 Dihedral symmetry in three dimensions

This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dih_{n} ( n ≥ 2 ).
See also point groups in two dimensions.
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} × C_{2})
 D_{nd} (or D_{nv}) (2*n) of order 4n – antiprismatic symmetry (abstract group D_{2n})
For a given n, all three have nfold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and 2fold 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 nfold 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 2nfold rotoreflection axis.
D_{nh} is the symmetry group for a regular nsided prisms and also for a regular nsided bipyramid. D_{nd} is the symmetry group for a regular nsided antiprism, and also for a regular nsided 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 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.
 D_{2} (222) of order 4 is one of the three symmetry group types with the Klein fourgroup as abstract group. It has three perpendicular 2fold 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).
Subgroups
For D_{nh}
 C_{nh}
 C_{nv}
 D_{n}
For D_{nd}
 S_{2n}
 C_{nv}
 D_{n}
D_{nd} is also subgroup of D_{2nh}.
See also cyclic symmetries
Examples
D_{nh} (*22n):
prismsD_{5h} (*225):
Pentagrammic prism
Pentagrammic antiprismD_{4d} (2*4):
Snub square antiprismD_{5d} (2*5):
Pentagonal antiprism
Pentagrammic crossedantiprism
pentagonal trapezohedronD_{17d} (*22(17)):
Heptadecagonal antiprismSee also
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