The symmetry group of an object (e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant), with composition as the operation. It is a subgroup of the isometry group of the space concerned. Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... An invariant in mathematics is something that does not change under a set of transformations. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...

(If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts, see below.) In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations and compositions of these) which leave the figure invariant is called proper symmetry group. Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...

Any symmetry group whose elements have a common fixed point can be represented as a subgroup of orthogonal group O(n) (by choosing the origin to be a fixed point). This is true for all finite symmetry groups, and also for the symmetry groups of bounded figures. Of course, the proper symmetry group is a subgroup of SO(n) then, and therefore also called rotation group of the figure. In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

Discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections - they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. In mathematics, a discrete group is a group G equipped with the discrete topology. ... A cherry lattice pastry A mathematical lattice is a type of partially ordered set. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Two dimensions

The discrete point groups in 2 dimensional space consist of two infinite families

• cyclic groups C₁, C₂, C₃, C₃,... where C_n consists of all rotations about a fixed point by multiples of the angle 2π/n
• dihedral groups D₁, D₂, D₃, D₄,... where D_n (of order 2n) consists of the rotations in C_n together with reflections in n axes that pass through the fixed point.

For the dihedral groups, this is a classification up to conjugacy. The n evenly spaced reflection axes can be rotated up to an angle of π/n, before coinciding with the original set. Two subgroups H₁, H₂ of a group G are conjugate, if there exists g ∈ G such that H₁=g^-1H₂g. This corresponds to a rotation, a reflection, and an opposite rotation, together amounting to a reflection in another axis. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

C₁ is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C₂ is the symmetry group of the letter Z, C₃ that of a triskelion, C₄ of a swastika, and C₅, C₆ etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four. The armoured triskelion on the flag of the Isle of Man Triskelion (or triskele, from Greek τρισκελης three-legged) is a symbol consisting of three bent human legs, or, more generally, three interlocked spirals, or any similar symbol with three protrusions exhibiting a symmetry of the cyclic group C3. ... The swastika (卐) is an equilateral cross with its arms bent at right angles either clockwise or anticlockwise. ...

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, for example the letter A. D₂, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D₃, D₄ etc. are the symmetry groups of the regular polygons. In biology, bilateral symmetry is a characteristic of multicellular organisms, particularly animals. ... In mathematics, the Klein four-group (or just Klein group or Viergruppe, often symbolized by the letter V), named after Felix Klein, is a group with four elements, the smallest non-cyclic group. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Providing the figure is bounded and topologically closed (so that the group is a complete point group) the only other possibility is the orthogonal group O(2), also called Dih(S¹) (where S¹ denotes the circle group: the multiplicative group of complex numbers of absolute value 1) 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. Its proper symmetry group is the circle group, the continuous equivalent of C_n, but there is no figure which has as full symmetry group the circle group. The closure condition here is a natural one for subsets of the plane that can be considered "figures", as it excludes non-drawable sets such as the set of all points on the unit circle with rational coordinates. The symmetry group of this set includes some, but not all, arbitrarily small rotations. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, the circle group is the group of all complex numbers on the unit circle under multiplication. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... Illustration of a unit circle. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

For non-bounded figures, the symmetry group can include translations, so that the seventeen wallpaper groups and seven frieze groups are possibilities. Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ...

Three dimensions

There are infinitely many 3D point groups. This should not be confused with the fact that in crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups. In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ...

The situation in 3-D is more complicated than in 2D, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group, and then there are three groups of order 2, called C_s (or C_1h), C_i, and C₂. These have the single symmetry operation of reflection in a plane, in a point, and in a line (equivalent to a rotation of π), respectively.

The last of these is the first of the uniaxial groups C_n, which are generated by a single rotation of angle 2π/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group C_nh, or a set of n mirror planes containing the axis, giving the group C_nv.

If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through π, so the group is no longer uniaxial. This new group is called D_nh. Its subgroup of rotations called D_n still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. There is one more group in this family, called D_nd (or D_nv), which has vertical mirror planes containing the main rotation axis but located halfway between the other 2-fold axes, so the perpendicular plane is not there. D_nh and D_nd are the symmetry groups for regular prisms and antiprisms, respectively. D_n is the symmetry group of a partially rotated prism. In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

There is one more group in this family to mention, called S_n. This group is generated by an improper rotation of angle 2π/n - that is, a rotation followed by a reflection about a plane perpendicular to its axis. For n odd, the rotation and reflection are generated, so this becomes the same as C_nh, but it remains distinct for n even. In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...

Of these 7 families of groups described above, it has been noted that there are some coincidences for small values of n. In particular

• C_s, C_1h and C_1v are all groups of order 2 with a single reflection
• C₂ and D₁ are groups of order 2 with a single 180° rotation
• C_i and S₂ are groups of order 2 with a single inversion
• D_1h and C_2v are groups of order 4 with a reflection in a plane π and a 180° rotation through a line in π
• D_1d and C_2h are groups of order 4 with a reflection in a plane π and a 180° rotation through a line perpendicular to π

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using C_n to denote an axis of rotation through 2π/n and S_n to denote an axis of improper rotation through the same, the groups are:

• T (tetrahedral). There are four C₃ axes, directed through the corners of a cube, and three C₂ axes, directed through the centers of its faces. There are no other symmetry operations, giving the group an order of 12. This group is isomorphic to A₄, the alternating group on 4 letters, and is the rotation group for a regular tetrahedron.

• T_d. This group has the same rotation axes as T, but with six mirror planes, each containing a single C₂ axis and four C₃ axes. The C₂ axes are now actually S₄ axes. This group has order 24, and is the symmetry group for a regular tetrahedron. T_d is isomorphic to S₄, the symmetric group on 4 letters.

• T_h. This group has the same rotation axes as T, but with mirror planes, each containing two C₂ axes and no C₃ axes. The C₃ axes become S₆ axes, and a center of inversion appears. Again, group has order 24. T_h is isomorphic to A₄ × C₂.

• O (octahedral). This group is similar to T, but the C₂ axes are now C₄ axes, and a new set of 12 C₂ axes appear, directed towards the edges of the original cube. This group of order 24 is also isomorphic to S₄, and is the rotation group of the cube and octahedron.

• O_h. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group has order 48, is isomorphic to S₄ × C₂, and is the symmetry group of the cube and octahedron.

• I, I_h (icosahedral) are the groups of symmetries of the icosahedron and the dodecahedron. The group of proper rotations, I, is a normal subgroup of index 2 in the full group of symmetries, with I having order 60 and I_h having order 120. The group I is isomorphic to A₅, the alternating group on 5 letters, and I_h to A₅ × C₂....

This completes the classification (up to conjugacy) of the finite subgroups of O(3). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces. ... A dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ... 5 (five) is the natural number following 4 and preceding 6. ...

The classification of the finite subgroups of SO(3), which occur as rotation groups of bounded figures, is easier: Up to conjugacy the finite subgroups of SO(3) are the following classes: the cyclic groups C₁, C₂, C₃ etc., the dihedral groups D₂, D₃, D₄ etc., and the rotation groups T, O and I of a regular tetrahedron, octahedron and icosahedron.

In particular, the dihedral groups D₃, D₄ etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.

Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...

Related topics

• symmetry
• Euclidean plane isometry
• group action
• point group
• crystallographic group
  • crystallographic point group
  • space group