Everything about The Crystallographic Restriction Theorem totally explained
The
crystallographic restriction theorem in its basic form is the observation that the
rotational
symmetries of a
crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. This is strictly true for the mathematical formalism, but in the physical world crystals are finite, and
quasicrystals occur with other symmetries, such as 5-fold.
In mathematics, a crystal is modeled as a discrete
lattice, generated by a list of
independent finite
translations . Because discreteness requires that the spacings between lattice points have a lower bound, the
group of rotational symmetries of the lattice at any point must be a
finite group. The force of the theorem is that
not all finite groups are compatible with a discrete lattice; in any dimension, we'll have only a finite number of compatible groups.
Dimensions 2 and 3
The special cases of 2D (
wallpaper groups) and 3D (
space groups) are most heavily used in applications, and we can treat them together.
Lattice proof
A rotation symmetry in dimension 2 or 3 must move a lattice point to a
succession of other lattice points in the same plane, generating a
regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts . (We might call this a proof in the style of
Busby Berkeley, with lattice
vectors rather than pretty ladies dancing and swirling in geometric patterns.)
(
Compatible: 6-, 4-, 3-, 2-fold) Consider a lattice built from
equilateral triangles. That is, the lattice basis vectors are two sides of an equilateral triangle, and all other displacements are sums of integer multiples of these. With 60° angles at each vertex, six of these triangles exactly fit (sum to 360°) around every lattice point, demonstrating 6-fold rotation symmetry. Instead building from squares, the vertex angles are 90°, four fit around each lattice point, and the rotation symmetry is 4-fold. These examples also exhibit 3-fold and 2-fold symmetry. Thus the possibilities included by the theorem exist.
(
Incompatible: k-fold, k>6) Now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon. If a displacement exists between any two lattice points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a single lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point. But this is
impossible, because the new octagon is about 80% smaller than the original. The significance of the shrinking is that it's unlimited. The same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like; thus no
discrete lattice can have 8-fold symmetry. The same argument applies to any
k-fold rotation, for
k greater than 6.
(
Incompatible: 5-fold) A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every
other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.
Thus the theorem is proved.
Matrix proof
For an alternative proof, consider
matrix properties. The sum of the diagonal elements of a matrix is called the
trace of the matrix. In 2D and 3D every rotation is a planar rotation, and the trace is a function of the angle alone. For a 2D rotation, the trace is 2 cos θ; for a 3D rotation, 1 + 2 cos θ.
Examples
- Consider a 60° (6-fold) rotation matrix with respect to an orthonormal basis in 2D. » :
