A scroll cannot be turned end for end

A sheet with two faces rolls into two tubes took the eighty layer groups, the symmetries of a sheet whose two faces may differ, and rolled each into a tube twice: once with face A outside and once with face B. An isometry of space that preserves a tube preserves distance from its axis, so no operation that exchanges the faces can be a symmetry of either tube. But it can carry one tube onto the other, and the census found that it does so for most layers. Seventeen layers always give two different tubes. Forty-six give one tube in every direction. Sixteen relate their two tubes only along special directions, and only as mirror images. One, whose faces are exchanged by a rotoinversion, never gives the same tube twice.

It ended on the other way a sheet can curl. A scroll is a sheet wound into a spiral of several turns with its edge left free, as carbon and several layered dichalcogenides do. A scroll has no closed circumference, and whatever symmetry it keeps is a group of motions of a spiral rather than of a cylinder. For a mirrored sheet the two scrolls were probably one object, and for the special layers perhaps a pair of hands again. That essay also asked whether the special directions survive the extra requirement a spiral imposes.

They do, some of them. But the larger change is elsewhere. Of the forty-six layers whose two tubes are one object in every direction, only seventeen still give one scroll in every direction, and eight of the rest never relate their two scrolls at all. The operations that joined their tubes are ones a spiral cannot accommodate.

What an isometry can do to a spiral #

The picture at the head of this essay shows the two scrolls in cross-section. A point of the sheet at distance $s$ along the rolling direction, $t$ across it and height $h$ above the sheet’s middle plane goes to the point of the spiral at arc length $s$, at height $t$ along the axis, and at radial offset $h$ outward if face A is out, inward if face B is.

A tube is a closed circle in cross-section, and the circle has every rotation about its centre and every reflection through it. That is why a tube’s group is a rod group, one of the seventy-five that a rolled sheet is never one of a pair reached by rolling the plane groups, and why a slide round the circumference becomes a turn: the tube has a screw no lattice allows found turns of order fourteen and ninety-eight made that way. A spiral of a few turns has neither. Its radius grows with arc length, so no rotation about the axis carries it onto itself, and its inner edge is not its outer edge, so nothing can run it backwards. What an isometry of space can do to a scroll is limited to three things. It can slide the scroll along its axis. It can reflect the scroll in a plane across the axis, which leaves the cross-section alone. And it can reflect the cross-section, which turns a spiral wound one way into a spiral wound the other way. In every case it keeps arc length along the spiral and it keeps radial offset.

Translated back to the sheet, that says exactly which of the sheet’s operations survive in a scroll. An operation survives when it leaves the rolling direction fixed and does not slide along it. It may reverse the direction across the roll, and it may turn the sheet over. What it becomes depends on which of those it does:

• a translation across the roll becomes the scroll’s own period along its axis;
• a mirror line of the sheet lying along the roll becomes a mirror across the scroll’s axis, and the scroll is then achiral;
• a mirror in the sheet itself, exchanging the faces, carries the scroll with face A out onto the one with face B out;
• a two-fold axis lying in the sheet along the roll carries face A out onto the mirror image of face B out.

And there is one relation every scroll has, whatever its sheet. Reflecting the cross-section turns a scroll into the same sheet wound the other way, with the same face out. A scroll and its opposite winding are always mirror images. So when face B out is the mirror image of face A out, it is face A out wound the other way.

The half-turns and screws that die #

The census of tubes sorted every operation that turns the sheet over into four kinds by what it does to the rolling direction and to the direction across it. A mirror or glide in the sheet keeps both, and in a tube it becomes a screw about the axis, since a slide along the circumference is a turn of the tube. A centre of inversion reverses both, and in a tube it becomes a half-turn about a line crossing the axis. A two-fold axis in the sheet along the roll keeps the roll and reverses the other, and becomes a mirror across the tube’s axis. A two-fold axis across the roll does the opposite and becomes a mirror containing the axis.

In a scroll the second and fourth kinds reverse the rolling direction, which would carry the inner edge onto the outer one, so they die. The first survives only when its slide can be taken at right angles to the roll, since a slide along the spiral moves every point to a place of different radius. In a tube it did not matter which way the slide pointed: a slide along the circumference was a rotation, and a rotation of a tube is a symmetry of it. The third survives only when it is a true axis and not a screw, for the same reason.

Across the census of 1,700 layer-directions, the operations of the first kind are used 668 times in tubes and 468 in scrolls. Centres of inversion are used 610 times in tubes and never in scrolls. Two-fold axes along the roll are used 112 times in tubes and 103 in scrolls, the difference being screw axes. Two-fold axes across the roll are used 112 times in tubes and never in scrolls.

Eighty layers as scrolls #

Rolled into scrolls along every primitive lattice direction with components up to four, the eighty layers fall into four kinds. Seventeen give one object with either face out in every direction. Thirteen do so only along special directions. Twenty-one give mirror images, and only along special directions. Twenty-nine never relate their two scrolls in any direction. The twenty-nine are the seventeen one-sided layers, which have no operation exchanging their faces at all, and twelve layers whose faces are exchanged but not in any way a spiral admits.

Set against the tubes, the picture shows where each kind comes from. The seventeen one-sided layers are one-sided either way. The sixteen layers whose tubes were related only along special directions keep that relation in thirteen cases and lose it in three. The three are the ones whose tubes were related only by two-fold axes lying across the roll, or by two-fold screw axes along it, and a spiral admits neither. The one layer whose tubes were never the same, turned over by a rotoinversion, gives unrelated scrolls too.

The forty-six layers whose tubes were one object in every direction divide four ways. Seventeen are the layers with a mirror in the sheet, and they give one scroll in every direction for the same reason they gave one tube: reflecting the sheet in its own plane exchanges its faces and moves nothing within it, so it carries face A out onto face B out whatever the spiral does. Thirteen give one scroll only along special directions, those at right angles to the slide of the glide that turns them over. Eight give mirror images along special directions, where a two-fold axis in the sheet lies along the roll. And eight lose every relation. The centring that turns into a screw found that rolling can turn a flat symmetry into a new kind of operation. Here it turns a symmetry of the tube into none at all.

A glide that joins tubes in every direction and scrolls in half #

The thirteen layers of the second kind show the rule most plainly. Take the layer on the square lattice of p4 whose faces are exchanged by a glide in the sheet, a reflection in the sheet’s own plane followed by a slide of half a lattice vector. Rolled into a tube along any lattice direction, the glide becomes a screw about the tube’s axis, and the two tubes are one object in all twelve directions of the census. Rolled into a scroll, the glide survives only where its slide, adjusted by a lattice vector of the sheet, can be taken at right angles to the roll, and that is true of six directions out of twelve. In the other six, every way of writing the glide slides partly along the spiral, and the two scrolls are different objects.

Eight layers that lose everything #

The eight layers of the last kind are the sharpest contrast. Each rolls into a single tube in every direction, and each rolls into two different scrolls in every direction the census reaches. Their faces are exchanged by a centre of inversion, which reverses the spiral, by two-fold screw axes lying in the sheet, which slide along the roll, by a rotoinversion, which fixes no direction, or by a glide whose slide is never at right angles to a lattice direction. The last is a statement about the lattice’s shape. On a general oblique or rectangular cell, a lattice vector at right angles to a diagonal slide does not exist, and only a cell with extra symmetry would supply one.

Six of the eight have a centre of symmetry. A centre is the operation that joined their tubes in every direction, as a half-turn across the axis, and it is the operation a scroll can least use, since it reverses everything. A layer is not a wallpaper counted the eighty by what they do to the sheet’s two faces. Its split of them into seventeen, seventeen and forty-six, by the sign each operation puts on the normal, is the split that decides a tube: forty-six of the eighty exchange their faces somehow. The count that matters for a scroll is narrower, and a centre of symmetry adds nothing to it.

Period and hand #

Two more properties of a scroll depend on its direction in the same way. A scroll repeats along its axis only if the sheet has a lattice vector at right angles to the roll. That is a question about lengths, and the lattice types answer it differently. The census asks it on two unrelated cells of each type and counts a direction only if it holds on both. On an oblique lattice no direction has a perpendicular lattice vector on a general cell, and none of the 148 oblique scrolls repeats: a scroll of an oblique sheet never repeats along its axis. A tube rolled the same way meets the same condition for a period, and most sheets roll into a tube that never repeats found that it usually fails. But a tube that does not repeat still has a helical symmetry, a screw carrying every lattice point of the sheet onto another, whereas a scroll that does not repeat has no symmetry along its axis at all. On a rectangular lattice only the two edge directions do, 58 of 592. On square and hexagonal lattices every rational direction has a rational perpendicular, and all 732 scrolls repeat.

A scroll is achiral only when its sheet has a mirror line along the roll, which becomes a mirror across the axis. That happens in 84 of the 1,700 layer-directions, never on an oblique lattice. Every other scroll is chiral, and its mirror image is the same sheet wound the other way. A tube can be achiral through a mirror containing its axis as well. A scroll cannot, since that mirror would reverse the spiral. Whether the sheet itself is chiral is a separate matter, decided by a sign the plane pattern leaves free, as a thread’s hand is not a choice found. A scroll can be chiral when its sheet is not, and usually is.

A sheet with a different atom on each face #

The distinction is not only formal. A sheet whose two faces carry different atoms, a layer of a transition-metal chalcogenide with sulphur on one face and selenium on the other, has no operation that exchanges its faces. Its layer group is one of the seventeen one-sided ones, and its two scrolls are different objects in every direction, as its two tubes are. That is the case in which the question of which face goes outside is a physical one, since the two scrolls have different curvature on each face, different strain, and in general different energy. A sheet with a mirror in its plane, such as a single layer of graphene or of molybdenum disulphide, rolls into one scroll whichever face is out.

Between those extremes the census separates two situations that the tube census joined. A sheet whose faces are exchanged by a centre of symmetry rolls into one tube, face A out or face B out, in every direction. The same sheet rolls into two different scrolls in every direction. Whether a physical sheet of that kind prefers one scroll to the other is not a symmetry question. Symmetry has stopped forbidding a preference, and has stopped deciding one.

A third situation lies in the two-fold axes along the roll. There the two scrolls are mirror images, which means that face B out is face A out wound the other way. A sheet of that kind that rolls with face A out can make scrolls of either winding, and the scroll with face B out of one winding is congruent to the scroll with face A out of the other. In such a sheet the choice of face and the choice of winding are one choice.

What the census does not decide #

The scroll is idealised. A real scroll’s turns press on each other, and a scroll of a two-sided sheet puts face A of one turn against face B of the next, whichever face is out. The interlayer contact is therefore the same in both scrolls, and so is any strain it causes. The spiral here has no thickness beyond the sheet’s own, and its turns do not interact.

The directions are rational ones up to four. A generic direction, at an irrational slope, can only keep operations that fix every direction, and those are the mirror in the sheet and nothing else. So the seventeen mirrored layers are the only ones whose scrolls are related in a generic direction, which the census confirms for the rational directions it reaches.

The verdicts depend on the cell only through its type. A scroll’s period and whether a glide slides at right angles to the roll are questions about lengths. The census takes the general cell of each lattice type. A rectangular cell that happens to be square, or an oblique one that happens to be rectangular, would have more perpendicular directions, and more of its scrolls would repeat.

Where the argument comes from #

The layer groups were enumerated in the 1920s and 1930s, and their use for rolled structures begins with the carbon nanotube, whose line groups Damnjanović and his colleagues worked out around 2000. Scrolls of carbon were described by Bacon in 1960 and have been made from graphene and from several transition-metal dichalcogenides. The classification of which sheet operations survive in a spiral, and the census above, are worked out on this page.

What the census must refuse #

The refused claim is the natural extension of the tube census, and it fails for twenty-nine of the forty-six layers it would have covered. The test that a scroll never relates its faces where the tube of the same direction does not is a consistency check between the two computations. A scroll keeps a subset of the operations a tube keeps, so any direction in which the scrolls were related and the tubes were not would mean one of the two computations was wrong.

Still open: the scroll that closes into a tube #

Between a scroll and a tube lies the case a real sheet often chooses: a scroll whose outer edge has wound round far enough to overlap its inner turns and bond to them, closing into a multi-walled tube with a seam. At the seam the spiral becomes a set of nested circles, and the operations a spiral forbade, the half-turns across the axis and the rotations, could return in part, since the walls far from the seam are circles. Which of them return, and whether a two-sided sheet’s two nested tubes, face A out and face B out, become related again when the seam is far away and not when it is near, is a question about the seam’s position rather than the sheet’s symmetry. A tube’s own closure already imposes arithmetic, the condition that the circumference be a lattice vector, and the seam of a closed scroll would impose more. It has not been worked out here.