Rotational symmetry, also known asradial symmetry ingeometry, is the property ashape has when it looks the same after somerotation by a partialturn. An object's degree of rotational symmetry is the number of distinctorientations in which it looks exactly the same for each rotation.

Certain geometric objects are partially symmetrical when rotated at certain angles such assquares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and otherspheroids.^[1][2]

Formally the rotational symmetry issymmetry with respect to some or allrotations inm-dimensionalEuclidean space. Rotations aredirect isometries, i.e.,isometries preservingorientation. Therefore, asymmetry group of rotational symmetry is a subgroup ofE ⁺(m) (seeEuclidean group).

Symmetry with respect to all rotations about all points impliestranslational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the wholeE(m). With themodified notion of symmetry for vector fields the symmetry group can also beE ⁺(m).

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the specialorthogonal groupSO(m), the group ofm ×morthogonal matrices with determinant 1. Form = 3 this is the rotation groupSO(3).

In another definition of the word, the rotation groupof an object is the symmetry group withinE ⁺(n), thegroup of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. Forchiral objects it is the same as the full symmetry group.

Laws of physicsare SO(3)-invariant if they do not distinguish different directions in space. Because ofNoether's theorem, the rotational symmetry of a physical system is equivalent to theangular momentum conservation law.

Rotational symmetry of order n, also calledn-fold rotational symmetry, ordiscrete rotational symmetry of thenth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of⁠${\displaystyle \frac{{360}^{\circ }}{n}}$ ⁠ (180°, 120°, 90°, 72°, 60°, 51 3⁄7°, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).

Thenotation forn-fold symmetry isC_n or simplyn. The actualsymmetry group is specified by the point or axis of symmetry, together with then. For each point or axis of symmetry, the abstract group type iscyclic group of order n,Z_n. Although for the latter also the notationC_n is used, the geometric and abstractC_n should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, seecyclic symmetry groups in 3D.

Thefundamental domain is asector of⁠${\displaystyle \frac{{360}^{\circ }}{n}.}$ ⁠

Examples without additionalreflection symmetry:

• n = 2, 180°: thedyad; letters Z, N, S; the outlines, albeit not the colors, of theyin and yang symbol; theUnion Flag (as divided along the flag's diagonal and rotated about the flag's center point)
• n = 3, 120°:triad,triskelion,Borromean rings; sometimes the termtrilateral symmetry is used;
• n = 4, 90°:tetrad,swastika
• n = 5, 72°:pentad,pentagram, regular pentagon; 5-fold symmetry is not possible in periodic crystals.
• n = 6, 60°:hexad,Star of David (this one has additionalreflection symmetry)
• n = 8, 45°:octad, Octagonalmuqarnas, computer-generated (CG), ceiling

C_n is the rotation group of a regularn-sidedpolygon in 2D and of a regularn-sidedpyramid in 3D.

If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, thegreatest common divisor of 100° and 360°.

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is apropeller.

Fordiscrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:

• In addition to ann-fold axis,n perpendicular 2-fold axes: thedihedral groupsD_n of order 2n (n ≥ 2). This is the rotation group of a regularprism, or regularbipyramid. Although the same notation is used, the geometric and abstractD_n should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, seedihedral symmetry groups in 3D.
• 4×3-fold and 3×2-fold axes: the rotation groupT of order 12 of a regulartetrahedron. The group isisomorphic toalternating groupA₄.
• 3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of acube and a regularoctahedron. The group is isomorphic tosymmetric groupS₄.
• 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of adodecahedron and anicosahedron. The group is isomorphic to alternating group A₅. The group contains 10 versions ofD₃ and 6 versions ofD₅ (rotational symmetries like prisms and antiprisms).

In the case of thePlatonic solids, the 2-fold axes are through the midpoints of opposite edges, and 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 is, in two dimensions,circular symmetry. The fundamental domain is ahalf-line.

In three dimensions we can distinguishcylindrical symmetry andspherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle usingcylindrical coordinates and no dependence on either angle usingspherical coordinates. The fundamental domain is ahalf-plane through the axis, and a radial half-line, respectively.Axisymmetric andaxisymmetrical areadjectives which refer to an object having cylindrical symmetry, oraxisymmetry (i.e. rotational symmetry with respect to a central axis) like adoughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).

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 theCartesian product of two rotationally symmetry 2D figures, as in the case of e.g. theduocylinder and various regularduoprisms.

2-fold rotational symmetry together with singletranslational symmetry is one of theFrieze groups. A rotocenter is thefixed, or invariant, point of a rotation.^[3] There are two rotocenters perprimitive cell.

Together with double translational symmetry the rotation groups are the followingwallpaper groups, with axes per primitive cell:

• p2 (2222): 4×2-fold; rotation group of aparallelogrammic,rectangular, andrhombiclattice.
• 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 theregular triangular tiling with the equilateral triangles alternatingly colored.
• p4 (442): 2×4-fold, 2×2-fold; rotation group of asquare lattice.
• p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of ahexagonal lattice.
• 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.
• 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${\displaystyle \frac{1}{3}\sqrt{3}}$ 
• 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${\displaystyle \frac{1}{2}\sqrt{2}}$ 
• 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 (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${\displaystyle 2\sqrt{3}}$  times their distance.

• Weyl, Hermann (1982) [1952].Symmetry. Princeton: Princeton University Press.ISBN 0-691-02374-3.

•   Media related toRotational symmetry at Wikimedia Commons
• Rotational Symmetry Examples fromMath Is Fun