The shapes a lattice in space can thin to

The sublattices that are the same shape asks which indices admit a sublattice that is the parent rotated and scaled. In the plane the answer is arithmetic and it is beautiful: the square lattice has one at every index that is a sum of two squares, the hexagonal lattice at every Loeschian number a² + ab + b², and deciding which integers those are is Fermat’s theorem about primes.

That essay ends with a sentence this one is about:

That is a richer arithmetic than the plane’s and it is not computed here.

The three-dimensional case. It turns out to be a different question rather than a harder one, and the reason is one line of algebra.

The index has no choice #

Write the similarity as an integer matrix M on the parent’s basis. The sublattice M·L is similar to L when M is a rotation times a scale, which in terms of the Gram matrix G is

Mᵀ G M = α² G.

Take determinants. (det M)² · det G = α⁶ · det G, so det M = ±α³ — and the index of the sublattice is |det M|.

Every similar sublattice in space has an index that is a perfect cube. No sums of squares, no Loeschian numbers, no theorem of Fermat’s: the representable indices are 1, 8, 27, 64, 125, … and that is the complete list, for every lattice in space whatever its shape.

That collapse is worth pausing on because it inverts what the plane suggests. In two dimensions the interesting question is which indices work and the answer is a piece of number theory. In three the answer is “the cubes”, immediately, and the interesting question becomes how many work at each one.

Counting them #

At scale α the columns of M are the images of the basis vectors, so their norms and their mutual inner products are all fixed by the condition. That turns the search from a sweep over nine entries into three lookups by norm: find the vectors of norm α²G₀₀, then those of norm α²G₁₁ orthogonal to the first in the right way, then the third. The radius the lookup needs is derived from the norms rather than chosen, which is what makes the enumeration complete.

Two matrices describe the same sublattice exactly when they differ by an automorphism of the parent on the left, so the count is a count of orbits — taken by marking rather than by dividing, since dividing by the order of the point group would be wrong if the action were not free and there is no reason given that it is.

For the cubic lattice the counts at scales one to ten are

1, 1, 5, 1, 7, 5, 9, 1, 17, 7

Two patterns are visible in that list and both are checked rather than admired.

The count is multiplicative. f(6) = f(2)·f(3) and f(10) = f(2)·f(5), so it is an arithmetic function of the scale rather than a list of ten numbers. That is the same structure the plane’s count has, and it comes from the same place: a similarity at a composite scale factors through similarities at the coprime parts.

And at an odd prime the count is the prime plus two. f(3) = 5, f(5) = 7, f(7) = 9. That is a small, sharp pattern and the enumeration produces it without being told.

What a factor of two does not buy #

The other pattern is the more striking one. f(1) = f(2) = f(4) = f(8) = 1: at every power of two there is exactly one similar sublattice, and it is the scaled copy.

The reason is the same parity that makes every coincidence index odd. A similar sublattice at scale α amounts to a rotation whose matrix has denominator α on the cubic lattice, and a rotation with a denominator of two is a rotation the cubic lattice already has — the denominators that produce something new are odd, and a factor of two in α contributes only the scaling that goes with it.

That is worth stating as a fact about superstructures rather than about arithmetic. A cubic crystal can be thinned to a sublattice of the same shape at index eight in exactly one way, and at index twenty-seven in five. Doubling a cell is not a choice and tripling it is.

That asymmetry has no counterpart in the plane, where the square lattice’s similar sublattices sit at every sum of two squares and the even indices among those — two, eight, ten, eighteen — are as populous as the odd ones. The parity that empties the powers of two in space is a three-dimensional fact, and it arrives from the same place as the odd coincidence indices rather than from anything about cubes.

The count as an arithmetic function #

Multiplicativity is worth taking seriously rather than noting, because it means the ten numbers are not ten numbers.

A multiplicative function is determined by what it does at prime powers, so the whole of f is fixed by f(2ᵏ), f(3ᵏ), f(5ᵏ) and so on. The enumeration supplies f(2ᵏ) = 1 for k up to three, f(p) = p + 2 at every odd prime it reaches, and one value at a higher odd prime power: f(9) = 17.

That last number is the one that says the function is not simply p + 2 extended. If f were determined by the prime alone in the obvious way, f(9) would be f(3)² = 25 — multiplicativity says nothing about prime powers, only about coprime factors — and it is 17. So the prime-power behaviour is its own recursion, and reading it off two values is guesswork rather than measurement. The honest statement is that the values are computed, that multiplicativity is checked on the coprime cases inside the bound, and that the prime-power pattern is one datum past the primes.

The corresponding function in the plane is worth putting beside it. The square lattice has 1 similar sublattice at every index that is a sum of two squares and 0 at the rest, which as an arithmetic function is a character sum: Σ χ(d) over the divisors, with χ the non-principal character modulo four. It is a criterion — most indices give nothing — and the interesting content is where the zeros are.

In space there are no zeros to find: every cube works. The interesting content moved to the size of the answer, and the function is a count rather than an indicator. Two questions that read identically in the two dimensions have different subjects, which is the sort of thing that only becomes visible when both are computed by the same routine.

There is a structure behind the counting and this collection has half of it already. A similarity at scale α is α times a rotation whose matrix has rational entries with denominator dividing α, and rational rotations of space are parametrised by integer quaternions — the same parametrisation the cubic coincidence series is enumerated from. Counting similar sublattices at scale α is then close to counting quaternions of norm α up to units, which is where both the multiplicativity and the powers of two come from: the quaternions of norm two are all associates of one another. Deriving f that way is not done here — what is done is the enumeration, and the quaternion picture is named because it says where the pattern comes from rather than being asked to produce it.

What a superstructure of index twenty-seven looks like in a pattern #

The practical form of all this is about diffraction, and the constraint is unusually easy to state.

A superlattice adds reflections: thinning the lattice by index n multiplies the reciprocal lattice by n, so new spots appear between the parent’s, and their positions say which sublattice was taken. For a shape-preserving sublattice in space the index is α³, so the new spots appear at thirds or halves of the parent’s spacing and never at, say, fifths of one axis alone — the sublattice is the same shape, so the subdivision is the same in every direction.

That is a filter a reader can apply to a pattern without computing anything. Superlattice spots that subdivide one axis and not the others cannot come from a shape-preserving superstructure, because a shape-preserving one subdivides all three equally. What they can come from is an ordinary superstructure that changes the cell’s proportions, and those are the common case: the three that stay cubic counts how few sublattices of a cubic lattice keep even the symmetry, and shape is a stronger condition than symmetry.

The other half of the filter is the count. At index eight there is one shape-preserving superstructure and at twenty-seven there are five, so a pattern that indexes on a tripled cell of the same shape has five candidate orientations to distinguish between and a pattern on a doubled cell has one. Distinguishing them is a question about intensities rather than positions, since all five give spots in the same places up to a rotation the pattern may not resolve — which is the same ambiguity twinning by merohedry produces from the other direction, and it is worth recognising when it appears.

A hexagonal lattice’s answer depends on its shape #

The three that stay cubic ends on a related admission — that its computation thins ℤ³ with the cubic metric, and that “a hexagonal parent has a different list and it is not computed here”. Here it is, and the answer has a shape the cubic case does not prepare a reader for.

A hexagonal lattice in space is a triangular net stacked at a spacing c, so its shape is one number: c²/a². That number decides the entire answer.

For most of the ratios tried, the only similar sublattices are the scalings: one at every cube, nothing else, no matter how far the search runs. For a handful — c²/a² equal to 1, 3, 1/2, 3/2, 2, 4, 6 — there are more, and the counts jump to four, seven or ten.

The line between the two groups is not the line a reader would guess from the cubic case, and one entry makes that plain. The ideal close-packing ratio c²/a² = 8/3 is in the group with nothing but scalings. That is the ratio a hexagonal metal sits at when its atoms are spheres touching their neighbours — magnesium, zinc, titanium, cobalt to within a few per cent — so the commonest hexagonal lattice in the periodic table has no similar sublattice that is not simply a scaled copy.

Why the three cases differ #

Putting the three side by side says something about what “similar sublattice” is measuring.

In the plane, every lattice with more than the minimum symmetry is described by a ring of algebraic integers — the Gaussian integers for the square lattice, the Eisenstein integers for the hexagonal one — and a similar sublattice is an ideal. The indices are norms, and which integers are norms is Fermat’s question. The arithmetic is rich because the ring is.

In the cubic case the corresponding object is the quaternions, and a similarity is a quaternion of norm α. Quaternion multiplication is not commutative, so there is no ideal theory of the same kind, and what survives is a count rather than a criterion. The index is forced to be a cube by the determinant and the count inherits its multiplicativity from the quaternions’.

In the hexagonal case there is no such ring at all unless the axial ratio makes one. A generic ratio gives a lattice whose only similarities are the obvious ones, and the special ratios are exactly the ones at which the lattice acquires more symmetry than a stack of triangular nets normally has. The special ratios are where the lattice is secretly something else — at c²/a² = 3 the stacking has a relationship to the base the generic case does not — and the extra sublattices are what that relationship produces.

The same question on a square base #

A hexagonal lattice is not the only one in space with a free shape parameter. A tetragonal lattice is a square net stacked at a spacing c, and it has the same single number c²/a². Running the identical sweep on it is worth the minute it costs, because otherwise the hexagonal finding is a fact about one family and a reader has no way to tell whether the ratios that admit extras are peculiar to the triangular net.

They are not, and the two families do not behave alike. On a square base almost every ratio tried admits similar sublattices beyond the scalings — 1/2, 1, 2, 3, 4 and 5 all do — and only 17/10 comes back with nothing. On a triangular base the position is reversed: most of the ratios tried give nothing and a handful give extras.

So the dependence on shape is general and the proportion of shapes that admit anything is not. What separates the two is which denominators the condition can absorb. Take the tetragonal Gram, with c²/a² = p/q: the third column of M needs norm α²p/q, and for that to be the norm of an integer vector the fraction has to clear — so a ratio with a small denominator has many scales at which it can, and 17/10 has few. The hexagonal Gram carries an off-diagonal term as well, and the two conditions together are harder to satisfy.

The asymmetry is worth stating carefully, because it is easy to turn into a rule it does not support. What has been measured is thirteen ratios on one base and seven on the other, at four scales each. That is enough to establish that both behaviours occur on both bases and that the majorities differ; it is not enough to say which ratios pass in general, and the essay does not.

The ratio at which a lattice is interesting is not the ratio at which it is common. The tetragonal sweep includes c²/a² = 1, which is the cubic lattice arriving as a special case of its own family, and the counts there are the cubic ones — 1, 1, 5, 1 — which is the sweep checking that it is measuring the lattice it thinks it is rather than a coincidence of parameters.

What this does not settle #

The search is inside a bound and says so. The radius is derived from the norms the columns must have, so it is complete for the scale being asked about rather than a sample; but the sweep over axial ratios covers twelve ratios and four scales, not all of either. What is established is that both behaviours occur and which of the twelve shows which, not a classification of the ratios.

And a similar sublattice is not a coincidence site lattice. The two are related and are not the same object: a coincidence lattice is the intersection of a lattice with a rotated copy of itself at the same scale, and a similar sublattice is a rotated copy at a smaller scale sitting inside. The arithmetic overlaps — both are about rotations with rational matrices — which is why the parity argument transfers, and the questions are different. The coincidence site lattice is about grain boundaries; this is about superstructures.

What a crystallographer takes from it #

A superstructure of the same shape as its parent is a real object: it is what a lattice does when an ordering doubles or triples a cell without changing the symmetry. In the plane, which indices allow it constrains surface reconstructions — a √7 × √7 R19.1° reconstruction exists because seven is a Loeschian number.

In space the constraint is much tighter and much simpler. The only shape-preserving superstructures have index a perfect cube, so a doubled cell that keeps the lattice’s shape does not exist, a tripled one does not either, and the first non-trivial one is at index eight. Anything else that looks like a shape-preserving superstructure is either not shape-preserving — the cell has changed proportions — or is a coincidence relationship rather than a sublattice one.

That is a strong constraint arrived at from a determinant, and it is the kind this collection likes: permission is not presence, but a permission this narrow does most of the work of a prediction.