
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 

Symmetryequivalent atoms are tabulated in the International Tables like this:
(x,y,z), (y, xy, z+1/3), (yx, x, z+2/3) (this is for spacegroup P3_{1}).
But it is useful to think of symmetry operators as 3x4 or 4x4 matrices and generate symmetry related
atoms thus. Then the symmetryrelated coordinates in P3_{1} 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}].