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. 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.
Discrete groups 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.
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:
C1, C2, C3, C4, C6, D1, D2, D3, D4, D6 The groups may be constructed as follows:
Cn. Generated by an element also called Cn, which corresponds to a rotation by angle 2π/n. Its elements are E (the identity), Cn, Cn2, ..., Cnn−1, corresponding to rotation angles 0, 2π/n, 4π/n, ..., 2(n − 1)π/n. Dn. Generated by element Cn and reflection σ. Its elements are the elements of group Cn, with elements σ, Cnσ, Cn2σ, ..., Cnn−1σ added. These additional ones correspond to reflections across lines with orientation angles 0, π/n, 2π/n, ..., (n − 1)π/n. Dn is thus a semidirect product of Cn and the group (E,σ). All of these groups have distinct abstract groups, except for C2 and D1, which share abstract group Z2. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are: D1 ~ Z2 and D2 ~ Z2×Z2. In fact, D3 is the smallest nonabelian group. For even n, the Hermann–Mauguin symbol nm is an abbreviation for the full symbol nmm, as explained below. The n in the H-M symbol denotes n-fold rotations, while the m denotes reflection or mirror planes.
More general groups 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:
R ( θ ) = [ cos θ − sin θ sin θ cos θ ] {\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}
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 Cn, elements Cnk = R(2πk/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:
S ( θ ) = [ cos θ sin θ sin θ − cos θ ] {\displaystyle S(\theta )={\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}}
where O(2) has any possible θ. However, the only abelian subgroups of O(2) with reflections are D1 and D2. For discrete dihedral groups Dn, elements Cnkσ = S(2πk/n) When one uses polar coordinates, the relationship of these groups to one-dimensional symmetry groups becomes evident. Types of subgroups of SO(2):
… excerpt ends here. Continue reading the full article.
