Every band group is a ribbon

A ribbon can be achiral where its edges are not cut sheets twice in one direction and asked whether the strip between the cuts has a hand. A ribbon keeps whatever its edges keep. It can also carry operations that exchange its two edges, a mirror parallel to the cut, an inversion centre and two kinds of half-turn. Those exist only when the ribbon’s centre line sits at a special position. Improper exchanging operations rescued seven of the eight sheets whose edges are all chiral. The essay sorted ribbons by hand and by kind of exchanging operation, and left their groups unnamed. 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 it asked which of the 31 the ribbons of the eighty layers reach.

This essay names every ribbon’s group. It first derives the list of 31 from three signs rather than quoting it. All 31 turn out to occur. Seven are also the groups of single edges, and twenty-four occur only as ribbons, which says exactly what the second cut adds.

Three signs, and half a period #

A ribbon lies in a sheet, so every operation of its group does one of two things to each of three directions: the direction along the ribbon, the direction across it within the sheet, and the direction normal to the sheet. It keeps each one or reverses it. Write $\delta$, $\varphi$ and $\varepsilon$ for the three signs. Composing two operations multiplies their signs, so the signs an operation carries form a homomorphism from the ribbon’s group onto a subgroup $S$ of the eight sign triples. That subgroup is the ribbon’s point group, and it is a subgroup of $\mathbb{Z}_2^3$.

The signs do not say everything. An operation that keeps the direction along the ribbon, $\delta = +1$, moves every point of the centre line along itself by some distance, and modulo the ribbon’s period that distance is either nothing or half a period. It is nothing for a mirror in the sheet and half a period for a glide in the sheet. It is nothing for a two-fold axis along the ribbon and half for a screw. The choice is a homomorphism $\tau$ from the part of $S$ with $\delta = +1$ to $\mathbb{Z}_2$. An operation that reverses the direction along the ribbon fixes a point of the centre line, so it carries no such part. A band group is therefore a pair $(S, \tau)$, and ten kinds of operation besides translations can appear, each one a sign triple with or without half a period.

The count is short enough to do in full, and the picture at the head of this essay does it. The subgroups of $\mathbb{Z}_2^3$ are the trivial one, seven of order two, seven of order four and the whole group. A subgroup contributes $2^k$ band groups, where $k$ is the dimension of its intersection with the plane $\delta = +1$. The trivial subgroup contributes one. Of the seven of order two, three lie in that plane and contribute two each, and four do not and contribute one each, ten in all. Of the seven of order four, one is the plane itself and contributes four, and six meet it in a line and contribute two each, sixteen in all. The whole group meets it in the plane and contributes four. The total is $1 + 10 + 16 + 4 = 31$. That is the number of band groups in the International Tables, derived here from the structure of the objects rather than from the table.

What an edge can have #

An edge is a single cut, and its group keeps the direction across the cut: whatever is on one side of the edge stays there. So every operation of an edge’s group has $\varphi = +1$, and the edge’s point group lies in the plane $\varphi = +1$ of the sign group. Cutting a sheet can give it a hand found that each edge’s group is one of the seven friezes, drawn in the plane perpendicular to the edge. The sign count agrees. The subgroups of the plane $\varphi = +1$ weighted in the same way give $1 + 2 + 1 + 1 + 2 = 7$, and those seven are the band groups an edge can have.

The correspondence can be written out element by element. Seen in the plane perpendicular to the edge, which contains the direction along the edge and the sheet’s normal, the mirror in the sheet is a frieze’s horizontal mirror and the glide in the sheet is its glide. The mirror across the ribbon is a vertical mirror, and the two-fold axis across the ribbon, which reverses both the direction along the edge and the normal, is the frieze’s half-turn. The seven sign subgroups with their glide choices are then $p1$, $p11m$, $p11g$, $p1m1$, $p2$, $p2mm$ and $p2mg$ in the profile plane, one for one. The edge census named its groups that way, and the sign count and the frieze count are the same count.

Every one of the seven is reached, and the census shows how often. Sixty-three layers give translations only in some direction, so the most common band group is the trivial one. The mirror across the ribbon and the two-fold axis across it, the two operations that reverse the ribbon’s direction while keeping its width, each occur for 25 layers. The mirror and the glide in the sheet, which turn the ribbon over, occur for 17 and 20.

What the second cut adds #

Everything else needs $\varphi = -1$, an operation that reverses the direction across the ribbon and so exchanges its edges. The sign count says how much that is: 31 band groups in all, seven with $\varphi = +1$ throughout, and twenty-four that need at least one exchanging operation. The census finds all twenty-four. None of them is ever an edge’s group, and each is reached only by a ribbon whose centre line sits on one of the special positions, where the earlier essay found the exchanging operations.

The drawing shows what the position does for a single layer and a single direction. The layer over $pmm$ with a mirror in the sheet has mirror lines perpendicular to the cut every half row. Cut an edge, or a ribbon centred between two of those lines, and the group has three operations besides translations: the mirror in the sheet, the mirror across the ribbon and the two-fold axis across it that is their product. Centre the ribbon on a mirror line and four more appear: the mirror along the centre line, the two-fold axis along it, the two-fold axis normal to the sheet, and an inversion centre. The group jumps from order four to order eight, from a band group an edge could have to one only a ribbon can. Moving the centre line a quarter of a row loses all four again.

The census does not need exotic directions to reach everything. Cutting only along the two lattice axes and the two diagonals, four directions, already reaches all 31. Longer directions add ribbons, and they add no new groups. Low-index directions pass through the most special positions, so they carry the most operations.

Five that have no hand-reversing operation #

The earlier essay was about hand, and the census carries a result about it that the sign count predicts. An operation is proper, preserving hand, when the product of its three signs is $+1$. A band group with no improper operation has its point group inside the four triples with that product, which form a subgroup of order four. Weighted as before, that gives seven chiral band groups: translations only, a two-fold axis in any one of the three directions, a screw along the ribbon, and the two groups with all three two-fold axes, one with a screw along the ribbon.

Two of the seven are edges’ groups: translations only, and the two-fold axis across the ribbon. The other five are only ribbons. The obvious expectation is that an operation exchanging a ribbon’s edges, which reverses the direction across it, must make the ribbon achiral. The earlier essay’s rescued sheets were rescued by improper exchanges, a mirror parallel to the cut or an inversion, and that is the refused claim below. Two of the four exchanging kinds are half-turns, about the sheet’s normal and about the centre line, and both are proper. A ribbon whose only exchanging operations are half-turns is exactly as chiral as its edges, with a larger group. The two-fold axis normal to the sheet is the commonest of all the ribbon-only groups, reached by 34 layers, and it is chiral.

A half-turn, and a ribbon that turns over as it goes #

Two of the ribbon-only groups show the proper exchanges in their simplest form. The first is the two-fold axis normal to the sheet alone. Take the one-sided layer over $p2$, a pattern with half-turns and nothing else, and cut it along a lattice row. Its edge has translations only, since a half-turn reverses the direction across every cut. A ribbon cut anywhere has the same group. A ribbon centred on a row of half-turn centres gains the half-turns, each exchanging the ribbon’s edges and reversing its direction, with the faces left alone. The ribbon’s group doubles and it stays chiral, since the half-turn preserves hand. The layer was chiral to begin with, and cutting it symmetrically enlarged its group without making it any less so.

The second is a screw along the ribbon with nothing else. The layer over $pg$ turned over by its glide shows it. In that layer the glide of the plane pattern is realised in space as a half-turn about a line in the sheet, carrying a translation of half a period, which turns the sheet over as it advances. A ribbon centred on that line has the line as its axis, and the operation is a two-fold screw along the ribbon. A strip with that symmetry shows one face and then the other at each half period, like a ribbon given a half twist every step, though the ribbon is flat. The screw is proper and the ribbon is chiral in the sense of this essay, although a two-fold screw is its own mirror image in the sense a thread’s hand is not a choice uses. Seven layers produce this band group, and seven band groups in all contain a screw along the ribbon.

These two cases also say something about chiral in the plane is not chiral in the room. A pattern drawn in a sheet is achiral in space whenever the sheet’s own mirror is a symmetry, since reflecting through the sheet leaves every point where it was. A band group containing the mirror in the sheet is always achiral for that reason, and fourteen of the 31 contain it or a glide in the sheet. Ten more are achiral through an inversion or a mirror across or along the ribbon, twenty-four achiral groups in all against seven chiral ones. The chiral band groups are the ones in which no operation reverses hand, whether by reflecting through the sheet or across it, and a ribbon’s hand is decided in space rather than in the drawing. A pattern that looks chiral on the page, a strip of commas all turning one way, is achiral as soon as the sheet it is printed on can be turned over onto itself.

How many one layer can reach #

No single layer comes close to the whole list. Four layers give one band group however they are cut: $p1$ and $p3$, each one-sided and with a mirror in the sheet. Their plane groups have no operation that reverses a direction across any cut, so every ribbon has its edge’s group. The most prolific layer gives eleven, the layer over $cmm$ turned over by an operation carrying a translation. Next come the layers over $pmm$ and $p4m$ turned over on a translation, with ten each. The rarest band group, with an inversion centre, a two-fold axis along the ribbon, a glide along the centre line and a glide in the sheet, is reached by one layer only, one of the layers over $pmm$ that turns the sheet over by an operation carrying a translation. The list of 31 is completed only by the eighty together.

The count has the same shape as eighty is seventeen, seventeen and forty-six, where the eighty layer groups come from the seventeen plane groups coloured by the sheet’s two faces: seventeen that keep the faces, seventeen grey ones with the sheet mirror, and forty-six two-colourings. A band group is a frieze coloured by the faces in the same way. Seven friezes keep the faces, seven are grey, and the seventeen two-colour friezes give the rest: $7 + 7 + 17 = 31$. The strip’s version of two colours and a swap is exactly those seventeen. The sign count reaches the same number without naming a frieze.

The count of 31 can also be checked against a layer is not a wallpaper, which makes the same point one dimension up: a sheet in space has more symmetry available than a pattern in a plane, because the sheet can be turned over. A ribbon has more again than an edge, because it can be turned end to end across its width. Each added freedom is one more sign, and the counts grow from seven friezes to 31 band groups by exactly the extra sign’s weight.

What the census assumes #

Band groups are identified by their sign subgroup and their glides and screws. Two ribbons with the same pair $(S, \tau)$ are counted as the same band group, and that is the right identification. Every band group has a representative with its operations on one centre line, and the pair records exactly which operations it has and which slide. That the count comes to 31, the published number, is evidence that no two distinct groups share a pair and no pair is impossible.

Ribbons are cut in 24 directions. Every primitive lattice direction with entries up to four is cut. All 31 groups appear with the four shortest, so the longer directions confirm the census and do not extend it.

A ribbon’s width is irrelevant. Only its centre line and direction matter to its group. A very narrow ribbon, narrower than the spacing of the lattice rows, is still a strip with the same symmetry, although no crystal would call it a ribbon.

What the census has to refuse #

The first test is the one the rest depend on. The sign count gives 31 abstractly, the census must find 31 distinct groups, and it must find no group outside them. Each ribbon’s operations are checked on the way: a translation of the ribbon must be a whole period, an operation keeping the direction must slide the centre line by nothing or half a period, and one kind of operation must have one slide. A layer whose period was misjudged would fail the second of these. The edge census that this essay builds on had exactly that fault in 240 of its 5,120 layer-directions, where a search too narrow to find twice the direction left the period undefined. The frieze names came out right by accident, and the period is now decided exactly.

Still open: rods, and the other kind of strip #

A band is a strip that keeps a plane. A rod keeps only a line, and its group is one of the 75 rod groups that seventy-five ways to be a thread counts. A rod’s operations can turn it by any angle its line allows about its axis, not only by half-turns, so its point group is not a group of signs. Cutting a rod from a three-dimensional crystal, as a ribbon is cut from a sheet, is a question with the same structure: which of the 75 are reached, how many only by cutting symmetrically about a special axis, and how many of those are chiral. The sign count does not extend to rods, since a four-fold or six-fold screw is not a sign. The census would need the space groups’ operations in place of the layers’, and it has not been run.