The sublattices that are the same shape

Counting sublattices asked how many ways there are to thin a lattice by a given factor, and answered with a divisor sum. Counting the ones that stay square asked how many of those keep the parent’s symmetry operations, which is a question about containment: does each operation of the parent still carry the sublattice to itself?

There is a stronger relation between a lattice and a sublattice, and it is the one a crystallographer means when a superstructure “looks like” its parent. A sublattice is similar when it is the parent rotated and scaled — when the two have the same shape and differ only in size and orientation.

The test, and the trap in stating it #

Similarity has an exact test. If the parent’s metric is the Gram matrix G and the sublattice has basis B in the parent’s basis, then the sublattice is similar exactly when

$$B^{\mathsf T} G B = n\,G, \qquad n = |\det B|$$

with n the index. The scale factor has to be n and nothing else: determinants give det(BᵀGB) = n² det G, so a scalar λ would force λ² = n².

Every quantity in that equation is an integer, so it decides. And stating it that way is a trap, which this file walked into and is worth recording because the wrong version produces a plausible answer.

Similarity is a property of the sublattice and the identity is a statement about a basis. A sublattice similar to its parent has some basis in which the identity holds, and generally not the one it was handed in. Testing the given basis asks whether that basis is a rotated copy of the parent’s, which almost none of them is — and the answer that comes back is that the only similar sublattices of the square lattice are those of square index, which looks like a result and is wrong.

The repair, which is the reduction #

What has to be compared is the two forms, up to a change of basis, and this collection already has the machinery: every positive definite binary form has one reduced representative, and two forms are equivalent exactly when their reduced representatives agree.

So the test is: reduce BᵀGB, reduce nG, compare. The reduction is the same walk into the modular region that the lattice-space essays draw, and running it here costs nothing.

That repair is a small instance of a general rule this collection keeps rediscovering: a question about a lattice must not be answered by a computation about a basis. The cell is a choice and the lattice is not; a similar sublattice is a lattice, and the test has to be blind to the description.

What the enumeration finds #

Running the corrected test over the sublattices of each index gives a list, and the list is recognisable at once.

For the square lattice: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29 … These are the sums of two squares, and the numbers missing — 3, 6, 7, 11, 12, 14, 15, 19 — are the ones with a prime congruent to 3 modulo 4 appearing to an odd power.

For the hexagonal lattice: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28 … These are the numbers of the form a² + ab + b², the Loeschian numbers, and the missing ones are those with a prime congruent to 2 modulo 3 to an odd power.

Neither list was put in. Both came out of an enumeration over Hermite normal forms with an integer test applied to each.

Why those lists, and not others #

The reason is that a similar sublattice is a multiplication.

The square lattice is the ring of Gaussian integers ℤ[i]: the lattice points are a + bi, and multiplying every point by a fixed Gaussian integer a + bi rotates the whole lattice through that number’s argument and scales it by its modulus. The image is a sublattice, it has the parent’s shape by construction, and its index is the number’s norm, a² + b².

Every similar sublattice arises that way. So the indices that admit one are exactly the norms of Gaussian integers, which are exactly the sums of two squares — and which numbers those are is Fermat’s theorem of 1640, proved by Euler a century later.

The hexagonal lattice is the ring of Eisenstein integers ℤ[ω], with ω a cube root of unity, and the same sentence holds with the norm a² + ab + b². Its missing primes are those congruent to 2 modulo 3, for the same reason: those primes stay prime in the ring, so they can enter a norm only squared.

What the enumeration looks at #

The enumeration runs over sublattices in Hermite normal form — bases written as [[a, b], [0, d]] with ad = n and 0 ≤ b < a — which is the canonical form the sublattice-counting essay established: one such basis per sublattice, no duplicates, no omissions.

That form makes the count of all sublattices of index n a divisor sum, which is where the earlier essay stopped. Here each of them is then tested for similarity, so the work per index is the divisor sum’s worth of tests and the test is a reduction of a binary form.

The whole census to index forty is a few thousand reductions and takes no measurable time — which is worth mentioning only because the naive alternative, searching for a rotation carrying the parent onto the sublattice, would be a search over angles with a tolerance in it. Reducing a form instead turns a geometric search into an integer comparison, and that substitution is the reason the answer here is exact rather than approximate.

Two routes, required to agree #

The enumeration and the factorisation are computed separately here and required to agree at every index up to forty. That is the collection’s standing habit — a claim gets a test it could fail — and it is worth saying what each route would catch about the other.

The factorisation checks the enumeration. A similarity test admitting a few extra sublattices would show up as an index with a similar sublattice that Fermat’s condition forbids, immediately and unmistakably.

The enumeration checks the factorisation. More precisely, it checks that the theorem being quoted is the right theorem. It would be easy to write down the wrong congruence — 1 modulo 4 rather than 3, or modulo 3 rather than modulo 4 — and the enumeration would refuse it at the third index.

And a third check is available that neither route supplies alone: the count is multiplicative on coprime indices, so the number of similar sublattices of index 15 must be the product of those at 3 and 5. That is a property of counting functions coming from rings of integers, it holds on all twenty-two coprime pairs below forty, and it would break long before the individual counts did if the similarity test were subtly wrong.

The counts themselves, and their shape #

The support of the count — which indices have any — is the part with a classical name, and the count’s values are worth a look too.

Index one has one, trivially. Index two has one. Index five has two, index thirteen two, index seventeen two: the primes that are sums of two squares each contribute two, being the two ways to write them up to the symmetries. Index twenty-five has three, because the prime five enters squared and the count of representations grows accordingly.

The pattern is the one number theory predicts for a counting function attached to the Gaussian integers: multiplicative, with a value at each prime power decided by how that prime factors in the ring. This collection does not derive it — that is a chapter of algebraic number theory rather than of crystallography — and it does measure it, and the measurement agrees with the multiplicativity test at every coprime pair below forty.

What a crystallographer takes from that is a small practical fact: the superstructures with the parent’s shape come in families, they are indexed by an arithmetic condition rather than by a geometric one, and the smallest ones are the ones with the smallest norms.

Keeping the symmetry is not keeping the shape #

The distinction between this essay’s relation and the previous rung’s is worth a paragraph, because both are called “symmetric sublattices” in places.

A sublattice keeps the symmetry when every operation of the parent’s point group maps it to itself. A sublattice is similar when it has the same shape as the parent.

Neither implies the other. The rectangular lattice halved along one axis keeps every operation — the mirrors and the two-fold all preserve it — and is emphatically not similar, because halving one axis changes the aspect ratio. In the other direction, a similar sublattice of the square lattice at index five is turned through an angle that is not a symmetry of the parent, so it is not mapped to itself by the parent’s rotations.

The two relations answer different questions. Containment of operations says whether the sublattice can carry the same pattern; similarity says whether a picture of it could be mistaken for a picture of the parent.

A picture worth two paragraphs #

The similar sublattice of index five is the one to look at, because it makes the definition physical.

The parent’s points are a square grid. The sublattice’s points are every fifth one, chosen so that they form a square grid again — turned through the angle whose tangent is a half, and spaced by the square root of five. Every property a lattice has that does not involve orientation is the same for both: the same shape, the same point group, the same position in the modular region.

A superstructure built on it is what a crystallographer writes as √5 × √5 R26.6° — the two lengths and the rotation, in a notation designed exactly for this situation. The rotation angle is not free: it is arctan(1/2), determined by the Gaussian integer 2 + i whose norm is five, and every √5 superstructure anybody has ever reported is at that angle.

Where this appears in a laboratory #

Similar sublattices are the arithmetic behind two things a crystallographer meets.

Coincidence orientations. The coincidence site lattice of two copies of a lattice rotated against one another is a sublattice of both, and it is similar to the parent by construction — which is why the coincidence indices of the square lattice are the odd sums of two coprime squares and the arithmetic in that essay is the arithmetic here. A grain boundary at such an orientation is cheap because the two crystals share a lattice of the same shape.

Superstructures that keep the cell’s shape. A superstructure whose cell is a similar sublattice of the parent’s has the same axial ratios and the same angles, so its diffraction pattern looks like the parent’s with extra spots inserted in the same geometry rather than a different one. Those are the superstructures a √5 × √5 or √7 × √7 label refers to, and the numbers in the labels are exactly this essay’s list.

Both are cases of arithmetic that was studied for its own sake in the seventeenth century being what decides a materials question in the twentieth.

The forms this does not reach #

Two lattices in the plane have no similar sublattices worth the name, and saying which is part of the answer.

A generic oblique lattice has similar sublattices only at square indices, where the sublattice is the parent scaled by a whole number. Its ring of “multipliers” is just the integers, because a generic lattice has no rotation carrying it to a scaled copy of itself.

A rectangular lattice of generic aspect ratio is the same. Only a lattice with extra symmetry has a ring bigger than ℤ behind it, and in the plane that means the square and hexagonal lattices and nothing else — which is a small echo of the crystallographic restriction: the rotations available are what makes the arithmetic rich.

In three dimensions, briefly #

The same question in space has the same shape and a harder answer. A similar sublattice of the cubic lattice is a rotation and scaling carrying ℤ³ into itself, and the indices that admit one are governed by the arithmetic of quaternions rather than of complex numbers — Lipschitz and Hurwitz quaternions, whose norms are sums of four squares.

Lagrange’s theorem says every positive integer is a sum of four squares, so every index admits something, and the interesting question becomes how many rather than whether. That is a richer arithmetic than the plane’s and it is not computed here; what is worth carrying is that the pattern in the plane — indices decided by norms in a ring — is the general shape and not a two-dimensional accident.

Where this goes #

The natural continuation is the count itself rather than its support: how many similar sublattices each index has, as a function of the index, which for the square lattice is a divisor sum over the Gaussian primes and is the sort of exact arithmetic this collection likes. The plan records it.

The nearer neighbour is the space every lattice lives in, where the reduction used here to repair the similarity test is the subject rather than the tool — and where the next essay measures how far apart two lattice shapes are, using the same walk into the same region.

The same question in space #

The plane’s answer is clean because a plane lattice with symmetry is a ring — the Gaussian integers or the Eisenstein ones — and a similar sublattice is a multiplication. Three dimensions has a version of that and it is a good deal less tidy.

The cubic lattice has an analogous algebraic description in terms of quaternions: multiplication by a suitable quaternion rotates and scales space, and the sublattices similar to the cubic lattice arise that way, exactly as the plane’s arise from complex multiplication. The counting function is again multiplicative, and again its support is decided by which integers are norms.

What does not carry over is generality. In the plane, every lattice with more than the minimum symmetry is one of two rings, so the description covers the interesting cases. In space the lattices with such a description are a handful — the cubic ones and a few others — and a general three-dimensional lattice has similar sublattices only at cube indices, where the sublattice is the parent scaled by a whole number and nothing has been learned.

That is the usual shape of things one dimension up, and it is worth stating rather than leaving as an exercise. The plane’s result is not the first case of a general pattern; it is a consequence of two dimensions being where lattices and rings coincide.

The superstructure that hides in its parent’s pattern #

There is a practical hazard attached to this relation, and it follows directly from the definition rather than from anything about a particular material.

A superstructure built on a similar sublattice has a cell of the same shape as the parent’s — the same axial ratios, the same angles, the same symmetry — differing only in scale and orientation. So its diffraction pattern is the parent’s pattern scaled and turned, with the same arrangement of spots and the same apparent symmetry.

The only evidence that the larger cell is the right one is the extra reflections, and those are superlattice reflections: weak, because they come from a small departure from the parent arrangement, and easy to miss. A data set collected without them is perfectly consistent with the parent cell, indexes cleanly, refines, and describes an average structure rather than the real one.

That is the sublattice ambiguity in its most deceptive form. In the ordinary case a missed superstructure gives a cell of the wrong shape as well as the wrong size, and the shape is a warning. Here it gives a cell of exactly the right shape, and the error is invisible in everything but the intensities. The similarity that makes the relation interesting is the same property that makes it dangerous, and the only defence is to look where the weak reflections would be.