Dihedral symmetry in three dimensions

There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: 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 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_n contains rotations only, not reflections. The other group is pyramidal symmetry C_nv of the same order, 2n.

With reflection symmetry in 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 prism 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 about a line in that plane.
• D_1d and C_2h: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.

For n = 2 there is not one main axis 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).

{{See also|Cyclic symmetry in three dimensions}}

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.

D_nh, [2,n], (*22n):

D_5h, [2,5], (*225):

D_4d, [8,2⁺], (2*4):

D_5d, [10,2⁺], (2*5):

D_17d, [34,2⁺], (2*17):

• List of spherical symmetry groups
• Point groups in three dimensions
• Cyclic symmetry in three dimensions

• {{cite book | author=Coxeter, H. S. M. and Moser, W. O. J. | title=Generators and Relations for Discrete Groups | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}}
• N.W. Johnson: Geometries and Transformations, (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups
• {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Huson | first2=Daniel H. | title= The Orbifold Notation for Two-Dimensional Groups | publisher=Springer Netherlands | doi=10.1023/A:1015851621002 | year=2002 | journal=Structural Chemistry | volume=13 | issue=3 | pages=247–257| s2cid=33947139 }}

• [http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Graphic overview of the 32 crystallographic point groups] – form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups

{{DEFAULTSORT:Dihedral Symmetry In Three Dimensions}}

Category:Symmetry

Category:Euclidean symmetries

Category:Group theory