Dihedral symmetry in three dimensions

Point groups in three dimensions
Sphere symmetry group cs.png
Involutional symmetry
Cs, (*)
[ ] = CDel node c2.png
Sphere symmetry group c3v.png
Cyclic symmetry
Cnv, (*nn)
[n] = CDel node c1.pngCDel n.pngCDel node c1.png
Sphere symmetry group d3h.png
Dihedral symmetry
Dnh, (*n22)
[n,2] = CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c1.png
Polyhedral group, [n,3], (*n32)
Sphere symmetry group td.png
Tetrahedral symmetry
Td, (*332)
[3,3] = CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group oh.png
Octahedral symmetry
Oh, (*432)
[4,3] = CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
Sphere symmetry group ih.png
Icosahedral symmetry
Ih, (*532)
[5,3] = CDel node c2.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c2.png

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).

TypesEdit

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, [n,2]+, (22n) of order 2ndihedral symmetry or para-n-gonal group (abstract group: Dihn).
Achiral
  • Dnh, [n,2], (*22n) of order 4nprismatic symmetry or full ortho-n-gonal group (abstract group: Dihn × Z2).
  • Dnd (or Dnv), [2n,2+], (2*n) of order 4nantiprismatic symmetry or full gyro-n-gonal group (abstract group: Dih2n).

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:

  • 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 about a line in that plane.
  • D1d and C2h: 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.

  • D2, [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.
  • D2h, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
  • D2d, [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 (D2d is a subgroup of Td; by scaling, we reduce the symmetry).

SubgroupsEdit

 
D2h, [2,2], (*222)
 
D4h, [4,2], (*224)

For Dnh, [n,2], (*22n), order 4n

  • Cnh, [n+,2], (n*), order 2n
  • Cnv, [n,1], (*nn), order 2n
  • Dn, [n,2]+, (22n), order 2n

For Dnd, [2n,2+], (2*n), order 4n

  • S2n, [2n+,2+], (n×), order 2n
  • Cnv, [n+,2], (n*), order 2n
  • Dn, [n,2]+, (22n), order 2n

Dnd is also subgroup of D2nh.

ExamplesEdit

D2h, [2,2], (*222)
Order 8
D2d, [4,2+], (2*2)
Order 8
D3h, [3,2], (*223)
Order 12
 
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

See alsoEdit

ReferencesEdit

  • 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, Springer Netherlands, 13 (3): 247–257, doi:10.1023/A:1015851621002

External linksEdit