In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. Point groups can exist in a Euclidean space of any dimension. In 2D, a discrete point group is sometimes called a rosette group, and is used to describe the symmetries of an ornament. The 3D discrete 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. See point groups in three dimensions. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... Electron atomic and molecular orbitals In quantum chemistry, the molecular electronic states, i. ... Covalently bonded hydrogen and carbon in a molecule of methane. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

There are infinitely many discrete point groups in each number of dimensions. However, only a finite number is compatible with translational symmetry. This is stated in the crystallographic restriction theorem. In 1D there are 2, in 2D 10, and in 3D 32. They are called crystallographic point groups. In physics and mathematics, translational symmetry is the invariance of an object or a system of equations under the translations - operations that change the coordinates of all objects by a constant. ... 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 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. ...

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

Point groups in 2D fall into two distinct families, according to whether they consist of rotations only, or include reflections. The cyclic groups, C_n (abstract group type Z_n), consist of rotations by 360°/n, and all integer multiples. For example, a swastika has symmetry group C₄, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of dihedral groups, D_n (abstract group type Dih_n), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S₁ is distinct from Dih(S₁) because it explicitly includes the reflections. Note that 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 application, homogeneity up to arbitrarily fine detail in transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored. See also point groups in two dimensions. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... Binomial name Bauhinia blakeana S. T. Dunn Bauhinia blakeana, sometimes called Hong Kong orchid tree, is a tree in the genus Bauhinia. ... Rotation of a planar figure around a point Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... 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. ... The swastika () (Sanskrit: स्वस्तिक) is an equilateral cross with its arms bent at right angles either clockwise or anticlockwise. ... The symmetry group of an object (e. ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... In physics, homogeneity is the quality of having all properties independent of the position. ... The term, transverse means side-to-side as opposed to longitudinal which means front-to-back. In automotive engineering, the term, transverse refers to an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle. ... 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. ...

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. Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ...

More complex symmetries arise in 3D, see point groups in three dimensions. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

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. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... 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. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

See also

• Crystallography
• Crystallographic point group
• Wallpaper group

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. ... 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. ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ...

External links

• Downloadable point group tutorial (Mac and Windows only)
• Molecular symmetry examples
• Web-based point group tutorial (needs Java and Flash)