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| {{refimprove|date=May 2009}} [[File:The armoured triskelion on the flag of the Isle of Man.svg|thumb|The [[triskelion]] appearing on the [[Flag of the Isle of Man|Isle of Man flag]].]]
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| Generally speaking, an object with '''rotational symmetry''', also known in biological contexts as '''radial 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 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.
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| == Formal treatment ==
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| {{See also|Rotational invariance}}
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| Formally, rotational symmetry is [[symmetry]] with respect to some or all [[rotation]]s in ''m''-dimensional [[Euclidean space]]. Rotations are [[Euclidean group#Direct and indirect isometries|direct isometries]], i.e., [[Isometry|isometries]] preserving [[Orientation (mathematics)|orientation]]. Therefore a [[symmetry group]] of rotational symmetry is a subgroup of ''E''<sup>+</sup>(''m'') (see [[Euclidean group]]).
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| 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''). With the [[Symmetry#Mathematical model for symmetry|modified notion of symmetry for vector fields]] the symmetry group can also be ''E''<sup>+</sup>(''m'').
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| 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 ''m''×''m'' [[orthogonal matrices]] with determinant 1. For {{math|''m'' = 3}} this is the [[rotation group SO(3)]].
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| In another meaning of the word, the rotation group ''of an object'' is the symmetry group within ''E''<sup>+</sup>(''n''), the [[Euclidean group|group of direct isometries]]; in other words, the intersection of the full symmetry group and the group of direct isometries. For [[chirality (mathematics)|chiral]] objects it is the same as the full symmetry group.
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| Laws of physics [[isotropy|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.
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| ===n-fold rotational symmetry===
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| '''Rotational symmetry of order ''n''''', also called '''''n''-fold rotational symmetry''', or '''discrete rotational symmetry of the ''n''th 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 {{frac|3|7}}°, 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".
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| The [[Crystal system#Overview of point groups by crystal system|notation]] for ''n''-fold symmetry is '''''C<sub>n</sub>''''' 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<sub>''n''</sub> of order ''n''. Although for the latter also the notation ''C''<sub>''n''</sub> is used, the geometric and abstract ''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]].
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| The [[fundamental domain]] is a sector of 360°/n.
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| Examples without additional [[reflection symmetry]]:
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| *''n'' = 2, 180°: the ''dyad'', [[quadrilateral]]s with this symmetry are the [[parallelogram]]s; other examples: letters Z, N, S; apart from the colors: [[yin and yang]]
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| *''n'' = 3, 120°: ''triad'', [[triskelion]], [[Borromean rings]]; sometimes the term ''trilateral symmetry'' is used;
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| *''n'' = 4, 90°: ''tetrad'', [[swastika]]
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| *''n'' = 6, 60°: ''hexad'', [[Raelism|raelian]] symbol, new version
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| *''n'' = 8, 45°: ''octad'', Octagonal [[muqarnas]], computer-generated (CG), ceiling
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| ''C''<sub>''n''</sub> is the rotation group of a regular ''n''-sided [[polygon]] in 2D and of a regular ''n''-sided [[pyramid]] in 3D.
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| 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°.
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| A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a [[propeller]].
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| ===Examples===
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| {| class="wikitable" width=600
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| ! C2 ([[commons:Category:2-fold rotational symmetry|more]])
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| ! C3 ([[commons:Category:3-fold rotational symmetry|more]])
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| ! C4 ([[commons:Category:4-fold rotational symmetry|more]])
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| ! C5 ([[commons:Category:5-fold rotational symmetry|more]])
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| ! C6 ([[commons:Category:6-fold rotational symmetry|more]])
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| |- valign=top
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| |[[File:Double_pendulum_flips_graph.png|120px]]<BR>[[Double Pendulum#Chaotic motion|Double Pendulum fractal]]
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| |[[File:Liikenneympyrä 166.svg|120px]]<BR>[[Roundabout]] [[traffic sign]]
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| |[[File:HinduSwastika.svg|120px]]<BR>Decorative [[Hindu]] form of the [[swastika]]
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| |[[File:United States Bicentennial star 1976 (geometry).svg|120px]]<BR>[[United States Bicentennial|US Bicentennial]] Star
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| |[[File:Crop_circles_Swirl.jpg|120px]]<BR>[[Crop circle]] in perspective
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| |- valign=top
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| |[[File:MacShogi.jpg|120px]]<BR>The starting position in [[shogi]]
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| |[[File:Snoldelev-three-interlaced-horns.svg|120px]]<BR>[[Snoldelev Stone]]'s interlocked [[drinking horn]]s design
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| |[[File:Op-art-4-sided-spiral-tunnel-7.svg|120px]]
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| |[[File:15crossings-decorative-knot.svg|120px]]
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| |[[File:Olavsrose.svg|120px]]
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| |}
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| ===Multiple symmetry axes through the same point===
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| For [[discrete symmetry]] with multiple symmetry axes through the same point, there are the following possibilities:
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| *In addition to an ''n''-fold axis, ''n'' perpendicular 2-fold axes: the [[dihedral group]]s ''D''<sub>n</sub> of order 2''n'' ({{math|''n'' ≥ 2}}). This is the rotation group of a regular [[prism (geometry)|prism]], or regular [[bipyramid]]. Although the same notation is used, the geometric and abstract ''D''<sub>n</sub> should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see [[Dihedral group#Dihedral symmetry groups in 3D|dihedral symmetry groups in 3D]].
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| *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]] ''A''<sub>4</sub>.
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| *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]] ''S''<sub>4</sub>.
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| *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 ''A''<sub>5</sub>. The group contains 10 versions of ''D<sub>3</sub>'' and 6 versions of ''D<sub>5</sub>'' (rotational symmetries like prisms and antiprisms).
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| In the case of the [[Platonic solid]]s, 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.
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| ===Rotational symmetry with respect to any angle===<!-- [[axisymmetric]], [[axisymmetrical]] and [[axisymmetry]] redirect to here -->
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| Rotational symmetry with respect to any angle is, in two dimensions, [[circular symmetry]]. The fundamental domain is a [[Line (mathematics)#Ray|half-line]].
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| In three dimensions we can distinguish '''cylindrical symmetry''' and '''spherical symmetry''' (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using [[Coordinates (elementary mathematics)#Cylindrical coordinates|cylindrical coordinates]] and no dependence on either angle using [[Coordinates (elementary mathematics)#Spherical coordinates|spherical coordinates]]. The fundamental domain is a [[half-plane]] through the axis, and a radial half-line, respectively. '''Axisymmetric''' or '''axisymmetrical''' are [[adjective]]s which refer to an object having cylindrical symmetry, or '''axisymmetry'''. An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
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| 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 [[duoprism]]s.
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| ===Rotational symmetry with translational symmetry===
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| {| class=wikitable align=right width=400
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| |[[File:Wallpaper group diagram p4.png|160px]]<BR>Arrangement within a [[primitive cell]] of 2- and 4-fold rotocenters. A [[fundamental domain]] is indicated in yellow.
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| |[[File:Wallpaper group diagram p6.png|240px]]<BR>Arrangement within a primitive cell of 2-, 3-, and 6-fold rotocenters, alone or in combination (consider the 6-fold symbol as a combination of a 2- and a 3-fold symbol); in the case of 2-fold symmetry only, the shape of the [[parallelogram]] can be different. For the case p6, a fundamental domain is indicated in yellow.
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| |}
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| 2-fold rotational symmetry together with single [[translational symmetry]] is one of the [[Frieze group]]s. There are two rotocenters per [[primitive cell]].
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| Together with double translational symmetry the rotation groups are the following [[wallpaper group]]s, with axes per primitive cell:
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| *p2 (2222): 4×2-fold; rotation group of a [[parallelogram]]mic, [[rectangle|rectangular]], and [[rhombus|rhombic]] [[Lattice (group)|lattice]].
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| *p3 (333): 3×3-fold; ''not'' the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the [[Tilings of regular polygons#Regular tilings|regular triangular tiling]] with the equilateral triangles alternatingly colored.
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| *p4 (442): 2×4-fold, 2×2-fold; rotation group of a [[Square (geometry)|square]] lattice.
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| *p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a [[hexagonal]] lattice.
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| *2-fold rotocenters (including possible 4-fold and 6-fold), 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.
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| *3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor <math>\frac{1}{3} \sqrt {3}</math>
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| *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 <math>\frac{1}{2} \sqrt {2}</math>
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| *6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.
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| 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.
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| 3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is 2√3 times their distance.
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| {| class=wikitable width=480
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| !Euclidean plane
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| !Hyperbolic plane
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| |- valign=top
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| |[[File:Tile V46b.svg|240px]]<BR>[[Hexakis triangular tiling]], an example of p6, [6,3]<sup>+</sup>, (632) (with colors) and p6m, [6,3], (*632) (without); the lines are reflection axes if colors are ignored, and a special kind of symmetry axis if colors are not ignored: reflection reverts the colors. Rectangular line grids in three orientations can be distinguished.
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| |[[File:Order-3 heptakis heptagonal tiling.png|240px]]<BR>[[Order 3-7 kisrhombille]], an example of [7,3]<sup>+</sup> (732) symmetry and [7,3], (*732) (without)
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| |}
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| == See also ==
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| {{Div col|3}}
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| * [[Ambigram]]
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| * [[Axial symmetry]]
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| * [[Crystallographic restriction theorem]]
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| * [[Lorentz symmetry]]
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| * [[Point groups in three dimensions]]
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| * [[Recycling symbol]]
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| * [[Screw axis]]
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| * [[Space group]]
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| * [[Three hares]]
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| {{Div col end}}
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| == References ==
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| {{reflist}}
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| *{{cite book |title=Symmetry |last=Weyl |first=Hermann |authorlink=Hermann Weyl |coauthors= |year=1982 |origyear=1952 |publisher=Princeton University Press |location=Princeton |isbn=0-691-02374-3 |pages= |url= |ref=Weyl 1982}}
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| == External links ==
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| * {{Commons category-inline|Rotational symmetry by order}}
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| *[http://www.mathsisfun.com/geometry/symmetry-rotational.html Rotational Symmetry Examples] from [[Math Is Fun]]
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| <!--Categories-->
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| [[Category:Rotational symmetry| ]]
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| [[Category:Symmetry]]
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| [[Category:Vision rivalry]]
| |
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