The plane that carries two glides

The five glide planes enumerates the kinds of glide a lattice permits — a, b, c, n and d — and treats a plane as carrying one of them. That is true of every primitive group and false of a good many centred ones, and the case it misses ended up changing the International Tables.

A plane can carry two glide operations at once. Not two planes at different heights; one plane, with two different slides along it, both genuine symmetries of the same crystal.

Why it happens, and exactly when #

The reason is one line and it decides the whole census.

Two reflections with the same plane have the same matrix and the same position, so they differ by a translation lying in that plane. If that translation is a lattice vector the two are the same operation, since a space group is taken modulo its lattice. So a plane carries two genuinely different operations exactly when the lattice has a centring vector lying in the plane — not perpendicular to it, which would merely move the plane to a new height, and not absent, which is the primitive case.

That is a statement about a lattice and a matrix, decidable in integers, and it is checked here on every plane of every group this site defines: eighty-nine pairs of operations share a plane, and in all eighty-nine the difference of the two slides is a centring translation. No primitive group has one.

Getting that statement into a form that could be checked took three attempts, and the two that failed are worth recording because both were wrong in the same way.

“Is the centring vector perpendicular to the plane’s normal?” uses a dot product in fractional coordinates, which is not an angle unless the axes are orthogonal. It misreported P3̅m1 and P6₃/mmc, whose hexagonal axes are at 120°.

“Is the centring vector fixed by the matrix, modulo the lattice?” needs no metric and admits vectors fixed only after a lattice translation — which move the plane rather than adding to it. It misreported Cm.

The difference of two slides is the quantity that actually appears in the operations, and it needs neither a metric nor a case. Both rejected criteria are still run, over all three hundred and fifteen planes, and scored against what the operations carry — because a criterion nobody exercises is a criterion nobody knows the cost of. The dot-product test and the exact-fixed-vector test each get thirty-three planes wrong, all of them in Fm3̅m and Im3̅m; the modulo-the-lattice test gets seventy-four wrong, across eight groups. Which particular groups a bad criterion fails on depends on what the census contains, and that is the point: the failures move and the theorem does not.

Three kinds of pair, and only one of them needed a letter #

A mirror with a glide. The symbol writes m, and nothing is lost: the glide is the mirror composed with a centring translation and names nothing new. Fm3̅m and Im3̅m are full of these.

Two glides with the same letter. Fddd’s planes carry two d glides whose slides differ by (½, ½, 0) — (¼, ¼, 0) and (¾, ¾, 0). Both are d, the symbol writes d, and again nothing is lost.

Two axial glides with different letters. A plane carrying a b glide and a c glide, neither with any claim over the other. This is the case with no winner, and every symbol printed before 1992 simply chose. Abm2 and Acm2 were the same group under two conventions; so were Cmca and Cmcb.

The International Tables’ 1992 revision gave the double plane its own letter, e, and five groups changed name: Abm2 → Aem2, Aba2 → Aea2, Cmca → Cmce, Cmma → Cmme, Ccca → Ccce.

What this site had to change to see it #

Nothing among the forty-four space groups defined here had a centring vector inside one of its own planes. So the symbol derivation had never met the case, and it would have written whichever letter it found first — a symbol depending on the order the operations came out of the closure.

Adding one group made that visible. Aem2, number 39: A-centring is (0, ½, ½), which lies in the plane perpendicular to a; the b glide there composed with it is a c glide on the same plane. The derivation then had to learn the Tables’ rule — no mirror on the direction, two distinct axial glides, write e — because the requirement that every group derive the symbol it was entered under is not met otherwise.

That requirement turned an addition into a repair, which is the whole argument for having it. A derivation that reproduces forty-four names is not evidence that it is right; it is evidence that it is right about those forty-four. The forty-fifth is what found the gap.

The second defect, which was in the element enumerator #

Adding the group also found a bug that every diagram on the site had been drawing since the first of them, and it is the kind this site’s checks are least able to see.

The routine that lists a group’s symmetry elements deduplicated them on kind, order, direction, normal and position — and not on the slide. Two glides sharing a plane therefore counted as one element, and the second was dropped.

Five of the site’s groups have such a plane: Aem2, Ccmm, Fddd, Fm3̅m and Im3̅m. Every diagram of them drew one line where there are two, and nothing could report it. The missing element is absent from the drawing, and every check the fleet has asks whether what is drawn is correct — whether a label fits, whether ink stays inside the canvas, whether a contrast is adequate. Nothing asks whether something that should be there is missing.

Fixing the key added a second line to every one of Fddd’s fourteen double planes, which is the change a reader can see. It is the shape of defect that a check on the drawing cannot have an opinion about: what is drawn is correct, and what is missing was never asked after.

Reading the census: where the double planes are #

The five groups here with double planes are worth naming, because the pattern in them is the theorem restated.

Aem2 has three, all perpendicular to a — which is the only direction whose planes contain the A-centring vector (0, ½, ½).

Ccmm has one, a mirror with an n glide, on the plane perpendicular to c at z = ¼ — the only direction whose planes contain (½, ½, 0).

Fddd has fourteen, all of them two d glides sharing a plane, because an F lattice has three centring vectors and every one of the coordinate planes contains one of them.

Fm3̅m and Im3̅m have the most — fifty-two and nineteen, seventy-one between them — because a cubic lattice has planes perpendicular to many directions and a centred cubic lattice has centring vectors in most of them. Fm3̅m has all three kinds at once: twenty-six mirrors with n glides, nineteen pairs of d glides, and seven genuine pairs of axial glides.

Read the other way, the theorem says which planes are safe. A plane whose normal is such that no centring vector lies in it carries exactly one operation, whatever the group. That is every plane of every primitive group, and a good many planes of centred ones — which is why the phenomenon can be missed for so long by anyone whose examples are primitive.

Where the exactness stops #

The theorem is exact. The difference of two slides is compared with the centring vectors as an equality modulo one, in rationals.

The census is over the groups this site defines, which is forty-five of the two hundred and thirty. It finds double planes in five of them and states the count as a fraction of what it looked at. The full census over all 230 is a different computation and is not made here; the five renamed groups are quoted from the Tables rather than enumerated, and named as quoted.

The letter is a convention, and the theorem is not. That a plane carries two operations is a fact about the group. That the symbol for it is e is a decision made by a committee in 1992, and a reader meeting Cmca in a paper from 1985 is meeting the same group under the older decision.

What a reader should do with an old symbol #

The rename has a practical edge, because both spellings are in the literature and databases hold both.

Cmca and Cmce are the same group, number 64, and a structure published under either is describing the same symmetry. The same holds for the other four pairs. Nothing about coordinates, absences or refinement changes; only the name does.

The old symbols are not wrong, they are under-determined. Abm2 says there is a b glide perpendicular to a, which is true; it does not say there is also a c glide there, which is also true. The new symbol says both by refusing to choose.

And the change is invisible to a computer that reads absences. The extinction conditions of Aem2 are those of an A-centred group with a b glide and those of one with a c glide, and the two are the same set. The reason is worth stating exactly, because the obvious version of it is wrong. It is not that the glide adds nothing and the centring’s own condition is the whole story: in the 0kl zone the A-centring leaves eighty-four of a hundred and sixty-eight reflections present and the group leaves forty-eight, so the double plane really does halve the zone again. What happens is that inside the set the centring already permits — where k + l is even — the b glide’s condition that k be even and the c glide’s that l be even are the same condition. Each spelling predicts the identical forty-eight, so no diffraction experiment could ever have distinguished them, which is why the ambiguity survived so long.

And this is the second of two independent ways for one group to have several names, which is worth separating because they are often run together. The first is the choice of cell: relabel the axes of Cc and the same group comes out as Cn, Cm or Ca depending on which pair of directions is called a and c, and one group three symbols is about exactly that. Every one of those names describes a different basis, and a reader given the cell can convert between them mechanically. The double plane is not that. There is one cell, one basis and no relabelling anywhere in it, and the ambiguity is a choice between two operations that both sit on one plane. So the two mechanisms compose rather than overlap: before 1992 a group with a double plane could be named several ways for its setting and several ways for its plane, and only the second of those was a decision nobody had to make. The letter e removes the second and leaves the first exactly where it was.

The same argument for an axis #

Nothing in the derivation mentioned that the element was a plane. Two operations with the same matrix and the same locus differ by a translation of the lattice that the locus contains — and an axis is a locus exactly as a plane is.

For a plane the condition is that a centring vector lies in it, which is a two-dimensional condition and therefore easy to satisfy. For an axis the condition is that a centring vector lies along it, which is one-dimensional and much harder: no centring vector of a C, A, B, F or I-centred lattice is parallel to any of a, b or c, which is why the phenomenon is invisible for the axes a monoclinic or orthorhombic group has.

The cubic system is where it appears, and it appears for the reason the symbol makes obvious. A body-centred cubic lattice has the centring vector (½, ½, ½), and that vector is parallel to [111] — which is a three-fold direction. So a three-fold axis of a body-centred cubic group carries a screw operation on the very same line, with an intrinsic translation of a third of the body diagonal, exactly as a screw’s pitch is a property no origin removes.

The notation never needed a letter for it, and that is a fact about the symbol rather than about the group. A Hermann–Mauguin symbol names one operation per symmetry direction, and for the cubic body diagonal it names the rotation; the screw riding on the same line is not written, is not lost, and is recovered by anybody who generates the group from its operations rather than reading its name. That is the same asymmetry the e letter was invented to remove one direction lower down, resolved in the opposite way — by leaving the symbol alone.

Which is why the repair to the element enumerator was the right shape. Deduplicating on the intrinsic translation rather than on the slide of a plane is a statement about operations, not about glides, so the axes were covered by the same one-line change. Had the key been patched to say slide, if the element is a plane, the screw axes would have kept the defect and nothing on the site would have said so — no diagram would have lost a line a reader could count, and the absences would have been unchanged, because an operation the enumerator failed to list is still an operation of the group.

Who changed it, and when #

The e glide was introduced by the IUCr’s Commission on Crystallographic Nomenclature and first appeared in the 1992 edition of International Tables for Crystallography, Volume A, becoming standard in the fifth edition of 2002.

The reasoning in the commission’s report is exactly the argument above: the two glides are both present, neither is a consequence of the other in any sense that would let a symbol omit it, and choosing one made the symbol depend on the setting rather than on the group. The report notes that the double planes had been known since Hermann’s work in the 1920s and had simply never been given a notation.

Five groups were renamed and a sixth notational change came with them — the reintroduction of the double glide notation in the diagrams, where a plane carrying two glides is now drawn with both dash patterns rather than one.

It is a small change and it is the kind that matters: a notation that cannot express something reports the strongest thing it can express instead, and looks right. That is the same failure this site’s own catalogue of extinction rules met, where Fddd’s quarter conditions had to be added because without them the group’s zonal conditions were reported as a weaker axial one — true, and not what the group does.

There is one more thing the episode says about how this site checks itself, and it belongs beside the two defects rather than after them. Both were found by adding a group, not by inspecting the forty-four already present — and neither would have been visible in any figure, any count or any gate reading the existing output. A defect that needs new input to appear is a defect that a stable site can carry indefinitely, which is the argument for the site’s own checks being run against every group it defines rather than against the ones a particular argument happened to need.

What it costs to have got this wrong #

It is worth being concrete about the damage, because “a diagram was missing a line” sounds like a typographical matter and is not.

A missing glide is a missing operation in an argument. Anyone reading the site’s Fddd diagram and counting glide planes would have counted half of them: every one of its fourteen double planes carries two d glides, and one of each pair was being dropped. A count of elements is exactly the kind of thing an essay states, and this site states such counts constantly.

And a symbol chosen by iteration order is a symbol that can change without the group changing. Before the rule was added, deriving Aem2’s symbol from its operations in one order gave one answer and in another order gave another — the derivation was reporting a property of the loop. That is worse than a wrong constant, because a wrong constant is at least stable.

Both defects were found by adding one group, and neither would have been found by any amount of checking the existing forty-four. The site’s own gates measure the tree as it is; a gap in coverage is invisible to all of them by construction, and the only thing that finds it is putting the missing case in.

What a diagram should show, once the elements are right #

The repair has a consequence for how a plan is drawn, and the Tables settled it at the same time as the letter.

A double plane is drawn with both patterns. Where a plane carries a mirror and a glide, the line is drawn solid with the glide’s dashes alongside; where it carries two glides, both dash patterns appear. That convention arrived with the 1992 revision and it exists because a single line is a claim that there is a single operation.

And the marks are located, not translated. An operation’s lattice translates are not its element’s lattice translates — composing a reflection at x = 0 with the translation a gives a reflection at x = ½, not at x = 1 — so a diagram built by copying marks across a cell shows the corners and misses everything between. Every plan on this site is built by re-locating each translated operation, which is where three quarters of a diagram’s content comes from and is why the missing slide mattered: the machinery that produces the marks correctly was dropping one of them at the last step.

Where the ladder goes next #

This rung closes a gap in what a plane can be. Two rungs sit above it.

The full census over 230 groups. Which groups have double planes, of which kind, and how many — a computation this site could make once it defines the whole list rather than a chosen forty-five. The answer includes the five renamed groups and a good many more of the mirror-and-glide kind.

The same question for axes, which turns out to have the opposite answer. A line cannot carry two: two operations sharing an axis differ by a translation along it, every lattice vector along a line is a whole multiple of the shortest one, and the intrinsic translation of a screw is measured in $n$-ths of exactly that — so the two have the same screw index and there is nothing for a symbol to choose. A line carries one screw is the argument and the count that goes with it: seventy-seven directions carry more than one kind of axis, on parallel lines, and no line anywhere carries two. The difference from the plane is a dimension count, since a slide is a vector in a two-dimensional element and an intrinsic translation is a vector in a one-dimensional one.