Two patterns laid over one another

Closing the plane from two centres composes two operations and asks what the closure is. The same question can be asked one level up: compose two groups.

Lay two patterns over one another and two questions arise, and only one of them is the usual one. The usual one is what symmetry they share — the intersection of the two groups, which is the symmetry of the superposition. The other is the group generated by both, which is the symmetry of nothing in particular and is a perfectly good group to ask about, because it is a closure and a closure either terminates or does not.

It does not terminate very often. Two copies of one plane group, one turned against the other, generate a discrete group at a sparse set of angles and a dense one everywhere else.

The first condition is the one two centres already forced #

If the two groups have rotations of orders n and m, the join contains both, so it contains rotations of every order dividing their least common multiple — and closing from two centres establishes that a group of motions with more than six distinct rotations manufactures translations without limit.

So a four-fold and a three-fold cannot both be in a discrete group, whatever else is arranged, and neither can a four-fold and a six-fold: the multiple is twelve and no lattice holds twelve rotations. That condition is necessary and it is not sufficient, which the same figure shows: a three-fold centre added to a square lattice’s translations makes the join dense even when the multiple is six, because a three-fold rotation cannot sit on a square lattice at all.

Both failures are the crystallographic restriction, applied to different things — once to the pair of orders and once to the pair of a rotation and a lattice. Neither is new. What is new is the condition that survives when both are satisfied.

The second condition is about the lattices #

Take two copies of the same group, so the orders agree and every rotation is compatible with its own lattice. What can still fail is the relation between the two lattices.

The join’s translations are generated by the two lattices together, so they are a lattice exactly when the turned basis vectors have rational coordinates in the original basis — that is, exactly when the rotation matrix has rational entries. A rational point on the unit circle is a Pythagorean one, ((p² − q²)/(p² + q²), 2pq/(p² + q²)) for coprime whole numbers p and q, and at every such angle the join’s translations are a lattice.

Everywhere else they are dense. Two lattices at an irrational relation generate translations as short as anyone asks, which is the classification of the subgroups of the plane’s translations in its simplest instance: a subgroup of the translations is a lattice, a line’s worth, a dense set or the trivial group, and nothing else.

And now Niven’s theorem of 1956 does something startling. It says the only rational values of the cosine of a rational number of degrees are 0, ±½ and ±1. Of those, only 0 and ±1 have a rational sine as well. So the only turns measured in whole degrees at which two square lattices join discretely are the multiples of ninety — where they coincide outright. Every angle that works is an irrational number of degrees.

The coincidence at (3, 1) is 36.870°, at (5, 1) it is 22.620°, at (4, 1) it is 28.072°. Not one of them is a round number and not one of them ever will be.

The shortest translation is one over the square root of Σ #

At a turn that does work, the join’s lattice is not an arbitrary one and its size is exactly predictable.

Write Σ for the odd part of $p^2 + q^2$. The measurement gives the shortest translation as 1.000000 at Σ = 1, 0.447214 at Σ = 5, 0.277350 at Σ = 13 and 0.242536 at Σ = 17 — and those are one over the square roots of 1, 5, 13 and 17 to six places.

That number has been met here before under another name. Σ is the index of the coincidence site lattice, the lattice of points the two agree on, whose cell is Σ times the original; and the lattice generated by both is its dual partner, whose cell is Σ times smaller. The coincidence site lattice is the same arithmetic asked about the intersection where this page asks about the join, and the two indices multiply to one: the shared lattice is Σ times coarser and the generated one Σ times finer, so their cells multiply to the original’s.

That also settles which Σ occur. Every coincidence index is odd establishes it for the plane’s lattices, and the definition of Σ as the odd part of $p^2 + q^2$ is where the oddness comes from — a factor of two in that sum is a rotation by a right angle in disguise, and dividing it out is refusing to count the same coincidence twice.

So the join is discrete exactly when the intersection is a lattice of finite index, which is the sentence the two questions share. What the join adds is the direction the failure runs in: an intersection that is not a lattice of finite index is a lattice of infinite index or worse, and the corresponding join is a set with no shortest element at all.

The intersection, which is the question everyone else asks #

It is worth spending a paragraph on the other question, because the two are easy to confuse and the literature is about the other one.

Superpose two patterns and the symmetries both have are the intersection of the two groups. That is the symmetry of the superposition, it is what an experiment on a bicrystal measures, and it is always a group — an intersection of groups is a group, with no closure required and no question of discreteness, since a subgroup of a discrete group is discrete.

The symmetries the superposition has are therefore never in doubt. What is in doubt is whether there are any interesting ones: at a general angle the intersection of two copies of p4 is the four-fold rotation about the one shared centre and nothing else, a finite group, and the coincidence site lattice is the measurement of when it contains translations too.

The join is the opposite question and it is not about the superposition at all. It asks what the two groups generate, which is a group larger than either and the symmetry of nothing either pattern has. Its interest is that it is a closure, so it can fail — and the conditions under which it fails are the conditions composition has been about all along, arriving one level up from operations.

The two questions share an answer because they share an arithmetic: the intersection contains translations exactly when the join’s translations are a lattice, since both are the statement that the two lattices are commensurate. One condition, two consequences, and the literature has needed only one of them.

Discrete or dense, with nothing between #

The heading of this section is the claim, and it is worth saying why there is no middle.

A group of motions of the plane has a subgroup of translations, and that subgroup is one of five things: trivial, infinite cyclic, a lattice, dense in a line, or dense in the plane. A group whose translations are a lattice and whose rotations are finite in number is one of the seventeen, by Bieberbach’s first theorem and the classification. A group whose translations are dense is not discrete and is the symmetry of nothing.

The cases in between — translations dense in a line, or infinite cyclic — cannot arise from a join of two plane groups, because each ingredient already supplies a full lattice and the join contains both. So the join of two plane groups has a lattice’s worth of translations or a dense set of them, and the five-way classification collapses to two.

That is the sharpest form of the answer, and it is why the question has a clean answer at all. The same question about two groups of a more general kind — two Fuchsian groups, say — has a much longer list of possible outcomes.

The angles, and how sparse they are #

A set of angles parametrised by coprime pairs is dense in the circle and has measure zero, and both halves matter.

Dense means there is a coincidence angle as close as anybody likes to any angle at all. So a boundary between two grains at 15.0° is arbitrarily near a coincidence — the pair (p, q) = (15, 2) gives 15.189° with Σ = 229 — and the practical question is never whether a coincidence is near but whether a low one is.

Measure zero means that a turn chosen at random is a coincidence with probability nought. So the discrete joins are the exception and the dense ones are the rule, and the exception is the whole subject: every grain boundary anybody names by its Σ is one of the exceptions.

The two facts together are why the sequence of Σ values matters more than the angles. Σ = 5 at 36.870°, Σ = 13 at 22.620°, Σ = 17 at 28.072° — and between any two of them an infinity of larger ones, each at a finer angle and each generating a finer lattice. The shortest translation of the join is one over the square root of Σ, so a high-Σ coincidence is a join whose lattice is very fine, which is a discrete group in the same sense that a group with translations a millionth of a cell apart is discrete: true, and not useful.

What a moiré is, and what it is not #

A reader who has seen two lattices laid over one another has seen a moiré pattern, and it is worth separating what that is from what is computed here.

A moiré is an artefact of intensity: where two patterns nearly agree the superposition looks light, where they disagree it looks dark, and the resulting large-scale pattern has a period that is the beat between the two. That period is a real thing and it is not a symmetry of anything — the superposition at a general angle has no translational symmetry at all, and the beat is a statement about how nearly it does.

The join is about exact agreement rather than near agreement. At a coincidence angle the two lattices share a genuine sublattice, the moiré period is finite in the ordinary sense, and the superposition is genuinely periodic. At every other angle the superposition is aperiodic — quasiperiodic, in fact, which is the subject of the aperiodic pages here — and the moiré’s apparent period is a beat rather than a repeat.

So the sparse set of angles this page finds is the set at which a moiré is a crystal. A beat is not a period is the same distinction drawn about two lattices directly, and what the join adds is that the failure is a failure of a group to be discrete rather than of a pattern to repeat.

That is worth one more sentence, because it is the reason the question belongs in this field rather than in the one about interfaces. A pattern that does not repeat is a fact about a picture; a group that is not discrete is a fact about a closure, and a closure is what composing two operations starts. The whole of this subject is one procedure — take what is given, compose, and see whether it stops — and the join is that procedure fed two groups instead of two motions.

The measurement is identical in the two cases and that is the point. A join is a closure and a closure is a closure; what differs is the generators fed to it, and the shortest translation is the one quantity that answers the question either way.

Where the exactness stops #

Computed here. For each of four coincidence rotations and four whole-degree ones, the group generated by two copies of p4 on a square lattice with one turned, closed to two search depths, with the shortest translation at each. For six pairs of rotation orders, the same measurement with the centres half a cell apart. And for each angle, whether a rational point on the unit circle sits there, by a search over coprime pairs up to two hundred.

The dense side is measured and not proved. A finite search cannot show that a set has no shortest element; what it shows is that the shortest found falls when the search is deepened, and that it is already below anything the coincidence lattices in range would give. The proof is the rationality argument, which is exact, and the search is the illustration.

Two copies of one group, on one lattice. Every join measured has the same group on both sides and the same lattice, differing only by a turn. Two different groups, or one group on two lattices of different shape, is a larger question and the condition would be correspondingly longer.

And no shift is varied. The two copies share an origin throughout. A relative translation between two lattices at a coincidence angle changes which points they share and not whether they share a lattice, so the answers above are unaffected; a relative translation between two groups moves the rotation centres and can break the join in ways nothing here measures.

The second refusal is the one worth having: two copies of p4 at fifteen degrees offered as a discrete join. Fifteen degrees is a natural-looking angle, both groups are the same group, the rotation orders are compatible in every way — and the shortest translation is 0.261 at one search depth and 0.141 at the next.

Who asked, and in what field #

The mathematics is old and is nobody’s in particular: the join of two lattices, the rational points of the circle, and the classification of the discrete subgroups of the plane’s translations are all nineteenth-century.

Niven’s theorem is Ivan Niven’s, from Irrational Numbers in 1956, and it is usually met as a curiosity about trigonometry. Meeting it as the reason two crystals almost never fit together is what makes it worth quoting here: the fact that 36.870° is irrational in degrees and 15° is rational is the whole of why the first is a grain boundary and the second is not.

The coincidence site lattice is Kronberg and Wilson’s, from 1949, and it is how the metallurgy of grain boundaries is organised — a boundary is named by its Σ, and the low-Σ ones are the ones with low energy. That literature asks about the intersection. Asking about the join instead is the same arithmetic from the other side and it is not, as far as can be told from that literature, a question the metallurgy has needed to ask.

The twisted bilayer is where it has been needed since 2018. Two sheets of honeycomb at a small angle have a moiré whose period is enormous, and the electronic behaviour at particular angles is the subject of a large literature; the angles at which the superposition is genuinely periodic are the coincidence ones, and the famous angles are not among them.

Still open: the join of two different groups #

Everything above joins a group to a copy of itself. The general question is larger and has more room in it.

Two different plane groups on the same lattice join to something whose rotations are the join of the two point groups, and the condition on the least common multiple applies to every pair of orders rather than to one pair — so the answer is a condition on a pair of point groups, and the seventeen make a hundred and fifty-three pairs to ask it of. Nothing above does that, and the answer would be a table rather than a sentence.

Two groups on lattices of different shape is harder still, because the coincidence condition is no longer a rational rotation: two lattices of different shape share a sublattice when the matrix taking one basis to the other is rational, which is a condition on four numbers rather than on an angle. The set of relative positions at which they do is dense and measure zero, which is the same shape of answer with none of the arithmetic.

And the question this page most obviously suggests is about the intersection rather than the join. At a coincidence angle the two copies of p4 share a group; which of the seventeen it is depends on where the rotation centres fall relative to one another, and that is a computation of exactly the kind two centres already need and this page has not asked for.