The crystallographic restriction

A pattern that repeats can be turned by half a turn, a third of a turn, a quarter of a turn or a sixth of a turn and land on itself. It cannot be turned by a fifth of a turn, or a seventh, or an eighth. Not because nobody has managed it — because it is impossible, and the proof takes one line.

This is the crystallographic restriction, and it is the single most consequential fact in the subject. Every finiteness result downstream — five lattices, seventeen wallpapers, two hundred and thirty space groups, thirty-two crystal classes — inherits its finiteness from here.

The argument #

Suppose a pattern has a lattice, and suppose a rotation $R$ through angle $\theta$ is a symmetry of that pattern.

Then $R$ maps the lattice onto itself. Write $R$ in coordinates along the lattice’s own basis vectors. Because $R$ sends each basis vector to a lattice vector, and lattice vectors have whole-number coordinates, every entry of $R$ in that basis is a whole number.

The trace of a matrix — the sum of its diagonal entries — does not change when the basis changes. In ordinary orthogonal coordinates a rotation by $\theta$ has trace $2\cos\theta$. In the lattice basis it is a sum of whole numbers. So

$$2\cos\theta \in \mathbb{Z}.$$

And that is the whole proof. Cosine lies between $-1$ and $1$, so $2\cos\theta$ lies between $-2$ and $2$, so it is one of $-2, -1, 0, 1, 2$. Those five values give $\theta = 180°, 120°, 90°, 60°, 0°$ — rotations of order two, three, four, six and one.

Nothing else is available. A fifth of a turn would need $2\cos 72° \approx 0.618$, which is not a whole number, and there the matter ends.

Why the trace is the right thing to look at #

The step that carries the argument is the invariance of the trace, and it is worth dwelling on because it is doing something slightly magical.

A change of basis replaces a matrix $M$ by $P^{-1}MP$. Traces are unchanged by that operation — a two-line calculation — so the trace is a property of the transformation rather than of any particular description of it. That means a quantity computed in the convenient coordinates (orthogonal, where the trace is $2\cos\theta$) must equal the same quantity computed in the awkward ones (the lattice basis, where every entry is a whole number).

The two computations are of the same number. One of them says it is $2\cos\theta$; the other says it is an integer. Setting them equal is the entire content of the theorem.

This is a pattern that recurs throughout mathematics: find a quantity that two different descriptions must agree on, compute it both ways, and read off the constraint. The proof does not construct anything or exhaust any cases. It notices that one number has to be two things at once.

What the five values look like #

Each permitted order corresponds to a lattice that can carry it, and the correspondence is tight.

Order one is the identity and imposes nothing. Order two is the half turn, which every lattice has: negating every vector maps any lattice onto itself. Order three and order six both require a hexagonal lattice — and having one implies a good deal about the other, since a threefold rotation composed with the half turn every lattice has gives a sixfold rotation. Order four requires a square lattice.

So the five permitted orders map onto three lattices, and those three plus the two with no rotation beyond a half turn are the five plane lattices. The restriction is not merely compatible with that enumeration; it is where the enumeration comes from.

The five orders, one at a time #

It is worth walking the five permitted values individually, because each has a slightly different character and the differences are informative.

Order one is the identity. It is permitted trivially and it is the group p1’s entire rotational content.

Order two is the half turn, and it is permitted on every lattice without exception. If $\mathbf{v}$ is a lattice vector then so is $-\mathbf{v}$, and negating everything is what a half turn does. This is why every one of the five lattices has a holohedry containing at least the identity and a half turn, and why the oblique lattice’s holohedry has order two rather than one.

Order three requires a hexagonal lattice, and it comes with a subtlety: a threefold rotation composed with the universally available half turn produces a sixfold rotation. So a lattice with threefold symmetry automatically has sixfold symmetry, and there is no separate trigonal lattice in the plane. A pattern on that lattice may still have only threefold symmetry, because the motif can decline what the lattice offers — which is the difference between p3 and p6.

Order four requires a square lattice and is the only order that does. The square lattice’s holohedry has order eight: four rotations and four mirrors.

Order six requires the hexagonal lattice, whose holohedry has order twelve and is the largest in the plane.

Notice what is absent from that list: five, seven, eight, nine and everything above. Eight is the one that surprises people, because an eightfold rosette is easy to draw and octagonal tilings are common in Islamic ornament. Those tilings are not periodic with an eightfold centre; they are periodic patterns containing octagonal motifs, and the eightfold symmetry is local to the motif rather than global to the pattern.

The other proof, and why it is worth having #

There is a second and entirely different argument for the special case of five-fold rotation, and it is geometric rather than algebraic: assume a five-fold centre, construct from it a lattice vector shorter than the shortest one, and contradict.

That proof is the subject of its own essay, and it is worth having for a reason beyond elegance. A fact this load-bearing should be reachable by more than one route. The trace argument is short and slightly slick; the shrinking-vector argument is longer and shows why the impossibility is about distances rather than about angles. A reader who has seen both is much less likely to misremember the result as a convention.

The same instinct governs the diffraction figures on this site, which recompute a pattern’s symmetry from what it would scatter rather than from its point set. Two routes that share nothing but the input are worth more than one route checked twice.

The trace, read backwards #

The correspondence between trace and order is useful in the other direction, and every generator here uses it.

Given an integer matrix that maps a lattice to itself and has determinant $+1$, its trace determines its rotation order immediately: trace $2$ is the identity, $1$ is order six, $0$ is order four, $-1$ is order three, $-2$ is order two. There is no need to compute an angle, take an inverse cosine, or compare a floating-point number against anything.

That is how the symmetry-element marks on every pattern figure are chosen. The detector finds an operation, reads its determinant to decide whether handedness is preserved, reads its trace to get the order, and draws a lens, triangle, square or hexagon accordingly. The classification is a lookup on two integers.

The value of that is not speed. It is that no threshold appears anywhere in the pipeline, so no figure on this site can be wrong because a tolerance was set badly.

How the enumeration is run here #

The bar chart at the top of this essay is not a drawing of a known result. It is the result, computed while the figure is being made.

The generator loops over rotation orders from one to a chosen ceiling, evaluates $2\cos(2\pi/n)$ for each, and asks whether the value is a whole number to within the precision at which the question is unambiguous. The orders that survive are collected, and the figure then asserts that the surviving list is exactly $1, 2, 3, 4, 6$ — throwing, and stopping the build, if it is not.

That is more work than typing five numbers into a template, and the reason for doing it is a rule this site applies everywhere: no count is a memory. Seventeen wallpaper groups, seven friezes, five lattices and five rotation orders are each produced by running an enumeration, and each is checked by a gate that fails if the enumeration changes.

The gate does one further thing that a generator cannot do from the inside: it feeds the machinery cases that ought to be rejected and requires the rejection. A matrix that is not integral in the lattice basis must be refused; a five-fold rotation written out in that basis must be refused. An assertion that has quietly stopped failing is worse than no assertion, because the build stays green and nobody looks again.

What the restriction does not say #

Three misreadings are common enough to be worth heading off, and the third is the one that matters.

It does not forbid five-fold objects. A starfish has five-fold symmetry, a pentagon has five-fold symmetry, and viruses are routinely icosahedral. The restriction applies to patterns with a lattice, and a finite object has no lattice to be restricted by.

It does not forbid five-fold local arrangements. A crystal structure may contain a cluster of five atoms arranged with local five-fold symmetry, provided that arrangement is not a symmetry of the whole infinite structure. Local and global symmetry are different claims, and the restriction is entirely about the global one.

It does not forbid five-fold diffraction. This is the one that caused the trouble. A sharp diffraction pattern with tenfold symmetry was measured in 1982, and the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered.

The resolution is that the theorem’s hypothesis is periodicity, and the material was not periodic. It had long-range order — enough to diffract sharply — without a lattice. Order and periodicity are different properties, and the 1982 measurement is what forced the distinction into the open.

The restriction as a filter on patterns #

Read forwards, the theorem forbids. Read backwards, it identifies, and the backwards reading is how the result is used in practice.

Confronted with an unfamiliar repeating pattern, the fastest way into its structure is to find its highest rotation order. If it is six, the lattice is hexagonal and the group is one of p6 or p6m. If it is four, the lattice is square and the group is one of p4, p4m or p4g. If it is three, the lattice is hexagonal again and the group is one of p3, p3m1 or p31m. If it is two, the lattice may be oblique, rectangular or centred, and five further groups are in play. If it is one, there are two candidates.

So the restriction converts an open-ended question — what symmetry does this have? — into a decision tree five branches wide. That is the practical content of a finiteness theorem, and it is why crystallographers reach for the rotation order before anything else. The same tree, with more branches, is how a space group is identified from a diffraction pattern.

There is one trap in the backwards reading, and it is the one this site’s round trip exists to catch. A pattern drawn with a symmetric motif can display a rotation order that belongs to the motif rather than to the pattern, and the decision tree then leads confidently to the wrong branch. Identifying the order by eye is fast and sometimes wrong; computing it is slower and is not.

In three dimensions, and beyond #

The argument transfers to space with no change of substance. A rotation in three dimensions has trace $1 + 2\cos\theta$, integrality forces $2\cos\theta$ to be an integer again, and the permitted orders are the same five.

That is a reassuring outcome rather than an obvious one. There was no guarantee that the extra dimension would not open up new possibilities, and the reason it does not is that a rotation in space acts as a plane rotation on the plane perpendicular to its axis, with a $1$ contributed by the axis itself. The extra dimension contributes a constant, and constants do not change integrality.

Above three dimensions the answer changes, and the way it changes is illuminating. In four dimensions, order five becomes available: there exist four-dimensional lattices with five-fold symmetry, because the trace condition now involves a sum of two cosines and a sum of two irrational numbers can be an integer even when neither is. In six dimensions, icosahedral symmetry becomes available.

That is not an idle curiosity. The standard modern description of a quasicrystal is as a three-dimensional slice through a six-dimensional periodic lattice, and the five-fold symmetry that is forbidden in three dimensions is inherited from a symmetry that is perfectly legal in six. The restriction is not evaded. It is satisfied in a higher-dimensional space and projected down.

Who noticed, and when #

The result is old and its attribution is diffuse, which is usually a sign that it was noticed independently by several people who each thought it obvious.

Something equivalent appears in the crystallographic literature of the mid-nineteenth century — it is implicit in Bravais’s 1848 enumeration, since the lattice types could not have come out at fourteen without it. Auguste Bravais and Camille Jordan both use it. Its modern one-line form, resting on the invariance of the trace, is a twentieth-century tidying-up of an argument that had been made in longer ways for decades.

The interesting part of the history is not who first stated it but how completely it was believed. By 1980 the restriction was so thoroughly established that a five-fold diffraction pattern was read as evidence of an experimental error rather than of a new kind of matter. That reading was reasonable. It was also wrong, and the correction took most of a decade.

Where the ladder goes next #

The natural companion is the second proof, which reaches the five-fold case by constructing a contradiction out of distances rather than by counting integers.

The natural consequence is the seventeen: the restriction leaves five possibilities for the highest rotation order in a plane group, and enumerating what can sit alongside each is the whole classification.

And the natural sequel is the exception that is not one. Order is not periodicity takes up what a quasicrystal actually does, why its diffraction is sharp, and why the theorem above survives it intact.

What the pictures here cannot show. The restriction is a statement about all patterns, and a figure shows one. The bar chart on this page is an enumeration of a formula, not a survey of examples — no drawing can demonstrate that no seven-fold pattern exists, because the claim is about a completed search rather than about anything visible.