | ||
| nmm | nm | Cnv | *nn | p1m1 | 2n | Dihn | Pyramidal symmetry; in biology, biradial symmetry | |
| n22 | n2 | Dn | 22n | + | p211 | 2n | Dihn | Dihedral symmetry |
| 2n2m | nm | Dnd, Dnv | 2*n | p2mg | 4n | Dih2n | Antiprismatic symmetry | |
| n/mmm | 2n2m | Dnh | *22n | p2mm | 4n | Dihn × Z2 | Prismatic symmetry | |
For odd n we have Z2n = Zn × Z2 and Dih2n = Dihn × Z2.
The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).
The simplest nontrivial ones have Involutional symmetry (abstract group Z2 ):
- Ci – inversion symmetry
- C2 – 2-fold rotational symmetry
- Cs – reflection symmetry, also called bilateral symmetry.
The second of these is the first of the uniaxial groups (cyclic groups) Cn of order n (also applicable in 2D), which are generated by a single rotation of angle 360°/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group Cnh of order 2n, or a set of n mirror planes containing the axis, giving the group Cnv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cn or Dn is a propeller.
If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4n is called Dnh. Its subgroup of rotations is the dihedral group Dn of order 2n, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections.
There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/n. Dnh is the symmetry group for a regular (n+2)-sided prisms and also for a regular (2n)-sided bipyramid. Dnd is the symmetry group for a regular (n+2)-sided antiprism, and also for a regular (2n)-sided trapezohedron. Dn is the symmetry group of a partially rotated prism.
Sn is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For n odd this is equal to the group generated by the two separately, Cnh of order 2n, and therefore the notation Sn is not needed; however, for n even it is distinct, and of order n. Like Dnd it contains a number of improper rotations without containing the corresponding rotations.
All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
- C1h and C1v: group of order 2 with a single reflection (Cs )
- D1 and C2: group of order 2 with a single 180° rotation
- D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
- D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane
S2 is the group of order 2 with a single inversion (Ci )
"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. S2n is algebraically isomorphic with Z2n.
The groups may be constructed as follows:
- Cn. Generated by an element also called Cn, which corresponds to a rotation by angle 2π/n around the axis. Its elements are E (the identity), Cn, Cn2, ..., Cnn−1, corresponding to rotation angles 0, 2π/n, 4π/n, ..., 2(n − 1)π/n.
- S2n. Generated by elements C2nσh, where σh is a reflection in the direction of the axis. Its elements are the elements of Cn with C2nσh, C2n3σh, ..., C2n2n−1σh added.
- Cnh. Generated by element Cn and reflection σh. Its elements are the elements of group Cn, with elements σh, Cnσh, Cn2σh, ..., Cnn−1σh added.
- Cnv. Generated by element Cn and reflection σv in a direction in the plane perpendicular to the axis. Its elements are the elements of group Cn, with elements σv, Cnσv, Cn2σv, ..., Cnn−1σv added.
- Dn. Generated by element Cn and 180° rotation U = σhσv around a direction in the plane perpendicular to the axis. Its elements are the elements of group Cn, with elements U, CnU, Cn2U, ..., Cnn − 1U added.
- Dnd. Generated by elements C2nσh and σv. Its elements are the elements of group Cn and the additional elements of S2n and Cnv, with elements C2nσhσv, C2n3σhσv, ..., C2n2n − 1σhσv added.
- Dnh. Generated by elements Cn, σh, and σv. Its elements are the elements of group Cn and the additional elements of Cnh, Cnv, and Dn.
Taking n to ∞ yields groups with continuous axial rotations:
| H–M | Schönflies | Orbifold | Coxeter | Limit of | Abstract group |
|---|---|---|---|---|---|
| ∞ | C∞ | ∞∞ | + | Cn | SO(2) |
| ∞, ∞/m | C∞h | ∞* | Cnh, S2n | SO(2) × Z2 | |
| ∞m | C∞v | *∞∞ | Cnv | O(2) | |
| ∞2 | D∞ | 22∞ | + | Dn | O(2) |
| ∞m, ∞/mm | D∞h | *22∞ | Dnh, Dnd | O(2) × Z2 |
Read more about this topic: Point Groups In Three Dimensions
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