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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a point group is a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of geometric symmetries
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 (isometries
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
) leaving a point fixed.

Overview
Point groups can exist in a Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 of any dimension. A discrete point groups in two dimensions
Point groups in two dimensions

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....
, also called a rosette group, is used to describe the symmetries of an ornament. The point groups in three dimensions
Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere....
 are heavily used in chemistry, especially to describe the symmetries of a molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
 and of molecular orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region....
s forming covalent bond
Covalent bond

A covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms, or between atoms and other covalent bonds....
s, and in this context they are also called molecular point groups.

There are infinitely many discrete point groups in each number of dimensions.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a point group is a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of geometric symmetries
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 (isometries
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
) leaving a point fixed.

Overview


Point groups can exist in a Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 of any dimension. A discrete point groups in two dimensions
Point groups in two dimensions

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....
, also called a rosette group, is used to describe the symmetries of an ornament. The point groups in three dimensions
Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere....
 are heavily used in chemistry, especially to describe the symmetries of a molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
 and of molecular orbital
Molecular orbital

In chemistry, a molecular orbital is a mathematical function that describes the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region....
s forming covalent bond
Covalent bond

A covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms, or between atoms and other covalent bonds....
s, 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
Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetry of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold....
 demonstrates that only a finite number are compatible with translational symmetry
Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
. In 1D there are 2, in 2D 10, and in 3D 32 such groups, called crystallographic point group
Crystallographic point group

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....
s.
Flag of Hong Kong

In two dimensions


Point groups in 2D
Point groups in two dimensions

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....
 fall into two distinct families, according to whether they consist of rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
s only, or include reflection
Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
s. The cyclic group
Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
s, Cn (abstract group type Zn), consist of rotations by 360°/n, and all integer multiples. For example, a swastika
Swastika

The swastika is an equilateral cross with its arms bent at Angle#Types of angles, in either right-facing form or its mirrored left-facing form....
 has symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 C4, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 belongs to the family of dihedral group
Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
s, Dn (abstract group type Dihn), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group
Circle group

In mathematics, the circle group, denoted by T , is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane....
 S1 is distinct from Dih(S1) because the latter explicitly includes the reflections.

An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/v2, which does not include rotation by 180°. Depending on its application, homogeneity
Homogeneity (physics)

In physics, homogeneous mixtures are mixtures that have definite, consistent composition and properties. Particles are uniformly spread. For example, any amount of a given mixture has the same composition and properties....
 up to an arbitrarily fine level of detail in a transverse
Transverse

Transverse may refer to:*Transversality, a concept related to the intersection of manifolds in topology*Transverse City, an album by Warren Zevon...
 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 group
Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry in the pattern. Such patterns occur frequently in architecture and decorative art....
s.

In three dimensions

More complex symmetries arise in 3D. See point groups in three dimensions
Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere....
.

Generalization

In any dimension d, the continuous group of all possible fixed point isometries is the orthogonal group
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
, 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
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 theory.

See also

  • Crystallography
    Crystallography

    Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals....
  • Crystallographic point group
    Crystallographic point group

    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....
  • Molecular symmetry
    Molecular symmetry

    Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical property, such as its molecular dipole moment and its allowed spectroscopy ....
  • Space group
    Space group

    The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the group , which is used to build the set of space groups....
  • X-ray diffraction
  • Bravais lattice
    Bravais lattice

    In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations....


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