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 Dihn (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. 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 Dn 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 Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, 2n. With reflection symmetry in a plane perpendicular to the n-fold rotation axis, we have Dnh, [n], (22n). Dnd (or Dnv), [2n,2+], (2n) has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2n-fold rotoreflection axis. Dnh is the symmetry group for a regular n-sided prism and also for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn is the symmetry group of a partially rotated prism. n = 1 is not included because the three symmetries are equal to other ones: For n = 2 there is not one main axis and two additional axes, but there are three equivalent ones.

Subgroups

For Dnh, [n,2], (22n), order 4n For Dnd, [2n,2+], (2n), order 4n Dnd is also subgroup of D2nh.

Examples

Dnh, [n], (22n): D5h, [5], (225): D4d, [8,2+], (24): D5d, [10,2+], (25): D17d, [34,2+], (2*17):