Eighty is seventeen, seventeen and forty-six

A sheet that repeats in two directions and lives in three — a monolayer, a slab, a woven cloth — has a symmetry group that is not one of the seventeen plane groups. A layer is not a wallpaper explained why: every operation either keeps the sheet’s two faces where they are or exchanges them, and that one bit per operation is extra structure a plane group has no room for. The literature records eighty such groups. That essay counted sixty-three, and said the remaining seventeen came from translations it had not attempted. What a cleave leaves named what would close the gap: the eighty, derived, so that the last count of sheet groups taken on trust becomes one computed from the plane groups up.

The derivation turns out to be three lines long, and it needs nothing that has not already been counted. Eighty is the seventeen plane groups, the seventeen again, and the forty-six two-colour plane groups. The gap of seventeen in the earlier count was not the translations as such but one particular kind of operation — a glide plane lying in the sheet itself — and the argument that excluded it was an argument that sounded right.

Project the layer onto its mid-plane #

Every operation of a layer group maps the sheet onto itself, so it maps the sheet’s mid-plane onto itself, and it either keeps the normal or reverses it. Forget the normal and what is left is a motion of the plane. Doing that to every operation projects the layer group onto a group of motions of the plane — and since the layer’s translations are a lattice in the plane and its operations are finitely many modulo them, the projection is one of the seventeen plane groups.

That projection loses exactly one piece of information per operation: whether it turned the sheet over. So a layer group is a plane group $G$ together with a rule $\varepsilon$ assigning a sign to each operation, and the rule has to respect composition — an operation that turns the sheet over, done twice, leaves it the right way up. There are only three ways that can go.

Nothing turns the sheet over. Then the layer group is its projection, one for each plane group: seventeen layers whose two faces are different and never exchanged.

The reflection in the mid-plane is itself an operation. Then every operation appears both ways up — once as it is, once followed by that reflection — and the layer group is its projection doubled: seventeen more, each with a mirror lying in the sheet.

Some operations turn the sheet over, and the mid-plane reflection is not one of them. Then no two operations have the same projection, so the layer group is its projection with a sign attached, and the sign is a homomorphism from $G$ onto $\{+1, -1\}$ — which is exactly a two-colouring of $G$, with “turned over” in place of “the other colour”. Two such layers are the same type when a change of description carries one onto the other, and a change of description of a layer is a change of description of its projection. So the layers of this kind correspond to the two-colouring classes of the seventeen plane groups, which seventy-four colourings, forty-six groups counted: forty-six.

$$17 + 17 + 46 = 80.$$

The layer that turns over on a translation #

The forty-six two-colour classes include twenty in which a translation of the pattern reverses the colour. In a coloured pattern that is a familiar thing: a chessboard’s diagonal translation keeps colours and its edge translation swaps them. Read as a layer, it is less familiar and it is the whole story of the missing seventeen.

Take p1 — a lattice of copies of one motif, nothing else — and its one two-colouring, which reverses colour on the translation along the first axis. As a layer, that is the operation $(x, y, z) \mapsto (x + 1, y, -z)$: a slide by one cell of the projected pattern, followed by a reflection in the mid-plane. It is a glide plane lying in the sheet. The copies alternate between the upper face and the lower along the first axis; seen from above the pattern is plain p1, and the layer’s own lattice is half as dense as the projection’s, because two steps along the first axis are needed to come back to the same face.

The earlier count excluded exactly these. Its argument was that the sign on the normal cannot depend on which cell an operation lands in — that a translation changes no direction and so cannot turn anything over. A pure translation cannot. But the projection of an operation can be a translation while the operation itself is not one: it is the translation composed with the mid-plane reflection. The earlier argument was about translations of the layer, and what the colouring assigns a sign to is translations of the projection. They are different groups, and the difference is twenty classes of layer.

Sixty-three, reproduced and corrected #

The earlier count is worth reproducing exactly, because it went wrong in two directions at once, and only one of them showed.

It added to the seventeen and seventeen every homomorphism onto a sign that leaves the lattice translations alone — twenty-nine of them over the seventeen plane groups — and reached sixty-three. The first correction removes three. Two homomorphisms that differ only by a change of description, such as moving the origin by half a cell, describe one layer group, and counted as classes the twenty-nine are twenty-six. The second correction adds twenty: the classes that turn the sheet over on a translation of the projection. Twenty-six and twenty are forty-six, the two-colour count, and the total is eighty.

The first error overcounted and the second omitted. A count that made only the overcount would have reached eighty-three and one that made only the omission sixty; making both gave sixty-three, and only making neither gives eighty. The literature’s number was the one check that could tell, and it could only say that something was wrong. That is the reason for reproducing the old count as a refusal rather than simply replacing it: a number that agrees with the literature only after two corrections is a number whose corrections have to be seen.

The distribution is uneven in an instructive way. p3 carries only its two obligatory layers, because it has no two-colouring at all: a three-fold rotation has odd order and cannot reverse anything, and no index-two sublattice of the hexagonal lattice is invariant under it. p1 carries three, its only turned-over layer being the one on a translation. pm, pmm, pmg, cmm and p4m carry seven, and pm is the extreme case of the new kind: four of its five turned-over layers turn over on a translation, because a pattern with one mirror direction has room for a sign on each of its two translations as well as on the mirror.

The seven layers over pmm #

The count is easiest to believe on one plane group taken apart. pmm is a rectangular pattern with mirror lines in two perpendicular directions and half-turns where they cross, and seven layer groups project onto it.

Two are the obligatory ones: the sheet whose faces differ and are never exchanged, and the sheet with a mirror in its own plane, where every operation occurs both ways up. The other five turn the sheet over with some of the pattern’s operations and not others.

In two of them the choice involves no translation. If both families of mirror lines turn the sheet over, each becomes a half-turn axis lying in the sheet — a mirror line of the pattern, done while exchanging the faces, is a rotation about that line — and the half-turn where they cross keeps the sheet the right way up. Every operation is then a proper motion of space, and this is the one sheet over pmm that is chiral: a pattern full of mirrors, and a sheet with a hand, exactly as the chirality of a sheet predicts for every achiral pattern. If only one family turns it over, that family becomes half-turn axes, the other family stays mirror planes standing across the sheet, and the half-turn where they cross — which now turns the sheet over — becomes a centre of inversion.

The last three are turned over by a translation of the projection, combined in different ways with the mirrors. In each of them some slide of the pattern, done while exchanging the faces, is a glide plane lying in the sheet, and the sheet’s own lattice is a sublattice of index two in the pattern’s. None of them is chiral, since a glide plane is a reflection of space. Seven in all: one, one, two and three, as the block diagram at the head of the page shows for pmm and as the same count shows for every plane group.

The strip, counted independently #

A derivation that ends on the literature’s number has passed one test. A second test, on an object that shares nothing with the first, is worth more.

The same construction applies to a strip with two faces — a ribbon, a length of tape — whose symmetry groups are the band groups. Project a band group onto the strip’s own plane and the projection is one of the seven frieze groups; the three cases give one band over each frieze with distinct faces, one with a mirror in the strip’s plane, and one for each two-colour class of the frieze group. The two-colour friezes are not taken from anywhere: each frieze group is written down by generators as motions of the strip, closed modulo twice its lattice, its homomorphisms onto a sign found by closure exactly as the sixteen candidates become seven, and those homomorphisms sorted into classes under the strip’s changes of description. They come to seventeen, and

$$7 + 7 + 17 = 31,$$

which is the number of band groups the literature records. The strip’s count uses no plane group, no two-colour plane group and no normaliser of a lattice in two dimensions; it agrees because the argument is the same argument, and it would have disagreed if the argument had a gap that only the plane hides.

Where a sign is enough, and where it is not #

The construction works because a strip and a sheet each have exactly one direction they do not extend in, and an operation can do only two things to one direction. A rod — a helix, a chain, a nanotube — has a whole plane of directions around it, and an operation can turn that plane through any angle the rod’s own repeat permits. No sign captures a rotation about the rod’s axis, so there is no three-line account of the seventy-five rod groups, and seventy-five ways to be a thread had to derive them class by class. The contrast is the useful part: the layer and band counts are short because their extra dimension is short.

The same sign has a second reading that makes the eighty familiar from elsewhere. A two-colouring in which one colour is “reversed in time” is a magnetic symmetry, and the operation that reverses time uses exactly this arithmetic to turn crystal classes into magnetic ones. So the eighty layer groups are also the eighty magnetic plane groups — seventeen ordinary, seventeen with time reversal as a symmetry by itself, and forty-six in which time reversal is combined with a spatial operation — and “turning a sheet over” and “reversing its moments” are the same piece of group theory, attached to different physics.

What the derivation rests on #

It rests on the forty-six. The layer count is exact only because the two-colour classes are, and those are found by searching each plane group’s affine normaliser for the changes of description that identify two colourings. That search is run at several widths and required to give the same answer, which is the check that it did not stop too soon; a search that stopped too soon would report too many classes and too many layers, silently.

The equivalence is affine, and it includes reflections. Two layers are counted as one type when any affine change of description — a shear, a change of axes, a change of origin, or a reflection of the whole description — carries one onto the other, which is the convention under which there are seventeen plane groups and forty-six two-colour ones. Counted instead up to changes of description that keep handedness, some types split into mirror-image pairs and the numbers grow; the eighty is the count with reflections allowed.

It assumes a layer has a mid-plane. A layer group’s operations all keep some plane, and the derivation projects onto it. That is true of every layer group — a finite thickness in one direction has a middle — but it is the step that fails for a rod, and it is worth naming because the whole argument hangs on it.

It says nothing about which layer a given sheet has. The count is of types. Deciding the layer group of a particular monolayer from its atoms is the same detection problem a plane pattern poses, with one more bit per operation to decide; the construction here builds each type from its projection and a sign and does not run that detection.

No figure shows a layer in three dimensions. The edge-on view of the glide plane is a side projection of one row, and the blocks and tables are counts. That every block is a distinct layer group is a statement about classes of homomorphisms, which a picture of motifs can only illustrate.

What the two correct counts change #

Two older essays stood on the sixty-three and have been corrected. A layer is not a wallpaper now derives the eighty rather than quoting it. Chiral in the plane is not chiral in the room counted sheets over the seventeen and found exactly one chiral in space over each plane group; that conclusion survives the correction untouched, because a sheet turned over by a translation always has an improper operation — the glide plane in its own plane — and so none of the twenty new classes is chiral. What changed there are the totals: of the eighty sheets, seventeen are chiral in space, fifty-two are chiral neither in the plane nor in space, eleven only in the plane and twelve only in space.

So seventeen of the eighty layer groups are chiral, one over each plane group, and the eighty are now derived rather than cited. The number still taken from the literature is two hundred and thirty.

Still open: the eighty by name #

The census establishes how many layer groups there are over each plane group and what kind each is. It does not name them in the notation of the International Tables, where each of the eighty has a number and a symbol — p11a for the glide in the sheet over p1, among others. Matching the forty-six classes to the Tables’ symbols is bookkeeping rather than arithmetic, but it is the bookkeeping that would let any layer described in these essays be cross-checked against a table, and it would say which of the Tables’ eighty correspond to which colouring of which plane group.

The other question is the one the rods raise. A rod has no mid-plane to project onto, but it has an axis, and projecting a rod group onto the axis gives a group of motions of a line with a finite group of rotations around it. Whether the seventy-five rod groups have an account of the same shape — a group of the line, a crystal class that keeps a direction, and a rule attaching one to the other — is what the thread’s own census did class by class without saying, and saying it would put the rods beside the layers and bands in one construction.