The symmetry of crystals. The crystallographic restriction theorem

Although objects themselves may appear to have 5-fold, 7-fold, 8-fold, or higher-fold rotation axes, these are not possible in crystals. Crystals can only show 2-fold, 3-fold, 4-fold or 6-fold rotation axes. The reason is that the external shape of a crystal is based on a geometric arrangement of atoms. In fact, if we try to combine objects with 5-fold and 8-fold apparent symmetry, we cannot combine them in such a way that they completely fill space, as illustrated below:

Packing of objects displaying 5- and 8-fold symmetry. Note that they do not fill completely the space and therefore these symmetry elements are not compatible with crystal lattices..

If we assume that 5-fold symmetry is possible, we would be able to draw the grid points shown below as small gray circles, defined by the shortest translation vectors t1 = t2 = t3 = t4 = t5. The sum of any two lattice vectors must also be a lattice vector, but no lattice vector can be shorter than the shortest lattice vectors we have just set up. In fact, if we take the sum of the t1 and t4 vectors we will get a new vector  (
t1 + t4, shown in red) whose magnitude is less than our “shortest” lattice vectors, what destroys the hypothesis.

Geometric proof of the impossibility of a lattice with 5-fold symmetry

Consider a grid line with lattice points (gray circles) separated by a translation vector t (see below). If we rotate the line around a symmetry axis of order n (perpendicular to the drawing),
+α (=360º/n), we will get the line with blue circles. And similarly, applying the reverse rotation, -α we will get the line of red circles.

Applying a rotation axis of order n to a grid line (gray circles)

If the
α rotation is a lattice symmetry operation, the blue circles will correspond to lattice points, and similarly will occur with the red circles. And if this is so, any distance between blue and red circles must be equal to an integral multiple of the lattice translation (t): m.t, m'.t, etc., where m is an integer.

In the isosceles triangle (obtained after rotating the grid line) the following expression can be written:

cos α = (1/2) m.t / t

that is:
cos α  = m / 2

And since the value of the cosine function must be between -1 and +1, only five possibilities will be allowed, corresponding to the rotation axes of order 2, 3, 4, 6 and 1 (rotation of 0º or 360º):

But let's go back...