Prismatic Symmetry
Get Prismatic Symmetry essential facts below. View Videos or join the Prismatic Symmetry discussion. Add Prismatic Symmetry to your PopFlock.com topic list for future reference or share this resource on social media.
Prismatic Symmetry
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

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 2n - dihedral symmetry or para-n-gonal group (abstract group Dihn)
Achiral
  • Dnh, [n,2], (*22n) of order 4n - prismatic symmetry or full ortho-n-gonal group (abstract group Dihn × Z2)
  • Dnd (or Dnv), [2n,2+], (2*n) of order 4n - antiprismatic 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 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 [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).

Subgroups

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

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
(ignoring directionality of seam)
BeachBall.jpg
Beach ball
(ignoring colors)

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

D5h, [5], (*225):

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

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

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

See also

References

  • 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.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 links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Prismatic_symmetry
 



 



 
Music Scenes