Dihedral symmetry in three dimensions

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This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dih_n ( n ≥ 2 ).

Chiral:

D_n (22n) of order 2n - dihedral symmetry (abstract group D_n)

Achiral:

D_nh (*22n) of order 4n - prismatic symmetry (abstract group D_n × C₂)
D_nd (or D_nv) (2*n) of order 4n - antiprismatic symmetry (abstract group D_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 about a perpendicular axis, hence about n of those. For n = ∞ they correspond to three frieze groups. Schönflies notation is used, and, in parentheses, Orbifold notation. 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 in 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.

With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have D_nh (*22n).

D_nd (or D_nv) 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 prisms 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:

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 through a line in that plane
D_1d and C_2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane

For n=2 there is not one main axes and two additional axes, but there are three equivalent ones.

D₂ (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 (*222) of order 8 is the symmetry group of a cuboid
D_2d (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).