A line carries one screw

The plane that carries two glides settles what a plane can be. Two reflections in the same plane differ by a translation lying in that plane; if the lattice has a centring vector there, the plane carries two genuinely different operations; and when both are axial glides — a slide of $b/2$ on one and $c/2$ on the other — neither has any claim on the symbol. Every symbol printed before 1992 chose one, Abm2 and Acm2 were the same group under two conventions, and the revision of the International Tables introduced the letter e to stop it.

That essay ends by putting the same question to axes, and guesses at the answer: a line can carry two screw operations whose intrinsic translations differ by a centring vector along it, so the symbol has the same choice to make and handles it by writing the lower screw index.

The guess is wrong, and the way it is wrong is the interesting part. A line cannot carry two.

The difference has to go somewhere, and for a line there is only one place #

Two operations with the same linear part sitting on the same axis differ by a pure translation. That translation must be parallel to the axis: a component across it would carry the line somewhere else, and then the two operations would not share an axis at all.

So the difference is a lattice vector lying along the line. And every lattice vector along a line is a whole multiple of the shortest one — that is what “shortest” means for a one-dimensional lattice. The intrinsic part of a screw is measured in $n$-ths of exactly that shortest vector, so two operations on one line have intrinsic parts differing by a whole number of repeats, and the same screw index.

The plane’s version of the argument fails at exactly the step this one passes, and the failure is a dimension count. A glide’s slide is a vector in the plane, and a plane is two-dimensional: a centring vector lying in it need not be a multiple of the slide, so two slides can differ by something that is not a whole repeat of either and the two glides are different letters. A line has one dimension to be ambiguous in, and the requirement that the difference be a lattice vector uses it up.

The arithmetic is worth writing once. An operation $(M, t)$ with $M$ a rotation of order $n$ has intrinsic part $\tfrac1n\sum_k M^k t$, which is the component of $t$ that no choice of origin removes and which is parallel to the axis. Adding a lattice vector $L$ to $t$ adds $\tfrac1n\sum_k M^k L$ to it — the component of $L$ along the axis. If $L$ is across the axis that component is zero and the operation moves to a parallel line without changing its intrinsic part at all; if $L$ is along the axis the line stays and the intrinsic part changes by $L$, which is a whole number of repeats. There is no third case, because a lattice vector splits into those two pieces and each is handled.

Checked over every axis of every group defined here — 3,388 of them, taking each operation with every lattice translation in a cell around it — not one line carries two operations of different screw index, and every pair of operations on one line differs by a whole number of that line’s repeats. Two statements of one fact, so that a mistake in the bookkeeping would break one of them.

What that means for a reader looking at a diagram is worth saying plainly. A symbol on a space-group diagram marks a line, and the mark says what kind of axis it is. The theorem says the mark is not hiding anything: whatever else of that linear part passes through that line is the same kind, differing only by how many cells it has already climbed. The corresponding statement about a plane is false, which is why the double planes had to be found rather than assumed.

What the check found while it was failing #

Asked in the obvious way, the answer came back the other way round: 284 axes across the two centred cubic groups appeared to carry two operations of different index. Every one was an artefact, and finding out which artefact repaired something.

A screw index is a fraction of the repeat along the axis, and the repeat is not always the integer direction. In a face-centred cubic lattice the shortest lattice vector along $[1\,1\,0]$ is $(\tfrac12, \tfrac12, 0)$ — half the integer direction, because the face centre sits on the diagonal. In a body-centred one the shortest along $[1\,1\,1]$ is $(\tfrac12,\tfrac12,\tfrac12)$, a third of it.

Measured against the direction rather than against the repeat, ten operations came back with the wrong index: every $\langle 110\rangle$ two-fold of Fm-3m and every $\langle 111\rangle$ three-fold of Im-3m, a rotation called a screw or a screw called a rotation depending on which coset representative the enumeration happened to hand over. Nothing drawn was in the wrong place; only its tail was — which is the same shape as the defect the diagonal screws of a tetragonal group once had, where a reduction destroyed the sign of an intrinsic translation and nothing looked wrong, because a mark that is drawn and correctly placed looks right.

The repair is to take the shortest lattice vector parallel to the axis, over the whole lattice including its centrings, and measure against that. For a primitive lattice it is the integer direction and nothing moves, which is what makes the change safe: all thirty-three primitive groups here are untouched.

The diagnosis is sharper than the symptom and is what makes it a repair rather than a patch. Every operation affected sits on an axis whose shortest lattice vector is a centring vector — not merely a group with centring, but an axis along which the centring vector points. That is a condition on the direction and the lattice together, decidable before any operation is examined, and it is the same shape of condition as the one that decides whether a plane carries two glides: there the centring vector must lie in the plane, here it must lie along the axis. The two questions are the same question asked of a two-dimensional element and a one-dimensional one, and they have different answers for the reason the first figure draws.

It is also a reminder of which quantities on a diagram are safe. The position of an axis comes out of solving $(I - M)p = t - \text{intrinsic}$, which is exact integer algebra and cannot be wrong in this way. The kind of axis is a ratio against a length, and a ratio needs the right unit. Every defect of this shape found here has been in a quantity that needed a unit.

One more consequence deserves stating, because it reverses the natural expectation about where trouble lives. A centred lattice is the source of the plane’s ambiguity — no primitive group has a plane carrying two — so a reader might expect centring to be the source of the axis’s trouble too. It is not: centring produces more kinds of axis along a direction, which is a harmless surplus, and it never puts two on one line. What centring does to an axis is the thing it does to a sublattice of the same shape: it supplies extra lattice vectors, and extra lattice vectors along a line only shorten the repeat.

What a direction does instead #

The choice the symbol makes is real. It is simply not a choice between two operations on one element.

A C-centred monoclinic group has two-folds and $2_1$ screws alternating along $b$, because the centring vector $(\tfrac12,\tfrac12,0)$ has a component across the axis and carries each rotation to a screw half a cell away. The two kinds are interleaved and neither is on the other’s line.

Seventy-seven directions across twenty groups carry two or three kinds of axis, and the pattern is the expected one. A centring vector with a component across a direction produces the rotation-and-screw pairs of C2, Aem2, Ccmm and Fddd. A primitive screw axis is accompanied by its own opposite hand — P4_1’s four-fold direction carries $4_1$ and $4_3$, which is the enantiomorphic pair showing itself in one group rather than between two. And the cubic groups carry three kinds along $\langle111\rangle$, a rotation and both hands of the three-fold screw.

The column that matters in that table is the one that is zero all the way down.

The counts have a pattern that is worth reading rather than skipping. The monoclinic and orthorhombic centred groups contribute one or two mixed directions each — the directions their centring vector has a component across. The primitive screw groups P4_1, P4_3, P3_1 and P3_2 contribute one each, and the mixture there is between hands rather than between a rotation and a screw: a $4_1$ axis and a $4_3$ axis in the same cell, which is the pair the eleven screws counts as enantiomorphic, arriving inside a single group because the group’s own operations carry one to the other. And the cubic groups contribute ten and sixteen, because a cubic group has many directions and a body diagonal collects three kinds.

The rhombohedral group is the one worth a second look. R3 has a three-fold along $c$ carrying a rotation, a $3_1$ and a $3_2$ — all three kinds on parallel lines — because the rhombohedral centring vectors $(\tfrac23,\tfrac13,\tfrac13)$ and $(\tfrac13,\tfrac23,\tfrac23)$ each carry a three-fold to a screw a third of a cell along and a third of a cell across. It is the clearest single picture of why a direction is not an element.

Planes and lines, counted the same way #

Running the count on both kinds of element makes the asymmetry a measurement rather than an argument. Planes carrying two reflection operations are ordinary among centred groups; lines carrying two rotation operations do not exist. And the reason is not a fact about crystals, it is a fact about how much room each element leaves: a plane’s slide lives in two dimensions and a line’s intrinsic translation in one.

That also settles why the Tables have a letter for the plane’s case and have never needed one for an axis. The letter e exists because a symbol naming one of two operations on one plane hides the other: a reader given Abm2 cannot tell that a c glide is there too, because it is on the same plane and the diagram draws one line. A symbol naming one of two kinds of axis hides nothing of the sort — the other kind is on a different line, in a different place, and the diagram draws it.

The convention is the one a reader would guess, and it is the plane’s convention for a mirror beside a glide: a rotation outranks a screw, and between two screws the lower index is written. C2 is written C2, not C2₁, although both axes are there; P4_1 is written P4_1, although $4_3$ axes run through the same cell. Nothing is concealed, because what the symbol leaves out is somewhere a reader can go and find.

And the enantiomorphic case shows how little the convention is costing. P4_1 and P4_3 are different space groups, mirror images of one another, and each contains axes of both kinds; what distinguishes them is not which kinds are present but where they sit relative to the origin and to each other. So the symbol’s choice of the lower index is not even choosing between the two hands — it is naming one axis of a structure that has both, and the group is fixed by the whole arrangement rather than by the label. That is exactly what the e case is not: there the two operations coincide in place as well as in direction, so a symbol naming one really does lose the other.

The comparison with what a closure adds is the useful one. Most of a space group’s operations arrive by composition rather than by being put in, and the ones on a given line are the composite of one operation with the translations. The theorem here says that composing with a translation along the line never changes the kind — so the whole family of operations a line carries is one operation repeated, and the line’s label is complete.

What the census does and does not settle #

The theorem holds for any lattice and the census does not need to. The dimension count uses nothing about which lattices exist — only that the translations along a line form a one-dimensional lattice, which is true of every space group in every dimension. So the statement “a line carries one screw” is a proof, and the 3,388 axes are a check on the bookkeeping rather than evidence for the claim. That is the right way round, and it is the opposite of the plane’s case, where whether a plane carries two genuinely depends on which centring the lattice has and has to be enumerated.

The theorem is about rotations and screws, not about every element. A rotoinversion has a point as well as an axis, and the argument above does not apply to it unchanged: the location of a rotoinversion is fixed by its point rather than by its line, so two rotoinversions with the same axis and different points are different elements rather than two operations on one. That case is not counted here and it is not the same question.

The theorem says nothing about how many operations a line carries, only about their kind. A line in a centred group carries twice as many operations as the same line in the primitive sublattice would, and in Fm-3m a $\langle110\rangle$ two-fold carries one for every half-diagonal along it. Counting operations on a line is a different and easier question, and it is answered by the index of the axis’s repeat lattice in the group’s translations along that direction.

And the count is over forty-five groups rather than two hundred and thirty. The theorem is a proof and does not need the census — the dimension count holds for any lattice — but the claim that seventy-seven directions carry two kinds, and the ten mislabelled operations, are measurements over the groups this collection defines. A group with a centring vector along a direction nothing turns about would add nothing; a group with a threefold along a body diagonal of a face-centred lattice might add a case, and there is none here.

The repair changes numbers elsewhere and they have not all been re-read. Anything that reads a screw index from a located element in a centred cubic group was reading ten wrong ones, and the diagrams of Fm-3m and Im-3m are the obvious places. What the repair cannot do is find the same defect in a quantity nobody computes twice, which is the standing limitation of catching a bug with a second route to one answer.

Still open: whether a plane’s two operations can differ in more than their letter #

The plane’s case has a sharper version that nothing here or in the earlier account asks. Two operations on one plane differ by a centring vector lying in it, and the census sorts them by the letters they earn — mirror beside glide, two axial glides, glide beside diagonal glide. What it does not ask is how many operations one plane can carry at once. A lattice with two independent centring vectors in one plane would give four, and a face-centred lattice has planes with exactly that.

Whether any of the two hundred and thirty has a plane carrying four reflection operations, and what a symbol could possibly do with it, is a question the same enumeration answers by being run over the whole list rather than over forty-five — and it is the reason the full census over all two hundred and thirty, already owed, is worth more than it looked.