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. 4647
Section 2.1.2. Symmetry^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 
A symmetry operation can be defined as an operation which, when applied, results in a structure indistinguishable from the original one. According to this definition, the periodic repetition along a, b and c represents translational symmetry.
In addition, rotational symmetry exists, but only rotational angles of 60, 90, 120, 180 and are allowed (i.e. rotation over 360/n degrees, where n is an integer). These correspond to nfold rotation axes, with and 1 (identity), respectively. Rotation axes with or are not found as crystallographic symmetry axes, because translations of unit cells containing these axes do not completely fill threedimensional space. Another type of rotational symmetry axis is the screw axis. It combines a rotation with a translation. For a twofold screw axis, the translation is over 1/2 of the unitcell length in the direction of the axis; for a threefold screw axis, it is 1/3 or 2/3 etc. In this way, the translational symmetry operators can be obeyed. The requirement that translations are 1/2, 1/3, 2/3 etc. of the unitcell length does not exist for individual objects that are not related by crystallographic translational symmetry operators. For instance, an αhelix has 3.6 residues per turn.
Besides translational and rotational symmetry operators, mirror symmetry and inversion symmetry exist. Mathematically, it can be proven that not all combinations of symmetry elements are allowed, but that 230 different combinations can occur. They are the space groups which are discussed extensively in IT A (2005). The graphical and printed symbols for the symmetry elements are also found in IT A (Chapter 1.4 ).
Biological macromolecules consist of building blocks such as amino acids or sugars. In general, these buildingblock structures are not symmetrical and the mirror images of the macromolecules do not exist in nature. Space groups with mirror planes and/or inversion centres are not allowed for crystals of these molecules, because these symmetry operations interchange right and left hands. Biological macromolecules crystallize in one of the 65 enantiomorphic space groups. (Enantiomorphic means the structure is not superimposable on its mirror image.) Apparently, some of these space groups supply more favourable packing conditions for proteins than others. The most favoured space group is (Table 2.1.2.1). A consequence of symmetry is that multiple copies of particles exist in the unit cell. For instance, in space group (space group No. 4), one can always expect two exactly identical entities in the unit cell, and one half of the unit cell uniquely represents the structure. This unique part of the structure is called the asymmetric unit. Of course, the asymmetric unit does not necessarily contain one protein molecule. Sometimes the unit cell contains fewer molecules than anticipated from the number of asymmetric units. This happens when the molecules occupy a position on a crystallographic axis. This is called a special position. In this situation, the molecule itself obeys the axial symmetry. Otherwise, the molecules in an asymmetric unit are on general positions. There may also be two, three or more equal or nearly equal molecules in the asymmetric unit related by noncrystallographic symmetry.

References
International Tables for Crystallography (2005). Vol. A. SpaceGroup Symmetry, edited by Th. Hahn. Heidelberg: Springer.