Point groups in two dimensions

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The Bauhinia blakeana flower on the Hong Kong flag has C₅ symmetry; the star on each petal has D₅ symmetry.

In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. It is a subgroup of the orthogonal group O(2), the group of all isometries which leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(2) itself is a subgroup of the Euclidean group E(2) of all isometries.

Abelian group SO(2), also called the circle group, is a subgroup of E⁺(2), which consists of direct isometries, i.e., isometries preserving orientation; it contains those which leave the origin fixed. The group is isomorphic to R/Z and to U(1), the first unitary group.

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 $\oplus$ 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) which 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 Z₂.

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

[edit] Symmetry 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).

[edit] Combinations with translational symmetry

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) $\cong$ 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 C₁ and C₂.

The isometry groups of p6m are each mapped to one of the point groups of type D₆. For the other 11 wallpaper groups, each isometry group is mapped to one of the point groups of the types D₁, D₂, D₃, or D₄. Also five frieze groups correspond to D₁ and D₂.

For a given hexagonal translation lattice there are two different groups D₃, giving rise to P31m and p3m1. For each of the types D₁, D₂, and D₄ 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 D₄.

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) which is a subgroup of the isometry group which only depends on the translation we started with and the point group associated with the isometry group. This is because e.g. 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 2n-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 2n ≤ 6, hence n = 1, 2, 3, 4, or 6 (the crystallographic restriction theorem).