A ribbon can be achiral where its edges are not

Cutting a sheet can give it a hand cut each of the eighty layer groups along a line and asked what symmetry the edge keeps. The answer was a frieze group in the plane across the cut. It depends only on the direction of the cut, not on where it is made. It is chiral whenever no mirror in the sheet and no mirror across the edge survives. Most achiral sheets turned out to have chiral edges in almost every direction, and eight of the eighty have no achiral edge at all, however they are cut. A centrosymmetric sheet, for example, cut anywhere, leaves two edges that are each other’s mirror images, one on each piece.

That essay ended on the object with two edges. A ribbon is a sheet cut twice along parallel lines, and its group can contain something neither edge’s group can: an operation that carries the ribbon onto itself while exchanging its two edges. This essay computes those operations for every layer group and every direction of cut. A ribbon can be achiral when both its edges are chiral, and whether it is depends on where the ribbon is cut, which an edge’s symmetry never does. Seven of the eight sheets with no achiral edge give achiral ribbons. One achiral sheet gives no achiral strip of any kind.

What a second cut adds #

Write the cut direction as $d$ and let $f$ be a linear function on the sheet that is constant along $d$, so that a cut is a line $f = c$ and a ribbon is the strip $c_1 \le f \le c_2$. An operation of the layer acts on the sheet as $x \mapsto Ax + t$, with a sign on the sheet’s normal saying whether it turns the sheet over.

The operations that keep each edge are those with $f\circ A = f$. After a translation of the layer brings $f(t)$ back to nought, they move every line $f = c$ onto itself. Those are the edge’s operations, found in the earlier census, and they are the same for every ribbon of direction $d$ whatever its width and position.

The new operations are those with $f\circ A = -f$. Such an operation turns the direction across the ribbon round and maps the strip $[c_1, c_2]$ to $[f(t) - c_2,\, f(t) - c_1]$. That is the same strip exactly when $f(t) = c_1 + c_2$, allowing a translation of the layer to add a whole row. So an operation of this kind carries the ribbon onto itself with its edges exchanged only if the ribbon’s middle line, $c = (c_1 + c_2)/2$, lies at $f(t)/2$, modulo half the spacing between rows. A ribbon has special middle lines, and only a ribbon centred on one of them can use an operation that exchanges its edges. A generic ribbon’s group is its edges’ group and nothing more.

The picture at the head of this essay is the plainest case. The sheet is centrosymmetric, each flag paired with a copy turned half round and moved to the underside, and a single cut in any direction leaves a chiral edge. Cut a ribbon whose middle line runs through a row of inversion centres, and each of those centres maps the ribbon onto itself, exchanging the edges. An inversion is improper, so the ribbon is achiral: its two chiral edges are mirror images of each other across it. Move the same ribbon a little sideways, keeping its width, and its middle line misses every centre. Nothing exchanges its edges, and it is as chiral as its edges are.

Four ways to swap the edges #

An operation that exchanges a ribbon’s edges reverses the direction across the ribbon. It can do either thing to the direction along the ribbon and either thing to the sheet’s normal, and the four combinations are four kinds of operation.

A mirror standing across the sheet, parallel to the cut, keeps the direction along the ribbon and the normal, and reverses only the direction across. A half-turn about the sheet’s normal reverses both directions in the sheet. A half-turn about the ribbon’s middle line keeps the direction along it and turns the sheet over. An inversion reverses everything. The first and last are improper. The two half-turns are proper, since they are rotations, and they cannot turn a chiral ribbon achiral, though they enlarge its group. A ribbon cut from a sheet with half-turns about lines in its plane, centred on one of them, is a strip that can be turned end over end onto itself, and it is still exactly as chiral as before.

The census finds a mirror or glide parallel to the cut doing the rescuing in 25 of the 63 achiral layer groups, and an inversion in 21. The glide needs a comment, because it is not an operation a reader expects to exchange the edges of a strip. A glide standing across the sheet, parallel to the cut, is a mirror followed by a slide along the cut. The slide runs along the ribbon and does no harm, so the glide exchanges the two edges just as a mirror would. That is why the one-sided sheets over pg and pgg have achiral ribbons in the directions of their glides, although none of their edges is achiral: the edge census found that a glide standing across the sheet shifts any cut at right angles to it by half a row, so it never keeps a single edge. A ribbon can use it where an edge cannot.

The census over the eighty #

For every achiral layer group and 24 primitive cut directions, the census computes the edge’s frieze and every special middle line with the operations that sit on it. A ribbon in a given direction counts as achiral if some middle line makes it so. Two conditions have to hold, and both are checked: a direction whose edge is already achiral gives an achiral ribbon at every middle line, and a generic ribbon’s group is exactly its edge’s.

The totals move a long way. Of the 1,512 combinations of an achiral layer and a direction, 787 give an achiral edge and 1,094 give an achiral ribbon, so more than three hundred directions are rescued by the second cut. Seventeen of the 63 achiral layers have an achiral edge in every direction, and they are the ones with a mirror in the sheet’s own plane, which every edge keeps. Thirty-eight have an achiral ribbon in every direction. The difference is twenty-one layers, the same number as the layers in which an inversion rescues some ribbon, and the reason is the one just given for the centrosymmetric sheet. An inversion reverses every direction, so it sits on a special middle line in every direction of cut, and every ribbon of such a sheet can be made achiral by centring it correctly.

The half-turns show up in the census only as enlarged ribbon groups, never as a change of hand. That is a check on the bookkeeping rather than a result, since a proper operation cannot reverse a hand. A census in which a half-turn made some ribbon achiral would have misclassified an operation.

Middle lines are the ribbon’s Wyckoff positions #

The edge census could not have seen any of this, and the reason is worth making exact, because it is the same distinction that separates an operation’s orientation from its location everywhere in the subject. An operation that keeps each edge satisfies $f\circ A = f$, and it shifts every line $f = c$ by the same amount. It has an orientation, the direction it keeps, but no location across the cut, so it acts on all cuts of that direction alike, and that is why an edge’s group does not depend on where it is made. An operation that exchanges the edges satisfies $f\circ A = -f$. It reflects the line of values of $f$ about the point $f(t)/2$, so it has a location across the cut, and it acts only on the ribbons centred there.

So the positions of a ribbon’s middle line behave like the positions of a point in a crystal. A general middle line is fixed by nothing but the edges’ own operations. A special one is fixed by more, and the extra operations are its site symmetry, just as a point on a mirror plane or an inversion centre has a site symmetry the general point lacks. The special middle lines are the Wyckoff positions of the one-dimensional problem the ribbon poses: a set of points on a line, spaced at halves or quarters of a row, each carrying a small group. Choosing where to cut a ribbon is choosing a Wyckoff position for its middle line, and an achiral ribbon from a sheet with chiral edges is a ribbon centred on a position whose site symmetry is improper.

Special middle lines are not rare, but they are not everywhere either. Across the 1,512 combinations of an achiral layer and a direction, a direction has on average 0.77 special middle lines in each half-row, and 0.54 of them carry an improper operation. In 455 combinations there is no special middle line at all, because no operation of the layer reverses that direction’s normal. In those directions a ribbon is exactly as chiral as its edges, wherever it is cut. The census therefore divides the directions of a sheet into three kinds: those whose edges are already achiral, those where the right placement of a ribbon makes it achiral, and those where no placement can.

Width plays no part in any of it, in either direction. The condition for an operation to exchange the edges involves only the sum $c_1 + c_2$, the middle line, and a ribbon one row wide and a ribbon a thousand rows wide centred on the same line have the same group. What width changes is how much of the ribbon’s structure is near an edge and how much is interior, which matters for a ribbon’s physics but not for its symmetry.

Seven of eight, and the one that stays chiral #

The eight layers the edge census found with no achiral edge are where the ribbon makes the most difference. Four of them, the centrosymmetric sheet over p2 and the sheets over pmg, pgg and p6 that turn over on an operation containing an inversion, give an achiral ribbon in every one of the 24 directions, through the inversion. Three, the one-sided sheets over pg and pgg and a sheet over p4g, give achiral ribbons only in the one or two directions of their glides. The eighth gives none. It is the sheet over p4 that turns over by its four-fold rotation, so that its four-fold axis becomes a four-fold rotoinversion, $\bar 4$, about the sheet’s normal.

The reason is short and geometric. The sheet’s only improper operations are its $\bar 4$ operations, and a $\bar 4$ turns every direction in the plane through a right angle while turning the sheet over. To keep a cut, an operation must send the cut’s normal to itself; to exchange a ribbon’s edges, it must send the normal to its opposite. A right-angle turn does neither, for any direction. So this sheet’s improper operations can touch no strip of it. Every edge and every ribbon, in every direction and at every position, has only proper operations and is chiral. It is an achiral sheet from which no achiral strip can be cut, and among the eighty layer groups it is the only one.

The same argument explains why the other rotoinversion sheet escapes. The sheet over p6 that turns over by its six-fold rotation has, as its generating operation, a turn through a sixth combined with turning the sheet over. Its cube is a half-turn combined with turning the sheet over, which reverses both directions in the plane and the normal: an inversion. The inversion reverses the normal of every cut, so it sits on a special middle line in every direction, and every ribbon of that sheet can be made achiral. The $\bar 4$ sheet has no power of that kind. The square of $\bar 4$ is a half-turn about the normal with the sheet kept the right way up, which is proper; its cube is $\bar 4$ turned the other way. Its only improper elements are the two quarter-turns with the sheet turned over, and neither reverses any direction.

The contrast is a small instance of a general fact about how an operation’s powers inherit its properties. Whether a group contains an inversion is decided by whether any of its elements acts as minus the identity on every direction. An operation that turns the sheet through $1/n$ of a circle and turns it over contains one among its powers exactly when $n/2$ is odd, because its power $n/2$ is a half-turn combined with turning over $n/2$ times. Six gives three and passes; four gives two and fails. The one achiral sheet with no achiral strip is therefore the one whose improper operations are all rotoinversions of order four.

Where the middle line has to sit #

The special middle lines are few and regularly spaced. For the sheet over pmg that turns over by its second homomorphism, drawn here, every direction has an inversion centre on the middle line at position nought in the row, and the direction along the first axis has a half-turn about the middle line a quarter of a row further on. The ribbon’s group therefore jumps as its middle line moves: achiral at the centre positions, chiral in between, with the half-turn adding a proper operation at its own position without changing the hand. That is the ribbon version of the observation, made for the sections of a cleaved crystal, that a symmetry group of a cut spikes at special positions. It gives a practical reading: a ribbon or nanobelt cut from a centrosymmetric layer is achiral only if the cut is placed symmetrically about a row of inversion centres. A strip cut by an uncontrolled process is almost always chiral, whatever the sheet it came from.

What the census has to refuse #

The refused claim is the natural inference from the edge census, that a ribbon with two chiral edges is chiral. It fails in 34 layer groups, in some direction, and the census names one: the centrosymmetric sheet over p2, whose ribbons are achiral in all 24 directions at the right middle line, although each of its edges in every direction is chiral. The first test is the control in the other direction: a chiral layer has no improper operation at all, so no placement of two cuts can produce one.

Still open: ribbons as band groups #

A ribbon’s symmetry group is a group of a strip in space with translations along one direction, one of the 31 band groups, and the census here classifies ribbons by their hand and by which exchanging operations they carry, not by naming the band group. The edge census named its friezes; naming the band groups needs the combination of the edge frieze with the exchanging operations and their positions, matched against the list derived as two-colour friezes in eighty is seventeen, seventeen and forty-six. Which of the 31 occur as ribbons of the eighty layers, and whether any band group can be reached only by a ribbon and never by a single edge, is the question the naming would answer. A ribbon can carry operations no edge can, so the ribbons might reach band groups the edges miss, and the half-turn about the middle line, which no edge ever keeps, is the obvious candidate for reaching them.