Thirty-two from fourteen matrices

Thirty-two and no others enumerates the crystal classes by taking subgroups of the two largest point groups, m3̅m and 6/mmm, quotienting by conjugacy and merging the two lists. It is a construction about groups, and the lattices appear only at the end, as the reason the list is finite.

This essay does it the other way about. It starts from the fourteen lattices, asks each one what its own symmetry is, and takes every subgroup of every answer. No point group is named anywhere in the construction, and the thirty-two arrive at the end as a count.

A lattice’s own group, from its metric #

A Bravais lattice is a shape, and a shape has a symmetry: the integer matrices U with

Uᵀ G U = G

for the lattice’s Gram matrix G. That is a finite set — the matrices are integral and preserve a positive definite form, so they preserve lengths and there are finitely many — and it is found by the same column-by-column search similar sublattices uses at scale one: each column has a norm and a set of inner products fixed by the condition, so the search is three lookups rather than a sweep over nine entries.

The fourteen have to be written as Gram matrices on a primitive basis, since a centred cell’s conventional basis does not span the lattice. And their parameters have to be generic — no two lengths equal unless the type requires it, no right angle unless the type requires it — or a lattice will report a symmetry it does not have and every count downstream will inherit it. That is asserted rather than trusted: each lattice’s group must come back at the order its Bravais type has.

The subgroups are enumerated by closing every subset of at most three elements. That gives 2, 5, 5, 16, 16, 16, 16, 35, 35, 54, 16, 98, 98, 98 — five hundred and ten point groups, one for each way a crystal on one of these lattices could be symmetric.

Two numbers name an operation #

Five hundred and ten is not the answer, because the same crystal class occurs on several lattices and in several orientations on each. Sorting them means deciding when two subgroups are the same class, and there is a cheap invariant that does it: what the subgroup is made of.

A three-dimensional crystallographic operation is named completely by its determinant and its trace. A rotation of order n has determinant one and trace 1 + 2cos(2π/n); a rotoinversion has determinant minus one and its rotation’s trace negated. Ten pairs, ten kinds of operation, and nothing else a crystallographic matrix can be.

So a subgroup is described by ten numbers: how many of each kind it contains. Sorting the five hundred and ten by that description gives thirty-two distinct answers.

The same thirty-two, from a construction sharing no step #

Thirty-two is the right number, and getting the right number is not the same as getting the right list. The check is that these thirty-two are the thirty-two the other construction produces — and they are matched by signature rather than by name, since this construction has no names to match with.

That other construction takes subgroups of m3̅m and of 6/mmm in fixed settings, quotients each by conjugacy inside its own group, and merges the two lists by comparing what the classes are made of. This one starts from fourteen metrics, never mentions m3̅m, and reaches the same thirty-two ten-number signatures. Every one matches and nothing is left over on either side.

Two things make that agreement worth having. It is a check on both: a wrong lattice parameter here or a missed conjugacy there would show as a mismatch rather than as a plausible number. And it says the crystal classes are not an artefact of either route — they are what comes out whether the starting point is the groups or the shapes.

Which classes a lattice can carry #

There is a table in the middle of this that the group-first construction does not produce, and it is the reason for doing it this way.

A class occurs on a lattice exactly when the lattice’s own group contains a copy of it. So each of the thirty-two has a number attached: how many of the fourteen it can sit on.

The extremes are the informative ones. 1 and 1̅ occur on all fourteen — every lattice contains the identity and, since a lattice is symmetric about any of its points, the inversion. A triclinic crystal can have any lattice at all, which is why the triclinic system is not a shape but the absence of a constraint.

The hexagonal classes occur on exactly one. A six-fold axis needs a lattice with a six-fold axis, and there is one such lattice; a crystal of class 6/mmm has no choice of Bravais type at all. The cubic classes occur on three, which is why a cubic crystal has a choice — primitive, body-centred or face-centred — and a hexagonal one does not.

This is the holohedry is the ceiling read as a table: a lattice’s group is the largest a crystal on it can have, so the classes it carries are its subgroups, and counting them the other way round says which lattices each class is allowed.

A class and a subgroup are different things #

Five hundred and ten subgroups and thirty-two classes is a ratio of sixteen, and where that factor goes is worth following, because it is the difference between a symmetry and a symmetry in a particular orientation.

A cubic lattice has ninety-eight subgroups. Its distinct classes are far fewer, because a mirror in a cubic lattice can face along an axis or along a face diagonal, and those are two subgroups and one class. The same is true of every axis: the class 2 occurs many times over on a cubic lattice, once for each direction a two-fold axis can point.

That ratio is not a nuisance to be divided away. It is the reason the classification of space groups needs more than a list of classes: a space group is a class in a setting, and which settings are genuinely different is exactly the question the arithmetic classes answer. One class, two names is the plane’s version of the same trouble, where 3m1 and 31m are the same class in two orientations relative to the lattice and are given different symbols because the difference is real.

On a triclinic lattice the ratio is one: two subgroups, two classes, no orientations to choose between because there are no axes to align with. The ratio measures how much the lattice’s own symmetry can move a subgroup around, so it is largest for the cubic lattices and smallest for the triclinic one — and a reader who wants a single number for “how much geometry is in a crystal class” could do worse than this one.

What the fourteen are for #

There is a question this construction answers as a by-product and it is one the collection has asked before: why fourteen?

Fourteen lattices and no others derives them, and the derivation is about which centrings of which cell shapes are genuinely different. Here they arrive as a list of Gram matrices, and what makes them the right list is visible in the output: the fourteen groups they produce are all the groups that can be a lattice’s own symmetry, and every subgroup of any of them is a possible crystal class.

Adding a fifteenth lattice would mean a Gram matrix whose group is not one of the fourteen already listed, or whose group is one of them but which is not related to the corresponding lattice by a change of basis. The first cannot happen — the groups here are the seven holohedries in their possible centrings, and the argument that there are no more is the classification. The second is what “no others” means.

So the fourteen are the fourteen distinct answers to “what can a lattice’s symmetry be, together with how the lattice sits inside it” — which is why the list is fourteen rather than seven, and why the extra seven are centrings rather than new shapes. A body-centred cubic lattice has the same group as a primitive one and is not related to it by a change of basis, so it is a separate entry; a C-centred tetragonal lattice has the same group and is a primitive tetragonal lattice on other axes, so it is not.

That last point is where a reader’s intuition usually fails and it is worth the sentence: there is no C-centred cubic lattice not because nobody drew one but because the thing drawn is a tetragonal lattice, and the census here would find its group to be of order sixteen rather than forty-eight.

Where this stops, and why #

The thirty-two are geometric classes: two subgroups are the same class when they contain the same operations, regardless of how those operations sit relative to the lattice.

The unit the space groups are actually built from is finer. It is the arithmetic class — a point group together with the lattice type it acts on — and there are seventy-three of them, one for each symmorphic space group. mm2 on a primitive orthorhombic lattice and mm2 on a C-centred one are one geometric class and two arithmetic ones, and they carry different numbers of space groups.

Getting from thirty-two to seventy-three looks like it means deciding when two of these five hundred and ten subgroups are conjugate in GL(3, ℤ), and that search does not finish. A brute force over integer matrices with entries in [−2, 2] is nearly two million candidates for a single comparison and there are hundreds of comparisons; the standard alternative — that a conjugating matrix must carry one group’s invariant quadratic form to the other’s — pins the search down when the group acts irreducibly and does not when it acts reducibly, because a reducible group has a whole family of invariant forms. The reducible cases are exactly the low-symmetry classes, which are exactly the ones with the most arithmetic classes to separate.

The way past it is to not ask that question at all, and seventy-three without a search is the essay that does. Every finite group of integer matrices has a canonical larger group attached to it — the symmetry of a generic form it fixes — and that group is one of the fourteen and is preserved by conjugation. So each of these five hundred and ten belongs to exactly one lattice, at the cost of one linear solve, and the only conjugacy left to decide is inside a set of at most sixteen. The count comes out at seventy-three and sorting those by operation content gives back these thirty-two.

The ten ways of being mm2 reaches the same place from the other side, and says so in a sentence this essay is the other half of: computing the normaliser of each of the seventy-three correctly is the content of the classification rather than an application of it.

What the two routes are each good for #

Both constructions produce thirty-two and they answer different questions, which is worth stating because the agreement can make them look interchangeable.

The group-first route answers what are the classes. It works inside two fixed groups, so it can be exhaustive without any geometry, and it produces the classes with their symbols, their orientations and their merges — everything about a class as an abstract object with a standard setting.

The lattice-first route answers where can a class sit. It produces the occurrence table, which the other route cannot: to know that 6/mmm occurs on one lattice and mmm on ten, one has to have the lattices.

And the second route is the one that generalises in the direction the classification needs. The arithmetic classes are pairs of a group and a lattice, so a construction that already has the lattices is a step from them — one hard step, but a step, where the group-first route would have to introduce the lattices from scratch.

What a Gram matrix knows #

Standing back, the striking thing about this route is how little goes in.

The input is fourteen symmetric three-by-three matrices of real numbers. No axes, no symbols, no crystal systems, no notion of a mirror or a rotation. Out of it come the fourteen holohedries at their correct orders, five hundred and ten point groups, the thirty-two crystal classes, and a table saying which class can sit on which lattice.

Everything that happened in between was arithmetic on those matrices: matching norms to find the automorphisms, multiplying matrices to close subgroups, and taking a determinant and a trace to say what each matrix is. The geometry never appeared. A reader who wanted to could run the whole construction without knowing that a 3 is a third of a turn.

That is the habit this collection is built on, and it is worth naming when it works this cleanly. Every pattern generated from its rule is the same principle for pictures; here it is the same principle for a classification. The classical route to the thirty-two is a sequence of geometric arguments about which axes can coexist, and it is a fine argument; the route here is a sequence of matrix operations that has no opinion about axes and reaches the same answer, which is what says the answer was about the matrices all along.

There is a limit to how far the observation goes. The step from thirty-two to seventy-three is not more of the same arithmetic — the invariant that separated the thirty-two, a count of operation kinds, is a geometric invariant and cannot see the difference between two ways of sitting on a lattice. A finer classification needs a finer invariant, and the one that works is not another count of operations but the group of a generic invariant form, which is what the companion essay computes.

The five hundred and ten, read as a warning #

One number in this census deserves a closing look, because it is the one a reader is most likely to mistake.

Five hundred and ten is not a count of anything a crystal can be. It is a count of descriptions: a point group, in a particular orientation, on a particular lattice. The thirty-two is the count of what a crystal can be as far as its operations go; the seventy-three is the count as far as the space groups care; and five hundred and ten is a count of the intermediate objects a computation happens to produce.

Every classification in this collection has such a number in front of it, and reporting it is not padding. Seventy-four two-colourings become forty-six designs. A hundred and sixty-four three-colourings become twenty-five. Five hundred and seventy-six descriptions of a misorientation become one disorientation. The enumeration always produces more than the classification does, and the difference is always a group acting — a normaliser, a relabelling, a change of setting.

So the question to ask of any such count is what is the group being quotiented by, and a count published without that group cannot be compared with anything. Here it is spelled out: the five hundred and ten are quotiented by “same operation content”, which gives the geometric classes; quotienting by GL(3, ℤ) conjugacy instead gives the arithmetic ones, which is a different quotient over the same five hundred and ten objects and needs a different invariant to take.

The one thing to carry #

A lattice’s symmetry is a property of six numbers, and everything else here follows from being able to compute it. The Gram matrix is the whole of a lattice’s shape; the matrices preserving it are found by matching norms; their subgroups are the point groups a crystal on that lattice may have; and sorting those by content gives the classes.

Nothing in that chain requires knowing that there are thirty-two, or fourteen, or that a crystal class is a thing. It requires a metric and a search, which is a good description of what this collection tries to do with every count it makes.