Dihedral symmetry in three dimensions
Involutional symmetry C_{s}, (*) [ ] = |
Cyclic symmetry C_{nv}, (*nn) [n] = |
Dihedral symmetry D_{nh}, (*n22) [n,2] = | |
Polyhedral group, [n,3], (*n32) | |||
---|---|---|---|
Tetrahedral symmetry T_{d}, (*332) [3,3] = |
Octahedral symmetry O_{h}, (*432) [4,3] = |
Icosahedral symmetry I_{h}, (*532) [5,3] = |
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 Dih_{n} ( n ≥ 2 ).
Types
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notation: Schönflies notation, Coxeter notation, and orbifold notation.
- Chiral
- D_{n}, [n,2]^{+}, (22n) of order 2n – dihedral symmetry or para-n-gonal group (abstract group Dih_{n})
- Achiral
- D_{nh}, [n,2], (*22n) of order 4n – prismatic symmetry or full ortho-n-gonal group (abstract group Dih_{n} × Z_{2})
- D_{nd} (or D_{nv}), [2n,2^{+}], (2*n) of order 4n – antiprismatic symmetry or full gyro-n-gonal group (abstract group Dih_{2n})
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 D_{n} 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 D_{n} contains rotations only, not reflections. The other group is pyramidal symmetry C_{nv} of the same order.
With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have D_{nh} [n], (*22n).
D_{nd} (or D_{nv}), [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.
D_{nh} is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. D_{nd} is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. D_{n} is the symmetry group of a partially rotated prism.
n = 1 is not included because the three symmetries are equal to other ones:
- D_{1} and C_{2}: group of order 2 with a single 180° rotation
- D_{1h} and C_{2v}: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
- D_{1d} and C_{2h}: 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.
- D_{2} [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.
- D_{2h}, [2,2], (*222) of order 8 is the symmetry group of a cuboid
- D_{2d}, [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 (D_{2d} is a subgroup of T_{d}, by scaling we reduce the symmetry).
Subgroups
D_{2h}, [2,2], (*222) |
D_{4h}, [4,2], (*224) |
For D_{nh}, [n,2], (*22n), order 4n
- C_{nh}, [n^{+},2], (n*), order 2n
- C_{nv}, [n,1], (*nn), order 2n
- D_{n}, [n,2]^{+}, (22n), order 2n
For D_{nd}, [2n,2^{+}], (2*n), order 4n
- S_{2n}, [2n^{+},2^{+}], (n×), order 2n
- C_{nv}, [n^{+},2], (n*), order 2n
- D_{n}, [n,2]^{+}, (22n), order 2n
D_{nd} is also subgroup of D_{2nh}.
Examples
D_{2h}, [2,2], (*222) Order 8 |
D_{2d}, [4,2^{+}], (2*2) Order 8 |
D_{3h}, [3,2], (*223) Order 12 |
---|---|---|
basketball seam paths |
baseball seam paths |
Beach ball (ignoring colors) |
D_{}nh, [n], (*22n):
prisms |
D_{5h}, [5], (*225):
Pentagrammic prism |
Pentagrammic antiprism |
D_{4d}, [8,2^{+}], (2*4):
Snub square antiprism |
D_{5d}, [10,2^{+}], (2*5):
Pentagonal antiprism |
Pentagrammic crossed-antiprism |
pentagonal trapezohedron |
D_{17d}, [34,2^{+}], (2*17):
Heptadecagonal antiprism |
See also
- List of spherical symmetry groups
- Point groups in three dimensions
- Cyclic symmetry in three dimensions
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.
- 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
- Graphic overview of the 32 crystallographic point groups – form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups