How many lattices share a determinant

A lattice has an area per point, and it has a shape. The first is a single number and the second is not, and the question of how much one decides about the other has a clean answer for the lattices whose metrics are integers: at determinant one there is exactly one lattice, at determinant five there are two, at eleven there are four, and the sequence is not increasing.

The number is a class number, computed here by enumeration, and the interest is less in the values than in what makes the count finite and how it is checked.

A lattice is a form, up to a change of basis #

Writing a lattice down means choosing a basis, and the choice is not part of the lattice. What survives the choice is the metric: the array of dot products of the basis vectors, which is a symmetric two-by-two array, and equivalently the quadratic form ax² + bxy + cy² giving the squared length of the vector with coordinates (x, y).

Changing the basis by an integer matrix of determinant ±1 changes the form by G → UᵀGU, and two forms related that way describe the same lattice. So a lattice is a form modulo that action, and the number of lattices of a given determinant is the number of equivalence classes of forms of that determinant.

That reframing is the whole of the essay. It replaces a geometric question — how many shapes are there at a given area — with an arithmetic one, and the arithmetic one is finite and decidable.

Reduction settles equivalence #

The classes are countable because there is a normal form: every form can be carried by a sequence of basis changes to one satisfying |b| ≤ a ≤ c, and two forms are equivalent exactly when their reduced forms are identical.

This collection has the reduction already — two moves, a swap and a shear, generate every basis change, and applying them greedily terminates. What is added here is the consequence: with a normal form in hand, counting classes is enumerating reduced forms, which is a finite search because |b| ≤ a ≤ c and 4ac − b² is fixed bounds all three coefficients.

The convention that makes the reduced form unique rather than merely reduced is one line and is easy to omit: when a = c or when |b| = a, the sign of b is fixed positive. Without it, [3, −2, 4] and [3, 2, 4] are two reduced forms for one lattice at some determinants and two genuinely different lattices at others, and a count that gets the convention wrong is wrong by a predictable amount.

Checked by scrambling #

A normal form is a claim that can fail quietly, so it is tested by the obvious means: take a form, apply a random sequence of basis changes, and reduce what comes back.

The scrambles are products of the two generators of the basis-change group, so every one of them is a legitimate change of basis. The scrambled forms have coefficients running into the hundreds. Reducing each of them must return the form it started from — exactly, coefficient by coefficient — and it does, on every form and every scramble tried.

That is the check with content. It would catch a reduction that terminated too early, a convention applied inconsistently, or a swap with a sign error, and any of those would produce a class count that was too large by a plausible-looking margin.

Counted twice #

The enumeration is one route to the count. The other is to reduce.

Every form with coefficients inside a box is generated, each is reduced, and the distinct results are collected. That is a search over all forms followed by an algorithm, where the first route is a search over reduced ones — different computations, and they must agree.

They do, at every determinant tried. The agreement is what makes the enumeration a count of lattices rather than a count of coefficient triples satisfying some inequalities: it says the inequalities pick out exactly one representative per class, which is the definition of a normal form and is otherwise an assertion.

The values, against a table nobody here wrote #

The class numbers of negative discriminants are a standard object in number theory, and this computation’s answers can be checked against values it did not produce.

For the discriminants −3, −4, −7, −8, −11 the class number is one — those are five of the nine famous discriminants with unique factorisation — and this enumeration returns one for each. For −15 and −20 it returns two; for −23 and −31, three; for −47, five; for −71, seven. Every one matches.

That is an external check of a kind this collection rarely gets. Most of what it computes is verified against a second computation of its own, which catches implementation errors and not conceptual ones. Here the answers are compared with a body of arithmetic developed for entirely different reasons, and agreement means the objects being counted are the ones the standard theory counts.

Two of those values are the lattices this site draws constantly. Discriminant −4 is the square lattice, x² + y², alone in its class; discriminant −3 is the hexagonal lattice, x² + xy + y², alone in its. The two most symmetric plane lattices are the two whose discriminants have class number one, which is a pleasing coincidence and — as far as this computation can say — nothing more.

The determinant, and the two conventions for it #

A caution about numbers, because two conventions are in use and they differ by a factor of four.

Crystallography measures a lattice by the determinant of its metric — the square of the cell area — so the square lattice has determinant one and the hexagonal one three quarters. Number theory measures a form by its discriminant b² − 4ac, which is negative for a positive-definite form, and the square lattice’s is −4 while the hexagonal lattice’s is −3.

The relationship is D = −4·det G, so the two disagree for the hexagonal lattice in exactly the way that matters: its metric determinant is not an integer, while its discriminant is. The form x² + xy + y² has integer coefficients and the metric it corresponds to has a half in it, because the off-diagonal entry of a metric is b/2.

This essay computes in the discriminant convention for that reason, and reports determinants where they are integers. A reader comparing with a table of class numbers should check which convention is in use before concluding anything, and this collection has made the same warning about cells: a number describing a lattice usually carries a convention with it.

The count as an orbit count #

There is a third way to see the number, and it connects this essay to the counting machinery elsewhere in the collection.

The forms of a given discriminant are a set; the basis-change group acts on it; and the class number is the number of orbits. So the whole computation is an orbit count, and reduction is a way of picking one representative from each orbit — a transversal, in the language of group actions.

Counting orbits by finding a normal form is one of two standard techniques and it is the one that gives a list rather than a total. The other, Burnside’s lemma, gives a total without a list and does not apply here, since the group of basis changes is infinite and its fixed-point counts are not finite.

That the infinite group has finitely many orbits on a set which is itself infinite is the surprising part, and it is exactly what reduction proves: the group is large enough to bring any form into a bounded region, and the bounded region contains finitely many integer points.

Why the count is not monotonic #

The sequence 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 4, 4, 2 — the class numbers at determinants one to thirteen — is not increasing, and the reason is worth naming because it is where the arithmetic stops being geometric.

A determinant with many divisors admits many forms, since 4ac − b² fixed leaves room for several factorisations of ac. A determinant that is prime and of the right residue admits few. So the count follows the factorisation of the determinant rather than its size, and a prime determinant can have a smaller class number than a composite one half as large.

That is the same dependence this collection meets in the crystallographic restriction: what decides the answer is a number’s arithmetic rather than its magnitude, and the two are unrelated in the way that matters. A reader expecting the number of lattice shapes to grow with the area is expecting geometry to control an arithmetic question.

Telling the classes apart by measurement #

Two lattices of the same determinant are genuinely different, and it is worth asking what would distinguish them experimentally.

The theta series — how many vectors have each squared length — is the natural candidate, and this collection has an essay on whether the lengths name the lattice. For the two lattices of determinant five the series differ immediately: x² + 5y² has vectors of squared length 1, 4, 5, 9…, and 2x² + 2xy + 3y² has 2, 3, 5, 8…. A powder pattern would separate them at the first line.

In two and three dimensions the theta series always separates — that is Schiemann’s theorem, imported and not proved here — so the classes counted in this essay are distinguishable by their lengths alone. In higher dimensions they are not, and the counterexample is the oldest one there is: two sixteen-dimensional lattices with identical theta series, which is the pair of tori that sound alike.

What “integral” is doing #

One restriction has been implicit throughout and it decides everything: the forms here have integer coefficients.

A general lattice’s metric has real entries and there is a continuum of them at any given area — the moduli region is two-dimensional, and a fixed determinant is a curve through it with uncountably many points. There is no class number for real lattices, only a shape parameter.

What makes the count finite is the demand that the squared lengths be integers, which is a genuine restriction on the lattice and a natural one where the lattice comes from arithmetic rather than from a crystal. A crystal’s lattice is not integral in this sense, and the class-number question does not apply to it directly.

Where it does touch crystallography is in sublattices: the sublattices of a given lattice at a given index have integral metrics relative to the parent, and counting them up to equivalence is a class-number computation. This collection has that count already, arrived at from Hermite normal forms rather than from reduction, and the two are the same enumeration in different coordinates.

Where a class number is a physical count #

The abstraction pays off in one place this collection has already built, and it is worth making the connection explicit.

The sublattices of a lattice at a given index are counted by a divisor sum, and that count is of sublattices as subsets. Asking instead how many of them are inequivalent — how many distinct shapes appear among them — is a class-number question, because a sublattice’s metric relative to the parent has integer entries and a fixed determinant.

The same question arises for coincidence-site lattices, where two grains share a sublattice of index sigma: the possible shapes of that shared lattice at a given sigma are a class of forms, and the count of them is a count of distinct boundary geometries.

Neither of those counts is computed here — this essay computes the class numbers themselves, and the connection is a route rather than a result. What the connection establishes is that the arithmetic in this essay is not exotic: it is the arithmetic a crystallographer already does when asking how many distinct ways a lattice can sit inside another.

What is checked and what is imported #

Checked. That reduction is a normal form, by scrambling forms with random basis changes and requiring the reduced result back. That the enumeration of reduced forms agrees with the reduction of every form in a box. That the class numbers agree with the standard values at twelve discriminants. And that the two lattices of determinant five have different theta series, which is the negative test: if the count were wrong by merging two classes, that comparison would find them identical.

Imported. That reduction terminates, which this collection proves in the two-moves essay. That the theta series determines a plane lattice, which is Schiemann’s theorem. And the standard class numbers themselves, which are checked against rather than derived.

Refused. An indefinite form — one whose discriminant is positive, so that it takes both signs — is refused rather than reduced, because the reduction here is for positive-definite forms and a lattice’s metric is positive definite. Handing it x² + 4xy + y² produces an error, not a reduced form.

What the pictures cannot show #

A class is not a shape a figure can be faithful to. The lattices drawn for a given determinant are drawn at one basis each — the reduced one — and every other basis of the same lattice would give a different-looking picture of the same object. The picture shows a representative, and the essay is about the class.

Nothing here is three-dimensional. Ternary quadratic forms have their own reduction, their own class numbers, and a good deal more structure; the Niggli reduction this collection computes for three-dimensional cells is the crystallographic version of the same idea, and it is not a class-number computation.

And no lattice here has a crystal in it. The forms are integral because that is what makes the count finite. A crystal’s metric is measured in ångströms and is integral only by accident, so the class number is a fact about lattices as arithmetic objects rather than about any material.

Why the count is finite at all #

There is an assumption in the question that is easy to read past, and stating it explains why a determinant can have a number of lattices attached to it rather than a continuum.

Lattices of a given area are not finite in number. Fix the area and the shape is still free: a lattice is determined up to rotation by two lengths and an angle, one relation removes a parameter, and what remains is a two-parameter family. Divide by the basis changes and the family does not become finite — it becomes a surface, the space in which every lattice lives, on which the hexagonal and square lattices are two marked points among infinitely many others.

What makes this essay’s count finite is integrality. The forms enumerated here have whole-number coefficients, so the metric’s entries are whole numbers, which means every inner product between lattice vectors is a whole number. That is a strong condition — it picks out a discrete scatter of points on the continuous surface, and only finitely many of them lie at any given determinant.

So the class number answers a narrower question than “how many lattices have this area”. It answers “how many lattices have this area and integer inner products”, and both halves are doing work. Drop the integrality and the answer is infinite for every determinant; drop the determinant and the answer is infinite for every integrality condition.

This is not a limitation peculiar to the computation. It is why the object is a class number in the first place: number theory counts equivalence classes of integral forms, and the finiteness theorem is a statement about integers rather than about geometry. The lattices this essay draws are the ones a crystallographer would call special — every distance commensurate with every other — and the generic lattice in a real crystal is not among them.

The nine, and the largest of them #

The external check above lands on a famous list, and the list is worth finishing because it is one of the few places where a question about lattice shapes has a hard, hard-won answer.

Exactly nine discriminants have class number one. They are $-3, -4, -7, -8, -11, -19, -43, -67$ and $-163$, and at each of them the enumeration returns a single lattice: every integral lattice of that determinant is the same lattice in a different basis.

Gauss conjectured that the list was complete in 1801, having found no tenth, and the conjecture stood for more than a century and a half. It is genuinely hard: knowing that a list of small cases is complete requires ruling out a tenth case arbitrarily far out, and nothing in the enumeration itself does that. Heegner gave a proof in 1952 that was not accepted at the time, and Baker and Stark settled it independently in the mid-1960s, at which point Heegner’s argument was recognised as essentially correct.

The class number does grow, which is what makes the list finite. It tends to infinity as the discriminant does, roughly like the square root of it — so lattices of large determinant come in many inequivalent shapes, and the bars in the figure above rise on average even though they jump about from one determinant to the next.

The last of the nine leaves a fingerprint anyone can check. The number $e^{\pi\sqrt{163}}$ is within $10^{-12}$ of a whole number — 262 537 412 640 768 744 — and the near-miss is a consequence of that discriminant having class number one rather than a coincidence. It is the most quotable thing to come out of a question this essay asks in the form “how many lattices share a determinant”, and it is a fair indication of how far the arithmetic behind these figures reaches.