The space every lattice lives in

Five lattices and no others is one of the first counts this collection makes, and it is a theorem. A repeating grid in the plane is oblique, rectangular, centred rectangular, square or hexagonal; there is no sixth; the argument closes.

It is also, read carelessly, a sentence that says something false. Five is the number of kinds. The number of plane lattices is not five and not any other number, because a lattice can be stretched by any factor and sheared by any amount and is still a lattice. What five counts is how many different symmetries a lattice can have, and the objects being counted are not five lattices but five families, of wildly different sizes.

So how large are the families, and what is the space they sit in? Both questions have an exact answer, and it is one picture.

The shape of a lattice is one number #

A lattice is fixed by two vectors, which is four numbers. Three of those four are not about the lattice’s shape.

Rotating the whole lattice changes both vectors and changes nothing about it — a lattice that has been turned is the same lattice seen from a different chair. Scaling it changes both vectors and changes nothing worth classifying either: the square lattice with a spacing of one and the square lattice with a spacing of a hundred are the same object at two magnifications, and every symmetry statement about one is a symmetry statement about the other. What is left, after a rotation and a scale have been used up, is one complex number.

The construction is short. Put the plane in the complex numbers, turn the lattice until its first basis vector u lies along the positive real axis, and scale until that vector has length one. The second basis vector is then a complex number all by itself, and it can be taken in the upper half-plane, since replacing v by −v is a change of basis and does not move the lattice. Write it $\tau = v/u$.

Every lattice, up to rotation and scale, is now a point of the upper half-plane. Every point of the upper half-plane is some lattice. The correspondence is exact and it has no slack in it.

Nothing on this site is decided with τ, and it is worth saying why before it is used. τ is irrational for almost every lattice and its imaginary part is a square root, so a decision made by comparing two of them would need a tolerance — which is the one thing this collection refuses to introduce. Everything decided below is decided on the integers underneath. Write $a = |\mathbf{u}|^2$, $b = 2\,\mathbf{u}\cdot\mathbf{v}$, $c = |\mathbf{v}|^2$, so that the squared length of a general lattice vector is

$$a x^2 + b x y + c y^2$$

with $x$ and $y$ whole numbers. That is a quadratic form with integer entries, it is the Gram matrix in disguise, and $\tau = (b + i\sqrt{4ac - b^2})/2a$ is read off it. The form decides; τ draws.

The same lattice, over and over #

The trouble with putting a lattice at a point is that a lattice does not have a basis. It has infinitely many, and each of them produces its own τ.

Two bases describe the same lattice exactly when an integer matrix of determinant ±1 carries one to the other — nothing else, because the change has to send lattice points to lattice points in both directions. Such a matrix acts on τ by

$$\tau \;\longmapsto\; \frac{p\tau + q}{r\tau + s},$$

so one lattice does not correspond to one point at all. It corresponds to a whole orbit of points, scattered right across the upper half-plane, and any two of them are the same lattice described twice.

That is the same difficulty the reduced cell exists to solve, met one level up. There the question was which of a lattice’s infinitely many cells to publish; here it is which of a lattice’s infinitely many points to call its place in the space. And the answer is the same procedure, which turns out to be doing something larger than picking a tidy basis.

Reduction is the division #

Gauss’s reduction takes a form and applies two moves until neither applies. T replaces v by v ± u, which changes b by ±2a and slides τ sideways by one. S exchanges u and v, which swaps a and c and sends τ to −1/τ. A final sign change makes b positive. It stops when

$$0 \le b \le a \le c,$$

and those three inequalities are not a tidiness convention. Written in τ they say: Re τ lies between 0 and a half, and |τ| is at least one. The reduced forms are exactly the lattices whose point is in one particular region of the upper half-plane, and reduction is the walk that puts every lattice into it.

So the space of lattice shapes is not the upper half-plane. It is the upper half-plane divided by the group of integer changes of basis, and the region above is a fundamental domain for that division — one point per lattice, every lattice reached. Two published cells describe the same crystal exactly when their forms reduce to the same triple of integers, which is a comparison of six whole numbers and is why a structural database can settle the question the moment an entry arrives.

What the five kinds turn out to be #

Now the count of five can be asked again, and the answer is a great deal more interesting than a list.

A lattice has a symmetry when its form has an equality in it. That is the whole of it: an integer matrix preserving a x² + bxy + cy² has to send vectors to vectors of the same length, so it exists precisely when two of the three quantities coincide or b vanishes. And an equality among a, b and c is a boundary of the region.

• b = 0 is the left edge, Re τ = 0. The form is a x² + c y², the two basis vectors are perpendicular, and the lattice is rectangular.
• a = c is the arc, |τ| = 1. The two basis vectors are the same length, so the perpendicular bisector between them is a mirror, and the lattice is centred rectangular.
• b = a is the right edge, Re τ = ½. This is the centred rectangular lattice again, in its other description — the rhombus seen along its diagonal rather than along its sides.
• b = 0 and a = c together are the single point τ = i: the square lattice.
• a = b = c is the single point τ = ρ = (1 + i√3)/2: the hexagonal lattice.
• Everything else — the interior — is oblique.

Five kinds, and they are one region, three arcs and two points. The count is right and the impression a plate of five pictures gives is wrong, because four of the five families have no area at all.

Two details in that list repay a second look. The first is that centred rectangular occupies two of the three edges, and the two meet at ρ. That is not a duplication in the arithmetic; it is the fact that a centred rectangular lattice has two natural descriptions — as a rhombus of two equal sides, and as a rectangle with a point in the middle — and the region cuts the one family in two places. The second is that the square lattice sits on the rectangular edge and on the centred arc at once, which is the algebraic form of a remark crystallographers make and rarely justify: a square lattice can be described as centred rectangular, and there is nothing wrong with doing so.

Symmetry is a property of the boundary #

The same picture answers a question that has no obvious home anywhere else: how much symmetry does a lattice picked at random have?

Two, everywhere in the interior. The identity and the inversion through the origin, which every lattice has because −v is a lattice vector whenever v is. Four on the edges, where a mirror has appeared. Eight at i and twelve at ρ. Those numbers are the holohedries the classification already knows, and what the region adds is where they live: symmetry is an edge phenomenon, and the interior — which is almost the whole of the space — has none of it.

A lattice with a symmetry is a lattice that has been given one. In a crystal it is given by the contents: atoms arranged with a three-fold axis force the lattice onto the hexagonal point, and a structure with no symmetry at all leaves it in the interior. The lattice does not acquire symmetry by accident, and the region says how unlikely an accident would be. That is the exact form of a warning this collection has issued twice already — that a lattice which nearly has a symmetry is a fact about the tolerance and not about the crystal — and here it is a statement about area.

Half of the usual region, and why the half matters #

One decision above was made quietly and is worth un-quieting, because getting it wrong has already cost this site a wrong number once.

The fundamental domain usually drawn for this construction runs from Re τ = −½ to +½. That is the right region for lattices up to rotation. A lattice and its mirror image sit at points reflected in the vertical axis, and no rotation carries one onto the other.

But a lattice and its mirror image are the same lattice. Nothing about a grid of points distinguishes a hand; handedness is a property of what is placed on the lattice, not of the lattice, and the two forms describe the same set of points seen from opposite sides of the paper. So the region this site uses is the half with Re τ ≥ 0, which is exactly Gauss’s condition b ≥ 0.

Taking the whole domain instead counts every asymmetric lattice twice. When the lengths were asked whether they name the lattice, an exhaustive search over forms found 1,140 pairs with identical theta series in a search whose correct answer is none — and every one of those pairs was a lattice paired with its own reflection. The computation was right; the equivalence was wrong. It is the same choice as here, made once in each direction.

The two corners, and the restriction #

The crystallographic restriction is usually stated about a matrix: the trace of an integer matrix is an integer, so 2cos(2π/n) is an integer, so n is 1, 2, 3, 4 or 6. The region states it about a space: the whole two-dimensional continuum of plane lattices contains exactly two points where the symmetry is more than fourfold, and they are the corners of the fundamental region.

The two statements are the same one read at different altitudes. The usual form counts the rotation orders an integer matrix can have — 1, 2, 3, 4 and 6, because 2cos(2π/n) has to be an integer and only those five values of n manage it — and it is a fact about matrices. The region’s form counts the lattices that carry each: a continuum of them for the orders 1 and 2, a curve for nothing at all, and exactly two isolated points for everything above. The five-element list is what a matrix can do; the two points are how much room the plane leaves for it, and the second is the sharper statement because it says the excess symmetry is not merely rare but isolated.

Those two corners are also the two points where the region has an angle rather than an edge, and that is not a coincidence: the extra symmetry is precisely what makes several copies of the region meet there. Six copies meet at ρ and four at i, which is the same arithmetic renamed — the order of the symmetry group and the number of copies are the same number, halved by the fact that the inversion acts trivially on τ. The next rung makes those copies visible.

What the region does not decide #

The region is a space of shapes, not of lattices. Scale has been divided out, so a lattice and the same lattice at twice the spacing are one point. That is the right quotient for every symmetry question and the wrong one for a diffraction experiment, which measures lengths and cares about nothing else; the theta series is a function of the lattice and not of its shape.

The counting is a count of integer forms, not a measure. The falling fraction above is evidence for the claim that four of the five types have measure zero, and it is not the proof. The proof is that a curve inside a two-dimensional region has no area, which is a sentence about the region rather than about any enumeration.

And it is the plane. The same construction exists in three dimensions and is far larger: the shape of a three-dimensional lattice needs five numbers rather than one, its reduction is Niggli’s rather than Gauss’s, and the fourteen Bravais types are its strata. Nothing above was proved there, and the fourteen are counted here by a different argument for exactly that reason.

The size of that gap is worth a sentence, because “the same construction, one dimension up” makes it sound like an exercise. In the plane a lattice’s shape is one complex number and the region it lives in can be drawn on a page; in space it is five numbers, so the region is a five-dimensional body and nothing about it can be looked at whole. The two corners here become a list of special strata there, and the two-and-three cone points become symmetry groups of orders up to forty-eight. The one part that carries over intact is the method: reduce a basis to a canonical one, and the reduced bases are a fundamental domain for the change-of-basis group acting on the space of shapes. Everything else — how many strata, what shape they are, where they meet — has to be redone, and the fourteen Bravais lattices are the answer to only the coarsest of those questions.

There is a second respect in which the region is smaller than it looks, and it is the one most likely to mislead. A point of the region is a shape and a shape is not a crystal. Two crystals whose lattices sit at the same point may have nothing else in common: the point fixes the grid up to rotation and scale, and says nothing whatever about what sits on it. All seventeen plane groups occur over the oblique interior — most of them at particular points of it, since a group with a three-fold forces its lattice to the corner — so the region classifies one of the two ingredients of a pattern and the group classifies the other.

What the region does settle is a question this collection had been answering in two incompatible ways without noticing: how many plane lattices are there? Five, if the question is about symmetry. A two-dimensional continuum, if it is about lattices. The region holds both answers at once, with the five sitting inside the continuum as its edges and corners — and the next essay asks what the copies of that region are, and how far a lattice has to walk to get home.

The corner that is not there #

The region has two corners and a third place where something happens, and the third is not a corner at all.

Follow the region upwards. The two vertical edges never meet; the region runs to infinity between them, narrowing to nothing in neither direction. A lattice high up in the region is one whose second basis vector is very long compared with the first — a lattice of two long rows far apart, or equivalently a lattice whose points cluster into widely separated lines.

That end is a cusp, and it is why the two questions the essay opens with have different answers. What shape is the space? — a region with two corners and an open end. How large are the families? — a question about measure, and the region has infinite area under the natural measure, so the answer has to be phrased as a proportion of something bounded rather than as an area.

The cusp also explains a fact about real lattices that would otherwise need its own argument. There is no most-elongated lattice. However far a lattice is from square or hexagonal, another is further, and the sequence never terminates in a special case — which is exactly what a non-compact end means. Every other feature of the region is reachable and bounded; this one is not.

What the region is, in the vocabulary the collection already has #

There is a reading of the picture that connects it to arithmetic elsewhere on this site, and it is worth making because it is the same arithmetic and it does not look it.

The region is a fundamental domain for a group acting on the upper half-plane, and the quotient — the region with its edges glued as the group requires — is a surface with two cone points and one cusp. The cone points have orders two and three, corresponding to the square lattice’s four-fold symmetry and the hexagonal lattice’s six-fold, each halved because the inversion every lattice has is already divided out.

In the notation this collection uses for such quotients that is an orbifold with corners of orders 2, 3 and infinity, and its cost is exactly two — which puts it, by the same accounting, on the boundary between the finite and the infinite lists. The infinity is the cusp: an order that is not a whole number at all, and the reason the group here is infinite while a wallpaper group’s point group is finite.

So the space of lattice shapes is a hyperbolic orbifold, and the two special lattices are its two cone points. That is a satisfying place for the square and the hexagonal to end up, because it says they are special for the same reason a rotation centre is special in a pattern: they are the points that are fixed by something.

Both readings are the same picture answering different questions, and the second is why the first is worth drawing at all. A list of five kinds is a list; a region with two corners and an open end says how the five sit relative to one another, which of them are limits of the others, and why exactly two of them are points rather than families.