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).
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.