Dihedral symmetry in three dimensions
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 Dihn.
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:
- Dn, +, of order 2n – dihedral symmetry or para-n-gonal group
- Dnh, , of order 4n – prismatic symmetry or full ortho-n-gonal group
- Dnd, , of order 4n – antiprismatic symmetry or full gyro-n-gonal group
In 2D the symmetry group Dn 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 Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order.
With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh ,.
Dnd, , has vertical mirror planes between the horizontal rotation axes, not through them. As a result the vertical axis is a 2n-fold rotoreflection axis.
Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn 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
- D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
- D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane
- D2 +, 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.
- D2h, , of order 8 is the symmetry group of a cuboid
- D2d, , 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.
Subgroups
- Cnh, ,, order 2n
- Cnv, ,, order 2n
- Dn, +,, order 2n
- S2n, ,, order 2n
- Cnv, ,, order 2n
- Dn, +,, order 2n
Examples
Dnh, , :D5h, , :
Pentagrammic prism | Pentagrammic antiprism |
D4d, , :
D5d, , :
Pentagonal antiprism | Pentagrammic crossed-antiprism | pentagonal trapezohedron |
D17d, , :