# 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} (for *n* ≥ 2).

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 *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*contains rotations only, not reflections. The other group is pyramidal symmetry

_{n}*C*of the same order, 2

_{nv}*n*.

With reflection symmetry in a plane perpendicular to the *n*-fold rotation axis, we have *D _{nh}*, [n], (*22

*n*).

*D _{nd}* (or

*D*), [2

_{nv}*n*,2

^{+}], (2*

*n*) has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2

*n*-fold rotoreflection axis.

*D _{nh}* is the symmetry group for a regular

*n*-sided prism and also for a regular n-sided bipyramid.

*D*is the symmetry group for a regular

_{nd}*n*-sided antiprism, and also for a regular n-sided trapezohedron.

*D*is the symmetry group of a partially rotated prism.

_{n}*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.