Dihedral symmetry in three dimensions

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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 abstract group is a dihedral group Dihn ( n ≥ 2 ).

Types[edit]

Chiral
  • Dn, [n,2]+, (22n) of order 2ndihedral symmetry or para-n-gonal group (abstract group Dn)
Achiral
  • Dnh, [n,2], (*22n) of order 4nprismatic symmetry or full ortho-n-gonal group (abstract group Dn × C2)
  • Dnd (or Dnv), [2n,2+], (2*n) of order 4nantiprismatic symmetry or full gyro-n-gonal group (abstract group D2n)

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 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 [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 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

For n = 2 there is not one main axes and two additional axes, but there are three equivalent ones.

  • D2 (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 (*222) of order 8 is the symmetry group of a cuboid
  • D2d (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).

Subgroups[edit]

Order 2 dihedral symmetry subgroup tree.png
D2h, [2,2], (*222)
Order 4 dihedral symmetry subgroup tree.png
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.

Examples[edit]

D2h, [2,2], (*222)
Order 8
D2d, [4,2+], (2*2)
Order 8
D3h, [3,2], (*223)
Order 12
Basketball.png
basketball seam paths
Baseball (crop).png
baseball seam paths
BeachBall.jpg
Beach ball
(ignoring colors)

Dnh, [n], (*22n):

Geometricprisms.gif
prisms

D5h, [5], (*225):

Pentagrammic prism.png
Pentagrammic prism
Pentagrammic antiprism.png
Pentagrammic antiprism

D4d, [8,2+], (2*4):

Snub square antiprism.png
Snub square antiprism

D5d, [10,2+], (2*5):

Antiprism5.jpg
Pentagonal antiprism
Pentagrammic crossed antiprism.png
Pentagrammic crossed-antiprism
Trapezohedron5.jpg
pentagonal trapezohedron

D17d, [34,2+], (2*17):

Antiprism17.jpg
Heptadecagonal antiprism

See also[edit]

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, (2015) Chapter 11: Finite symmetry 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]