If a discrete group of displacements in the plane has more than one center of rotation, then the only rotations that can occur are
by 2, 3, 4, and 6. This can be shown as follows. It must be true that the sum of
the interior angles divided by the number of sides is a divisor of 
.
 |
where 
is an integer. Therefore, symmetry will be possible only
for
 |
where 
is an integer. This will hold for 1-, 2-, 3-, 4-, and
6-fold symmetry. That it does not hold for 
is seen by noting that 
corresponds to 
. The 
case requires that 
(impossible), and the 
case requires that 
(also impossible).
The point groups that satisfy the crystallographic restriction are called crystallographic point groups.
Although 
-fold
rotations for 
differing from 2, 3, 4, and 6 are forbidden in the strict
sense of perfect crystallographic symmetry, there are exotic materials called quasicrystals
that display these symmetries. In 1984, D. Shechtman discovered a class of aluminum
alloys whose X-ray diffraction patterns display 5-fold symmetry. Since this was long
known to be crystallographically forbidden, this came as quite a shock initially,
until it became apparent that materials exist which are not exact crystals but very
nearly so which display symmetries forbidden by actual crystals. Many known quasicrystals
can be thought of as three-dimensional analogs of the aperiodic tiling produced by
Penrose tiles.
