Concept

Point groups in two dimensions

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. 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.

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Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.
Isometry group
In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space. Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space.
One-dimensional symmetry group
A one-dimensional symmetry group is a mathematical group that describes 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 only nontrivial point group in 1D is a simple reflection. It can be represented by the simplest Coxeter group, A1, [ ], or Coxeter-Dynkin diagram . Affine symmetry groups represent translation. Isometries which leave the function unchanged are translations x + a with a such that f(x + a) = f(x) and reflections a − x with a such that f(a − x) = f(x).
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