Every coincidence index is odd, and in the plane most of them do not exist

Enumerate the rotations that bring a cubic lattice into coincidence with itself and list their indices. The list begins 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 — and it continues with every odd number and none of the even ones.

That is not a feature of the range. There is no Σ2, no Σ4, no Σ6, no Σ8, ever. A coincidence site lattice of even index does not exist, and the reason is a short argument about integers that has nothing to do with crystals — of the same kind, and with the same feel, as the trace argument that decides which rotations a lattice may have.

Where the rotations come from #

Every rotation of three-dimensional space with rational matrix entries comes from an integer quaternion. It is the same style of parametrisation the companion matrix supplies for rotations of a given order, and it is exact for the same reason: integer data in, integer matrix out. That is the parametrisation the enumeration uses, and it is worth stating carefully because everything below depends on it.

A quaternion is a quadruple (m, n₁, n₂, n₃). It gives a rotation about the axis [n₁n₂n₃] through an angle θ with tan(θ/2) = |n|/m, and the matrix of that rotation is A/N where N = m² + n₁² + n₂² + n₃² and A is an integer matrix built from products of the four components:

$$A_{11} = m^2 + n_1^2 - n_2^2 - n_3^2, \qquad A_{12} = 2(n_1 n_2 - m n_3), \qquad \dots$$

The construction guarantees AᵀA = N²I, which is exactly the statement that A/N is orthogonal, and the enumeration asserts that identity entry by entry for every quaternion it tries rather than trusting the algebra.

So enumerating rational rotations means enumerating integer quaternions, which is a search over four small integers. That is a finite search once a bound on the index is fixed, and finiteness is the property that makes the result a list rather than an observation about the examples somebody happened to try. The bound this site uses reaches every relation up to index twenty-five and would reach further at the cost of nothing but time.

What the search does not produce is a list of relations, and the gap between the two is worth seeing before any count is quoted. A rotation and the same rotation composed with a symmetry of the cube on either side describe the identical relationship between two grains, so the sweep finds each relation many times over and has to be reduced before it can be counted.

The two lines that rule out even indices #

Suppose N is even. The claim is that every entry of A is then even, so A/N is not in lowest terms and can be divided through by two.

The components cannot all be even — that would mean a common factor, and the quadruple is taken coprime. So among the four, either two are odd or all four are odd, since the sum of their squares is even and a square is odd exactly when its root is.

Two odd, two even. Say m and n₁ are odd. Then A₁₁ = m² + n₁² − n₂² − n₃² is odd + odd − even − even, which is even. The other diagonal entries are the same expression with signs rearranged and come out even too. Every off-diagonal entry has an explicit factor of two. So A is entirely even.

All four odd. Then A₁₁ = m² + n₁² − n₂² − n₃² is a sum and difference of four odd squares, which is even. Again the off-diagonals carry a factor of two. So A is entirely even.

In both cases divide A and N by two and repeat. The process terminates when N is odd, and the index is that odd number. Σ is the odd part of N, and no even index survives the reduction.

That is the whole proof, and it is worth noticing what kind of argument it is. It is a parity argument on integers, of exactly the kind the crystallographic restriction is: no geometry, no measurement, no appeal to what crystals do. The constraint is arithmetic and the crystals have to obey it.

Two routes to Σ, kept apart #

The formula “Σ is the odd part of N” is what every table uses, and this site does not use it to compute anything. It uses it to check.

The computation goes through the definition: Σ is the index of the sublattice {x : Aᵀx ≡ 0 mod N}, obtained from the elementary divisors of Aᵀ, which is a piece of integer linear algebra with the formula nowhere in it. The formula is then evaluated separately and the two required to agree.

They agree on every relation in the enumeration. If they ever failed to, one of the two would be wrong — and the figure would not appear at all, rather than appearing with the answer the two happened to share.

This is the arrangement the site puts under everything it counts, for a specific reason. A single computation that produces the right answer is evidence about that answer and no evidence at all about the method. Two computations sharing only their input are evidence about both.

The plane is a different question with a different answer #

Ask the same thing about a square lattice and the machinery is the same: enumerate rational rotations, compute the index, check the parity. What comes out is not the same list.

The plane’s rational rotations come from pairs rather than quadruples: for coprime m and k the rotation with half-angle tangent k/m has N = m² + k². So Σ is the odd part of a sum of two coprime squares, and Fermat’s theorem on sums of two squares decides which numbers those are.

The result is that the plane’s coincidence indices are 5, 13, 17, 25, 29, 37 and so on — the odd numbers all of whose prime factors are congruent to 1 modulo 4.

Σ3 does not exist in the plane. Nor does Σ7, nor Σ11. Three, seven and eleven are primes congruent to 3 modulo 4, no sum of two coprime squares is divisible by such a prime, and so no rotation of the square lattice shares one point in three with it.

That is a striking asymmetry between the two dimensions, and it comes from the difference between two squares and four. Every odd number is a sum of four squares — Lagrange’s theorem — so every odd index occurs in three dimensions. The step from two dimensions to four squares is the same widening that makes five-fold symmetry legal in four dimensions and not in three: more room, more solutions, a longer list. Only some odd numbers are sums of two, so only some occur in the plane.

Why the same index can happen more than one way #

Five of the twelve indices in the cubic list carry two relations each. Σ13 has one at 22.62° about ⟨100⟩ and another at 27.80° about ⟨111⟩, and Σ17, Σ19, Σ21 and Σ25 are the other four. Two things sharing a number and differing in everything else is a situation this site has met before, and the response is the same: distinguish them, and say what the shared number does and does not settle.

These are genuinely different misorientations. They share an index, so they share the density of coincidence sites, and they differ in every other respect — different axes, different angles, different coincidence lattices, and in general different boundary energies.

The tables distinguish them with letters — the same problem two group symbols differing in one position solve by encoding the difference rather than by appending a letter, and the letters are not arbitrary decoration: a paper reporting “Σ13 boundaries” without saying which is reporting an ambiguous measurement. The enumeration here counts the relations rather than the indices, which is why it reports seventeen relations across twelve indices — and it counts them rather than knowing them, which is why the caption on the plate above states five without that number appearing anywhere in the code.

Counting them correctly needs one piece of care. A rotation and the same rotation composed with a symmetry of the cube on either side describe the same relationship between two grains, so a naive enumeration over quaternions produces each relation up to 576 times. The relations are therefore reduced to a canonical representative under that double action before being counted, and the smallest angle in the resulting set is reported — the disorientation, which is the angle the tables print. Σ3 appears in the enumeration at 180° about ⟨111⟩ and at 60° about ⟨111⟩, and those are the same boundary.

The primes that are sums of two squares, and why they matter here #

Fermat’s theorem deserves a sentence of its own, because the plane result is entirely its doing.

An odd prime p is a sum of two squares exactly when p ≡ 1 (mod 4). Five is 4 + 1, thirteen is 9 + 4, seventeen is 16 + 1, twenty-nine is 25 + 4. Three, seven, eleven, nineteen and twenty-three are congruent to 3 modulo 4 and are not sums of two squares in any way at all.

The property extends multiplicatively: a product of two sums of two squares is another, by the identity behind complex multiplication, so 25 = 5 × 5 and 65 = 5 × 13 both qualify. And a number divisible by a prime congruent to 3 modulo 4 to an odd power does not.

So the plane’s coincidence indices are the odd numbers built only from primes congruent to 1 modulo 4. That is a purely number-theoretic description of a purely geometric question, and the bridge between them is one line: the rotation matrix has entries with denominator m² + k², and a rotation exists exactly when the denominator can be written that way.

A metallurgist asking why there is no two-dimensional annealing twin is asking a question about the primes. That is not a metaphor and not an analogy; it is the same statement, and the fact that no rotation of a square lattice shares one point in three with it is a consequence of three being congruent to three modulo four.

What the odd-index result does not mean #

Two cautions, and the second is the one worth carrying.

It is a statement about the lattice, not about the structure. A crystal is a lattice with a motif on it, and two grains related by a coincidence rotation share lattice points without necessarily having the same atoms at those points. For a structure with several atoms per cell the shared sites may hold different species on the two sides, and the boundary is not as good as the index suggests.

A low index is a necessary condition for a cheap boundary and not a sufficient one. The previous rung said this and it bears the repetition, because the odd-index theorem is exact and invites the exactness to be transferred to a claim that is not. Boundary energy depends on the boundary plane as well as on the misorientation, on the structure as well as on the lattice, and on the temperature. What the arithmetic delivers is a list of candidates, which is permitted rather than present in the form this whole field keeps arriving at.

There is a third caution that belongs to the plane result rather than to the cubic one. A square lattice is not a common thing in three-dimensional metallurgy, and the plane figures here are a model of the question rather than a case of it. What they demonstrate is the mechanism — an integer congruence deciding which points are shared — and what they cannot demonstrate is any particular metal’s boundaries.

The list is worth having precisely because it is short and closed. Seventeen relations up to Σ25, twelve indices, every one of them odd, all of it computed from four small integers at a time — and any boundary reported at an index not on the list is a boundary that has been mis-indexed.

Why the list stops where the tables stop #

The enumeration could run to any index. The metallurgical literature stops at twenty-nine, and the reason is worth stating because it is the same shape as every stopping rule in this field.

As Σ grows, the coincidence lattice thins out. At Σ3 one atom in three across the interface is shared, which is a large fraction and a real physical effect. At Σ29 it is one in twenty-nine, which is under four per cent, and the difference between such a boundary and a general one is within the scatter of measured energies.

There is also the tolerance. The rule of thumb allowing a deviation of about 15°/√Σ shrinks as Σ grows, so a high-index coincidence is a smaller and smaller target in orientation space — and the chance that a real boundary lies within the tolerance of one of them falls faster than the number of available indices rises. Past about Σ29 there are so many candidates so close together that almost every misorientation is near one, and a classification that fits everything has stopped classifying.

So the cut-off is a statement about the model’s usefulness rather than about the arithmetic. The arithmetic goes on for ever, produces an odd index at every step, and stops meaning anything about a real interface long before it stops producing answers. Saying where the second happens is not something the enumeration can do, and this site does not claim to.

The plane lattice that does have a Σ3 #

The claim that Σ3 has no plane analogue is a claim about the square lattice, and it is worth completing, because the hexagonal lattice answers differently and the difference is the same arithmetic seen through a different ring.

A rotation of thirty degrees carries the hexagonal lattice onto a sublattice keeping one point in three — the √3 × √3 R30° arrangement that surface science meets constantly. That is a coincidence relation of index three, and it exists.

So Σ3 does occur in the plane, on the hexagonal lattice, and the full list of hexagonal coincidence indices is the odd numbers built from three and from primes congruent to 1 modulo three — the Loeschian numbers, values of a² + ab + b², exactly as the square lattice’s are values of a² + b².

The two lists overlap and neither contains the other. Seven is hexagonal and not square; five is square and not hexagonal; thirteen is both. Which indices a plane lattice permits is a question about the ring it belongs to — the Gaussian integers for the square lattice, the Eisenstein integers for the hexagonal one — and the answer is the norms of that ring’s elements.

That reframes this essay’s asymmetry. It is not that the plane has fewer coincidences than space; it is that each plane lattice has its own list, and the cubic lattice’s list is the union of what four squares can do rather than what two can.

The lattices with almost no coincidences at all #

The cubic case is unusually generous and it is worth saying why, because most lattices are not.

A coincidence rotation must have a rational matrix in the lattice basis, and for a cubic lattice that condition is satisfied by a four-parameter family of quaternions — an infinite supply, giving an index at every odd number.

For a lattice with a free metric parameter the condition is far tighter. A hexagonal lattice in three dimensions has an axial ratio c/a, and a rotation about an axis not perpendicular to c has a matrix whose entries involve that ratio. Unless the ratio is rational — or the square root of a rational — such a matrix cannot be rational, and no coincidence exists about those axes at all.

So a general hexagonal, tetragonal or orthorhombic lattice has coincidence rotations only about its principal axis, where the metric does not enter, and nothing else. Its coincidence list is a one-parameter family rather than the cubic lattice’s four, and it is correspondingly short.

That is why the coincidence literature is overwhelmingly about cubic metals, and it is a fact about which lattices have rational rotations rather than about which materials anybody studied. A hexagonal metal with an axial ratio close to a simple fraction has near-coincidences instead, with an obliquity, which returns the question to how far one lattice is from another and out of exact arithmetic entirely.

Where the ladder goes next #

Everything in this anchor so far has been about one lattice meeting a rotated copy of itself. The indices came out of arithmetic, the answers were exact, and no measurement entered anywhere.

The last rung takes the case where the two lattices are different — a film of one substance grown on a crystal of another. Now the two spacings are in a ratio that is not a ratio of small integers and is not a ratio of any integers at all, because no measured ratio is rational. Coincidence in the exact sense is unavailable, the question becomes how nearly two lattices can be made to agree over how long a repeat, and the answer stops being a theorem and becomes a measurement with a threshold in it.