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. 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 objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.
The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.
The point groups in three dimensions are heavily 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.
Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups.
Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries H in O(3) and all groups K of isometries in O(3) that contain inversion:
For instance, if H is C2, then K is C2h, or if H is C3, then K is S6. (See lower down for the definitions of these groups.)
If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion:
where isometry ( A, I ) is identified with A. An example would be C4 for H and S4 for M.
Thus M is obtained from H by inverting the isometries in H ∖ L. This group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.
In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (Cn) is normal both in the group (Cnv) obtained by adding to (Cn) reflection planes through its axis and in the group (Cnh) obtained by adding to (Cn) a reflection plane perpendicular to its axis.
The isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as follows:
The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.
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 ).
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.)
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. 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.
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. Note that any physical object having infinite rotational symmetry will also have the symmetry of mirror planes through the axis.
There are seven continuous groups which are all 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.
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.
According to the crystallographic restriction theorem, a limited number of point groups is 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 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:
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:
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.
There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/n. Dnh is the symmetry group for a "regular" n-gonal prism and also for a "regular" n-gonal bipyramid. Dnd is the symmetry group for a "regular" n-gonal antiprism, and also for a "regular" n-gonal trapezohedron. Dn is the symmetry group of a partially rotated ("twisted") prism.
The groups D2 and D2h are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular 2-fold axes. D2 is a subgroup of all the polyhedral symmetries (see below), and D2h is a subgroup of the polyhedral groups Th and Oh. D2 can occur in homotetramers such as Concanavalin A, in tetrahedral coordination compounds with four identical chiral ligands, or in a molecule such as tetrakis(chlorofluoromethyl)methane if all the chlorofluoromethyl groups have the same chirality. The elements of D2 are in 1-to-2 correspondence with the rotations given by the unit Lipschitz quaternions.
The group Sn is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For n odd this is equal to the group generated by the two separately, Cnh of order 2n, and therefore the notation Sn is not needed; however, for n even it is distinct, and of order n. Like Dnd it contains a number of improper rotations without containing the corresponding rotations.
All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. S2n is algebraically isomorphic with Z2n.
The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Here, Cn denotes an axis of rotation through 360°/n and Sn denotes an axis of improper rotation through the same. On successive lines are the orbifold notation, the Coxeter notation and Coxeter diagram, and the Hermann–Mauguin notation (full, and abbreviated if different) and the order (number of elements) of the symmetry group. The groups are:
As noted above for the infinite isometry groups, any physical object having K symmetry will also have Kh symmetry.
This can also be applied for wallpaper groups and frieze groups: for them, the sum of the feature values is 2, giving an infinite order; see orbifold Euler characteristic for wallpaper groups
The reflective point groups in three dimensions are also called Coxeter groups and can be given by a Coxeter-Dynkin diagram and represent a set of mirrors that intersect at one central point, and bound a spherical triangle domain on the surface of a sphere. Coxeter groups with less than 3 generators have degenerate spherical triangle domains, as lunes or a hemisphere. In Coxeter notation these groups are tetrahedral symmetry [3,3], octahedral symmetry [4,3], icosahedral symmetry [5,3], and dihedral symmetry [p,2]. The number of mirrors for an irreducible group is nh/2, where h is the Coxeter group's Coxeter number, n is the dimension (3).
The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups Cn (the rotation group of a canonical pyramid), the dihedral groups Dn (the rotation group of a uniform prism, or canonical bipyramid), and the rotation groups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron.
In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore, it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.
The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.
As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:
They all correspond to a rotation group H and a subgroup L of index 2 in the sense that they are obtained from H by inverting the isometries in H \ L, as explained above:
There are two discrete point groups with the property that no discrete point group has it as proper subgroup: Oh and Ih. Their largest common subgroup is Th. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively.
There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: Oh and D6h. Their maximal common subgroups, depending on orientation, are D3d and D2h.
The smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), Z3 × Z3 (of order 9), the dicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16.
The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C2, Ci, Cs. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.
Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 4n + 1 elements of order 2, and there are three with 4n + 3 elements of order 2 (for each n ≥ 8 ). There is never a positive even number of elements of order 2.
The symmetry group for n-fold rotational symmetry is Cn; its abstract group type is cyclic group Zn, which is also denoted by Cn. However, there are two more infinite series of symmetry groups with this abstract group type:
Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the crystallographic restriction applies:
In 2D dihedral group Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside.
However, in 3D the two operations are distinguished: the symmetry group denoted by Dn contains n 2-fold axes perpendicular to the n-fold axis, not reflections. Dn is the rotation group of the n-sided prism with regular base, and n-sided bipyramid with regular base, and also of a regular, n-sided antiprism and of a regular, n-sided trapezohedron. The group is also the full symmetry group of such objects after making them chiral by e.g. an identical chiral marking on every face, or some modification in the shape.
The abstract group type is dihedral group Dihn, which is also denoted by Dn. However, there are three more infinite series of symmetry groups with this abstract group type:
Thus we have, with bolding of the 12 crystallographic point groups, and writing D1d as the equivalent C2h:
C2n,h of order 4n is of abstract group type Z2n × Z2. For n = 1 we get Dih2, already covered above, so n ≥ 2.
Thus we have, with bolding of the 2 cyclic crystallographic point groups:
Dnh of order 4n is of abstract group type Dihn × Z2. For odd n this is already covered above, so we have here D2nh of order 8n, which is of abstract group type Dih2n × Z2 (n≥1).
Thus we have, with bolding of the 3 dihedral crystallographic point groups:
The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):
The fundamental domain of a point group is a conic solid. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.
For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the disdyakis triacontahedron one full face is a fundamental domain of icosahedral symmetry. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.
Also the surface in the fundamental domain may be composed of multiple faces.
The map Spin(3) → SO(3) is the double cover of the rotation group by the spin group in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.) By the lattice theorem, there is a Galois connection between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3). (Note that Spin(3) has alternative descriptions as the special unitary group SU(2) and as the group of unit quaternions. Topologically, this Lie group is the 3-dimensional sphere S3.)
The preimage of a finite point group is called a binary polyhedral group, represented as ⟨l,n,m⟩, and is called by the same name as its point group, with the prefix binary, with double the order of the related polyhedral group (l,m,n). For instance, the preimage of the icosahedral group (2,3,5) is the binary icosahedral group, ⟨2,3,5⟩.
For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.
Note that this is a covering of groups, not a covering of spaces – the sphere is simply connected, and thus has no covering spaces. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under spin representations or other representations they may stabilize other polyhedra.
This is in contrast to projective polyhedra – the sphere does cover projective space (and also lens spaces), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.