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

Polyhedral group, [n,3], (*n32) | |||
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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 an abstract group is a dihedral group Dih_{n} (for *n* ≥ 2).

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]^{+}, (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 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

With reflection symmetry in a plane perpendicular to the *n*-fold rotation axis, we have *D _{nh}*, [n], (*22

*D _{nd}* (or

*D _{nh}* is the symmetry group for a regular

*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 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.

*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}

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

*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_{2h}, [2,2], (*222)Order 8 |
D_{2d}, [4,2^{+}], (2*2)Order 8 |
D_{3h}, [3,2], (*223)Order 12 |
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basketball seam paths |
baseball seam paths (ignoring directionality of seam) |
Beach ball (ignoring colors) |

*D _{nh}*, [

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 |

- 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.`{{cite book}}`

: CS1 maint: multiple names: authors list (link) - 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*,**13**(3), Springer Netherlands: 247–257, doi:10.1023/A:1015851621002, S2CID 33947139

- 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