In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind. That is, an infinite crystal would look exactly the same before and after any of the operations in its point group. In the classification of crystals, each point group corresponds to a crystal class. Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Sphere symmetry group o. ... Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid...

There are infinitely many 3D point groups, however, in crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups. The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ...

The point group of a crystal, among other things, determines some of the crystal's optical properties, such as whether it is birefringent, or whether it shows the Pockels effect. Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... The Pockels effect, or Pockels electro-optic effect, is the production of birefringence in an optical medium induced by a constant or varying electric field. ...

Notation

The point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, mineralogists, and physicists.

For the correspondence of the two systems below, see crystal system. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

Schönflies notation

Main article: Schoenflies notation

For more details see Point groups in three dimensions.

In Schönflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... Arthur Moritz Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ...

• The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (O_h) or without (O) improper operations (those that change handedness).
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper operations, T excludes improper operations, and T_h is T with the addition of an inversion.
• C_n (for cyclic) indicates that the group has an n-fold rotation axis. C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_nv is C_n with the addition of a mirror plane parallel to the axis of rotation.
• S_n (for Spiegel, German for mirror) denotes a group that contains only an n-fold rotation-reflection axis.
• D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a two-fold axis perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nv has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.

Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6. An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... A mirror, reflecting a vase. ... In geometry, the dihedral is the angle between two planes. ... The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...

Hermann-Mauguin notation

An abbreviated form of the Hermann-Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are In crystallography, Hermann-Mauguin notation is used to represent the symmetry elements in point- and space groups. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

• 1, 1
• 2, m, ²⁄_m
• 222, mm2, mmm
• 4,4, ⁴⁄_m, 422, 4mm, 42m, ⁴⁄_mmm
• 3, 3, 32, 3m, 3m
• 6, 6, ⁶⁄_m, 622, 6mm, 62m, ⁶⁄_mmm
• 23, m3, 432, 43m, m3m

See also

• Point group
• Space group
• Point groups in three dimensions
• Crystal system

In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

External links

• Pictorial overview of the 32 groups
• Property overview of the 32 groups