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 *D*_{2}, D_{3}, D_{4},D_{6},
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.