A lattice cannot have all its vectors long

The shortest basis finds the shortest vectors a lattice has and uses them as axes. The shortest vector and where it stops being easy asks what it costs to find them in high dimensions. Neither asks the question this essay is about, which is how long a lattice’s vectors are allowed to be.

The answer is not what a first guess suggests, and the reason is that the right quantity is a product.

The naive expectation is worth stating so that it can be seen to fail. A lattice of a given cell volume plainly cannot have all its vectors long — the points would be too sparse for the volume — so one expects a bound on each direction separately. There is one for the shortest, and there is none at all for the others: a lattice of unit volume can have a second and third minimum arbitrarily large, provided the first is small enough to pay for them. What is bounded is not any single length but a product, and stating it as a product is the difference between a true theorem and a plausible false one.

The three radii #

Fix a lattice and grow a ball centred on the origin. At some radius it first contains a non-zero lattice vector; call that radius λ₁. Keep growing: at some larger radius the ball contains two lattice vectors that are not parallel, and that is λ₂; at some larger one still it contains three that are not coplanar, and that is λ₃. Those are the successive minima.

Two things about the definition are worth fixing before it is used. The minima are radii and not the lengths of a basis: three vectors achieving λ₁, λ₂, λ₃ need not generate the lattice. In three dimensions they always do, which is why the distinction can be ignored here and cannot be ignored in general — from five dimensions upwards there are lattices whose minima are achieved only by vectors that generate a proper sublattice.

And the minima are increasing by construction, so λ₃/λ₁ measures how far a lattice is from being as round as its volume permits.

How they are computed is worth a sentence, because the procedure is what makes the numbers here measurements rather than recollections. Every lattice vector inside a radius fixed by Minkowski’s own bound is enumerated, sorted by length, and taken in order; a vector is accepted when it raises the rank of the matrix of vectors accepted so far, and rejected when it does not. The first accepted vector gives λ₁, the second λ₂, the third λ₃, and the radius is large enough that three are always found — which is asserted rather than assumed, so a search that came back with two would fail loudly instead of reporting a smaller answer.

The seven were chosen to spread that ratio rather than to be interesting individually. Three of them are the cubic lattices, whose minima are all equal because the holohedry carries every axis to every other; the hexagonal and tetragonal ones have two equal and one apart; the triclinic one has three different values with no relation between them; and the flattened one is the extreme case. So the column reads as a measure of symmetry as well as of shape, and the two coincide because a symmetry that carries one axis to another forces their minima to agree.

One property of the minima is worth naming because it governs how they behave across a family of lattices rather than at one. As functions on the space of lattice shapes they are continuous — a small change to the cell moves every minimum a little — but they are not smooth. The vector achieving a minimum can change identity as the shape varies, and at the shape where two candidates are equally short the function has a corner. Those corners are not defects of the definition; they are where the symmetry is highest, since a shape at which two different vectors tie is a shape with an operation carrying one to the other. That is why the cubic entries in the table have three equal minima and the triclinic one has none.

Scaling deserves a sentence, since without it none of the numbers below would be comparable. Multiplying a lattice by a factor multiplies every minimum by the same factor and the covolume by its cube, so the quantity that is a property of the shape rather than the size is λ₁ divided by the cube root of the covolume. Every lattice in this essay has been scaled to covolume one, which is the same normalisation the space every lattice lives in uses when it treats shape as a point in a moduli space and forgets size.

What Minkowski bounds #

The first minimum alone is bounded, and the bound is one of the oldest results about lattices.

Minkowski’s convex body theorem says that a centrally symmetric convex body whose volume exceeds 2ⁿ times a lattice’s covolume must contain a non-zero lattice point. Applied to a ball, that gives λ₁ a ceiling: no lattice of covolume one has a shortest vector longer than 1.24070.

That bound is not the best possible.

The square in the definition is not cosmetic and says where the constant comes from. A lattice’s squared lengths are the values of a positive definite quadratic form on integer arguments, and its covolume is the square root of that form’s determinant, so the whole question is a question about forms: among ternary forms of determinant one, how large can the smallest non-zero value be? Stating it that way is what makes it arithmetic rather than geometric, and it is why the answer is found by classifying reduced forms rather than by searching over shapes. It is also why the constant is squared — the natural quantity is a form value, and a length is its square root.

Hermite’s constant is defined as the maximum of λ₁² over lattices of covolume one, so the statement that it equals the cube root of two in three dimensions is a statement that a maximum is achieved and where. Its known values run out quickly: exact in dimensions one to eight and in twenty-four, and unknown in every other. That is a striking gap for a quantity with a two-line definition, and it is worth carrying as a calibration of how hard lattice questions become when the dimension rises.

The looseness is worth dwelling on rather than apologising for. Minkowski’s argument is three lines and holds in every dimension; Hermite’s constant is known exactly in eight dimensions and in no others, and each value took a separate theorem. A bound that is loose everywhere and provable everywhere is a different kind of object from one that is exact in the cases somebody has managed.

There is a second reason to compute Hermite’s constant rather than quote it. The claim “the face-centred cubic lattice has the longest shortest vector at fixed volume” and the claim “the face-centred cubic lattice is the densest lattice packing of spheres” are the same claim, since the sphere radius is half the shortest vector and the density is the sphere volume over the covolume. Anybody who has met the second has met the first, and the constant is the conversion between them.

One remark about how the bound is proved, because the argument is short enough to state and it explains why the body has to be symmetric. Take the body K, shrink it by half, and consider its translates by every lattice point. If they never overlapped, their total volume in a large region would be at most the region’s volume, giving vol(K)/2ⁿ ≤ covol. So if vol(K) > 2ⁿ covol two translates overlap, and the difference of the two lattice points is a non-zero lattice vector in K — provided K is symmetric, so that the difference of two of its points is in K doubled. Symmetry is the whole of what the hypothesis is doing.

The statement that is about all three #

The useful theorem is Minkowski’s second, and it is about the product.

λ₁λ₂λ₃ · vol(K) ≤ 2ⁿ · covol

for any centrally symmetric convex body K whose minima these are. With K the unit ball the right-hand side is eight times the covolume, and the left is the product of the three minima times 4π/3.

The cap is 2ⁿ times the covolume, and the reason it is exactly that number is the same halving argument as before. If the body scaled by 1/λᵢ in the i-th minimal direction is packed by lattice translates without overlap, its volume is at most the covolume — and unpicking the scaling gives the product bound. So the two theorems are one argument applied twice, once to a single body and once to a body squashed along the directions the minima pick out.

How far under the cap the seven actually sit is the next thing to ask, and the answer is: a long way, and by an amount that varies. The cap is derived by packing a squashed body without overlap, and a ball packed by a lattice leaves gaps — the face-centred cubic lattice, the best case, still leaves a quarter of space empty — so the inequality inherits every one of those gaps. A cap that is loose by a factor of three is still a cap. What it cannot do is tell two good lattices apart, which is why the second theorem is used to rule structures out and never to rank them.

That is a strong statement about structure and it is easy to underrate because the cap is not tight. What it forbids is a lattice that is uniformly sparse — one in which every direction has to be travelled a long way before a lattice point is met. A lattice of a given cell volume can be long in two directions or in one, and never in all three.

There is a lower bound in the theorem too, and it is worth knowing exists even though nothing here uses it. The full statement is (2ⁿ/n!) covol ≤ λ₁λ₂λ₃ vol(K) ≤ 2ⁿ covol, so the product cannot be too small either — a lattice cannot have all its vectors short at fixed cell volume, which is obvious for a different reason and is the same theorem. The two halves together say the product is pinned within a factor of n!, and in three dimensions that is a factor of six.

One more property of the product cap is worth extracting, because it is what makes it usable. The cap depends on the lattice only through its covolume — a single number — so it can be applied to a structure whose cell parameters are known and whose vectors are not. Given a cell volume, the product of the three minima is at most 8 covol / (4π/3), which is a number a reader can compute in a line, and any claim about a structure that violates it is wrong before the structure is examined.

What it says about a crystal #

Three things, and the second is the one that is used without being stated.

The first is that “the shortest interatomic distance” is not a free parameter of a structure. Given a cell volume, it is bounded above by Hermite’s constant, and given an anisotropic cell it is bounded much more severely. A proposed structure with a large cell volume and no short vectors is not merely unusual; it is impossible, and the impossibility is arithmetic.

The second is that reduction theory relies on this. The shortest basis works by finding short vectors and replacing longer ones, and it terminates because there is a shortest vector to find and because the minima are bounded in terms of the covolume. Without Minkowski’s theorem there would be no guarantee that a search over a bounded region finds anything, and every reduction algorithm would need a different argument.

The third is about density. Two stackings, one density computes the packing fraction of the close-packed arrangements, and the reason the face-centred cubic lattice is the densest lattice packing is exactly that it attains Hermite’s constant: the packing radius is half the shortest vector, so the longest possible shortest vector at fixed cell volume is the densest possible lattice packing at fixed density of centres. The two statements are the same statement, and the densest lattice in the plane is the two-dimensional case where the constant is 2/√3.

The second of those is the one that carries the essay’s claim about tightness. Saying a bound is the best possible means exhibiting something that attains it, and the face-centred cubic lattice’s shortest vector agreeing with the cube root of two to nine decimal places is that exhibit. A bound quoted without an attaining example is a bound nobody has checked is sharp.

One more use worth naming, because it is where the bound stops being about lattices at all. A pair of successive minima that are very unequal is the signature of a structure with a layer or a chain in it — a very short direction and two long ones is a chain compound, one long direction and two short is a layered one — and the product cap says those are the only two ways to be anisotropic at a given cell volume. So the classification of low-dimensional structural motifs has an arithmetic shadow, and the shapes a lattice in space can thin to is the same question asked about sublattices instead of about vectors.

There is one honest limitation in the tables worth pointing at. The seven lattices were chosen to span the shapes rather than sampled, so nothing in the census is evidence about typical lattices — the search is the part that samples, and it samples a family of Gram matrices built from a triangular basis with bounded entries. A different family would find a different best, and the argument that none of them can beat the face-centred cubic answer is the theorem rather than the search.

A last observation about the two bounds and what separates them. Minkowski’s argument uses only that the body is convex and symmetric; Hermite’s constant uses the specific geometry of spheres in three dimensions, and there is no reason to expect a general argument to be sharp for a particular body. The gap between 1.24070 and 1.12246 is the price of generality, and the fact that it is only ten per cent is the surprise — a bound proved in three lines for every dimension at once lands within a tenth of an answer that took a century to pin down.

One consequence for reduction is worth stating in its own right, since it is what makes the whole subject computable. Because the minima are bounded in terms of the covolume, the set of lattices of a given covolume with all minima below a given bound is compact — so an optimisation over lattice shapes has a maximum to find rather than an infimum to approach. Every statement in this collection of the form “the lattice that minimises such-and-such” depends on that compactness, and it comes from Minkowski.

Where this stops #

The search here is a search and not a proof. That the face-centred cubic lattice maximises the shortest vector at fixed covolume in three dimensions is a theorem of Gauss, and reproducing it means classifying reduced ternary forms and optimising over the resulting region — which this collection does for the plane, in the densest lattice in the plane, and not for space. What is here is the bound, the attaining lattice, and forty thousand attempts that fail to beat it.

The dual lattice is absent too, and it is where the subject goes next. A lattice’s minima and the minima of its dual are not independent: a lattice with a very short vector has a dual with a very long one, and the transference theorems make that precise by bounding the product λ₁(L) · λ₃(L*). Nothing here computes those bounds, and the reason for naming them is that the dual of a crystal lattice is its reciprocal lattice, so a transference statement is a statement relating a structure’s short interatomic distances to its widely spaced diffraction planes. That is a real connection between the arithmetic here and what a diffractometer measures, and it is a separate essay.

And the minima are computed by enumeration inside a radius fixed by Minkowski’s own bound. That is circular-looking and is not: the bound is proved independently and is used here only to decide how far to look, so a bug in the enumeration would show up as fewer than three independent vectors found, which is asserted against rather than assumed.