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In geometry, a **two-dimensional point group ** or **rosette group** is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

- Discrete groups
- More general groups
- Symmetry groups
- Combinations with translational symmetry
- See also
- External links

The two-dimensional point groups are important as a basis for the axial three-dimensional point groups, with the addition of reflections in the axial coordinate. They are also important in symmetries of organisms, like starfish and jellyfish, and organism parts, like flowers.

There are two families of discrete two-dimensional point groups, and they are specified with parameter *n*, which is the order of the group of the rotations in the group.

Group | Intl | Orbifold | Coxeter | Order | Description | |
---|---|---|---|---|---|---|

C_{n} | n | n• | [n]^{+} | n | Cyclic: n-fold rotations. Abstract group Z_{n}, the group of integers under addition modulo n. | |

D_{n} | nm | *n• | [n] | 2n | Dihedral: n-fold rotations and n-fold reflections. Abstract group Dih_{n}, the dihedral group. |

Intl refers to Hermann–Mauguin notation or international notation, often used in crystallography. In the infinite limit, these groups become the one-dimensional line groups.

If a group is a symmetry of a two-dimensional lattice or grid, then the crystallographic restriction theorem restricts the value of *n* to 1, 2, 3, 4, and 6 for both families. There are thus 10 two-dimensional crystallographic point groups:

- C
_{1}, C_{2}, C_{3}, C_{4}, C_{6}, - D
_{1}, D_{2}, D_{3}, D_{4}, D_{6}

The groups may be constructed as follows:

- C
_{n}. Generated by an element also called C_{n}, which corresponds to a rotation by angle 2π/*n*. Its elements are E (the identity), C_{n}, C_{n}^{2}, ..., C_{n}^{n−1}, corresponding to rotation angles 0, 2π/*n*, 4π/*n*, ..., 2(*n*− 1)π/*n*. - D
_{n}. Generated by element C_{n}and reflection σ. Its elements are the elements of group C_{n}, with elements σ, C_{n}σ, C_{n}^{2}σ, ..., C_{n}^{n−1}σ added. These additional ones correspond to reflections across lines with orientation angles 0, π/*n*, 2π/*n*, ..., (*n*− 1)π/*n*. D_{n}is thus a semidirect product of C_{n}and the group (E,σ).

All of these groups have distinct abstract groups, except for C_{2} and D_{1}, which share abstract group Z_{2}. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are: D_{1} ~ Z_{2} and D_{2} ~ Z_{2}×Z_{2}. In fact, D_{3} is the smallest nonabelian group.

For even *n*, the Hermann–Mauguin symbol *n*m is an abbreviation for the full symbol *n*mm, as explained below. The *n* in the H-M symbol denotes *n*-fold rotations, while the m denotes reflection or mirror planes.

Parity of n | Full Intl | Reflection lines for regular polygon |
---|---|---|

Even n | nmm | vertex to vertex, edge center to edge center (2 families, 2 m's) |

Odd n | nm | vertex to edge center (1 family, 1 m) |

These groups are readily constructed with two-dimensional orthogonal matrices.

The continuous cyclic group SO(2) or C_{∞} and its subgroups have elements that are rotation matrices:

where SO(2) has any possible θ. Not surprisingly, SO(2) and its subgroups are all abelian; addition of rotation angles commutes.

For discrete cyclic groups C_{n}, elements C_{n}^{k} = R(2π*k*/*n*)

The continuous dihedral group O(2) or D_{∞} and its subgroups with reflections have elements that include not only rotation matrices, but also reflection matrices:

where O(2) has any possible θ. However, the only abelian subgroups of O(2) with reflections are D_{1} and D_{2}.

For discrete dihedral groups D_{n}, elements C_{n}^{k}σ = S(2π*k*/*n*)

When one uses polar coordinates, the relationship of these groups to one-dimensional symmetry groups becomes evident.

Types of subgroups of SO(2):

- finite cyclic subgroups
*C*_{n}(*n*≥ 1); for every*n*there is one isometry group, of abstract group type Z_{n} - finitely generated groups, each isomorphic to one of the form Z
^{m}Z_{n}generated by*C*_{n}and*m*independent rotations with an irrational number of turns, and*m*,*n*≥ 1; for each pair (*m*,*n*) there are uncountably many isometry groups, all the same as abstract group; for the pair (1, 1) the group is cyclic. - other countable subgroups. For example, for an integer
*n*, the group generated by all rotations of a number of turns equal to a negative integer power of*n* - uncountable subgroups, including SO(2) itself

For every subgroup of SO(2) there is a corresponding uncountable class of subgroups of O(2) that are mutually isomorphic as abstract group: each of the subgroups in one class is generated by the first-mentioned subgroup and a single reflection in a line through the origin. These are the (generalized) dihedral groups, including the finite ones *D*_{n} (*n* ≥ 1) of abstract group type Dih_{n}. For *n* = 1 the common notation is *C*_{s}, of abstract group type Z_{2}.

As topological subgroups of O(2), only the finite isometry groups and SO(2) and O(2) are closed.

These groups fall into two distinct families, according to whether they consist of rotations only, or include reflections. The * cyclic groups *, C_{n} (abstract group type Z_{n}), consist of rotations by 360°/*n*, and all integer multiples. For example, a four-legged stool has symmetry group C_{4}, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of * dihedral groups *, D_{n} (abstract group type Dih_{n}), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S_{1} is distinct from Dih(S_{1}) because the latter explicitly includes the reflections.

An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/√2, which does not include rotation by 180°. Depending on its application, homogeneity up to an arbitrarily fine level of detail in a transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored.

C_{n} and D_{n} for *n* = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups.

The 2D symmetry groups correspond to the isometry groups, except that symmetry according to O(2) and SO(2) can only be distinguished in the generalized symmetry concept applicable for vector fields.

Also, depending on application, homogeneity up to arbitrarily fine detail in transverse direction may be considered equivalent to full homogeneity in that direction. This greatly simplifies the categorization: we can restrict ourselves to the closed topological subgroups of O(2): the finite ones and O(2) (circular symmetry), and for vector fields SO(2).

These groups also correspond to the one-dimensional symmetry groups, when wrapped around in a circle.

*E*(2) is a semidirect product of *O*(2) and the translation group *T*. In other words, *O*(2) is a subgroup of *E*(2) isomorphic to the quotient group of *E*(2) by *T*:

*O*(2)*E*(2)*/ T*

There is a "natural" surjective group homomorphism *p* : *E*(2) → *E*(2)*/ T*, sending each element *g* of *E*(2) to the coset of *T* to which *g* belongs, that is: *p* (*g*) = *gT*, sometimes called the *canonical projection* of *E*(2) onto *E*(2) */ T* or *O*(2). Its kernel is *T*.

For every subgroup of *E*(2) we can consider its image under *p*: a point group consisting of the cosets to which the elements of the subgroup belong, in other words, the point group obtained by ignoring translational parts of isometries. For every *discrete* subgroup of *E*(2), due to the crystallographic restriction theorem, this point group is either *C*_{n} or of type *D*_{n} for *n* = 1, 2, 3, 4, or 6.

C_{n} and *D*_{n} for *n* = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups, and the four groups with *n* = 1 and 2, give also rise to 7 frieze groups.

For each of the wallpaper groups p1, p2, p3, p4, p6, the image under *p* of all isometry groups (i.e. the "projections" onto *E*(2) */ T* or *O*(2) ) are all equal to the corresponding *C*_{n}; also two frieze groups correspond to *C*_{1} and *C*_{2}.

The isometry groups of p6m are each mapped to one of the point groups of type *D*_{6}. For the other 11 wallpaper groups, each isometry group is mapped to one of the point groups of the types *D*_{1}, *D*_{2}, *D*_{3}, or *D*_{4}. Also five frieze groups correspond to *D*_{1} and *D*_{2}.

For a given hexagonal translation lattice there are two different groups *D*_{3}, giving rise to P31m and p3m1. For each of the types *D*_{1}, *D*_{2}, and *D*_{4} the distinction between the 3, 4, and 2 wallpaper groups, respectively, is determined by the translation vector associated with each reflection in the group: since isometries are in the same coset regardless of translational components, a reflection and a glide reflection with the same mirror are in the same coset. Thus, isometry groups of e.g. type p4m and p4g are both mapped to point groups of type *D*_{4}.

For a given isometry group, the conjugates of a translation in the group by the elements of the group generate a translation group (a lattice)—that is a subgroup of the isometry group that only depends on the translation we started with, and the point group associated with the isometry group. This is because the conjugate of the translation by a glide reflection is the same as by the corresponding reflection: the translation vector is reflected.

If the isometry group contains an *n*-fold rotation then the lattice has *n*-fold symmetry for even *n* and 2*n*-fold for odd *n*. If, in the case of a discrete isometry group containing a translation, we apply this for a translation of minimum length, then, considering the vector difference of translations in two adjacent directions, it follows that *n* ≤ 6, and for odd *n* that 2*n* ≤ 6, hence *n* = 1, 2, 3, 4, or 6 (the crystallographic restriction theorem).

In group theory, the **symmetry group** of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space 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*).

In mathematics, a **dihedral group** is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

In linear algebra, an **orthogonal matrix** is a real square matrix whose columns and rows are orthogonal unit vectors.

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In physics and mathematics, the **Lorentz group** is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In 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 symmetries of a frieze pattern, specifically the set of isometries of the pattern, that is geometric transformations built from rigid motions and reflections that preserve the pattern. The mathematical study of frieze patterns reveals that they can be classified into seven types according to their symmetries.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.

In mathematics, the **circle group**, denoted by , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the **unit complex numbers**

In geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.

In mathematics, a **Euclidean group** is the group of (Euclidean) isometries of an Euclidean space 𝔼^{n}; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension *n* of the space, and is commonly denoted E(*n*) or ISO(*n*).

**Rotational symmetry**, also known as **radial symmetry** in biology, 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.

In geometry, a **point group** is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension *d* is a subgroup of the orthogonal group O(*d*). Point groups can be realized as sets of orthogonal matrices *M* that transform point *x* into point *y*:

In geometry, a **Euclidean plane isometry** is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

The **crystallographic restriction theorem** in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

In mathematics, the group of **rotations about a fixed point in four-dimensional Euclidean space** is denoted **SO(4)**. The name comes from the fact that it is the special orthogonal group of order 4.

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 of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

In a group, the **conjugate** by *g* of *h* is *ghg*^{−1}.

In mathematics, the **generalized dihedral groups** are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group *O*(2).

A **line group** is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations.

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