CCP4 web logo Basic Maths for Protein Crystallographers
Symmetry operators
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Since crystals are repeating lattices with straight edges, the only symmetry elements compatible with crystallinity are:

  rotation axis perpendicular to a crystal plane
2fold tick
3fold tick
4fold tick
5fold ×
6fold tick
screw axis with translation steps
tick 1/2
tick 1/3 or 2/3
tick 1/4, 2/4, or 3/4
× ×
tick 1/6,2/6,3/6,4/6,or 5/6

Symmetry-equivalent atoms are tabulated in the International Tables like this:
(x,y,z), (-y, x-y, z+1/3), (y-x, -x, z+2/3) (this is for spacegroup P31). But it is useful to think of symmetry operators as 3x4 or 4x4 matrices and generate symmetry related atoms thus. Then the symmetry-related coordinates in P31 are symmetry matrix notation,    i.e.    P31 symmetry operators.

If the [Si] is augmented to [S'i], a 4x4 matrix, by the addition of a fourth row, [0 0 0 1], the augmented symmetry operators make a closed group, i.e. for some i,j and k:
[S'i][S'j]=[S'k]and [S'i]-1=[S'k].