Dihedral symmetry in three dimensions

Updated on Dec 24, 2023

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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 abstract group is a dihedral group Dih_n ( n ≥ 2 ).

Types

There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notation: Schönflies notation, Coxeter notation, and orbifold notation.

Chiral

D_n, [n,2]⁺, (22n) of order 2n – dihedral symmetry or para-n-gonal group (abstract group Dih_n)

Achiral

D_nh, [n,2], (*22n) of order 4n – prismatic symmetry or full ortho-n-gonal group (abstract group Dih_n × Z₂)

D_nd (or D_nv), [2n,2⁺], (2*n) of order 4n – antiprismatic symmetry or full gyro-n-gonal group (abstract group Dih_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, 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 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 [n], (*22n).

D_nd (or D_nv), [2n,2⁺], (2*n) 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 [2,2]⁺, (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, [2,2], (*222) of order 8 is the symmetry group of a cuboid

D_2d, [4,2⁺], (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, [n,2], (*22n), order 4n

C_nh, [n⁺,2], (n*), order 2n

C_nv, [n,1], (*nn), order 2n

D_n, [n,2]⁺, (22n), order 2n

For D_nd, [2n,2⁺], (2*n), order 4n

S_2n, [2n⁺,2⁺], (n×), order 2n

C_nv, [n⁺,2], (n*), order 2n

D_n, [n,2]⁺, (22n), order 2n

D_nd is also subgroup of D_2nh.

Examples

D_nh, [n], (*22n):

D_5h, [5], (*225):

D_4d, [8,2⁺], (2*4):

D_5d, [10,2⁺], (2*5):

D_17d, [34,2⁺], (2*17):

References

Dihedral symmetry in three dimensions Wikipedia

(Text) CC BY-SA

Contents

Types

Subgroups

Examples

References