point group
(crystallography) A group consisting of the symmetry elements of an object having a single fixed point; 32 such groups are possible.
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(crystallography) A group consisting of the symmetry elements of an object having a single fixed point; 32 such groups are possible.
(DOD, NATO) A geographically or electronically defined location used in stationing aircraft in flight in a predetermined pattern in accordance with air traffic control clearance. See also orbit point.
In mathematics, a point group is a group of geometric symmetries (isometries) leaving a point fixed.
Point groups can exist in a Euclidean space of any dimension. A discrete point group in 2D is sometimes called a rosette group, and is used to describe the symmetries of an ornament. The 3D point groups are heavily used in chemistry, especially to describe the symmetries of a molecule and of orbitals forming covalent bonds, and in this context they are also called molecular point groups.
There are infinitely many discrete point groups in each number of dimensions. However, the crystallographic restriction theorem demonstrates that only a finite number are compatible with translational symmetry. In 1D there are 2, in 2D 10, and in 3D 32 such groups, called crystallographic point groups.
Point groups in 2D fall into two distinct families, according to
whether they consist of rotations only, or include
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.
Cn and Dn 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.
More complex symmetries arise in 3D, see point groups in three dimensions.
In any dimension d, the continuous group of all possible fixed point isometries is the orthogonal group, denoted by O(d); and its continuous subgroup of all possible rotations is the special orthogonal group, denoted by SO(d). This is not Schönflies notation, but the conventional names from Lie group theory.
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