Cutting a sheet can give it a hand

A sheet of graphene, a clay platelet, a monolayer of molecules on water: each repeats in two directions and lives in three, and each has a symmetry group that is one of the eighty layer groups. None of them is infinite. Every real sheet ends somewhere, and where it ends it has an edge, which is a line with the sheet on one side of it and nothing on the other. The edge is where a nanoribbon’s electronic states live and where a growing crystal adds its next row, and it has a symmetry group of its own.

A thread’s hand is not a choice sorted the seventy-five rod groups by their handedness and closed on the observation that a sheet’s edge is a thread — something that repeats along one line in three-dimensional space — which left open the question of how the handedness of a sheet and the handedness of its edges are related. The answer computed here is lopsided. A chiral sheet can only have chiral edges, but an achiral sheet usually has chiral edges too, in all but finitely many directions, and eight of the eighty layer groups have no achiral edge at all. Cutting a sheet is one of the ordinary ways of giving an object a hand it did not have.

What an edge keeps #

Cut the sheet along a line in some lattice direction $d$ and throw away one side. What remains is a half-sheet, and its symmetry group is the set of the sheet’s operations that carry the half-sheet onto itself. Chiral in the plane is not chiral in the room described every operation of a layer as a motion of the plane, $x \mapsto Ax + t$, together with a sign $\varepsilon$ saying whether it keeps the sheet’s two faces where they are or exchanges them. The question is which of those survive the cut.

Describe the half-sheet by a linear function $f$ on the plane that vanishes along $d$, so that the half-sheet is the set where $f(x) \ge c$. An operation carries this set onto itself exactly when two things hold. It must not turn the direction across the cut round, which says $f(Ax) = f(x)$ for every $x$. And it must not shift the cut sideways by anything the sheet cannot undo, which says that $f(t)$ is a value $f$ takes on a lattice translation, so that composing with that translation brings the cut back. Then

$$f(Ax + t) = f(x) + f(t),$$

and the right-hand side does not mention $c$ at all. An operation that keeps one half-sheet of direction $d$ keeps every half-sheet of that direction, wherever the cut is made. The edge’s symmetry group depends on the direction of the cut and not on its position.

That contrasts with the section of a crystal. What a cleave leaves found that a plane through a three-dimensional crystal keeps a group that changes with its height, spiking at special heights and flat between them. The spikes came from operations that exchange the two sides of the plane, which return the plane to itself at two heights per cell and nowhere else. An edge has only one side. The side-exchanging operations are exactly the ones that cannot survive, and what is left is the flat part of the profile: the operations that survive at every position or at none.

Seven edges, two with a hand #

An operation that passes the first condition fixes the direction across the cut, so it can only act on the two directions left: the direction along the edge and the sheet’s normal. It is a motion of the plane that contains those two, the plane standing across the sheet along the line of the cut — the profile of the edge. The edge’s translations run along one line of that plane, so its group is a group of motions of a plane with translations in one direction, and that is exactly what the seven frieze groups classify. Every edge of every sheet is one of the seven friezes, drawn in the profile plane.

The dictionary between the frieze’s operations and the sheet’s is short. A mirror across the strip reverses the edge direction and leaves the normal alone: it is a mirror plane standing across the edge. A mirror along the strip reverses the normal and leaves the edge alone: it is a mirror in the sheet’s own plane, and a glide along the strip is a glide in that plane. A half-turn reverses both, and in the sheet it is a half-turn about the line lying in the sheet at right angles to the cut. The first two kinds reverse orientation in space. The half-turn does not.

So an edge is chiral exactly when its frieze contains no mirror and no glide, which leaves two of the seven: p1, with translations only, and p2, with a half-turn. The other five all carry a mirror image of the edge onto itself. The frieze plate is then also a census of what can happen at an edge, and it is complete in the sense that matters: all seven occur among the eighty layers, each as the edge of at least seven of them in some direction.

Seen as a thread in space, an edge’s group is one of the seventy-five rod groups, and it is a special one: its operations all fix one direction across the thread, the direction pointing into the sheet. A rod group that fixes a transverse direction cannot contain any rotation or screw about its own axis, since every one of them turns that direction round. That is why the seventy-five collapse to seven here, and why no edge of any sheet can be a helix. An edge can be chiral, but only in the flat way a scalene row of flags is chiral, never by winding.

A chiral sheet has chiral edges; the converse fails #

The first result costs nothing. A chiral sheet has no improper operations — it is the layer version of the groups a single hand may sit in — and the edge group is a subgroup of the sheet’s group, so it has none either: the seventeen chiral layers, one over each plane group, have only chiral edges, in every direction. What is interesting is the converse, which fails badly, and the reason it fails is visible in a single picture.

The left-hand sheet is achiral because of its mirrors, and a mirror plane standing across the sheet survives a cut only if it stands across the cut as well. That fixes the direction: the cut must run at right angles to the mirror. In any other direction the mirror carries the cut to a cut in a different direction, so it is not in the edge’s group, and nothing else is either. The diagonal edge on the left is chiral, and its mirror image is the edge along the other diagonal, which the sheet also has. The sheet as a whole is achiral because it contains both hands of edge, in two directions.

The right-hand sheet is the sharper case. Its only improper operation is the inversion, and inversion reverses every direction in the plane, including the direction across the cut, so it fails the first condition in every direction at once. No cut of a centrosymmetric sheet keeps the inversion, and every edge of it is chiral. Cut it through an inversion centre and the two pieces carry edges that are each other’s mirror image, since the inversion carries one piece onto the other. An achiral sheet can be cut into two chiral halves of opposite hand, in every direction.

The general statement follows from listing which improper operations can survive at all. An operation that reverses the direction across the cut never survives: inversion, a half-turn about the normal, a mirror plane standing along the cut. A mirror or glide standing across the sheet survives only on the cut at right angles to it, which is finitely many directions. That leaves the mirrors and glides lying in the sheet’s own plane, which reverse the normal and nothing in the plane, and so pass the first condition in every direction whatever it is.

The eighty, sorted #

Running the two conditions over every layer group and every lattice direction with entries up to four sorts the eighty into five kinds, and the sorting has no exceptions.

The seventeen layers with a mirror in their own plane are the sheets printed identically on both faces. The mirror reverses the normal and nothing else, needs no translation, and survives every cut, so every edge of such a sheet is achiral and there is nothing more to say about them. They are exactly the seventeen “mirror in the sheet” layers, one over each plane group, and the census confirms the correspondence one for one rather than assuming it.

The twenty in the third row are the layers counted as anti-translation classes in the derivation of the eighty: sheets in which some lattice translation of the flat pattern also turns the sheet over. In space that operation is a glide lying in the sheet’s plane, and the census finds it is the only in-plane glide any layer has. Whether it survives a cut depends on the direction in an arithmetic way. The glide moves the cut sideways by $f(t)$, and whether that is a value $f$ takes on the sheet’s own translations depends on the parity of the direction’s indices, so achiral and chiral directions interleave rather than clustering.

The two roses show two different kinds of scarcity. On the square sheet with mirrors across it, achiral edges are a handful of directions against a continuum of chiral ones; if the directions up to four are extended to every rational direction, the four stay four and the chiral ones become infinitely many. On the oblique sheet with a glide in it, both kinds are infinite and they alternate: a cut in direction (1, 1) keeps the glide, a cut in direction (2, 1) does not, a cut in direction (3, 1) does again. The glide’s translation is a whole lattice vector of the flat pattern, and the edge keeps it when that vector has no component across the cut that the sheet’s halved lattice cannot absorb. Fifteen of the twenty-four directions keep it here, and between fifteen and twenty across the twenty glide layers.

The eighteen in the fourth row have mirrors or glides standing across the sheet, possibly with other operations, but nothing in the sheet’s own plane. Their achiral edges are the few directions at right angles to a mirror — one for a single mirror, two for the rectangular groups, three for the triangular ones, up to six for the hexagonal sheet with mirrors — and every other edge they have is chiral.

Eight achiral sheets with only chiral edges #

The last row of the census is the one the question was really about. These are achiral sheets, with improper operations, none of which survives any cut.

Each row fails for one of the reasons already listed. The centrosymmetric sheet over p2 has only the inversion. The sheets over p4 and p6 that turn over at their principal rotation have rotoinversions, which rotate the plane by a quarter or a sixth of a turn while exchanging the faces: they carry every direction to a different direction, and the inversion contained in the six-fold one reverses every direction. The one-sided sheets over pg and pgg have glides standing across the sheet but no mirrors, and a glide plane standing across the sheet shifts any cut at right angles to it by half a lattice row, which no translation can undo. A true mirror across the sheet would have survived on that cut; a glide never does. The remaining rows combine these.

The result is exact rather than a sample. The census tested twenty-four directions, but the argument that governs which improper operations can survive holds in every direction: an improper operation must be a mirror or glide in the sheet’s plane, or a true mirror standing across it with the cut at right angles, and these eight have neither. So a crystal grower, or a chemist tearing a monolayer, working with any of these eight sheet types, gets a chiral edge whatever direction the tear takes.

And the other hand is always there to be had. The census checks, for every chiral edge of every achiral sheet in every direction it tested, where its mirror image is: 307 of them lie on the other side of a cut in the same direction, 360 along a cut in another direction, and 58 along a parallel cut a fraction of a row further on — the last being the glides standing across the sheet, which carry a cut to its neighbour half a row away rather than to itself. Not one chiral edge of an achiral sheet is without its enantiomorph in the same sheet. That is the sense in which cutting gives a hand without taking the sheet’s achirality away: the sheet’s improper operations still act, but they act between edges rather than on any one of them.

The situation is the reverse of the one in a hand made of pieces that have none, where achiral tetrahedra assembled into a chiral quartz crystal because the arrangement did not keep their mirrors. There the whole had a hand its parts lacked. Here a part has a hand the whole lacks: the sheet is achiral, the edge cut from it is not, and the sheet’s mirror operations survive only as a relation between one edge and another. Both are the same fact about subgroups seen from opposite ends. Handedness belongs to a group, a subgroup can lose the improper operations of the group it sits in, and a larger group can be generated by pieces none of which has one.

What the computation has to refuse #

Each claim above is computed from the eighty layer groups’ operations, not from a table of edges, and each is tested against a case built to break it.

Two of these deserve a sentence. The test that the edge group does not move with the origin rebuilds every operation of all eighty layers about an origin placed at an arbitrary irrational-looking point and requires every edge verdict to stay put. Position-independence was derived above in one line, and the test is how a slip in that line — a lattice translation forgotten on a centred sheet, say — would show. The refusal that an operation reversing the cut can be kept tests the inversion and the half-turn about the normal on their own in every direction, since the whole argument for the eight rests on neither ever surviving.

What is left at the edge #

Every edge here was a straight cut through an ideal sheet, and the atoms along it were those of the sheet, no more and no fewer. A real edge reconstructs. Its atoms move and bond differently, and a reconstructed edge can have less symmetry than the frieze computed here, never more, since the frieze is what the unreconstructed sheet imposes on the line. So the chirality found here is a floor. A chiral edge stays chiral under any reconstruction, and an achiral one may become chiral.

A ribbon, a strip with two parallel edges, is the next object, and the computation for it is different in a way this essay’s first section already names. A ribbon has two sides that an operation may exchange, so its group includes operations that carry one edge to the other, and those survive only when the ribbon’s centre line sits at special positions, exactly as a cleave’s section spikes at special heights. A ribbon of a centrosymmetric sheet cut symmetrically about a row of inversion centres keeps the inversion and is achiral, although each of its two edges is chiral: the inversion exchanges the edges, and they are each other’s mirror image. Which ribbons of which layers are chiral, as a function of their width and their centring, is a census over the same eighty and the same directions with that second condition added. The friezes inside the seventeen is the flat version of it, the row of a plane pattern kept with both its sides.