Basic Maths for Protein Crystallographers
	Symmetry operators

Since crystals are repeating lattices with straight edges, the only symmetry elements compatible with crystallinity are:

rotation axis perpendicular to a crystal plane 2fold 3fold 4fold 5fold × 6fold

screw axis with translation steps 1/2 1/3 or 2/3 1/4, 2/4, or 3/4 × × 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 ,    i.e.    .

If the [S_i] 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].