Rotational symmetry

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Rotational symmetry

Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotationRotation Summary

Rotation is the movement of an object in a circular motion....
. An object may have more than one rotational symmetrySymmetry

Symmetry is a characteristic feature of geometrical shapes, systems, equations, and other real or conceptual objects —t...
; for instance, if reflections or turning it over are not counted, the triskelionTriskelion

Triskelion is a symbol consisting of three bent human legs, or, more generally, three interlocked spirals, or any similar sy...
appearing on the Isle of ManIsle of Man

The Isle of Man or Mann , is an island located in the Irish Sea at the geographical centre of Great Britain and Irela...
's flag (see opposite) has three rotational symmetries (or "a threefold rotational symmetry"). More examples may be seen below.

Formal treatment

Formally, rotational symmetry is symmetrySymmetry

Symmetry is a characteristic feature of geometrical shapes, systems, equations, and other real or conceptual objects —t...
with respect to some or all rotationRotation

Rotation is the movement of an object in a circular motion....
s in m-dimensional Euclidean spaceEuclidean space

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh...
. Rotations are direct isometriesIsometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism be...
, i.e., isometries preserving orientationOrientation (mathematics)

In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which...
. Therefore a symmetry groupSymmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operatio...
of rotational symmetry is a subgroup of E⁺(m) (see Euclidean groupEuclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Eucli...
).

Symmetry with respect to all rotations about all vertices implies translational symmetryTranslational symmetry

In geometry, a translation "slides" an object by a vector a: Ta = p + a....
with respect to all translations, and the symmetry group is the whole E⁺(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal groupOrthogonal group

In mathematics, the orthogonal group of degree n over a field F) is the group of n-by-n orthogonal matrices ...
SO(m), the group of m×m orthogonal matrices with determinant 1. For m=3 this is the rotation groupFacts About Rotation group

In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space,...
.

In another meaning of the word, the rotation group of an object is the symmetry group within E⁺(n), the group of direct isometriesEuclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Eucli...
; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiralChirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to 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 theoremNoether's theorem

Noether's theorem is a central result in theoretical physics that expresses the one-to-one correspondence between continuous...
, rotational symmetry of a physical system is equivalent to the angular momentumAngular momentum

In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the...
conservation law. See also Rotational invarianceRotational invariance

In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not cha...
.

n-fold rotational symmetry

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 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 mean "more than basic".

The notationCrystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translati...
for n-fold symmetry is C_n or simply "n". The actual symmetry groupSymmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operatio...
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 groupCyclic group

In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense...
Z_n of order 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 3DPoint groups in three dimensions

In geometry a point group in 3D is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, a...
.

The fundamental domainFundamental domain

In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possib...
is a sector of 360°/n.

Examples without additional reflection symmetryReflection symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respec...
:

n = 2, 180°: the dyadDyad

Etymology: Late Latin dyad-, dyas, from Greek, from dyo...
,quadrilateralQuadrilateral

In geometry, a quadrilateral is a polygon with four sides and four vertices....
s with this symmetry are the parallelogramParallelogram

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....
s; other examples: letters Z, N, S; apart from the colors: yin and yangYin and yang

The concepts of Yin and Yang originate in ancient Chinese philosophy and metaphysics, which describes two primal opposing bu...
n = 3, 120°: triadTriad

Triad or is a collective term that describes many branches of an underground society and organizations based in Hong Kong an...
, triskelionTriskelion

Triskelion is a symbol consisting of three bent human legs, or, more generally, three interlocked spirals, or any similar sy...
, Borromean ringsFacts About Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked despite the fact that no two of th...
; sometimes the term trilateral symmetry is used;
n = 4, 90°: tetradTetrad

Tetrad has a variety of meanings*For the tetrad or vierbein theory in differential geometry, see Cartan connection ...
, swastikaSwastika

he swastika is an equilateral cross with its arms bent at right angles in either left-facing or right-facing direction....
n = 6, 60°: hexad , raelianRaëlism

Ralism is the philosophical belief system promoted by the Ralian Movement, a humanist new religious movement founded i...
symbol, new version

C_n is the rotation group of a regular n-sided polygonPolygon

A polygon is a closed planar path composed of a finite number of sequential line segments....
in 2D and of a regular n-sided pyramidPyramid

Pyramids are among the largest man-made constructions as well as one of the great Wonders of the ancient world....
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 divisorGreatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common fact...
of 100° and 360°.

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

A propeller is a device which transmits power by converting it into thrust for propulsion of a vehicle such as an aircraft, ...
.

Examples

C2

C3


C4

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 groupDihedral group

In mathematics, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group i...
s D_n of order 2n (n=2). This is the rotation group of a regular prismPrism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces ...
, or regular bipyramidBipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image ...
. 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 3DFacts About Dihedral group

In mathematics, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group i...
.
4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedronTetrahedron Overview

A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex....
. The group is isomorphic to alternating groupAlternating group

In mathematics, an alternating group is the group of even permutations of a finite set....
A₄.
3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cubeCube

A cube is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each ver...
and a regular octahedronOctahedron

An octahedron is a polyhedron with eight faces....
. The group is isomorphic to symmetric groupSymmetric group

In mathematics, the symmetric group on a set X, denoted by SX or Sym, is the group whose underlying set is the set o...
S₄.
6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedronDodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid comp...
and an icosahedronIcosahedron

An icosahedron noun isa polyhedron having 20 faces, but usually a regular icosahedron is meant, which has face...
. The group is isomorphic to alternating group A₅. The group contains 10 versions of D₃ and 6 versions of D₅ (rotational symmetries like prisms and antiprisms).

In the case of the Platonic solidPlatonic solid

In geometry, a Platonic solid is a convex regular polyhedron....
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.

Rotational symmetry with respect to any angle

Rotational symmetry with respect to any angle is, in two dimensions, circular symmetryCircular symmetry

Circular symmetry in mathematical physics applies to a 2-dimensional field which can be expressed as a function of distance ...
. The fundamental domain is a half-lineLine (mathematics)

A line, or straight line, can be described as an infinitely thin, infinitely long, perfectly straight curve....
.

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 and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric or axisymmetrical are adjectiveAdjective

An adjective is a part of speech which modifies a noun, usually describing it or making its meaning more specific....
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).

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 productCartesian product

In mathematics, the Cartesian product of two sets X and Y, denoted X Y, is the set of all possible ordered...
of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinderDuocylinder

The duocylinder is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two dis...
and various regular duoprismDuoprism

A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean spa...
s.

Geometry, architecture and furniture

Rotational symmetry is a perfectly symmetrical shape wherein a two dimensional object is necessarily circular, and a three dimensional object may be considered as a stack of discs of differing radiiRADIUS

Remote Authentication Dial In User Service is an AAA protocol for applications such as network access or IP mobility....
.

Rotational symmetry with translational symmetry

2-fold rotational symmetry together with single translational symmetryTranslational symmetry

In geometry, a translation "slides" an object by a vector a: Ta = p + a....
is one of the Frieze groupFrieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one directi...
s. There are two rotocenters per primitive cellPrimitive cell

In geometry, solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell, is a minimu...
.

Together with double translational symmetry the rotation groups are the following wallpaper groupWallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the p...
s, with axes per primitive cell:

p2 (2222): 4×2-fold; rotation group of a parallelogramParallelogram

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....
mic, rectangularRectangle

In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles....
, and rhombicRhombus

In geometry, a rhombus is a quadrilateral in which all of the sides are of equal length, i.e., it is an equilateral quadrang...
latticeLattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn wh...
.
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 squareSquare (geometry)

In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides....
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

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
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 2v3 times their distance.

External links

from Math Is FunMath Is Fun

Math Is Fun is an educational website maintained by Rod Pierce devoted to the concept that mathematics is, indeed, fun....

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