# Dihedral symmetry in three dimensions

﻿
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 Dihn ( n ≥ 2 ).

Chiral:

• Dn (22n) of order 2ndihedral symmetry (abstract group Dn)

Achiral:

• Dnh (*22n) of order 4nprismatic symmetry (abstract group Dn × C2)
• Dnd (or Dnv) (2*n) of order 4nantiprismatic symmetry (abstract group D2n)

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 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 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 Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order.

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

Dnd (or Dnv) 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 prisms 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:

• D1 and C2: group of order 2 with a single 180° rotation
• D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
• D1d and C2h: 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.
• D2h (*222) of order 8 is the symmetry group of a cuboid
• D2d (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 (D2d is a subgroup of Td, by scaling we reduce the symmetry).

## Subgroups

For Dnh

• Cnh
• Cnv
• Dn

For Dnd

• S2n
• Cnv
• Dn

Dnd is also subgroup of D2nh.

## Examples

Dnh (*22n):

D5h (*225):

D4d (2*4):

D5d (2*5):

D17d (*22(17)):

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Dimensions - получить на Академике рабочий купон на скидку Paper Shop или выгодно dimensions купить с бесплатной доставкой на распродаже в Paper Shop

• Point groups in three dimensions — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… …   Wikipedia

• Dihedral — or polyhedral may refer to: Dihedral angle, the angle between two mathematical planes Dihedral (aircraft), the upward angle of a fixed wing aircraft s wings where they meet at the fuselage, dihedral effect of an aircraft, longitudinal dihedral… …   Wikipedia

• Dihedral (disambiguation) — Dihedral may refer to: *Dihedral the angle of a fixed wing aircraft s wings *Dihedral angle, is the angle between two planes, in geometry *Dihedral group, is the group of symmetries of the n sided polygon in abstract algebra **Also Dihedral… …   Wikipedia

• Dihedral group — This snowflake has the dihedral symmetry of a regular hexagon. In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.[1] Dihedr …   Wikipedia

• Symmetry group — Not to be confused with Symmetric group. This article is about the abstract algebraic structures. For other meanings, see Symmetry group (disambiguation). A tetrahedron can be placed in 12 distinct positions by rotation alone. These are… …   Wikipedia

• Symmetry combinations — This article discusses various symmetry combinations.In 2D, mirror image symmetry in combination with n fold rotational symmetry, with the center of rotational symmetry on the line of symmetry, implies mirror image symmetry with respect to lines… …   Wikipedia

• Molecular symmetry — in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule s chemical… …   Wikipedia

• Rotational symmetry — Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the …   Wikipedia

• List of spherical symmetry groups — List of symmetry groups on the sphere = Spherical symmetry groups are also called point groups in three dimensions. This article is about the finite ones.There are four fundamental symmetry classes which have triangular fundamental domains:… …   Wikipedia

• Point groups in two dimensions — In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. It is a subgroup of the orthogonal group O(2), the group of all isometries which… …   Wikipedia