Rotational Symmetry
Rotational Symmetry
Generally speaking, an object with rotational symmetry is an object that looks the
same after a certain amount of rotation. An object may have more than one
rotational symmetry; for instance, if reflections or turning it over are not counted,
the triskelion appearing on the Isle of Man's flag (see opposite) has three
rotational symmetries (or "a threefold rotational symmetry").
More examples may be seen below. The degree of rotational symmetry is how
many degrees the shape has to be turned to look the same on a different side or
vertex. It can not be the same side or vertex.
Formal treatment :- Formally, 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+(m) (see Euclidean group).
Symmetry with respect to all rotations about all points implies translational
symmetry with respect to all translations, so space is homogeneous, and the
symmetry group is the whole E(m).
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With the modified notion of symmetry for vector fields the symmetry group can
also be E+(m). For symmetry with respect to rotations about a point we can take
that point as origin. These rotations form the special orthogonal group SO(m), the
group of mxm orthogonal matrices with determinant 1. For this is the rotation
group SO(3).
In another meaning of the word, the rotation group of an object is the symmetry
group within E+(n), 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(3)-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.
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete
rotational symmetry of the nth order, with respect to a particular point (in 2D) or
axis (in 3D) means that rotation by an angle of 360/n (180, 120, 90, 72, 60,
51 37, etc.) does not change the object. Note that "1-fold" symmetry is no
symmetry, and "2-fold" is the simplest symmetry, so it does not 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 Zn 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.
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Examples without additional reflection symmetry:
n = 2, 180: the dyad, quadrilaterals with this symmetry are the parallelograms;
other examples: letters Z, N, S; apart from the colors: yin and yang
n = 3, 120: triad, triskelion, Borromean rings; sometimes the term trilateral
symmetry is used;
n = 4, 90: tetrad, swastika
n = 6, 60: hexad, raelian symbol, new version
n = 8, 45: octad, Octagonal muqarnas, computer-generated (CG), ceiling
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 (possibly also with perpendicular
axes) but no mirror symmetry is a propeller.


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