Dihedral symmetry in three dimensions

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.

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, ⁺, of order 2n – dihedral symmetry or para-n-gonal group.
;Achiral:D_nh,, of order 4n – prismatic symmetry or full ortho-n-gonal group.D_nd,, of order 4n – antiprismatic symmetry or full gyro-n-gonal group.
For a given n, all three have n-fold rotational symmetry about one axis, 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 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,,.
D_nd,, 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 2n-sided bipyramid. D_nd is the symmetry group for a regular n-gonal antiprism, and also for a regular n-gonal trapezohedron. D_n is the symmetry group of an n-gonal twisted prism and n-gonal twisted trapezohedron.
n = 1 is not included because the three symmetries are equal to other ones:D₁ and C₂: 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.D₂, ⁺, 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,, of order 8 is the symmetry group of a cuboid.D_2d,, 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.

Subgroups

For D_nh,,, order 4nC_nh,,, order 2nC_nv,,, order 2nD_n, ⁺,, order 2n
For D_nd,,, order 4nS_2n,,, order 2nC_nv,,, order 2nD_n, ⁺,, order 2n
D_nd is also subgroup of D_2nh.

Examples

D_nh,, :

Prisms

D_5h,, :

Pentagrammic prism

Pentagrammic antiprism

D₄d'',, :

Snub square antiprism

D₅d'',, :

Pentagonal antiprism

Pentagrammic crossed-antiprism

Pentagonal trapezohedron

D_17d,, :

Heptadecagonal antiprism