crystallographic groups

One of the first scientific applications of group theory was to the study of the structure of crystals. In fact the work on groups (simultaneously by Fëderov and Schoenflies in 1890) made predictions about crystals that were not confirmed experimentally until much later by von Laue (1912) using X-rays.

The basic result is that the symmetry group of an infinite three-dimensional repeating pattern is one of 230 distinct groups. The point being that the pattern is determined by its symmetry group. Thus this solves a basic problem in crystallography, what crystal lattices can occur.

The same problem in dimension two amounts to classifying the different types of wallpaper patterns. This time it turns out that there are exactly seventeen types.

One of the key ingredients in proving these results. is the Crystallographic restriction.

Theorem A rotation of an infinite repeating two or three-dimensional pattern has order 1,2,3,4 or 6.

This is important because it narrows the range of possibilities for the point group of the crystal or wallpaper pattern. The point group of a symmetry group is is the quotient group by the normal subgroup of all translations it contains.

Very roughly speaking if you think of a crystal being composed of atoms at points on a lattice in 3-space then the point group gives the internal atomic symmetry, whereas the symmetry group tells you about the entire crystal.

In dimension three, the crystallogrphic restriction forces that the subgroup of the point group consisting of rotations is either cyclic of order 1,2,3,4,6, a dihedral group D₂, D₃, D₄,D₆, or the rotational symmetry group of the cube or tetrahedron (see also symmetry groups of the Platonic solids).

We finish with an example in dimension two. Consider the infinite wallpaper pattern a finite segment of which is shown below:

-           -          -          -
 |           |          |          |
                                                A

 |           |          |          |
-           -          -          -
   B

       -          -          -
      |          |          |


      |          |          |
       -          -          -


-           -          -          -
 |           |          |          |


 |           |          |          |
-           -          -          -


       -          -          -
      |          |          |


      |          |          |
       -          -          -

In this case the pattern has symmetry group generated by right translation by one unit, upwards translation by one unit, rotation about B through pi and reflection in the horizontal line through A.