Editing Rotational symmetry (section)

===Discrete rotational symmetry===
'''Rotational symmetry of order&nbsp;{{mvar|n}}''', also called '''{{mvar|n}}-fold rotational symmetry''', or '''discrete rotational symmetry of the {{mvar|n}}th order''', with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of {{tmath|\tfrac{360^\circ}{n} }} (180°, 120°, 90°, 72°, 60°, 51&nbsp;{{frac|3|7}}°, etc.) does not change the object. A  "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).

The [[Crystal system#Overview of point groups by crystal system|notation]] for {{mvar|n}}-fold symmetry is {{mvar|'''C<sub>n</sub>'''}} or simply {{mvar|n}}. The actual [[symmetry group]] is specified by the point or axis of symmetry, together with the {{mvar|n}}. For each point or axis of symmetry, the abstract group type is [[cyclic group]] of order&nbsp;{{mvar|n}}, {{mvar|Z<sub>n</sub>}}. Although for the latter also the notation {{mvar|C<sub>n</sub>}} is used, the geometric and abstract {{mvar|C<sub>n</sub>}} should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see [[Point groups in three dimensions#Cyclic symmetry groups in 3D cyclic symmetry groups|cyclic symmetry groups in 3D]].

The [[fundamental domain]] is a [[Circular sector|sector]] of {{tmath|\tfrac{360^\circ}{n}.}}

Examples without additional [[reflection symmetry]]:
*{{math|1=''n'' = 2}}, 180°: the ''dyad''; letters Z, N, S; the outlines, albeit not the colors, of the [[yin and yang]] symbol; the [[Union Flag]] (as divided along the flag's diagonal and rotated about the flag's center point)
*{{math|1=''n'' = 3}}, 120°: ''triad'', [[triskelion]], [[Borromean rings]]; sometimes the term ''trilateral symmetry'' is used;
*{{math|1=''n'' = 4}}, 90°: ''tetrad'', [[swastika]]
*{{Math|1=''n'' = 5}}, 72°: ''pentad,'' [[pentagram]], regular pentagon; 5-fold symmetry is not possible in periodic crystals.
*{{math|1=''n'' = 6}}, 60°: ''hexad'', [[Star of David]] (this one has additional [[reflection symmetry]])
*{{math|1=''n'' = 8}}, 45°: ''octad'', Octagonal [[muqarnas]], computer-generated (CG), ceiling

{{mvar|C<sub>n</sub>}} is the rotation group of a regular {{mvar|n}}-sided [[polygon]] in 2D and of a regular {{mvar|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]].