Dihedral symmetry in three dimensions
Regular polygonal symmetry From Wikipedia, the free encyclopedia
Regular polygonal symmetry From Wikipedia, the free encyclopedia
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 Dihn (for n ≥ 2).
![]() Involutional symmetry Cs, (*) [ ] = ![]() |
![]() Cyclic symmetry Cnv, (*nn) [n] = ![]() ![]() ![]() |
![]() Dihedral symmetry Dnh, (*n22) [n,2] = ![]() ![]() ![]() ![]() ![]() | |
Polyhedral group, [n,3], (*n32) | |||
---|---|---|---|
![]() Tetrahedral symmetry Td, (*332) [3,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Octahedral symmetry Oh, (*432) [4,3] = ![]() ![]() ![]() ![]() ![]() |
![]() Icosahedral symmetry Ih, (*532) [5,3] = ![]() ![]() ![]() ![]() ![]() |
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation.
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 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 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 Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, 2n.
With reflection symmetry in a plane perpendicular to the n-fold rotation axis, we have Dnh, [n], (*22n).
Dnd (or Dnv), [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.
Dnh is the symmetry group for a regular n-sided prism 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:
For n = 2 there is not one main axis and two additional axes, but there are three equivalent ones.
![]() D2h, [2,2], (*222) |
![]() D4h, [4,2], (*224) |
For Dnh, [n,2], (*22n), order 4n
For Dnd, [2n,2+], (2*n), order 4n
Dnd is also subgroup of D2nh.
D2h, [2,2], (*222) Order 8 |
D2d, [4,2+], (2*2) Order 8 |
D3h, [3,2], (*223) Order 12 |
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![]() basketball seam paths |
![]() baseball seam paths (ignoring directionality of seam) |
![]() Beach ball (ignoring colors) |
Dnh, [n], (*22n):
![]() prisms |
D5h, [5], (*225):
![]() Pentagrammic prism |
![]() Pentagrammic antiprism |
D4d, [8,2+], (2*4):
![]() Snub square antiprism |
D5d, [10,2+], (2*5):
![]() Pentagonal antiprism |
![]() Pentagrammic crossed-antiprism |
![]() pentagonal trapezohedron |
D17d, [34,2+], (2*17):
![]() Heptadecagonal antiprism |
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