The crystallographic restriction

A snowflake has six-fold symmetry: turn it by 60^\circ and it drops back onto itself, six times in a full circle. Wallpaper, brick walls, honeycombs and salt crystals repeat with two-, three-, four- and six-fold symmetry too. But hunt through every crystal that has ever been catalogued and you will never find one with clean five-fold symmetry — nor seven-fold, nor eight-fold. Nature seems to have banned them.

This is not an accident or a gap in the collection. It is a theorem. Any pattern that repeats — that has translational symmetry, so a copy of it can be slid along and land exactly on itself — is allowed only the rotation orders 2, 3, 4 and 6. Five-fold symmetry is mathematically impossible for a repeating pattern. That single fact is the hinge this whole strand turns on.

What "periodic" means

A periodic pattern — a tessellation of the plane, or the atoms in a crystal — is one built from a repeating lattice: pick any two directions in which the pattern slides onto itself, and the whole thing is just one tile stamped over and over on that grid. The set of all those slide-vectors forms the lattice, and this is exactly what a crystal is made of.

Now suppose the pattern also has a rotational symmetry of order n — turning it by \tfrac{360^\circ}{n} about some centre leaves it unchanged. The astonishing thing is that the lattice and the rotation cannot get along for just any n. Requiring both at once forces n into a tiny list.

The proof — a shortest arrow can't be beaten

Here is the classic argument, and it is beautifully short. Among all the slide-vectors of the lattice, pick a shortest one; call its length a. Draw it as an arrow from one lattice point A to a neighbour B.

Worked example 1 — testing each order

Let's just try the condition 2\cos\!\left(\tfrac{2\pi}{n}\right) on each candidate and see which give a whole number:

The lone failure at n = 5 is the whole story. (Try n = 7: 2\cos\tfrac{360^\circ}{7} \approx 1.247, again not whole — every order except 2, 3, 4, 6 falls at exactly this hurdle.) The value 0.618 is no ordinary number, by the way: it is \tfrac{1}{\varphi}, the reciprocal of the golden ratio — the same irrational number that haunts five-fold shapes everywhere.

Worked example 2 — pentagons leave a gap

There is a second way to see why five is forbidden, with no trigonometry at all. To tile the plane you must fit shapes snugly around every point, and the angles meeting at a point must add to exactly 360^\circ. A regular pentagon's interior angle is 108^\circ. So:

Because 108^\circ does not divide 360^\circ a whole number of times, regular pentagons can never surround a point cleanly — so they can never be the repeating cell of a periodic tiling. Squares (4 \times 90^\circ), triangles (6 \times 60^\circ) and hexagons (3 \times 120^\circ) all fit perfectly; pentagons alone are locked out. Step through the figure below and watch the gap appear.

Allowed and forbidden centres on a lattice

On a triangular lattice, the six nearest neighbours of any point sit at the corners of a regular hexagon, so a 60^\circ turn carries every dot onto another dot — order 6 works (and with it orders 2 and 3). Then watch a five-armed star try the same trick: its arms fall between the dots, never on them.

See the pentagon gap

Three regular pentagons packed around a point use 324^\circ, and the leftover 36^\circ wedge is highlighted. There is simply no way to close the ring.

For most of the twentieth century the crystallographic restriction was treated as iron law: no crystal could show five-fold symmetry, full stop. Then in 1982 the materials scientist Dan Shechtman shot a beam of electrons through a rapidly cooled aluminium–manganese alloy and got a diffraction pattern with ten bright spots in a ring — clean, sharp five-fold symmetry. Colleagues told him it was impossible; one handed him a textbook and suggested he re-read it.

He was right and the textbook was incomplete. The material was ordered but not periodic — a quasicrystal. It obeys the theorem by escaping its one assumption. Shechtman won the 2011 Nobel Prize in Chemistry for the discovery.

The theorem forbids five-fold symmetry only for periodic patterns — ones with a repeating lattice. That assumption is doing all the work. Drop it, and five-fold symmetry becomes not just possible but famous:

So the crystallographic restriction is exactly what makes Penrose tilings and quasicrystals remarkable, not what makes them impossible. They are the beautiful exception that only exists because they threw away the repeating grid.

The one idea to keep