Point groups in three dimensions
All isometries of a bounded (finite) 3D object have one or more common fixed points.
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.
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e., they need not have the same center.
(The structure is chiral for 11 pairs of space groups with a screw axis.)
We may create non-cyclical abelian groups by adding more rotations around the same 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.
Up to conjugacy, the set of finite 3D point groups consists of: 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.
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;[4] they can be interpreted as frieze-group patterns repeated n times around a cylinder.
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).
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.
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.
Dn is the symmetry group of a partially rotated ("twisted") prism.
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.
The following table gives the five continuous axial rotation groups.
They are limits of the finite groups only in the sense that they arise when the main rotation is replaced by rotation by an arbitrary angle, so not necessarily a rational number of degrees as with the finite groups.
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.
The mirror planes bound a set of spherical triangle domains on the surface of a sphere.
Coxeter groups having fewer than 3 generators have degenerate spherical triangle domains, as lunes or a hemisphere.
Therefore, it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.
The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C2, Ci, Cs.
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.
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.
The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.
