### The crystallographic restriction theorem generally states that the rotational symmetries of a crystal are limited.

### We have seen from a previous section that a lattice is formed by repeating lattice points that have the same environment. It is this fundamental geometric property of a lattice that restricts the number of rotational symmetries of a space lattice and hence the types of unit cells.

To illustrate the ** crystallographic restriction theorem**, let’s consider a row of lattice points (indicated by the line connecting blue dots in figure I of the below diagram) that are equally spaced by the basis vector

**.**

*a*Assuming that the lattice has a rotational symmetry given by the angle , a rotation about the lattice point *X* by *+*, around an axis perpendicular to the plane of this page, maps the blue lattice point *X-* to the pink lattice point *Y*. Similarly, a rotation about the same lattice point *X* by *–* maps the blue lattice point *X+* to the green lattice point *Z*.

Since the rotations are symmetry operations, *Y* and *Z* are also lattice points. As mentioned in an earlier article, the lattice points in a two-dimensional lattice are described by the position vector ** r** =

*u*+

**a***v*where

**b***u, v ∈*. Therefore,

**must be a lattice vector separated by an integer multiple of**

*YZ***, i.e.**

*a**+*

**YZ**= u**a***0*or

**b****=**

*YZ**u*, where

**a***u ∈*

*.*With reference to figure II of the above diagram,

Since* -1 ≤ cos ≤ 1*,

Substituting eq5 in eq4, the possible values of are 180^{o}, 120^{o}, 90^{o}, 60^{o}, 0^{o} (or 360^{o}). As we have assumed that a rotation of the lattice by the angle transforms the lattice into a state that is indistinguishable from the starting state, the possible values of correspond to 2-, 3-, 4-, 6- and 1-fold symmetry of the two-dimensional lattice.

For a space lattice in three dimensions, consider a three-dimensional lattice vector * r = ua + vb + wc* that is neither parallel nor perpendicular to the axis of rotation (see diagram below). A rotation by maps

*onto another lattice vector*

**r****that has the same magnitude as**

*r’***. Since both**

*r***and**

*r***are lattice vectors, their difference**

*r’**is also a lattice vector, which lies on a plane that is perpendicular to the axis of rotation.*

**r’**–**r**By varying the integers *u*, *v* and *w*, we can extend *m* to a row of points *mno* that are spaced apart. This reduces a three-dimensional lattice proof of the crystallographic restriction theorem to a two-dimensional one, which is already given above.

We have proven the crystallographic restriction theorem with trigonometry. The proof can also be made with linear algebra (see this article).