Rotational symmetry

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Rotational symmetry

appearing on the Isle of Man flag

Flag of the Isle of Man

File:Flag of the Isle of Man.svgFile:Civil Ensign of the Isle of Man.svgThe flag of the Isle of Man shows a triskelion, the Three Legs of Mann emblem, in the centre of a red flag....
.]] Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
. An object may have more than one rotational symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
; for instance, if reflections or turning it over are not counted, the triskelion

Triskelion

A triskelion or triskele is a symbol consisting of 3 #In human culture interlocked spirals, or three bent human legs, or any similar symbol with three protrusions and a threefold rotational symmetry....
appearing on the Isle of Man

Isle of Man

The Isle of Man , or Mann , is a self-governing Crown dependency, located in the Irish Sea at the geographical centre of the British Isles....
's flag (see opposite) has three rotational symmetries (or "a threefold rotational symmetry"). More examples may be seen below.

ally, rotational symmetry is symmetry

Symmetry

Rotation

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
.

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Formal treatment

Formally, rotational symmetry is symmetry

Symmetry

Rotation

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
. Rotations are direct isometries

Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
, i.e., isometries

Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
preserving orientation

Orientation (mathematics)

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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
of rotational symmetry is a subgroup of E⁺(m) (see Euclidean group

Euclidean group

Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
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 group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
SO(m), the group of m×m orthogonal matrices with determinant 1. For m=3 this is the rotation group

Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
.

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

Euclidean group

Chirality (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 its mirror image by rotations and translations alone....
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

Noether's theorem states that any derivative Symmetry in physics of the action of a physical system has a corresponding conservation law. The action of a physical system is an integral of a so-called Lagrangian function, from which the system's behavior can be determined by the principle of least action....
, rotational symmetry of a physical system is equivalent to the angular momentum

Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
conservation law. See also Rotational invariance

Rotational invariance

In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument....
.

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 notation

Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
for n-fold symmetry is C_n or simply "n". The actual symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
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

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
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 3D

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....
.

The fundamental domain

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern....
is a sector of 360°/n.

Examples without additional reflection symmetry

Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
:

n = 2, 180°: the dyad
Dyad

Etymology: Late Latin dyad-, dyas, from Greek d?a?, from dyodyad may refer to:*dyad , according to the Pythagoreans, is the principle of "twoness" or "otherness"...
, quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
s with this symmetry are the parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s; other examples: letters Z, N, S; apart from the colors: yin and yang
Yin and yang

In Chinese philosophy, the concept of yin yang is used to describe how seemingly disjunct or opposing forces are interconnected and interdependent in the natural world, giving rise to each other in turn....
n = 3, 120°: triad
Triad

Triad is a term that describes many branches of China underground society and/or organizations based in Taiwan, Hong Kong, Malaysia, Singapore, Macau and also operating in mainland China, and countries and cities worldwide with significant Han Chinese populations such as San Francisco, California....
, triskelion
Triskelion

A triskelion or triskele is a symbol consisting of 3 #In human culture interlocked spirals, or three bent human legs, or any similar symbol with three protrusions and a threefold rotational symmetry....
, Borromean rings
Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are link ed and form a Brunnian link, i.e., removing any ring results in two unlinked rings....
; sometimes the term trilateral symmetry is used;
n = 4, 90°: tetrad
Tetrad

Tetrad may refer to:* Tetrad , a chromosomal formation in meiosis* Tetrad , a mathematical frame of reference in general relativity* Tetrad , the four spores of a yeast...
, swastika
Swastika

The swastika is an equilateral cross with its arms bent at Angle#Types of angles, in either right-facing form or its mirrored left-facing form....
n = 6, 60°: hexad, raelian
Raëlism

Ra?lism, or The Ra?lian movement, is a UFO religion founded by a former French sports-car journalist and test driver named Claude Vorilhon....
symbol, new version
n = 8, 45°: octad, Octagonal muqarnas
Muqarnas

Muqarnas is a type of corbel used as a decorative device in traditional Islamic architecture and Persian architecture. The term is similar to moc?rabe, but moc?rabe only refers to designs with formations resembling stalactites, by the use of elements known as alveole....
, computer-generated (CG), ceiling

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

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
in 2D and of a regular n-sided pyramid

Pyramid

A pyramid is a building where the outer surfaces are triangular and converge at a point. The base of pyramids are usually quadrilateral or trilateral , meaning that a pyramid usually has four or five faces....
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

Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
of 100° and 360°.

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

Propeller

A propeller is a type of fan which transmits power by converting rotational motion into thrust. It can be used to drive an fixed-wing aircraft, ship, or the fluid within a pump....
.

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 group
Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
s D_n of order 2n (n=2). This is the rotation group of a regular prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
, or regular bipyramid
Bipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
. 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
Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
.
4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
. The group is isomorphic to alternating group
Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt....
A₄.
3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
and a regular octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
. The group is isomorphic to symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
S₄.
6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
and an icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. 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 solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
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 symmetry

Circular symmetry

Circular symmetry in mathematical physics applies to a 2-dimensional field which can be expressed as a function of distance from a central point only....
. The fundamental domain is a half-line

Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
.

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 adjective

Adjective

In grammar, an adjective is a word whose main syntax role is to grammatical modifier a noun or pronoun, giving more information about the noun or pronoun's definition....
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 product

Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder

Duocylinder

The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disk s of radius r:...
and various regular duoprism

Duoprism

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
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 radis

radii

RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
.

Rotational symmetry with translational symmetry

of 2- and 4-fold rotocenters. A fundamental domain

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern....
is indicated in yellow.]] 2-fold rotational symmetry together with single translational symmetry

Translational symmetry

Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetry in the pattern....
s. There are two rotocenters per primitive cell

Primitive cell

In geometry, solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell, is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2D, 3D, or other dimensions....
.

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

Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry in the pattern. Such patterns occur frequently in architecture and decorative art....
s, with axes per primitive cell:

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

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
mic, rectangular
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
, and rhombic
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
lattice
Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
.
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
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
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

can be different. For the case p6, a fundamental domain is indicated in yellow.]]

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.

, an example of p6 (with colors) and p6m (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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