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group theory 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 ...
, the symmetry group of a geometric object is the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of all
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transf ...
s under which the object is invariant, endowed with the group operation of
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
. Such a transformation is an invertible mapping of the
ambient space An ambient space or ambient configuration space is the space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, al ...
which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
space, its symmetries form a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of the
isometry group In , the isometry group of a is the of all (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the as operation. Its is the . The elements of the isometry group are sometimes called s of the space. Every isomet ...
of the ambient space. This article mainly considers
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 ...

symmetry
groups 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 consists in assuming a sma ...
, but the concept may also be studied for more general types of geometric structure.


Introduction

We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a
scalar field 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). ...

scalar field
, a function of position with values in a set of colors or substances; as a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
; or as a more general function on the object.) The group of isometries of space induces a
group action 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). ...
on objects in it, and the symmetry group Sym(''X'') consists of those isometries which map ''X'' to itself (as well as mapping any further pattern to itself). We say ''X'' is ''invariant'' under such a mapping, and the mapping is a ''symmetry'' of ''X''. The above is sometimes called the full symmetry group of ''X'' to emphasize that it includes orientation-reversing isometries (reflections,
glide reflection In 2-dimensional 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 propert ...

glide reflection
s and
improper rotation 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), as long as those isometries map this particular ''X'' to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. An object is
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 ...
when it has no
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 ...
-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group. Any symmetry group whose elements have a common fixed point, which is true if the group is finite or the figure is bounded, can be represented as a subgroup of the
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 ...
O(''n'') by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(''n''), and is called the rotation group of the figure. In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit point. That is, every
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...
of the group (the images of a given point under all group elements) forms a
discrete set Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
. All finite symmetry groups are discrete. Discrete symmetry groups come in three types: (1) finite
point group 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 ...
s, which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(''n''); (2) infinite lattice groups, which include only translations; and (3) infinite
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 ...
s containing elements of both previous types, and perhaps also extra transformations like
screw displacement 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 rotat ...
s and glide reflections. There are also
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...
groups (
Lie 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 a ...
s), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(''n'') (the isometry group of R''n''). Two geometric figures have the same ''symmetry type'' when their symmetry groups are ''
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
'' subgroups of the Euclidean group: that is, when the subgroups ''H''1, ''H''2 are related by for some ''g'' in E(''n''). For example: *two 3D figures have mirror symmetry, but with respect to different mirror planes. *two 3D figures have 3-fold
rotational symmetry 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 ...
, but with respect to different axes. *two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction. In the following sections, we only consider isometry groups whose
orbits In celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical obje ...
are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.


One dimension

The isometry groups in one dimension are: *the trivial
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 ...

cyclic group
C1 *the groups of two elements generated by a reflection; they are isomorphic with C2 *the infinite discrete groups generated by a translation; they are isomorphic with Z, the additive group of the integers *the infinite discrete groups generated by a translation and a reflection; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D (which is a
semidirect product 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 ...
of Z and C2). *the group generated by all translations (isomorphic with the additive group of the real numbers R); this group cannot be the symmetry group of a Euclidean figure, even endowed with a pattern: such a pattern would be homogeneous, hence could also be reflected. However, a constant one-dimensional vector field has this symmetry group. *the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih(R). See also
symmetry groups in one dimension A one-dimensional symmetry group is a group (mathematics), mathematical group that describes symmetry, symmetries in one dimension (1D). A pattern in 1D can be represented as a function ''f''(''x'') for, say, the color at position ''x''. The on ...
.


Two dimensions

Up to Two mathematical 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 t ...
conjugacy the discrete point groups in two-dimensional space are the following classes: *cyclic groups C1, C2, C3, C4, ... where C''n'' consists of all rotations about a fixed point by multiples of the angle 360°/''n'' *
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 D1, D2, D3, D4, ..., where D''n'' (of order 2''n'') consists of the rotations in C''n'' together with reflections in ''n'' axes that pass through the fixed point. C1 is the
trivial groupIn 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 ha ...
containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C2 is the symmetry group of the letter "Z", C3 that of a
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 ...

triskelion
, C4 of a
swastika The swastika symbol, 卐 (''right-facing'' or ''clockwise'') or 卍 (''left-facing'', ''counterclockwise'', or sauwastika), is an ancient religious icon An icon (from the Greek language, Greek 'image, resemblance') is a religious work ...

swastika
, and C5, C6, etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four. D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of
bilateral symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pla ...

bilateral symmetry
, for example the letter "A". D2, which is isomorphic to the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes. D3, D4 etc. are the symmetry groups of the
regular polygon In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ...
s. Within each of these symmetry types, there are two
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors. The remaining isometry groups in two dimensions with a fixed point are: *the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the
circle 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 g ...
S1, the multiplicative group of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s of
absolute value 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 ...

absolute value
1. It is the ''proper'' symmetry group of a circle and the continuous equivalent of C''n''. There is no geometric figure that has as ''full'' symmetry group the circle group, but for a vector field it may apply (see the three-dimensional case below). *the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S1) as it is the generalized dihedral group of S1. Non-bounded figures may have isometry groups including translations; these are: *the 7
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 *the 17
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 *for each of the symmetry groups in one dimension, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction *ditto with also reflections in a line in the first direction.


Three dimensions

Up to Two mathematical 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 t ...
conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "cold drop, frozen drop" ...

crystallography
, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This crystallographic restriction of the infinite families of general point groups results in 32
crystallographic point group In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "col ...
s (27 individual groups from the 7 series, and 5 of the 7 other individuals). The continuous symmetry groups with a fixed point include those of: *cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example for a beer
bottle A bottle is a narrow-necked container made of an impermeable material (clay, glass, plastic, aluminium etc.) in various shapes and sizes to store and transport liquids (water, milk, beer, wine, ink, cooking oil, medicine, soft drinks, shampoo ...

bottle
*cylindrical symmetry with a symmetry plane perpendicular to the axis *spherical symmetry For objects with
scalar field 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). ...

scalar field
patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
patterns: for example, in
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 ...

cylindrical coordinates
with respect to some axis, the vector field \mathbf = A_\rho\boldsymbol + A_\phi\boldsymbol + A_z\boldsymbol has cylindrical symmetry with respect to the axis whenever A_\rho, A_\phi, and A_z have this symmetry (no dependence on \phi); and it has reflectional symmetry only when A_\phi = 0. For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry. The continuous symmetry groups without a fixed point include those with a
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 ...
, such as an infinite
helix A helix (), plural helixes or helices (), is a shape like a corkscrew or spiral staircase. It is a type of smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the ...

helix
. See also subgroups of the Euclidean group.


Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or
automorphism 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 t ...

automorphism
group. Each type of
mathematical structure 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 ...
has
invertible mappings
invertible mappings
which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the
Erlangen programme In mathematics, the Erlangen program is a method of characterizing geometry, geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' ...
. For example, objects in a hyperbolic
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
have Fuchsian symmetry groups, which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of Escher.) Similarly, automorphism groups of finite geometries preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space. Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any
finitely presented group In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...
is the symmetry group of its
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

Cayley graph
; the
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
is the symmetry group of an infinite
tree graph In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only wood plants with secondary ...
.


Group structure in terms of symmetries

Cayley's theorem In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group (mathematics), group ''G'' is group isomorphism, isomorphic to a subgroup of the symmetric group acting on ''G''. This can be understood as an example of ...
states that any abstract group is a subgroup of the permutations of some set ''X'', and so can be considered as the symmetry group of ''X'' with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries. For example, let ''G'' = Sym(''X'') be the finite symmetry group of a figure ''X'' in a Euclidean space, and let ''H'' ⊂ ''G'' be a subgroup. Then ''H'' can be interpreted as the symmetry group of ''X''+, a "decorated" version of ''X''. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to ''X'' so as to break all symmetry, obtaining a figure ''X''# with Sym(''X''#) = , the trivial subgroup; that is, ''gX''# ≠ ''X''# for all non-trivial ''g'' ∈ ''G''. Now we get: : X^+ \ = \ \bigcup_ hX^ \quad\text\quad H = \mathrm(X^+).
Normal subgroup 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 (mathematics), ...
s may also be characterized in this framework. The symmetry group of the translation ''gX'' + is the conjugate subgroup ''gHg''−1. Thus ''H'' is normal whenever: : \mathrm(gX^+) = \mathrm(X^+) \ \ \text \ g\in G; that is, whenever the decoration of ''X''+ may be drawn in any orientation, with respect to any side or feature of ''X'', and still yield the same symmetry group ''gHg''−1 = ''H''. As an example, consider the dihedral group ''G'' = ''D''3 = Sym(''X''), where ''X'' is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure ''X''#. Letting τ ∈ ''G'' be the reflection of the arrowed edge, the composite figure ''X''+ = ''X''# ∪ τ''X''# has a bidirectional arrow on that edge, and its symmetry group is ''H'' = . This subgroup is not normal, since ''gX''+ may have the bi-arrow on a different edge, giving a different reflection symmetry group. However, letting H = ⊂ ''D''3 be the cyclic subgroup generated by a rotation, the decorated figure ''X''+ consists of a 3-cycle of arrows with consistent orientation. Then ''H'' is normal, since drawing such a cycle with either orientation yields the same symmetry group ''H''.


See also


Further reading

* * * *


External links

* * {{MathWorld , urlname=TetrahedralGroup , title=Tetrahedral Group
Overview of the 32 crystallographic point groups
- form the first parts (apart from skipping ''n''=5) of the 7 infinite series and 5 of the 7 separate 3D point groups Geometry Symmetry Group theory