TheInfoList Rotational symmetry, also known as radial symmetry in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ... , 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
squares In geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in ... rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other
spheroids A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ... .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 Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ... with respect to some or all
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ... s in ''m''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. Rotations are direct isometries, i.e.,
isometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
preserving
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
. Therefore, a
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
of rotational symmetry is a subgroup of ''E''+(''m'') (see
Euclidean group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). Symmetry with respect to all rotations about all points implies
translational symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
SO(''m''), the group of ''m''×''m''
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is :Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose In linear algebra, t ...
with determinant 1. For this is the
rotation group SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
. 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 important in several branches of science. The word ''chirality'' is derived from the (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from its ; that is, 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 In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
, the rotational symmetry of a physical system is equivalent to the
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ... 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 In linguistics and semiotics, a notation is a system of graphics or symbols, Character_(symbol), characters and abbreviated Expression (language), expressions, used (for example) in Artistic disciplines, artistic and scientific disciplines to repr ...
for ''n''-fold symmetry is ''Cn'' or simply "''n''". The actual
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
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 In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... 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 Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
. The
fundamental domain Given a topological space and a group (mathematics), group Group action (mathematics), acting on it, the images of a single point under the group action form an Group action (mathematics)#Orbits_and_stabilizers, orbit of the action. A fundamental ...
is a sector of 360°/n. Examples without additional
reflection symmetry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... : *''n'' = 2, 180°: the ''dyad''; letters Z, N, S; the outlines, albeit not the colors, of the
yin and yang In Ancient Chinese philosophy Chinese philosophy originates in the Spring and Autumn period () and Warring States period (), during a period known as the "Hundred Schools of Thought", which was characterized by significan ... symbol; the
Union Flag The Union Jack, or Union Flag, is the de facto national flag of the United Kingdom. Though no law has been passed officially making the Union Jack the national flag of the United Kingdom, it has effectively become the national flag through prec ... (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 a motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral n ... ,
Borromean rings In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; sometimes the term ''trilateral symmetry'' is used; *''n'' = 4, 90°: ''tetrad'',
swastika The swastika symbol, 卐 (''right-facing'' or ''clockwise'') or 卍 (''left-facing'', ''counterclockwise'', or sauwastika), is an ancient religious icon in various Eurasian cultures. It is used as a symbol of divinity and spirituality in Ind ... Star of David The Star of David, known in Hebrew Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasiatic languages, Afroasiatic language family. Historically, it is regarded as the language of the Israelite ... *''n'' = 8, 45°: ''octad'', Octagonal
muqarnas Muqarnas ( ar, مقرنص; fa, مقرنس), also known in Iranian architecture Iranian architecture or Persian architecture (Persian language, Persian: معمارى ایرانی, ''Memāri e Irāni'') is the architecture of Iran and par ...
, computer-generated (CG), ceiling ''C''''n'' is the rotation group of a regular ''n''-sided
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... in 2D and of a regular ''n''-sided
pyramid A pyramid (from el, πυραμίς ') is a structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act ... 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ... of 100° and 360°. A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a
propeller . A propeller is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral, that, when rotated, exerts linear thrust upon a working fluid, such as water or air. Propellers are used to pump fluid through a ... .

## 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 In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
s ''D''n of order 2''n'' (). This is the rotation group of a regular
prism A prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...
, or regular
bipyramid A (symmetric) ''n''-gonal bipyramid or dipyramid is a polyhedron In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the old ...
. 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ... . The group is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... to
alternating group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
''A''4. *3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group ''O'' of order 24 of a
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ... and a regular
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral tri ... . The group is isomorphic to
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
''S''4. *6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group ''I'' of order 60 of a
dodecahedron In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... and an
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non-similarity (geometry), similar shapes of icosahedra, some of them ... . 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 solid In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...
s, 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
. The fundamental domain is a
half-line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ... . 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 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ... and no dependence on either angle using
spherical coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... . The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric or axisymmetrical are
adjective In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most langu ...
s which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a
doughnut A doughnut or donut (IPA IPA commonly refers to: * India pale ale, a style of beer * International Phonetic Alphabet The International Phonetic Alphabet (IPA) is an alphabetic system of phonetic notation based primarily on the Lati ... (
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ... ). 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular
duoprism In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
s.

## Rotational symmetry with translational symmetry

2-fold rotational symmetry together with single
translational symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
is one of the
Frieze groupIn mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction. Such patterns occur frequently in architecture and decorative art. A frieze group is the set of symmetry, symmetries of a frieze ...
s. There are two rotocenters per
primitive cell In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
. Together with double translational symmetry the rotation groups are the following
wallpaper groupImage:Wallpaper group-p4m-5.jpg, 250px, Example of an Egyptian design with wallpaper group #Group p4mm, ''p''4''m'' A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional rep ...
s, with axes per primitive cell: *p2 (2222): 4×2-fold; rotation group of a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ... mic,
rectangular In Euclidean plane geometry, a rectangle is a quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for ... , 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 In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ... lattice. *p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a
hexagonal In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regul ... 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.

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Ambigram An ambigram is a calligraphic design that has several interpretations as written. Etymology The word ambigram was coined by Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, phys ... *
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 Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some ro ...
*
Lorentz symmetry In relativistic mechanics, relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers t ...
*
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 (mathematics), group o ...
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Screw axis A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rot ...
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Space group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Translational symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

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