View source for Dihedral symmetry in three dimensions

{{Short description|Regular polygonal symmetry}}
{{3d_point_group_navigator}}
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<sub>''n''</sub> (for ''n''&nbsp;≥&nbsp;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]].

;Chiral:
*''D<sub>n</sub>'', [''n'',2]<sup>+</sup>, (22''n'') of order 2''n'' – '''dihedral symmetry''' or '''para-n-gonal group''' (abstract group: [[Dihedral group|''Dih<sub>n</sub>'']]).
;Achiral:
*''D<sub>nh</sub>'', [''n'',2], (*22''n'') of order 4''n'' – '''prismatic symmetry''' or '''full ortho-n-gonal group''' (abstract group: ''Dih<sub>n</sub>'' × ''Z''<sub>2</sub>).
*''D<sub>nd</sub>'' (or ''D<sub>nv</sub>''), [2''n'',2<sup>+</sup>], (2*''n'') of order 4''n'' – '''antiprismatic symmetry''' or '''full gyro-n-gonal group''' (abstract group: ''Dih''<sub>2''n''</sub>).

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 group]]s. [[Schoenflies notation|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<sub>n</sub>'' 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<sub>n</sub>'' contains rotations only, not reflections. The other group is [[pyramidal symmetry]] ''C<sub>nv</sub>'' of the same order, 2''n''.

With [[reflection symmetry]] in a plane perpendicular to the ''n''-fold rotation axis, we have ''D<sub>nh</sub>'', [n], (*22''n'').

''D<sub>nd</sub>'' (or ''D<sub>nv</sub>''), [2''n'',2<sup>+</sup>], (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<sub>nh</sub>'' is the symmetry group for a regular ''n''-sided [[Prism (geometry)|prism]] and also for a regular n-sided [[bipyramid]]. ''D<sub>nd</sub>'' is the symmetry group for a regular ''n''-sided [[antiprism]], and also for a regular n-sided [[trapezohedron]]. ''D<sub>n</sub>'' is the symmetry group of a partially rotated prism.

''n''&nbsp;=&nbsp;1 is not included because the three symmetries are equal to other ones:
*''D''<sub>1</sub> and ''C''<sub>2</sub>: group of order 2 with a single 180° rotation.
*''D''<sub>1''h''</sub> and ''C''<sub>2''v''</sub>: group of order 4 with a reflection in a plane and a 180° rotation about a line in that plane.
*''D''<sub>1''d''</sub> and ''C''<sub>2''h''</sub>: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.

For ''n''&nbsp;=&nbsp;2 there is not one main axis and two additional axes, but there are three equivalent ones.

*''D''<sub>2</sub>, [2,2]<sup>+</sup>, (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''<sub>2''h''</sub>, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
*''D''<sub>2''d''</sub>, [4,2<sup>+</sup>], (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''<sub>2''d''</sub> is a subgroup of [[tetrahedral symmetry|''T<sub>d</sub>'']]; by scaling, we reduce the symmetry).

== Subgroups ==
{| class=wikitable align=right
|- align=center
|[[File:Order 2 dihedral symmetry subgroup tree.png|200px]]<BR>''D<sub>2h</sub>'', [2,2], (*222)
|[[File:Order 4 dihedral symmetry subgroup tree.png|200px]]<BR>''D<sub>4h</sub>'', [4,2], (*224)
|}
{{See also|Cyclic symmetry in three dimensions}}

For ''D<sub>nh</sub>'', [n,2], (*22n), order 4n
* ''C<sub>nh</sub>'', [n<sup>+</sup>,2], (n*), order 2n
* ''C<sub>nv</sub>'', [n,1], (*nn), order 2n
* ''D<sub>n</sub>'', [n,2]<sup>+</sup>, (22n), order 2n

For ''D<sub>nd</sub>'', [2n,2<sup>+</sup>], (2*n), order 4n
* ''S''<sub>2''n''</sub>, [2n<sup>+</sup>,2<sup>+</sup>], (n×), order 2n
* ''C<sub>nv</sub>'', [n<sup>+</sup>,2], (n*), order 2n
* ''D<sub>n</sub>'', [n,2]<sup>+</sup>, (22n), order 2n

''D<sub>nd</sub>'' is also subgroup of ''D''<sub>2''nh''</sub>.

== Examples ==
{| class=wikitable width=600
!D<sub>2h</sub>, [2,2], (*222)<BR>Order 8
!D<sub>2d</sub>, [4,2<sup>+</sup>], (2*2)<BR>Order 8
!D<sub>3h</sub>, [3,2], (*223)<BR>Order 12
|- align=center
|[[File:Basketball.png|200px]]<BR>[[basketball]] seam paths
|[[File:Baseball (crop).png|200px]]<BR>[[baseball]] seam paths <BR>(ignoring directionality of seam)
|[[File:BeachBall.jpg|200px]]<BR>[[Beach ball]]<BR>(ignoring colors)
|}

'''''D<sub>nh</sub>'', [''n''], (*22''n''):'''

{| class="wikitable"
| [[Image:Geometricprisms.gif|250px]]<br>[[prism (geometry)|prisms]]
|}

'''''D''<sub>5''h''</sub>, [5], (*225):'''

{| class="wikitable"
|[[Image:Pentagrammic prism.png|100px]]<br>[[Pentagrammic prism]]
|[[Image:Pentagrammic antiprism.png|100px]]<br>[[Pentagrammic antiprism]]
|}

'''''D''<sub>4''d''</sub>, [8,2<sup>+</sup>], (2*4):'''

{| class="wikitable"
| [[Image:Snub square antiprism.png|100px]]<br>[[Snub square antiprism]]
|}

'''''D''<sub>5''d''</sub>, [10,2<sup>+</sup>], (2*5):'''

{| class="wikitable"
| [[Image:antiprism5.jpg|100px]]<br>[[Pentagonal antiprism]]
| [[Image:Pentagrammic crossed antiprism.png|100px]]<br>[[Pentagrammic crossed-antiprism]]
| [[Image:Trapezohedron5.jpg|60px]]<br>[[pentagonal trapezohedron]]
|}

'''''D''<sub>17''d''</sub>, [34,2<sup>+</sup>],  (2*17):'''

{| class="wikitable"
| [[Image:antiprism17.jpg|100px]]<br>[[antiprism|Heptadecagonal antiprism]]
|}

==See also==
* [[List of spherical symmetry groups]]
* [[Point groups in three dimensions]]
* [[Cyclic symmetry in three dimensions]]

== References ==
*{{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}}
* [[Norman Johnson (mathematician)|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 }}

== External links ==
*[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]]