Crystallography Restriction

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 360 degrees.

 (180 degrees(n-2))/n=(360 degrees)/m,

where m is an integer. Therefore, symmetry will be possible only for


where m is an integer. This will hold for 1-, 2-, 3-, 4-, and 6-fold symmetry. That it does not hold for n>6 is seen by noting that n=6 corresponds to m=3. The m=2 case requires that n=n-2 (impossible), and the m=1 case requires that n=-2 (also impossible).

The point groups that satisfy the crystallographic restriction are called crystallographic point groups.

Although n-fold rotations for n 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.

See also

Crystallographic Point Groups, Penrose Tiles, Point Groups, Symmetry

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Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 5, 1999.Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., p. 5, 1999.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 304, 1999.Yale, P. B. Geometry and Symmetry. New York: Dover, p. 104, 1988.

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Crystallography Restriction

Cite this as:

Weisstein, Eric W. "Crystallography Restriction." From MathWorld--A Wolfram Web Resource.

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