# Dihedral symmetry in three dimensions

Selected point groups in three dimensions :
| Involutional symmetry C$_s$, (*) [ ] = | Cyclic symmetry C$_{nv}$, (*nn) [n] = | Dihedral symmetry D$_{nh}$, (*n22) [n,2] = |
| Polyhedral group, [n,3], (*n32) |||
| Tetrahedral symmetry T$_d$, (*332) [3,3] = | Octahedral symmetry O$_h$, (*432) [4,3] = | Icosahedral symmetry I$_h$, (*532) [5,3] = |
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih$_n$ (for n ≥ 2).

## Types

There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation.

### Chiral

* D$_n$, [n,2]$^+$, (22n) of order 2n – dihedral symmetry or para-n-gonal group (abstract group: Dih$_n$).

### Achiral

* D$_{nh}$, [n,2], (*22n) of order 4n – prismatic symmetry or full ortho-n-gonal group (abstract group: Dih$_n$ × Z$_2$).
* D$_{nd}$ (or D$_{nv}$), [2n,2$^+$], (2*n) of order 4n – antiprismatic symmetry or full gyro-n-gonal group (abstract group: Dih$_{2n}$).
For a given n, all three have n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and 2-fold rotational symmetry about a perpendicular axis, hence about n of those. For n = ∞, they correspond to three Frieze groups. Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation.
In 2D, the symmetry group D$_n$ includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection through a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group D$_n$ contains rotations only, not reflections. The other group is pyramidal symmetry C$_{nv}$ of the same order, 2n.
With reflection symmetry in a plane perpendicular to the n-fold rotation axis, we have D$_{nh}$, [n], (*22n).
D$_{nd}$ (or D$_{nv}$), [2n,2$^+$], (2*n) has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2n-fold rotoreflection axis.
D$_{nh}$ is the symmetry group for a regular n-sided prism and also for a regular n-sided bipyramid. D$_{nd}$ is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. D$_n$ is the symmetry group of a partially rotated prism.
n = 1 is not included because the three symmetries are equal to other ones:
* D$_1$ and C$_2$: group of order 2 with a single 180° rotation.
* D$_{1h}$ and C$_{2v}$: group of order 4 with a reflection in a plane and a 180° rotation about a line in that plane.
* D$_{1d}$ and C$_{2h}$: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.
For n = 2 there is not one main axis and two additional axes, but there are three equivalent ones.
* D$_2$, [2,2]$^+$, (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation.
* D$_{2h}$, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
* D$_{2d}$, [4,2$^+$], (2*2) of order 8 is the symmetry group of e.g.:
** A square cuboid with a diagonal drawn on one square face, and a perpendicular diagonal on the other one.
** A regular tetrahedron scaled in the direction of a line connecting the midpoints of two opposite edges (D$_{2d}$ is a subgroup of T$_d$; by scaling, we reduce the symmetry).

## Subgroups

| D$_{2h}$, [2,2], (*222) | D$_{4h}$, [4,2], (*224) |
For D$_{nh}$, [n,2], (*22n), order 4n
* C$_{nh}$, [n$^+$,2], (n*), order 2n
* C$_{nv}$, [n,1], (*nn), order 2n
* D$_n$, [n,2]$^+$, (22n), order 2n
For D$_{nd}$, [2n,2$^+$], (2*n), order 4n
* S$_{2n}$, [2n$^+$,2$^+$], (n×), order 2n
* C$_{nv}$, [n$^+$,2], (n*), order 2n
* D$_n$, [n,2]$^+$, (22n), order 2n
D$_{nd}$ is also subgroup of D$_{2nh}$.

## Examples

| D$_{2h}$, [2,2], (*222) Order 8 | D$_{2d}$, [4,2$^+$], (2*2) Order 8 | D$_{3h}$, [3,2], (*223) Order 12 |
| basketball seam paths | baseball seam paths (ignoring directionality of seam) | Beach ball (ignoring colors) |
D$_{nh}$, [2,n], (*22n):
* prisms
D$_{5h}$, [2,5], (*225):
| Pentagrammic prism | Pentagrammic antiprism |
D$_{4d}$, [8,2$^+$], (2*4):
* Snub square antiprism
D$_{5d}$, [10,2$^+$], (2*5):
| Pentagonal antiprism | Pentagrammic crossed-antiprism | pentagonal trapezohedron |
D$_{17d}$, [34,2$^+$], (2*17):
* Heptadecagonal antiprism