The degrees that name the restriction

The crystallographic restriction is usually proved in one of two ways in this collection. Either the trace of an integer matrix is an integer, and 2cos(2π/n) is an integer for exactly five values of n; or a five-fold rotation of a lattice produces a shorter vector than the shortest, which is a contradiction. Both arguments start from a lattice on their first line.

There is a third route, and it is odd enough to be worth an essay. Count the invariant polynomials of the reflection group with an n-fold rotation. The counting needs no matrix and no lattice — it is a pairing of monomials — and it says that the group’s invariants are generated in degrees 2 and n, for every n whatever. Then ask which of those groups can be written in integer matrices, and the restriction is the answer to a question that has nothing to do with the counting.

Counting without a matrix #

Write the plane as one complex coordinate z. The dihedral group Dₙ acts on it in the plainest possible way: the rotation multiplies z by an n-th root of unity, and the reflection replaces z by its conjugate.

A polynomial in the plane is a polynomial in z and z̄, and the monomials are zᵃ z̄ᵇ. The rotation multiplies such a monomial by ζᵃ⁻ᵇ, so the monomial is invariant under the rotation exactly when a ≡ b modulo n. The reflection exchanges a and b, so it identifies the monomial zᵃ z̄ᵇ with zᵇ z̄ᵃ.

Counting the invariants of each degree is then counting the unordered pairs {a, b} with a + b = d and a ≡ b mod n. That is a small combinatorial computation, exact, and available for every n — five, seven, seventeen, whatever — because nothing in it asks for a matrix with integer entries or for a lattice to preserve.

The counts that come back are the expansion of

$$\frac{1}{(1 - t^2)(1 - t^n)}$$

which says the ring is generated in degrees 2 and n, and the agreement between the combinatorial count and that series is checked term by term rather than asserted.

Where the two degrees come from #

They are visible in the complex picture. The product z z̄ is the squared modulus: invariant under the rotation, since the phases cancel, and under the conjugation, since it is real. That is the generator of degree two, and it is the metric — the invariant quadratic form any finite group of matrices has.

The other is zⁿ + z̄ⁿ. The rotation multiplies zⁿ by ζⁿ = 1, so both terms are separately invariant under it; the conjugation exchanges them, so the sum survives. Its degree is n, and its level curves are the n-lobed shapes a reader would draw for an n-fold pattern.

Every invariant is a polynomial in those two. Chevalley’s theorem says so for any reflection group, and the counting above confirms it in this case: the sequence of dimensions is exactly the number of ways to write each degree as 2a + nb.

So the pair (2, n) is the complete invariant-theoretic description of Dₙ, and the product 2n is the order of the group, which is the numerical identity the free rings essay checks on the crystallographic cases.

The restriction is the second question #

Nothing above has excluded any n. D₅ exists, acts on the plane, and has an invariant ring generated in degrees 2 and 5; so does D₇, and D₁₇. The counting is indifferent.

What a crystal requires is not that the group exist but that it preserve a lattice — that its operations be integer matrices in some basis. That is a different demand, and the arithmetic that decides it is the one this collection has made twice: an integer matrix has an integer trace, the trace of a rotation by 2π/n is 2cos(2π/n), and that number is an integer only for n in {1, 2, 3, 4, 6}.

So the picture is two-layered. The invariant theory says what the ring of each Dₙ looks like, for all n; the lattice says which Dₙ may act on a crystal. Neither statement contains the other, and putting them side by side is the point of the figure above: the ring exists in every row, and the middle column is empty in three of them.

The check the machinery runs is the agreement of two counts. For the five crystallographic orders, the dimensions computed from the complex-monomial pairing must equal the dimensions computed from the Molien series of the actual integer matrices — a trace recursion on 2 × 2 integer matrices in a lattice basis. They agree at every degree in each of the five cases. For the other rows there is nothing to compare against, and the routine says so rather than inventing a matrix.

What is special about the five #

Read through the invariant degrees, the crystallographic orders are the ones whose second degree is small — 1, 2, 3, 4 and 6 against a first degree of 2. That framing is unhelpful on its own, since 5 sits inside that range and is excluded.

The useful framing is the cyclotomic one, and this collection has it already. The rotation of order n satisfies its cyclotomic polynomial, whose degree is Euler’s totient φ(n), and an integer matrix of size 2 can only satisfy a polynomial of degree at most 2. So the condition is φ(n) ≤ 2, which holds for n in {1, 2, 3, 4, 6} and fails for 5, where φ(5) = 4.

That is why five-fold symmetry becomes legal in four dimensions: there φ(5) = 4 fits, and the four-dimensional integer matrix of order five exists. And it is why the invariant-degree picture cannot decide the restriction on its own — the degrees say nothing about the size of the matrix needed, which is the whole of the condition.

The totient is the right test here and it is not the general one, which is worth saying because the plane makes it look like a definition. φ(n) ≤ d is the condition for one irreducible block. A matrix may be a direct sum of blocks, and the order of the sum is the least common multiple of its blocks’ orders — so an order may be reached more cheaply by splitting than by taking the whole cyclotomic polynomial at once. The first order where that happens is fifteen: a three-fold block costs two dimensions and a five-fold block costs four, so order fifteen lives in six, while φ(15) = 8. Twenty is the same story at the same numbers.

Below four dimensions the two never differ, which is why the plane’s argument may be stated with the totient and nothing is lost. The distinction only becomes visible where a rotation order has two coprime factors big enough to be worth separating, and the plane has no room for one.

Two arguments that do not overlap #

It is worth being explicit about what each half establishes, because they are easy to run together.

The invariant-theoretic half is about the group. It says that Dₙ has a free invariant ring with degrees 2 and n, and it is proved by counting monomials in complex coordinates. No lattice appears in it. Its content is that a function invariant under an n-fold reflection group is a polynomial in the modulus and in the n-lobed form.

The restriction is about the representation. It says that the group can be written in integer matrices only for five values of n. No polynomial appears in it. Its content is that a lattice’s automorphisms have integer traces.

Putting them together gives a statement neither makes alone: the invariant rings a crystal’s point symmetry can have, in the plane, are exactly those with second degree 1, 2, 3, 4 or 6. That is the restriction expressed in the vocabulary of invariants, and reaching it required both halves.

A test the machinery has to fail #

The site’s habit is to feed the machinery something it must refuse, and here the natural candidate is a five-fold group.

Asked for the invariant dimensions of D₅, the combinatorial routine answers — 1, 0, 1, 0, 1, 1, 1, 1, 1, … — and it is right to. That is a genuine sequence about a genuine group. Asked for the same group’s integer matrices, the machinery has nothing to hand back, and the row in the table reports no lattice rather than producing a matrix that is nearly right.

The failure mode being guarded against is a numerical one. A rotation by 2π/5 written as a floating-point matrix has a trace of 0.618…, and a routine that rounded traces to integers would report it as either 0 or 1 and hand back a fictitious four-fold or six-fold group. Nothing here rounds anything, so the question of which orders are permitted is answered by an exact comparison and never by a tolerance.

The same statement for the rotation groups #

Everything above is about reflection groups, which are the ones with free invariant rings. The rotation group Cₙ — the same rotations without the mirror — is worth a paragraph because its answer differs in a way that matters.

Its invariants are the monomials zᵃ z̄ᵇ with a ≡ b mod n, without the identification of a with b. So zⁿ and z̄ⁿ are separately invariant, and taking real and imaginary parts gives two independent invariants of degree n rather than one. Three generators, one relation, and the hypersurface ring this ladder has already computed for the four crystallographic cases.

The extra generator is the one that changes sign under a reflection — it distinguishes a pattern from its mirror image — and it exists precisely because the group has no reflection to kill it. That is the invariant-theoretic form of a fact this collection meets everywhere: the chiral groups are the ones in which handedness is a property of an orbit rather than a property of a description.

So the two-layer picture holds for the rotation groups too, with a different first layer. The ring of Cₙ has three generators for every n; five values of n give integer matrices; and those five are the same five, because the restriction is about the rotation and not about the mirror.

Reading the two arguments as one #

There is a way to put the two layers together that is more than a juxtaposition, and it is worth stating because it explains why the degrees are the right thing to look at.

The second invariant degree of Dₙ is n, which is the order of the rotation. The condition for an integer matrix is that the rotation satisfy a polynomial of degree at most two over the integers, which is φ(n) ≤ 2. So the question a crystal asks is not “how large is the second degree” but “how large is the totient of the second degree”, and those are different questions about the same number.

For the crystallographic orders they happen to run together — the five orders with φ(n) ≤ 2 are also the five smallest useful degrees — and it is easy to conclude that small degrees are what a lattice permits. The five-fold case is the counterexample sitting inside the range: degree 5 is smaller than degree 6, and 6 is crystallographic while 5 is not.

A number’s size and its totient are unrelated in the way that matters here, and that is the whole reason the restriction is a surprising theorem rather than an obvious bound. The invariant degrees do not explain it; they relabel it, and the relabelling makes visible that the arithmetic doing the work is about factorisation rather than about magnitude.

A five-fold ring with no crystal, and a crystal with no ring #

The two halves come apart in both directions, and each direction has an example this collection has already built.

A five-fold invariant ring with no crystal: D₅ has degrees 2 and 5 and its invariants are perfectly ordinary polynomials. No plane lattice admits it. What does exist is a quasicrystal, which has five-fold symmetry in its diffraction and no lattice at all, so the restriction never applies to it — and the invariant ring of D₅ is as relevant to its local order as any other group’s is to a crystal’s.

A crystal with no reflection group: the four rotation classes 2, 3, 4 and 6 preserve lattices and have no free ring. Their quotients are surfaces with a singular point rather than planes, and every physical property count for them carries the correction the relation imposes.

Neither example weakens the pairing. They mark where each half is doing its own work, which is the useful thing to know about an argument assembled from two independent pieces.

What the degrees do for a crystal #

The degrees are not only bookkeeping. They decide, directly, how many independent components a crystal’s physical properties may have.

For a class with degrees 2 and n, a fully symmetric property of rank r has as many independent components as there are ways to write r as 2a + nb. A hexagonal crystal — degrees 2 and 6 — permits one component at ranks 0, 2 and 4, and two at rank 6. A square crystal — degrees 2 and 4 — permits one at ranks 0 and 2, and two at rank 4. The difference between those two rows is the difference between the two lattices, expressed in a way a laboratory can measure.

So the second degree is a physically visible number. It is the lowest rank at which a crystal’s properties start to show its rotational symmetry rather than being isotropic, and the fact that it equals the rotation order is the reason a hexagonal crystal is isotropic in more of its properties than a square one is.

What the pictures cannot show #

The five-fold ring is not drawn. Its generators are z z̄ and z⁵ + z̄⁵, and both are perfectly good real polynomials — but the group they belong to has no integer matrices, so this site’s drawing machinery, which works in a lattice basis throughout, cannot produce a figure of them by the same route as everything else. Rather than draw them by another route with different guarantees, the figure reports the counts and leaves the curves undrawn.

Nothing here is three-dimensional, and the three-dimensional case is elsewhere. The degrees of a three-dimensional reflection group come in triples, the crystallographic ones are the point groups of the fourteen lattices and their subgroups, and the corresponding statement — which triples belong to a crystal — is a longer list. It is computed in twelve of the thirty-two are free, by matching each class’s Molien series against the product of three geometric series and reading the exponents off the match: twelve of the thirty-two classes have such a triple and twenty do not, and the twelve are exactly the classes generated by their reflections.

And the restriction is not proved here. It is imported from the trace argument and used. What this essay adds is a translation: the same five orders, named by the degrees of an invariant ring instead of by the traces of a matrix, and the observation that the ring itself never noticed the lattice.

Drop the mirrors and the ring stops being free #

The argument above rests on the group being generated by reflections, and it is worth seeing what happens when that hypothesis fails, because the failure is immediate and it is instructive about what the two degrees were doing.

Take the rotations alone. The cyclic group of order $n$ acts on the complex coordinate by $z \mapsto \omega z$ with $\omega$ a primitive $n$-th root of unity. A monomial $z^a \bar z^{\,b}$ is invariant when $a - b$ is a multiple of $n$, which is a weaker condition than the one the dihedral group imposes, so there are more invariants rather than fewer.

Three generators are needed, not two. The invariants are generated by $u = z\bar z$ of degree two, and by $v = z^n$ and $w = \bar z^{\,n}$, both of degree $n$. Nothing smaller will do: $z^n$ is invariant and is not a polynomial in $u$ alone.

And the three are not independent. They satisfy $vw = u^n$, one relation, exactly. So the ring of invariants is not a polynomial ring in disguise; it is a polynomial ring in three variables cut by one equation, and no change of generators removes the relation.

The numerical test says so immediately. For a reflection group the product of the invariant degrees equals the order of the group, which is the identity $2 \times n = 2n$ that closes the section above. Here the candidate degrees are $2, n, n$ and their product is $2n^2$ against a group of order $n$. The mismatch is the arithmetic signature of a ring that is not free.

That is Chevalley’s theorem doing real work rather than decorating a computation. Being generated by reflections is not a convenient extra property of these groups; it is the exact condition under which the invariants form a polynomial ring, and the crystallographic restriction as derived here is a statement about the degrees of that ring. Take the mirrors away and the derivation has no degrees to name.

What the second degree is worth to a physicist #

The pair $(2, n)$ is an algebraic fact, and it has a reading in the laboratory that makes the size of $n$ matter in a way the classification alone does not suggest.

A property that depends on direction is an invariant polynomial. Write the energy of a magnetised crystal as a function of the direction the magnetisation points, and require it to be unchanged by every operation of the crystal’s group: the result is a polynomial in the invariants, so it is a polynomial in the two generators.

The first generator carries no direction. $z\bar z$ is the squared distance from the axis, which is the same whichever way the magnetisation points in the plane. So it contributes a constant to the in-plane energy and nothing else.

The whole of the in-plane anisotropy therefore starts at degree $n$. Restricted to a direction, the second generator $z^n + \bar z^{\,n}$ is $2\cos n\theta$, so the leading term that distinguishes one in-plane direction from another varies as the cosine of $n$ times the angle — the first harmonic that a group of this order permits.

Which is why the anisotropy weakens as the symmetry rises. A twofold crystal has a term in $\cos 2\theta$, already large. A fourfold crystal starts at $\cos 4\theta$. A sixfold crystal starts at $\cos 6\theta$, and terms of that order in a magnetic energy are typically orders of magnitude smaller than the second-order ones — which is the standing reason that hexagonal magnets have magnetisations that turn almost freely within the basal plane while their axial anisotropy is enormous.

So the two degrees are a prediction about measurements. They say that a crystal’s in-plane response is flat to every order below $n$, and the restriction that limits $n$ to five values is simultaneously a limit on how flat that response can be made by symmetry alone.