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Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.Topological Bound States in the Continuum in Arrays of Dielectric Spheres. By Dmitrii N. Maksimov, LV Kirensky Institute of Physics, Krasnoyarsk, Russia
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Formal treatment

Formally the 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 In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
of rotational symmetry is a subgroup of ''E''+(''m'') (see
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations) ...
). 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 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 ''m''×''m'' orthogonal matrices with determinant 1. For this is the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
. In another definition 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
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 Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
, the rotational symmetry of a physical system is equivalent to the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
conservation law.


Discrete rotational symmetry

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 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°). The notation for ''n''-fold symmetry is ''Cn'' or simply "''n''". The actual
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
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 of order ''n'', Z''n''. Although for the latter also the notation ''C''''n'' is used, the geometric and abstract ''C''''n'' 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: *''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) *''n'' = 3, 120°: ''triad'',
triskelion A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of ...
,
Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
; sometimes the term ''trilateral symmetry'' is used; *''n'' = 4, 90°: ''tetrad'', swastika *''n'' = 6, 60°: ''hexad'', Star of David (this one has additional reflection symmetry) *''n'' = 8, 45°: ''octad'', Octagonal
muqarnas Muqarnas ( ar, مقرنص; fa, مقرنس), also known in Iranian architecture as Ahoopāy ( fa, آهوپای) and in Iberian architecture as Mocárabe, is a form of ornamented vaulting in Islamic architecture. It is the archetypal form of ...
, computer-generated (CG), ceiling ''C''''n'' 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.


Examples


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 ''D''n of order 2''n'' (). This is the rotation group of a regular
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentar ...
, or regular
bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does n ...
. Although the same notation is used, the geometric and abstract ''D''n 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 In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
''A''4. *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''4. *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''5. The group contains 10 versions of ''D3'' and 6 versions of ''D5'' (rotational symmetries like prisms and antiprisms). In the case of the Platonic 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

Rotational symmetry with respect to any angle is, in two dimensions,
circular symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
. The fundamental domain is a
half-line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segm ...
. 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
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
and no dependence on either angle using
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric or axisymmetrical are
adjective In linguistics, an adjective ( abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun. Traditionally, adjectives were considered one of the ma ...
s which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut ( 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 the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms.


Rotational symmetry with translational symmetry

2-fold rotational symmetry together with single translational symmetry is one of the
Frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...
s. 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 (2222): 4×2-fold; rotation group of a parallelogrammic,
rectangular In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
, and rhombic lattice. *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 regular triangular tiling with the equilateral triangles alternatingly colored. *p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice. *p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal 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 \frac \sqrt *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 \frac \sqrt *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 2\sqrt times their distance.


See also

*
Ambigram An ambigram is a calligraphic design that has several interpretations as written. The term was coined by Douglas Hofstadter in 1983. Most often, ambigrams appear as visually symmetrical words. When flipped, they remain unchanged, or they mut ...
*
Axial symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.
*
Crystallographic restriction theorem The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction ...
*
Lorentz symmetry In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the sam ...
*
Point groups in three dimensions In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries th ...
*
Screw axis A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a scr ...
*
Space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unch ...
* Translational symmetry


References

*


External links

* {{Commons category-inline, Rotational symmetry
Rotational Symmetry Examples
from Math Is Fun Symmetry Binocular rivalry