The operations nobody put in

Ccmm names three operations: a c glide, a mirror, a second mirror. Its group has sixteen.

Twelve of the sixteen were produced by composing those three with the centring translation the letter C stands for, and among them are operations of kinds the symbol does not mention at all — screw axes, an inversion centre, a diagonal glide. None of that is hidden or subtle. It is what a group is, and it is the reason a set of generators and a group are different objects.

That gap is the subject here, and the reason every group in this collection is produced by running a closure rather than by writing its operations down.

What a closure is #

Start with a set of generators and the identity. Compose every pair; keep whatever is new; repeat until nothing new appears. Modulo the lattice translations the result is finite, so the process terminates, and what it terminates on is the group.

That is all, and it is the definition of a group doing its job: closure under composition is one of the axioms, so a set of operations that is not closed is not a group, and the only way to know what group a set of generators produces is to close it.

The alternative — writing out the operations — sounds equivalent and is not, for a reason that is entirely practical. Composition in three dimensions is hard to do in the head. A 3×3 matrix product with a translation attached is four multiplications and three additions per component, twelve of each per operation, and the translations have to be reduced modulo one at the end. Doing that for sixteen operations by hand, correctly, is an afternoon; doing it for a hundred and ninety-two is not something anybody does.

The compositions that produce something new #

There are three mechanisms and each has a signature.

A mirror plus a translation in its own plane gives a glide. This is the cm mechanism, and in space it fires whenever a centring vector lies in a mirror plane. A C-centred group with a mirror perpendicular to c acquires an n glide, because the centring vector (½, ½, 0) lies in that plane and the mirror composed with it slides half a face diagonal.

A rotation plus a translation along its own axis gives a screw. The same argument one dimension over: the translation’s component along the axis survives the composition and becomes an intrinsic rise. This is why a body-centred group with a fourfold axis has 4₂ screws in it — the (½, ½, ½) centring has a half along c, and the fourfold composed with it advances by that half.

Two operations with intrinsic parts compose to something with the sum. Two perpendicular glides give a two-fold screw when exactly one of them slides along their line of intersection, which is the mm2 mechanism and produces four of the ten groups in that class.

The error this prevents, and it is a real one #

A group’s operation list is the list modulo lattice translations. That is the right object and it has a consequence that catches every hand-drawn diagram.

P2₁/c has four operations. Its cell contains four 2₁ axes: at x = 0 and x = ½, and at z = ¼ and z = ¾. All four belong to the single screw entry, differing by lattice translations.

Now, where does the axis go when the screw is composed with the translation a? The obvious answer is “one cell along”, and it is wrong. Composing (M, t) with a translation v gives (M, t + v), and the operation’s axis is where $(I - M)\mathbf{x} = \mathbf{t} + \mathbf{v} - \mathbf{t}_\text{intrinsic}$ — so the axis moves by $(I - M)^{-1}\mathbf{v}$, not by $\mathbf{v}$. For a two-fold that inverse halves; for a fourfold it halves the diagonal; for a threefold it produces thirds.

Every element in every plan diagram in this field is located by solving for the operation’s fixed set, for the operation and each of its lattice translates — which is a different and larger computation than locating the generators. It is also exactly solvable and exactly singular, since a rotation’s fixed set is a line rather than a point, so a solver that assumes invertibility places nothing at all.

Counting how much is composition #

It is worth putting a number on it, because the proportion is higher than intuition suggests.

P2₁/c has two generators — a screw and an inversion — and four operations. Half of the group is composition, and the composed half includes the glide the symbol is named for. A reader who took the symbol at face value would expect a screw and a glide and an inversion, three things, and would find that one of the three is a product of the other two.

Pnma has three generators and eight operations: five composed.

Ccmm has three generators plus a centring translation and sixteen operations: twelve composed.

Fm3̅m has three generators plus three centring translations and a hundred and ninety-two: a hundred and eighty-six composed, which is 97%.

The pattern is that composition dominates as soon as a group is large, which is exactly when writing the operations out becomes impossible. The two facts are the same fact: a group is big because composition produces a lot, and what composition produces is what nobody would have written down.

Where the closure has to be told to stop #

Running a closure needs one safeguard, and the reason is instructive.

Give the closure a translation with the wrong denominator — a rise of a fifth on a fourfold axis, say — and it does not fail. It generates: a fifth, two fifths, three fifths, four fifths, a whole cell, and then it has a new operation to compose with everything already found, and it keeps going. Modulo the lattice the set never closes, because the translations generated are dense in the cell.

So the closure carries a cap, and the cap is a tripwire rather than a limit. The largest group here has 192 operations; a closure that passes a few hundred has been handed generators that do not close, and stopping with an error is the only correct behaviour.

That property is doing real work in the class enumerations. Every candidate assignment of translations to generators is closed, and kept only if the result has exactly the order the point group has. An assignment that violates the consistency condition produces a runaway closure and gets discarded — so the condition is tested by running it rather than by deriving it, which is the version that cannot be got subtly wrong.

What the closure does not add #

Two things worth stating, because a closure sounds more powerful than it is.

It does not produce operations from nothing. Everything the closure finds is a product of the generators, so the group it produces is the smallest one containing them. If a real crystal has a symmetry the generators do not generate, the closure will not find it — that is the detector’s job, and it is the other half of the round trip for exactly this reason.

It does not know whether the generators were the right ones. Hand it the generators of P2 and it produces P2, correctly, even if the crystal it was meant to describe was P2₁. The closure answers “what group do these generate”; the round trip answers “does this point set have that group”; and only the two together are worth anything.

The order it works in, and why it does not matter #

A closure has an obvious free choice: which pairs to compose first. It turns out not to matter, and the reason is worth a sentence because it is what makes the algorithm trustworthy.

The set of products of generators is well-defined regardless of the order they are formed in — that is associativity, and it is one of the group axioms. So any procedure that keeps composing until nothing new appears arrives at the same set. What the order affects is how long it takes and in what sequence the operations come out, and neither of those is part of the answer.

The one place it shows is in what gets reported. An operation list has an order and the order is an artefact, so a figure that draws “the first four operations” is drawing an arbitrary four. This site’s figures avoid depending on that: where a subset is drawn, it is chosen by a property rather than by position, and where a whole list is printed the print is truncated with a count of what was left out rather than silently cut.

That is a small discipline and it exists because the alternative fails quietly. A figure showing the first eight operations of a sixteen-operation group is a correct figure of the wrong thing, and nothing about it looks incomplete.

The habit, and where it came from #

The habit of closing rather than listing started in the plane, on the seventeen wallpaper groups, where it was cheap enough to establish and just about verifiable by eye. p4m has eight operations and a determined reader can list them.

By the time the same code is producing groups of a hundred and ninety-two, the value of the habit has changed in kind. In the plane, closing rather than listing was tidy. Here it is the only option: no one is going to write out the operations of Fm3̅m, and a diagram of it drawn from a hand-written list would be wrong in ways nothing would surface.

The specific thing that changed is that the failure stopped being visible. A wallpaper figure missing two of its eight operations looks slightly wrong — the pattern has a hole in its symmetry that a reader can feel. A space-group diagram missing twelve of its sixteen elements looks like a space-group diagram. The check has to be mechanical because the eye has nothing to offer.

That is the same argument this site’s standard pass made about accidental symmetry and the same one the round trip makes about detection, and the three of them are one argument seen from three sides: in this subject, a wrong picture is indistinguishable from a right one, so the correctness has to live somewhere other than in the looking.

The plane’s version, and why it was easier to miss #

Everything above has a two-dimensional counterpart and the counterpart is smaller, which is why it took until this field for the point to be worth an essay.

The plane’s compositions produce exactly one surprising thing: a mirror composed with a centring translation gives a glide, which is cm’s whole story. Beyond that, the seventeen wallpaper groups have between one and twelve operations, and closing three generators into twelve is a computation a person can follow.

The interesting difference is in what a reader can verify. Given a wallpaper pattern and a claim about its group, a patient reader can find the operations: put a finger on a motif, look for where its reflection went, count the rotation centres. It is tedious and it works. The orbit essay is built on the fact that this is possible.

Given a space-group diagram, none of that is available. The operations relate points at different heights, the heights are printed as numbers, and the composition of two of them is a matrix product. So the plane could afford a habit of computing its groups and space cannot afford anything else.

What is left #

There is one class of composition this field does not compute, and it is worth naming as a boundary rather than leaving as an absence.

Composing operations between different groups — asking which space groups sit inside which, and what happens to the symmetry elements at a phase transition where one becomes another — is the subject of maximal-subgroup relations, and it is genuinely different work. The plane’s version is here, computed for the seventeen; the space-group version is a lattice of 230 nodes with the same question asked at each edge, and nothing in this field attempts it.

Why the closure stops #

A procedure that keeps composing until nothing new appears needs an argument that something stops it, and the argument here is short and is not available in general.

Modulo the lattice, a space group’s operations are its point group, and a point group is a finite subgroup of the symmetries of the lattice — so there are at most forty-eight of them, times the centring index. The closure works modulo the lattice, so the set it is building is bounded by that number before it starts, and it must terminate within it.

That is a bound computable in advance from the lattice type alone, which is what makes the cap a tripwire rather than a limit. A closure that exceeds it has been handed something that is not a crystallographic operation — a translation with a fifth in it, an angle the lattice does not permit — and stopping is the right response, because continuing would produce an infinite set one element at a time.

The contrast worth drawing is with the general problem. Coset enumeration — the Todd–Coxeter algorithm — does the same job for an arbitrary finitely presented group, and it is guaranteed to terminate only when the answer is finite, with no bound computable in advance from the presentation. A run that has produced ten thousand cosets and is still going might be about to finish or might never.

Here that uncertainty is absent, and it is absent for the same reason everything else in this collection is decidable: the point group is finite and bounded, the translations are ℤ³, and the two together leave nothing for a procedure to run away into.

An operation and an element are different objects #

The essay’s warning about lattice translates rests on a distinction the tables are careful about and casual writing is not, and naming it makes the warning easier to apply.

A symmetry operation is a motion: a matrix and a translation, one element of the group. A symmetry element is a geometric locus — a line, a plane, a point — together with all the operations that share it.

The two are not in correspondence. A four-fold axis is one element and carries three non-identity operations, the quarter turns and the half turn. A screw axis and a rotation axis can be the same line and are different elements, because their operations have different intrinsic translations — which is the double plane’s argument about planes, applied to axes.

A diagram draws elements and a closure produces operations, and going from one to the other is the location step the essay describes: solve for each operation’s fixed set, gather the operations sharing a locus, and draw one mark per gathered set. A diagram with one mark per operation would be wrong in one direction and one mark per generator wrong in the other, and both errors produce a plausible picture.

One more thing composition does #

There is a consequence of all this for the symbols that is easy to miss and worth ending on.

A space-group symbol names one operation per direction, and which one it names is a choice — the Tables name the generating operation, which is the one of highest order with the smallest intrinsic part. Everything else on that direction is composition and does not appear.

So a symbol systematically under-reports. P4₁ names a fourfold screw and its axis also carries a 4₃ and a 2₁; P6₃/mmc names a sixfold screw and its axis also carries a threefold rotation and a 2₁. A reader who takes a symbol as an inventory will always find more in the diagram than the symbol led them to expect, and they should.

The compression is deliberate and it is the right choice: a symbol that listed every operation on every direction would be unreadable and would carry no extra information, since the extras are determined. But it means the symbol and the diagram are answering different questions — the symbol says what generates this group and the diagram says what is in it — and comparing the two without knowing that is how a reader concludes that one of them is wrong.

What this field does have is the diffraction consequence of everything above: screws and glides extinguish reflections, the pattern of extinctions is computable from the group’s operation list, and since the operation list is what the closure produced, the composed operations extinguish just as thoroughly as the ones anybody chose.