# Dihedral symmetry in three dimensions

Involutional symmetry C _{s}, (*)[ ] = |
Cyclic symmetry C _{nv}, (*nn)[n] = |
Dihedral symmetry D _{nh}, (*n22)[n,2] = | |

Polyhedral group, [n,3], (*n32) | |||
---|---|---|---|

Tetrahedral symmetry T _{d}, (*332)[3,3] = |
Octahedral symmetry O _{h}, (*432)[4,3] = |
Icosahedral symmetry I _{h}, (*532)[5,3] = |

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[edit]

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]^{+}, (22*n*) of order 2*n*–**dihedral symmetry**or**para-n-gonal group**(abstract group*Dih*)_{n}

- Achiral

*D*, [_{nh}*n*,2], (*22*n*) of order 4*n*–**prismatic symmetry**or**full ortho-n-gonal group**(abstract group*Dih*×_{n}*Z*_{2})*D*(or_{nd}*D*), [2_{nv}*n*,2^{+}], (2**n*) of order 4*n*–**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*contains rotations only, not reflections. The other group is pyramidal symmetry

_{n}*C*of the same order.

_{nv}With reflection symmetry with respect to 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 prisms 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:

*D*_{1}and*C*_{2}: 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.

*D*_{2}[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*, by scaling we reduce the symmetry)._{d}

## Subgroups[edit]

D, [2,2], (*222)
_{2h} |
D, [4,2], (*224)
_{4h} |

For *D _{nh}*, [n,2], (*22n), order 4n

*C*, [n_{nh}^{+},2], (n*), order 2n*C*, [n,1], (*nn), order 2n_{nv}*D*, [n,2]_{n}^{+}, (22n), order 2n

For *D _{nd}*, [2n,2

^{+}], (2*n), order 4n

*S*_{2n}, [2n^{+},2^{+}], (n×), order 2n*C*, [n_{nv}^{+},2], (n*), order 2n*D*, [n,2]_{n}^{+}, (22n), order 2n

*D _{nd}* is also subgroup of

*D*

_{2nh}.

## Examples[edit]

D_{2h}, [2,2], (*222)Order 8 |
D_{2d}, [4,2^{+}], (2*2)Order 8 |
D_{3h}, [3,2], (*223)Order 12 |
---|---|---|

basketball seam paths |
baseball seam paths (ignoring directionality of seam) |
Beach ball (ignoring colors) |

*D _{nh}*, [

*n*], (*22

*n*):

prisms |

*D*_{5h}, [5], (*225):

Pentagrammic prism |
Pentagrammic antiprism |

*D*_{4d}, [8,2^{+}], (2*4):

Snub square antiprism |

*D*_{5d}, [10,2^{+}], (2*5):

Pentagonal antiprism |
Pentagrammic crossed-antiprism |
pentagonal trapezohedron |

*D*_{17d}, [34,2^{+}], (2*17):

Heptadecagonal antiprism |

## See also[edit]

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

## References[edit]

- Coxeter, H. S. M. and Moser, W. O. J. (1980).
*Generators and Relations for Discrete Groups*. New York: Springer-Verlag. 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 - Conway, John Horton; Huson, Daniel H. (2002), "The Orbifold Notation for Two-Dimensional Groups",
*Structural Chemistry*, Springer Netherlands,**13**(3): 247–257, doi:10.1023/A:1015851621002

## External links[edit]

- 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