The restriction, with no lattice assumed

The crystallographic restriction is this collection’s founding theorem: a periodic pattern can have rotations of order one, two, three, four and six, and of no other order. Its proof takes one line.

A rotation that maps a lattice to itself is an integer matrix in the lattice basis. Its trace is therefore an integer. The trace of a rotation by 2π/n is 2cos(2π/n), which lies between −2 and 2. An integer in that interval is one of five numbers, and each corresponds to exactly one n.

Read that proof again and notice what it does not use. It does not use a length. It does not use an angle. It does not use the fact that the lattice sits in a plane with a metric on it. Every step is arithmetic on an integer matrix, and the only geometric-sounding word in it — rotation — is doing no work that its matrix does not do.

The theorem, restated for graphs #

A periodic graph is an infinite graph with a free action of ℤ² whose quotient is finite. That definition mentions no distances; the translations are ℤ² because that is what the definition asks for, not because anything was measured.

Now take an automorphism of the net — a permutation of its vertices carrying edges to edges. It normalises the translation subgroup, since the translations are characterised by the action rather than by any geometry, so it acts on ℤ² and it acts by an integer matrix. Everything the original proof needs is now present, and the argument runs unchanged:

• the matrix is integral, so its trace is an integer;
• if the automorphism has finite order n on the translations, that matrix is a rotation of order n in any realisation, so its trace is 2cos(2π/n);
• an integer in [−2, 2] is one of five values.

So the restriction is a theorem about periodic graphs, and the lattice was never load-bearing. What was load-bearing is that the translations form a rank-two free abelian group and that symmetries normalise them.

Where the measurement is taken #

The claim above is a theorem and this collection’s habit is to measure rather than to quote, so it is measured.

For every net in the registry, the symmetry group is detected from the barycentric placement — the drawing in which every vertex sits at the average of its neighbours — and the linear parts of the operations with determinant one are the integer matrices the theorem is about. Their orders are computed by multiplying the matrix by itself until it comes back to the identity.

The orders that turn up are one, two, three, four and six. Five never appears; nor does seven, eight or twelve.

That is a measurement over nine objects rather than a proof over all of them, and it is worth having for the same reason every measurement here is: an argument nobody has run on anything is an argument nobody has tested.

The group being measured is also not a fact about one drawing, and that is worth stating because it is what makes the orders a property of the graph. A net has no metric, so every plane lattice type is available to it, and the symmetry reported is the largest any of them permits — found by asking each type in turn and keeping the best. So the operations counted are the ones that survive over every metric the net could be given, and a reader who drew the triangular net on a sheared cell and found only a two-fold would have measured their own drawing rather than the net.

Why the graph version is worth having #

A theorem restated is worth something only if the restatement does work the original did not, and this one does three pieces of work.

It separates what the theorem needs from what it was proved about. A reader meeting the restriction in the usual form could reasonably conclude that periodicity and distance are both involved, since a lattice is defined in terms of both. Removing the distances shows they were never used, which is a fact about the theorem rather than about lattices.

It makes the theorem available to objects that have no metric. A crystal net is one such object; so is an abstract group acting on a space nobody has drawn. The statement the translations are ℤ² and the symmetries normalise them is available whenever there is a group and a normal free abelian subgroup of rank two, whatever the group is acting on.

And it locates the failure precisely. Quasicrystals have five-fold symmetry, and the usual explanation — they are not periodic, so the restriction does not apply — is right and slightly glib. The graph version says exactly which line of the proof fails: there is no ℤ² for the symmetry to act on, so there is no integer matrix, so there is no trace to be constrained. Everything after that line was never reached.

Degree five, which is not a rotation of order five #

The commonest confusion in this area is between two numbers that have nothing to do with each other, and a net makes the distinction unusually clean.

A vertex’s degree is how many edges meet at it. A rotation’s order is how many times it has to be applied to get back to the identity. Nothing connects them.

A net may perfectly well have every vertex of degree five, and one in this collection does. There is no five-fold rotation anywhere in it, and there was never any reason to expect one: the restriction is a statement about which rotations an integer matrix can be, and a count of edges is not a rotation.

But the two numbers are not quite as unrelated as that, and the way they touch is worth having, because it runs through neither of the places one would look for it. The group of the net above is p2 and it could not have been p4m. With two vertices in the repeating cell, a group of order eight has to put a four-fold axis on a vertex; a four-fold sorts the edges meeting there into orbits of four, since no direction in the plane is fixed by a quarter-turn; so a vertex standing on a four-fold axis has a degree divisible by four. Five is not.

That is a real constraint linking a degree to a rotation order, and notice what it is not. It is not the crystallographic restriction, which has already permitted the four-fold and would permit it here at any degree. It is the site symmetry of one point, which is a local statement, and it constrains the two numbers together only where a vertex happens to stand on an axis. Move the vertices off the axes and the constraint evaporates while the group stays exactly what it was.

The same confusion appears in the older literature about pentagonal tilings, where the plane cannot be tiled by pentagons is used to mean several incompatible things. Fifteen convex pentagons tile the plane; a net can have faces of five sides; what it cannot have is a rotation of order five.

A five-fold automorphism, and what happens to it #

The sharpest way to see what the restriction restricts is to build something that has a five-fold symmetry and watch the theorem not object.

Take a quotient graph whose five vertices sit in a ring, each carrying the same pair of loops. The cyclic relabelling that sends every vertex to the next carries every edge to an edge: it is an automorphism of the graph of order five, in a collection whose central theorem forbids five-fold symmetry.

There is no contradiction, and the reason is exactly what the theorem says. The automorphism acts on the translations as the identity — it permutes the quotient vertices and moves no translation at all — so the integer matrix whose trace the theorem constrains is the identity matrix, whose order is one. The five lives entirely in the permutation of vertices, which the restriction has nothing to say about.

The second reason, which is the more interesting one #

That net has a further property, and it is the one that closes the argument.

Its barycentric placement puts all five vertices at the same point. Every vertex has the same neighbourhood — the two next round the ring, and its own two loops — so the equations that place them are the same equation, and the solve collapses them. The net is unstable, it has no faithful drawing, and there is therefore no plane for the automorphism to be a motion of.

So the five-fold symmetry fails to become a rotation twice over: it acts trivially on the translations, and the placement it would have had to act on does not exist. The graph may rotate; the plane may not receive it.

That is the distinction the restriction is really about, and stating it in terms of nets makes it sharper than the usual statement in terms of lattices. A lattice has a plane built into it, so the question of whether a symmetry can be realised never arises. A net does not, and the question is separate and answerable.

The measurement’s own limits #

Three things about the measurement should be said, because a table of orders is exactly the kind of evidence that reads as stronger than it is.

It is the nets drawn here, not all nets. The theorem covers every periodic graph in the plane; the table covers the ones this collection draws, which is a dozen and will be more. What the table can do is fail — a rotation of order five or seven appearing in it would mean the machinery was wrong somewhere — and it does not.

The orders come from a detection rather than from the graph directly. The linear parts are those of operations found on the equilibrium placement and then required to carry every edge to an edge. So the measurement is of the automorphisms that are realised as motions, which is the right set for this theorem and is a subset of all automorphisms.

And the distinction between the two sets is the whole subject of the next section. An automorphism that is not realised as a motion is exactly the case the theorem does not constrain, and building one is the only way to show that the constraint is on the action and not on the group.

What the restriction is a restriction on #

Pulling the strands together, the theorem constrains exactly one thing: the action of a symmetry on the translation subgroup.

It does not constrain the automorphism group of the quotient graph, which may have elements of any order at all. It does not constrain the degrees. It does not constrain the faces — the Euler accounting does that, and it is a different theorem with a different proof. And it does not constrain the local environment of a vertex, which is why a five-fold axis can sit in an ordinary crystal as a property of a molecule without being a symmetry of the structure.

Stated that way the theorem is less about crystals than it looks. Any group acting properly on any space, whose translations form a rank-two free abelian group, obeys it. Crystals are simply where the situation arises.

Two other framings of the same arithmetic are worth having, and neither needs a figure because each is a line. The original argument in its original form tabulates the rotation orders and the traces they would require: order five wants a trace of 2cos(72°) = 0.618…, order seven wants 1.247…, order eight wants 1.414…, and none of those is a whole number, while orders one, two, three, four and six want 2, −2, −1, 0 and 1. That is the whole of the restriction, and every non-integer in the column is an order that cannot occur. And the cyclotomic form says the same thing from the other end: an automorphism of order n acting faithfully on ℤ^d satisfies the n-th cyclotomic polynomial, whose degree is Euler’s φ(n), and that degree cannot exceed d. The companion matrix of the fifth cyclotomic polynomial is a perfectly good integer matrix of order five — it simply needs four rows to exist, which is why a five-fold symmetry of a lattice is legal in four dimensions and costs exactly the two extra dimensions φ(5) = 4 demands.

The two proofs, side by side #

This collection carries two proofs of the restriction and it is worth putting the graph version beside them, because the three fail in different places when periodicity is removed.

The arithmetic proof is the one above: integer matrix, integer trace, five values. It needs the translations and nothing else, and it is the version that survives the removal of the metric intact.

The shrinking-vector proof is geometric: assume a five-fold centre, use it to construct a shorter lattice vector than the shortest, contradiction. That argument does use lengths — the whole of it is about one vector being shorter than another — and it does not survive the removal of the metric at all. It is the more vivid of the two and the less portable.

And the cyclotomic version generalises rather than restates: an automorphism of order n acting faithfully on ℤ^d satisfies the n-th cyclotomic polynomial, whose degree is φ(n), and φ(n) must be at most d. In two dimensions that gives the same five orders; in four it admits five-fold, which is where a five-fold symmetry becomes legal and what it costs.

Three arguments, one conclusion, and only two of them are about arithmetic. That the geometric one is the one usually drawn in textbooks is a fact about teaching rather than about the theorem.

Where the same argument fails, and why #

The essay’s point is that the proof needs a free action of ℤ² with a finite quotient and nothing else. Testing that claim means finding a setting where everything else is present and that is absent, and the hyperbolic plane supplies one.

A tiling of the hyperbolic plane by regular heptagons, three at a vertex, has a symmetry group acting freely with a compact quotient — everything the argument seems to need — and it has rotations of order seven. The restriction plainly does not hold there.

What is missing is the abelian translation group. The symmetry group of a hyperbolic tiling has no normal subgroup of translations isomorphic to ℤ²; its analogue of a translation subgroup is a surface group, which is not abelian, has no basis, and provides no matrix for an automorphism to be written in. The step that fails is the first one — there is nothing for the rotation to be an integer matrix of — and every later step is about that matrix.

So the theorem’s hypothesis is exactly ℤ² and not periodicity in any looser sense. That is a sharper statement than “the pattern must be periodic”, and it is the one that explains both exceptions this collection carries: a quasicrystal has a translation group that is not discrete, and a hyperbolic tiling has one that is not abelian. Two different hypotheses failing, two different escapes, and the same list of orders forbidden in neither case.

The other half of the classification #

A theorem forbidding orders is half a statement. The other half is that the permitted orders occur, and it is worth saying that this collection establishes it by exhibiting rather than by arguing.

Every one of 1, 2, 3, 4 and 6 is the order of a rotation of some net in the registry, and the nets are drawn rather than described: the square net has a four-fold, the triangular and honeycomb nets have six-folds and three-folds, and every net has the identity and something of order two. So the list is attained, and no shorter list would be correct.

That matters because a prohibition alone is compatible with the answer being shorter. An argument establishing that no order above six is possible, with three and four never occurring in practice, would leave a theorem that is true and a classification that is wrong — and the difference between the two is exactly the exhibition.

It is the same arrangement this collection uses everywhere. A count is a prohibition together with a construction: seventeen plane groups means no eighteenth and all seventeen drawn; thirty-two classes means no thirty-third and all thirty-two enumerated. A number with only one half behind it is not a classification, and which half is missing is worth saying whenever it is.

What this does not extend to #

It does not extend to aperiodic patterns, and this collection has said so at every opportunity. A Penrose tiling has a five-fold symmetry and no translations at all, so the theorem’s first line is unavailable and nothing it says applies. The same is true of a net with no periodicity: the definition of a periodic graph asks for the ℤ² action, and a graph without one is outside the theorem.

It does not extend to the quotient’s own symmetry. The five-fold example is the demonstration.

And it is not a claim about three dimensions here. The argument runs in any dimension and the answer changes: in three the orders are the same five, in four a five-fold becomes possible, and the machinery for those cases sits in a different part of this collection. Nothing about nets was needed for that and nothing in this essay computes it.

One more place the same argument appears #

The pattern of this essay — a theorem restated with its geometric assumptions removed, and found still to hold — is worth naming because it recurs.

Bieberbach’s second theorem says that two crystallographic groups of the same dimension that are isomorphic as abstract groups are conjugate by an affine map. That is the same kind of statement: a purely algebraic hypothesis forcing a geometric conclusion, with the translations doing the work because they are a characteristic subgroup. The restriction is the same phenomenon one level down — the translations are characteristic, so any symmetry acts on them, and the action is integral.

In both cases the thing that makes the algebra bite is that ℤ² is rigid: it has very few automorphisms of finite order, and those few are the whole of the constraint. A group with a large abelian normal subgroup is a group whose symmetries are severely limited, and crystallography is the study of what is left.

The same rigidity is what makes the measurement above short. There are only four finite-order conjugacy classes of matrices in GL(2, ℤ) that a rotation can be — of orders two, three, four and six, beside the identity — so once a net’s symmetry has been detected, the order of each of its rotations is read off a list with five entries rather than computed against an open-ended set of possibilities. A rotation of order five would not merely be a surprise; there would be nowhere in the list to put it, and the machinery would have to have produced a matrix that is not conjugate to any of them.

Where this goes #

The restriction is the constraint this collection is built around, and seeing it hold with the lattice removed is a small piece of evidence that the constraint is arithmetic rather than geometric. The Euler accounting of the essay beside this one is the other constraint of that kind: a local count forced by a global fact about a quotient, with no lengths in the derivation.

Between them they say most of what can be said about a plane net without doing any geometry: how many edges may meet at a vertex on average, how many may bound a face, and what orders of rotation the whole thing may have.