Every plane lattice is its own dual

The dual lattice is built rather than defined here: one point per family of lattice rows, at the inverse of the spacing, perpendicular to them. Its Gram matrix is the inverse of the original’s, which is another way of saying the dual basis pairs with the direct one to give the identity — and that pairing is the reason a zone axis and a face index multiply to nothing with no metric anywhere in the expression.

In two dimensions there is a second thing true of the inverse, and it is short enough to write in one line.

For a two-by-two matrix, the inverse is the adjugate over the determinant. And the adjugate of a symmetric two-by-two matrix is what a particular change of basis does to it:

UᵀGU = adj(G) when U is the right-angle rotation (0 −1; 1 0).

U has determinant one, so UᵀGU is the Gram of the same lattice written on a different basis. So the dual’s Gram is the original’s, rotated, divided by the determinant — which is to say the dual lattice is the original one turned through a right angle and scaled.

Every plane lattice, with no condition attached. Not the hexagonal one especially, not the square one; the oblique lattice with Gram (3 1; 1 5) is similar to its own dual exactly as the square lattice is, and the scale factor is its determinant.

What the identity is saying geometrically #

The algebra is two lines and the picture is one sentence.

The dual basis vector a* is perpendicular to b and b* is perpendicular to a — that is what “one point per family of rows, perpendicular to them” means. Perpendicular to b is b turned through a right angle; perpendicular to a is a turned through a right angle. So the dual basis is the direct basis turned through a right angle, with the two vectors swapped and one of them reversed, and scaled by one over the area of the cell.

Swapping two basis vectors and reversing one is a change of basis of determinant one. A change of basis does not change a lattice. So the dual is the lattice itself, rotated and scaled, and the whole result is that sentence.

The reason this is a two-dimensional accident and not a general fact is equally short. In n dimensions the adjugate of a matrix has entries that are (n − 1)-by-(n − 1) minors, so it is a polynomial of degree n − 1 in the entries. At n = 2 that degree is one, and a linear map on the entries is exactly what a change of basis can be. At n = 3 the adjugate is quadratic, and no change of basis is.

The determinant is the whole of the scale #

The identity carries a factor and it is worth reading, because it is the only quantity in the result that is not pure shape.

UᵀGU = adj(G) = det(G) · G⁻¹, so the dual’s Gram is the original’s rotated divided by the determinant — and the determinant of a Gram is the square of the cell’s area. So the dual lattice’s cell has area one over the original’s, which is the ordinary reciprocal-space statement, and its basis vectors are shorter by the area rather than by any length.

That gives the two halves of the result their proper weight. The shape is unchanged and the scale is inverted, and every consequence below divides along that line: anything about angles, ratios and shells transfers untouched, and anything with a length in it picks up the area. A square lattice of side two has a dual that is a square lattice of side one half; a lattice of area one is a lattice whose dual is congruent to it and not merely similar.

The self-dual lattices in the strictest sense — congruent to their duals rather than similar — are therefore the plane lattices of unit area, one for every shape. That is a two-parameter family again, and it is the same statement with the scale nailed down.

In space the dual is a different lattice #

The cubic lattices are the cleanest case and they are read off a count rather than quoted.

The primitive cubic lattice has six shortest vectors and its dual has six: it is its own dual. The face-centred lattice has twelve and its dual has eight; the body-centred lattice has eight and its dual has twelve. The two are duals of each other, which is the fact behind every calculation in which a face-centred crystal’s reflections sit on a body-centred reciprocal lattice.

The count of shortest vectors is used here as the name, for the same reason the cubic sublattices are named by it: a lattice arrives as a Gram matrix, a Gram matrix is a basis rather than a lattice, and the shell counts are what survive a change of basis. Six, twelve and eight separate the three cubic lattices completely, and nothing about the answer depends on which cell anybody chose.

The one ratio a family gets #

Self-duality in space is a condition, and a one-parameter family of lattices satisfies it at a point rather than along an interval.

A tetragonal lattice with axes 1, 1, c has a dual with axial ratio 1/c. So the dual’s ratio falls as the original’s rises, the two cross exactly once, and the crossing is at c = 1 — the primitive cubic lattice. The only self-dual tetragonal lattice is the cubic one, which is a sentence worth pausing on: a family with a free parameter contains exactly one member with this property, and it is the member with the extra symmetry.

A hexagonal lattice with ratio c has a dual with ratio √3 / 2c. The same argument gives one crossing, and it is at the fourth root of three quarters, about 0.9306 — an irrational number, and not one that any extra symmetry marks out.

The contrast with the plane is the point of the whole essay. In two dimensions self-duality is not a property some lattices have — it is a fact about the dimension, holding for the whole two-parameter family of lattice shapes. In three dimensions the family of shapes has five parameters and the self-dual ones form a subset of lower dimension, which is another way of saying that a lattice picked at random in space is not its own dual and one picked in the plane always is.

What the identity does to an argument #

Before the repair below, it is worth saying what having this identity in hand changes about reasoning in the plane, because it is more than a curiosity.

Any statement of the form “this lattice is good at X and its dual is good at Y” is, in two dimensions, a statement about one lattice. There is no trade to make and no pair to compare: the dual is the same shape, so a property of the lattice is a property of its dual and back again. That collapses a whole class of arguments that are informative in space into tautologies in the plane.

It also means a plane calculation cannot be used to build intuition about the direct-versus-reciprocal distinction. A crystallographer’s feel for “the reciprocal lattice of a squashed cell is a stretched one” is right in space and vacuous in the plane, where squashing a lattice and dualising it give the same shape back. The plane is the wrong place to learn what duality does, which is worth knowing on a site that does most of its work there.

And it explains a small puzzle in the two-dimensional diffraction figures elsewhere in this collection: the reciprocal lattice of a hexagonal lattice is drawn as a hexagonal lattice rotated by thirty degrees, and of a square lattice as a square lattice. Those are not two facts about two lattices; they are one fact about the plane, and the thirty degrees is the ninety-degree rotation seen through the hexagonal lattice’s own six-fold symmetry.

What it repairs #

This identity was needed for something, and the something was a claim on another page of this collection that turned out to be wrong.

Covering and packing want different lattices argues that a lattice cannot be extremal for both problems unless it is close to self-dual, since dualising exchanges the roles of the shortest vector and the deepest hole. That argument is sound. What it said next was that “the hexagonal lattice is the only plane lattice of that kind” — and there is no such thing as being the only plane lattice of that kind.

The repair strengthens the argument rather than weakening it. If self-duality were rare in the plane, the hexagonal lattice’s winning both prizes would be a coincidence that self-duality happened to permit. Since every plane lattice is self-dual, the duality argument has nothing to push against anywhere in the plane, and the fact that some single lattice wins both is exactly what the argument predicts. In space, where self-duality is a condition and the extremal lattices do not meet it, the two prizes go to a dual pair — which is what is observed.

That is the useful shape of a correction: the wrong version made a true conclusion rest on a false premise, and the right version supports it better.

The second is the one that keeps the identity honest. UᵀGU = adj(G) is a claim about a specific U, and a check that only ever ran that one matrix would not distinguish “this rotation does it” from “any basis change does it”. Handed a shear instead, the left-hand side comes out as something else entirely, and the test requires that it does.

The fourth and fifth are a pair. The primitive cubic lattice being self-dual is what stops the space half of the essay from being a test that always says no — a procedure that returned “not self-dual” for everything would agree with the face-centred case for the wrong reason, and the primitive one is the control.

A caution about what “dual” means here #

Two things go by the name and only one of them is the subject.

The dual lattice is the one this essay is about: the set of vectors whose inner product with every lattice vector is an integer, which is the reciprocal lattice with a factor of 2π in or out depending on the convention. That is a lattice of the same dimension, and self-duality is a statement about its shape.

The dual tiling is a different construction entirely — the eleven duals of the uniform tilings, or a polyhedron and the solid built on its face centres. That duality exchanges vertices with faces and has nothing to do with Gram matrices. Nothing in this essay says anything about it, and the two are related only by sharing a word.

The confusion is worth heading off because both appear in this collection within a few pages of each other, and because the plane’s result sounds like the kind of thing that ought to be about tilings and is not.

Why the plane’s two-parameter family has no exceptions #

One more way of seeing the result, for a reader who finds the adjugate argument too quick.

The shape of a plane lattice, up to similarity, has two parameters — the ratio of the two basis lengths and the angle between them, or equivalently a point in the moduli space this collection draws. Dualising is a map from that space to itself, since the dual of a lattice is a lattice and similar lattices have similar duals.

The identity says that map is the identity map. Not “has a fixed point”, not “has a few fixed points” — every point of the moduli space is fixed, and dualising is a relabelling that moves nothing.

Compare the two other natural maps on the same space that this collection has already met. Two moves reach every basis generates the modular group acting on that space, and its elements move points around vigorously; the map that sends a lattice to a sublattice of given index moves points too, and the tree that results is infinite. Dualising is the one natural operation on plane lattices that does nothing at all.

In space the same map is not the identity, and the self-dual lattices are its fixed points — a subset of a five-dimensional space, which is why they are special there and why the question “which lattices are self-dual” has an answer worth asking for.

What a crystallographer uses it for #

The identity is not only decorative, and it is worth naming the two places a working reader meets it without being told.

A two-dimensional diffraction pattern has the same lattice shape as the crystal. Electron diffraction from a thin film, a surface reconstruction seen by low-energy electrons, a two-dimensional layer’s own reciprocal net — in every one of them the pattern’s lattice is the sample’s lattice, rotated a right angle and scaled. So indexing a two-dimensional pattern is reading a shape off it directly, and the axial ratio and the angle come out of the picture with no inversion at all. In three dimensions the same reading gives the reciprocal cell and the direct one has to be computed from it.

And a plane lattice’s own reciprocal is a poor guide to a crystal’s. The rule of thumb that a long axis gives a short reciprocal axis is right in space and empty in the plane, where the whole shape comes back unchanged: a plane lattice with one axis twice the other has a dual with one axis twice the other, because the dual is the same lattice turned. A reader who learned the rule from a plane figure has learned something about a scale factor rather than about a shape.

Both of those are consequences of one line of matrix arithmetic, which is the sort of thing this collection is for.

Where the exactness stops #

Computed here: the identity UᵀGU = adj(G) on the five named plane Grams and on eight more with entries generated from a stated seed and kept when positive definite; the shortest three shells of each cubic lattice and of its dual, by an exhaustive scan over a bounded box; the axial ratio of the dual of a tetragonal and of a hexagonal lattice across a ladder of ratios; and the similarity of each family’s self-dual member to its own dual, by comparing shells rather than by evaluating the formula that produced the ratio.

Similar means similar. A lattice and its dual in the plane are the same shape at different sizes and in different orientations, which is what “similar” means and is not the same as equal. A physicist working in reciprocal space is working in a space with different units, and the identity here says the shape transfers, not that the two spaces are the same one.

The shells are counted inside a box. Three shells out of a box of three cells is enough to separate six, eight and twelve, and it would not be enough to distinguish two lattices agreeing to that depth — which is a real possibility in high dimensions and is not one here, since the three cubic lattices are separated by their first shell alone.

And the space case is not a classification. The self-dual lattices in three dimensions are a subject with a literature; what is computed here is the three cubic ones and the crossing of two one-parameter families, which is enough to establish that self-duality is a condition and not enough to say which lattices satisfy it.

Where the ladder goes next #

Back, to the construction: the dual lattice, where the reciprocal is built one row family at a time, and the reciprocal lattice, where the same object is what a diffraction pattern is.

Sideways, to the argument this repairs: covering and packing want different lattices, where duality is what separates the two optima in space and fails to separate them in the plane.

Onward, to the lattices whose shells do not name them: the lengths do not name the lattice, where a theta series determines a plane lattice and stops determining one in sixteen dimensions — the same shape of statement as this essay’s, with the dimension doing the work.