International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 
International Tables for Crystallography (2012). Vol. F, ch. 2.1, pp. 4752
Section 2.1.3. Point groups and crystal systems^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 
If symmetry can be recognized in the external shape of a body, like a crystal or a virus molecule, corresponding symmetry elements have no translations, because internal translations (if they exist) do not show up in macroscopic properties. Moreover, they pass through one point, and this point is not affected by the symmetry operations (pointgroup symmetry). For idealized crystal shapes, the symmetry axes are limited to one, two, three, four and sixfold rotation axes because of the spacefilling requirement for crystals. With the addition of mirror planes and inversion centres, there are a total of 32 possible crystallographic point groups.
Not all combinations of axes are allowed. For instance, a combination of two twofold axes at an arbitrary angle with respect to each other would multiply to an infinite number of twofold axes. A twofold axis can only be combined with another twofold axis at . A third twofold axis is then automatically produced perpendicular to the first two (point group 222). In the same way, a threefold axis can only be combined with three twofold axes perpendicular to the threefold axis (point group 32).
For crystals of biological macromolecules, point groups with mirrors or inversion centres are not allowed, because these molecules are chiral. This restricts the number of crystallographic point groups for biological macromolecules to 11; these are the enantiomorphic point groups and are presented in Table 2.1.3.1.

Although the crystals of asymmetric molecules can only belong to one of the 11 enantiomorphic point groups, it is nevertheless important to be aware of the other point groups, especially the 11 centrosymmetric ones (Table 2.1.3.2). This is because if anomalous scattering can be neglected, the Xray diffraction pattern of a crystal is always centrosymmetric, even if the crystal itself is asymmetric (see Sections 2.1.7 and 2.1.8).

The protein capsids of spherical virus molecules have their subunits packed in a sphere with icosahedral symmetry (532). This is the symmetry of a noncrystallographic point group (Table 2.1.3.3). A fivefold axis is allowed because translation symmetry does not apply to a virus molecule. Application of the 532 symmetry leads to 60 identical subunits in the sphere. This is the simplest type of spherical virus (triangulation number ). Larger numbers of subunits can also be incorporated in this icosahedral surface lattice, but then the subunits lie in quasiequivalent environments and T assumes values of 3, 4 or 7. For instance, for particles there are 180 identical subunits in quasiidentical environments.

On the basis of their symmetry, the point groups are subdivided into crystal systems as follows. For each of the point groups, a set of axes can be chosen displaying the external symmetry of the crystal as clearly as possible, and, in this way, the seven crystal systems of Table 2.1.3.4 are obtained. If no other symmetry is present apart from translational symmetry, the crystal belongs to the triclinic system. With one twofold axis or screw axis, it is monoclinic. The convention in the monoclinic system is to choose the b axis along the twofold axis. The orthorhombic system has three mutually perpendicular twofold (screw) axes. Another convention is that in tetragonal, trigonal and hexagonal crystals, the axis of highest symmetry is labelled c. These conventions can deviate from the guide rules for unitcell choice given in Section 2.1.1.
^{†}A rhombohedral unit cell can be regarded as a cube extended or compressed along the body diagonal (the threefold axis) (see Fig. 2.1.3.2).

The seven crystal systems are based on the pointgroup symmetry. Except for the triclinic unit cell, all other cells can occur either as primitive unit cells or as centred unit cells (Section 2.1.1). A total of 14 different types of unit cell exist, depicted in Fig. 2.1.3.3. Their corresponding crystal lattices are commonly called Bravais lattices.