The lengths do not name the lattice

A lattice’s theta series counts its vectors at each length. It is a measurement, and the question every measurement raises is whether it determines what it measures.

Can two different lattices have the same lengths, with the same multiplicities, at every distance?

In the plane the answer is no, and the search that says so is exhaustive as far as it reaches. In sixteen dimensions the answer is yes, and the pair is famous.

Why anyone should expect either answer #

Before the search, both answers are plausible, and it is worth seeing why.

In favour of “the lengths determine the lattice”: a plane lattice has three parameters — two lengths and an angle — and the theta series has infinitely many coefficients. A map from three numbers to infinitely many is enormously over-determined, and the generic expectation for such a map is that it is injective.

Against it: the coefficients are not independent. They are the coefficients of a modular form, and the space of forms of a given weight and level is finite-dimensional, so infinitely many coefficients carry only a few numbers’ worth of information. Once that is known, the over-determination argument evaporates and the question becomes whether the number of parameters or the dimension of the space of forms is larger.

In two dimensions the parameters win; in sixteen the forms do. Neither is obvious, which is why the two-dimensional case took until 1990 to be proved and the sixteen-dimensional counterexample was found first.

What a search over lattices has to be a search over #

A lattice has infinitely many bases, so a search over bases would count the same lattice countless times and report collisions that are not collisions. What is needed is one representative per lattice, and reduction supplies it: a positive definite binary form is equivalent to exactly one form with 0 ≤ b ≤ a ≤ c.

So the search runs over reduced forms, and two distinct reduced forms are two distinct lattices. Seventeen hundred and fifty of them, up to a coefficient bound of twenty. Every theta series distinct.

The sign of b, which very nearly ruined it #

The first version of this search found eleven hundred and forty isospectral pairs, which would have been an extraordinary result and was a bug.

Reduction has two versions. The narrow one allows b to be negative, so [a, b, c] and [a, −b, c] are different forms. They represent exactly the same numbers, because (x, y) ↦ (x, −y) carries one to the other — so their theta series are identical, and the search dutifully reported every one of them as a pair.

They are not two lattices. A lattice and its mirror image are congruent: a reflection is an isometry, so the two describe the same lattice differently oriented, and no measurement of lengths could possibly tell them apart. The fix is one character — b runs from 0 rather than from −a — and the number of forms drops from 2,890 to 1,750 with no collisions left.

What makes this worth recording is that the wrong answer was self-consistent. Every “pair” it reported had genuinely equal theta series, at every term, exactly. The error was not in the arithmetic; it was in what the objects were taken to be. That is the characteristic shape of the mistakes this collection keeps finding: the computation is right and the equivalence relation is wrong, which is also what an oriented and an unoriented crystal class differ by, and what makes two settings one group.

How close two lattices come #

An exhaustive search that finds nothing raises a fair objection: perhaps the cutoff was too generous and a collision is waiting past it.

So the search also measures the closest approach: the longest prefix at which any two distinct forms still agree. The answer is thirty-eight terms — two lattices whose shortest vectors both have squared length twenty, which agree for as long as they do because neither has any vector at all before then, and which part at the next length.

Thirty-eight against a cutoff of a hundred and twenty is a margin of a factor of three. It does not prove there is no collision further out; it says the search was not stopped anywhere near where a collision would have to hide.

The pair that gets closest is worth naming, because it says what “close” means here. They are the forms [20, 0, 20] and [20, 1, 20] — a square lattice of edge √20 and a very slightly sheared one. Both have four shortest vectors at squared length twenty and nothing whatever below it, so every coefficient from one to nineteen is zero in both and the agreement over that stretch is an agreement about emptiness. They part at forty: the square lattice puts four vectors there and the sheared one puts two, with the missing pair reappearing at forty-two. So the longest agreement in seventeen hundred and fifty lattices is mostly a long shared silence, and the first length at which either of them has anything interesting to say is the length at which they disagree. That is the opposite of a near miss, and it is why the margin is more reassuring than the number alone suggests.

And the scale is doing none of the work. [20, 0, 20] is the ordinary square lattice with every length multiplied by twenty, so this near-collision is a scaled copy of the near-collision between [1, 0, 1] and its own slight shear — which parts at n = 2 rather than at n = 40. The search sees each scaling as a separate lattice, which is correct for the question being asked, and the effect is that the same geometric near-miss is counted many times over at many sizes. The longest prefix therefore grows with the bound on the coefficients, and reading it as evidence about high-dimensional behaviour would be reading an artefact of allowing large forms.

Schiemann proved in 1990 that there is none, in two dimensions and in three: the theta series determines a positive definite ternary form up to equivalence. The search here confirms the two-dimensional half as far as it reaches, and confirming is what a computation can do to a theorem.

It is worth being exact about the direction of that. The theorem settles the question and the search cannot; what the search supplies is a quantitative statement the theorem does not make — how nearly two plane lattices manage to agree before they fail to, which is thirty-eight terms out of the first hundred and twenty for the closest pair among seventeen hundred and fifty. A theorem saying no two agree says nothing about whether the margin is comfortable, and the margin is what decides whether a numerical test on measured data could ever be trusted.

What “the same lattice” has to mean #

The whole essay turns on an equivalence relation, so it is worth stating precisely which one, since three plausible candidates give three different answers.

Congruent — related by an isometry of the ambient space, including reflections. This is the relation used here, and it is the one a theta series could possibly determine, since lengths are all it knows.

Equal as subsets — the same set of points, which is far too strict: translating or rotating a lattice gives the same lattice by any sensible reckoning and a different set.

Similar — congruent after scaling. This is too coarse for the question: scaling multiplies every squared length by a constant, so a lattice and its double have theta series that are the same sequence stretched out, and they would count as agreeing under any reasonable reading. The search here keeps them apart, and the forms [1, 0, 1] and [2, 0, 2] appear as two entries.

The narrow-versus-wide reduction confusion above is exactly a slip between the first and a fourth relation — orientation-preserving congruence — which is meaningful for oriented objects and not for lattices. A lattice has no handedness. A crystal structure does, and that is why the same slip in that context is not a slip at all.

Sixteen dimensions, where it fails #

The counterexample is Milnor’s, from a one-page paper in 1964, and it is the answer to a question Mark Kac later made famous as can one hear the shape of a drum? — two flat tori whose Laplacians have the same spectrum, which is the same statement as two lattices with the same theta series.

The two lattices are:

E₈ ⊕ E₈ — two copies of the eight-dimensional root lattice side by side. E₈ is the integer vectors with even coordinate sum, together with the same set shifted by (½, …, ½); it has 240 vectors of norm 2 and 2,160 of norm 4, and the enumeration here finds both by direct search.

D₁₆⁺ — the sixteen-dimensional checkerboard lattice D₁₆ (integer vectors with even coordinate sum) together with one glue coset shifted by (½, …, ½). That coset is a lattice only because sixteen is a multiple of eight, which is the whole reason the pair exists in this dimension and not in another.

Their norm-2 counts are 480 and 480. Their norm-4 counts are 61,920 and 61,920. They agree at every norm — which is a theorem rather than a computation, and this essay computes two shells of it.

Validated at eight before being used at sixteen #

A box in sixteen dimensions is not a search anybody finishes. The counts above are therefore combinatorial: D₁₆’s norm-2 vectors are the 4·C(16,2) sign-and-position choices of (±1, ±1, 0…), its norm-4 vectors are 2·16 of the form (±2, 0…) plus 16·C(16,4) of the form (±1⁴, 0…), and the glue coset contributes 2¹⁵ vectors of norm 4 because all sixteen coordinates are ±½.

Formulas like that are exactly where an off-by-one hides, and there is nothing to compare them against in sixteen dimensions.

So they are compared in eight, where D₈⁺ is E₈ — a coincidence of the same construction, and the reason E₈ is sometimes defined as D₈⁺ — and where the direct enumeration runs. The formula gives 240 and 2,160; the enumeration gives 240 and 2,160. Only then is the formula used at sixteen, and if the eight-dimensional check ever failed the sixteen-dimensional figure would not be drawn at all.

That is the arrangement this collection puts under every count it cannot check directly: a construction that can be verified where verification is possible, and used where it is not.

Hearing the shape of a drum #

Milnor’s pair is usually told as a story about sound, and the translation is exact enough to be worth doing.

A flat torus is a plane — or a sixteen-dimensional space — with a lattice’s translations glued up. The vibrations of such a torus have frequencies determined by the lattice: each reciprocal-lattice vector gives a mode, and its frequency is that vector’s length. So the spectrum of the torus, as a list of frequencies with multiplicities, is the theta series of the reciprocal lattice.

Two lattices with the same theta series therefore give two tori that sound identical: every frequency present in one is present in the other, with the same multiplicity. They are different shapes with the same sound.

Kac asked the question for plane drums in 1966 and it stayed open for domains in the plane until 1992, when Gordon, Webb and Wolpert produced a pair of eight-sided regions that sound alike. Milnor’s tori came first by two years and are far simpler, and the reason they were not thought of as an answer to Kac’s question is that Kac had not asked it yet.

For this collection the interesting direction is the other one. In two and three dimensions the shape can be heard, for tori: Schiemann’s theorem says the spectrum determines the lattice, so a flat torus of a dimension a crystallographer cares about is decided by its sound. The failure is a high-dimensional phenomenon, and knowing where it starts is what makes the low-dimensional result a theorem rather than an assumption.

What the round trip checked, and how #

No two reduced forms may share a series, and the search reports the number of distinct series against the number of forms.

The margin is measured, not asserted. How far two different lattices agree before parting is computed and printed, so the cutoff can be seen to be past it.

An unreduced form must not enter the search, or the count of lattices is wrong before anything is compared.

The sixteen-dimensional formula must reproduce an eight-dimensional enumeration before it is believed.

Why sixteen, and not fifteen or seventeen #

The dimension is not arbitrary and the reason is worth having, because it says what kind of coincidence this is.

An even unimodular lattice is one whose vectors all have even norm and whose determinant is one. Such lattices exist only in dimensions that are multiples of eight — a fact about quadratic forms over the integers, not about geometry — and the count of them per dimension goes 1, 2, 24, and then upward very fast. In dimension 8 there is exactly one, E₈. In dimension 16 there are exactly two, and they are E₈ ⊕ E₈ and D₁₆⁺.

The theta series of an even unimodular lattice of rank n is a modular form of weight n/2 for the full modular group. For n = 16 that space is one-dimensional, so any two such lattices have theta series that are multiples of one another — and both start with a 1, so they are equal.

So the coincidence is forced. There are two lattices, there is only one series available, and therefore the two lattices share it. Nothing about either lattice had to be arranged; the arithmetic of modular forms left no room for them to differ.

In dimension 24 there are twenty-four even unimodular lattices and the space of weight-12 forms is two-dimensional, so the theta series is determined by one further number — the kissing number — and lattices with different kissing numbers are separated. The Leech lattice is the one with none at all, and it is the only one of the twenty-four with no vectors of norm two.

Where the exactness stops #

The plane search is bounded and says so. Coefficients up to twenty, series to a hundred and twenty terms. Schiemann’s theorem is what covers the rest, and it is quoted rather than proved.

Two shells are not a theta series. E₈ ⊕ E₈ and D₁₆⁺ agree at every norm, and this essay verifies two of them. The full statement follows from both lattices being even unimodular of rank sixteen, so their theta series are modular forms of weight eight, and that space is one-dimensional — a proof that needs machinery this collection does not carry and does not pretend to.

Nothing here is crystallography. No crystal has a sixteen-dimensional lattice, and the plane and space cases — where the answer is that the lengths do determine the lattice — are the ones a crystallographer meets. What the counterexample supplies is the knowledge that the two-dimensional result is a theorem about small dimensions and not an obvious fact about measurement.

The search is over lattices, not over crystals. A crystallographic database’s question — are these two reported cells the same cell? — is answered by reduction rather than by theta series, because reduction is cheap, canonical and exact. The theta series is the wrong tool for that job and this essay is not proposing it; what it settles is whether the measurement is faithful, which is a different question with a different use. The use is in powder work, where the line positions are a partial theta series and nothing else is available.

And a determined lattice is not a determined structure. Even in three dimensions, knowing the lattice exactly leaves the contents of the cell entirely open, which is the whole of the phase problem. The theta series is a measurement of the box, not of what is in it.

The pattern this essay is an instance of #

Three of this collection’s threads have now produced the same shape of result, and it is worth naming.

A measurement is made of an object — the lengths of a lattice, the vector set of a structure, the intensities of a diffraction pattern. The question is whether the measurement determines the object. The answer is nearly always yes in the cases that arise and no in general, and the boundary between them is where the subject’s interesting theorems live.

That is not a coincidence of subject matter. A measurement discards something — orientation, phase, position — and whether what is left suffices depends on how much room the discarded part had to vary in. In low dimensions there is little room and the measurement determines; add dimensions and the room grows faster than the measurement does.

The practical form of the lesson is that “the data determine the answer” is a theorem and not an assumption, and it has to be checked in each case rather than assumed from the abundance of the data.

Where the ladder goes next #

Two lattices that cannot be told apart by lengths is one kind of indistinguishability. The sharper kind, and the one that matters at a diffractometer, is two structures that cannot be told apart by a diffraction pattern at all — which is a finite search over small point sets, and it finds pairs immediately.

What this says about a powder pattern #

The theta series has a physical reading, and it turns Schiemann’s theorem into a statement about an experiment — which is worth extracting, because it is one of the few places where an abstract result settles a question a laboratory asks.

A powder pattern is a theta series. Grinding a crystal up discards every direction and keeps the spacings with their multiplicities, so what the instrument records is exactly the count of reciprocal lattice vectors at each length — the theta series of the reciprocal lattice.

So the question do the lengths determine the lattice is the question does a powder pattern determine a cell, and the answer is Schiemann’s: in three dimensions, yes. Two different lattices cannot produce identical lists of spacings with identical multiplicities. The cell is recoverable in principle from the peak positions alone.

In principle is doing real work in that sentence. The theorem is about exact, complete, infinite lists; a measurement gives a few dozen spacings with errors, and indexing them is the practical difficulty this collection has already described at length. The theorem says the difficulty is one of measurement rather than of information — there is a unique answer, and finding it is the problem.

And in sixteen dimensions there would not be. Milnor’s pair would give identical powder patterns to any resolution, so no measurement of spacings could ever separate them. That is a hypothetical for crystallography and it is exactly the situation for the higher-dimensional descriptions of aperiodic structures, where the relevant lattices are five- and six-dimensional and the theorem is not known to hold.