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. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.
Symmetry groups of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded (finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the origin as one of them.
The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection of its full symmetry group with E+(3), which consists of all direct isometries, i.e., isometries preserving orientation. For a bounded object, the proper symmetry group is called its rotation group. It is the intersection of its full symmetry group with SO(3), the full rotation group of the 3D space. The rotation group of a bounded object is equal to its full symmetry group if and only if the object is chiral.
The point groups that are generated purely by a finite set of reflection mirror planes passing through the same point are the finite Coxeter groups, represented by Coxeter notation.
The point groups in three dimensions are widely used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups.
3D isometries that leave the origin fixed
The symmetry group operations (symmetry operations) are the isometries of three-dimensional space R3 that leave the origin fixed, forming the group O(3). These operations can be categorized as: The direct (orientation-preserving) symmetry operations, which form the group SO(3): The identity operation, denoted by E or the identity matrix I. Rotation about an axis through the origin by an angle θ. Rotation by θ = 360°/n for any positive integer n is denoted Cn (from the Schoenflies notation for the group Cn that it generates). The identity operation, also written C1, is a special case of the rotation operator. The indirect (orientation-reversing) operations: Inversion, denoted i or Ci, that is, rotation by 180° about a coordinate axis followed by a reflection in the orthogonal coordinate plane. The matrix notation is −I. Reflection in a plane through the origin, denoted σ. Improper rotation, also called rotation-reflection: rotation about an axis by an angle θ, combined with reflection in the plane through the origin perpendicular to the axis. Rotation-reflection by θ = 360°/n for any positive integer n is denoted Sn (from the Schoenflies notation for the group Sn that it generates if n is even). Inversion is a special case of rotation-reflection (i = S2), as is reflection (σ = S1), so these operations are often classified as improper rotations.
A circumflex is sometimes added to the symbol to indicate an operator, as in Ĉn and Ŝn.
Conjugacy When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e., they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1 = g−1H2g ).
For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second. (In fact there will be more than one such rotation, but not an infinite number as when there is only one mirror or axis.) The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure is chiral for 11 pairs of space groups with a screw axis.)
Infinite isometry groups
There are many infinite isometry groups; for example, the "cyclic group" (meaning that it is generated by one element – not to be confused with a torsion group) generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical abelian groups by adding more rotations around the same axis. The set of points on a circle at rational numbers of degrees around the circle illustrates a point group requiring an infinite number of generators. There are also non-abelian groups generated by rotations around different axes. These are usually (generically) free groups. They will be infinite unless the rotations are specially chosen.
All the infinite groups mentioned so far are not closed as topological subgroups of O(3). We now discuss topologically closed subgroups of O(3).
The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The other infinite isometry groups consist of all rotations about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry. Any 3D shape (subset of R3) having infinite rotational symmetry must also have mirror symmetry for every plane through the axis. Physical objects having infinite rotational symmetry will also have the symmetry of mirror planes through the axis, but vector fields may not, for instance the velocity vectors of a cone rotating about its axis, or the magnetic field surrounding a wire.
There are seven continuous groups which are all in a sense limits of the finite isometry groups. These so called limiting point groups or Curie limiting groups are named after Pierre Curie who was the first to investigate them. The seven infinite series of axial groups lead to five limiting groups (two of them are duplicates), and the seven remaining point groups produce two more continuous groups. In international notation, the list is ∞, ∞2, ∞/m, ∞mm, ∞/mm, ∞∞, and ∞∞m. Not all of these are possible for physical objects, for example objects with ∞∞ symmetry also have ∞∞m symmetry. See below for other designations and more details.
Finite isometry groups Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also spherical symmetry groups.
Up to conjugacy, the set of finite 3D point groups consists of: , which have at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite cylinder, or equivalently, those on a finite cylinder. They are sometimes called the axial or prismatic point groups. , which have multiple 3-or-more-fold rotation axes; these groups can also be characterized as point groups having multiple 3-fold rotation axes. The possible combinations are: Four 3-fold axes (the three tetrahedral symmetries T, Th, and Td) Four 3-fold axes and three 4-fold axes (octahedral symmetries O and Oh) Ten 3-fold axes and six 5-fold axes (icosahedral symmetries I and Ih) According to the crystallographic restriction theorem, only a limited number of point groups are compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 of the 7 others. Together, these make up the 32 so-called crystallographic point groups.
The seven infinite series of axial groups The infinite series of axial or prismatic groups have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e., symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry (see cyclic symmetries) and three with additional axes of 2-fold symmetry (see dihedral symmetry). They can be understood as point groups in two dimensions extended with an axial coordinate and reflections in it. They are related to the frieze groups; they can be interpreted as frieze-group patterns repeated n times around a cylinder.
The following table lists several notations for point groups: Hermann–Mauguin notation (used in crystallography), Schönflies notation (used to describe molecular symmetry), orbifold notation, and Coxeter notation. The latter three are not only conveniently related to its properties, but also to the order of the group. The orbifold notation is a unified notation, also applicable for wallpaper groups and frieze groups. The crystallographic groups have n restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer. The series are: Intl Schoenflies Orbifold Coxeter Frieze Struct. Order Example Comments Even n Odd n (cylinder) n Cn nn [n]+p1 Znn n-fold rotational symmetry S2n n× [2n+,2+] p11g Z2n2n 2n-fold rotoreflection symmetry n/m Cnh n* [n+,2] p11m Zn×Z22n nmm nm Cnv *nn [n] p1m1 Dihn2n Pyramidal symmetry;in biology, biradial symmetry n22 n2 Dn 22n [n,2]+ p211 Dihn2n Dihedral symmetry 2m m Dnd 2*n [2n,2+] p2mg Dih2n4n Antiprismatic symmetry n/mmm 2m Dnh *22n [n,2] p2mm Dihn×Z24n Prismatic symmetry
For odd n we have Z2n = Zn × Z2 and Dih2n = Dihn × Z2.
The groups Cn (including the trivial C1) and Dn are chiral, the others are achiral.
The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).
The simplest nontrivial axial groups are equivalent to the abstract group Z2: Ci (equivalent to S2) – inversion symmetry C2 (equivalent to D1) – 2-fold rotational symmetry Cs (equivalent to C1hand C1v) – reflection symmetry, also called bilateral symmetry.
The second of these is the first of the uniaxial groups (cyclic groups) Cn of order n (also applicable in 2D), which are generated by a single rotation of angle 360°/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group Cnh of order 2n, or a set of n mirror planes containing the axis, giving the group Cnv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cn or Dn is a propeller.
If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4n is called Dnh. Its subgroup of rotations is the dihedral group Dn of order 2n, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes.
Note: in 2D, Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of frontside and backside; but in 3D, the two operations are distinguished: Dn contains "flipping over", not reflections.