Rotational symmetry
Rotational symmetry is
symmetry with respect to some or all
rotations in
m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a
symmetry group of rotational symmetry is a subgroup of
E+ .
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole
E+. This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
For symmetry with respect to rotations about a point we can take that point as origin.
Encyclopedia
Rotational symmetry is
symmetry with respect to some or all
rotations in
m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a
symmetry group of rotational symmetry is a subgroup of
E+ .
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole
E+. This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO, the group of
m×
m orthogonal matrices with determinant 1. For
m=3 this is the rotation group.
In another meaning of the word, the rotation group
of an object is the symmetry group within
E+, the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
Laws of physics are SO-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the
angular momentum conservation law. See also rotational invariance.
n-fold rotational symmetry
Rotational symmetry of order n, also called
n-fold rotational symmetry, or discrete rotational symmetry of
nth order, with respect to a particular point or axis means that rotation by an angle of 360°/n does not change the object.
Note that "1-fold" symmetry is no symmetry, and "2-fold" is the simplest symmetry, so it does mean "more than basic".
The
notation for
n-fold symmetry is
Cn or simply "
n". The actual
symmetry group is specified by the point or axis of symmetry, together with the
n. For each point or axis of symmetry the abstract group type is
cyclic group Z
n of order
n. Although for the latter also the notation
Cn is used, the geometric and abstract
Cn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see
cyclic symmetry groups in 3D.
The
fundamental domain is a sector of 360°/n.
Examples without additional
reflection symmetry:
Cn is the rotation group of a regular
n-sided
polygon in 2D and of a regular
n-sided
pyramid in 3D.
If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.
A typical 3D object with rotational symmetry but no mirror symmetry is a
propeller.
Examples
C2
C3
C4
Mixed
Multiple symmetry axes through the same point
For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:
- In addition to an n-fold axis, n perpendicular 2-fold axes: the dihedral groups Dn of order 2n . This is the rotation group of a regular prism, or regular bipyramid. Although the same notation is used, the geometric and abstract Dn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D.
- 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron. The group is isomorphic to alternating group A4.
- 3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron. The group is isomorphic to symmetric group S4.
- 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron
...
. The group is isomorphic to alternating group
A5. The group contains 10 versions of
D3 and 6 versions of
D5 .
In the case of the
Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.
Rotational symmetry with respect to any angle
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The fundamental domain is a half-line.
In three dimensions we can distinguish
cylindrical symmetry and
spherical symmetry . That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a
half-plane through the axis, and a radial half-line, respectively. An example of approximate spherical symmetry is the Earth .
In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular
duoprisms.
Rotational symmetry together with translational symmetry
2-fold rotational symmetry together with single translational symmetry is one of the
Frieze groups. There are two rotocenters per primitive cell.
Together with double translational symmetry the rotation groups are the following
wallpaper groups, with axes per primitive cell:
- p2 : 4×2-fold; rotation group of a parallelogrammic, rectangular, and rhombic lattice.
- p3 : 3×3-fold; not the rotation group of any lattice ; it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored.
- p4 : 2×4-fold, 2×2-fold; rotation group of a square lattice.
- p6 : 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice.
- 2-fold rotocenters , if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not apply.
- 3-fold rotocenters , if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° , and scaled by a factor
- 4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor
- 6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.
Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.
3-fold rotational symmetry at one point and 2-fold at another one implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point . The translation distance for the symmetry generated by one such pair of rotocenters is 2v3 times their distance.
See also
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External links
