Chiral in the plane is not chiral in the room

How chiral, as a number makes a remark in passing that deserves more than a sentence. “A scalene triangle is chiral in the plane and achiral in space, because three points always lie in a plane and that plane is a mirror.” Handedness is not a property of an object alone. It is a property of an object in a space, and the same object can have a hand in one space and none in another.

For a flat figure the effect runs one way: lifting it into space adds a mirror, the plane it lies in, so everything chiral in the plane becomes achiral. For a crystal the effect is richer, because the objects of interest are not flat. A monolayer, a slab, the atoms of a surface, one sheet of a layered mineral: each repeats in two directions and has thickness in the third, and its atoms sit above and below its middle. A layer is not a wallpaper, and the difference turns out to decide its handedness.

The claim this essay computes runs both ways. A pattern that is achiral in the plane — mirrors everywhere — can be a sheet that is chiral in space; a pattern that is chiral in the plane can be a sheet that is not. And over each of the seventeen plane groups there is exactly one chiral sheet.

Two signs for every operation #

Take a sheet whose pattern, seen from above, has plane group G. Each operation of G does something to the plane, and as an operation of space it must also do something to the direction perpendicular to the sheet: leave it alone, or reverse it. Reversing the normal is turning the sheet over — exchanging its top face with its bottom.

So each operation carries two signs. One is δ, the determinant of what it does to the plane: +1 for a rotation or translation, −1 for a mirror or glide line. The other is ε, what it does to the normal: +1 if the sheet stays the right way up, −1 if it is turned over. As a motion of space the operation is a 3 × 3 matrix whose determinant is the product δε.

That product is the whole of the arithmetic. An operation reverses handedness in the plane when δ = −1, and in space when δε = −1, and those are different conditions. A mirror line of the pattern that also turns the sheet over has δ = −1 and ε = −1, and as a motion of space it is proper: it is a half-turn about the mirror line, lying in the sheet. A half-turn of the pattern that also turns the sheet over has δ = +1 and ε = −1, and as a motion of space it is improper: it is the inversion, through the half-turn’s centre.

The table is easiest to believe by holding a sheet of paper. Reflect it in a line drawn on it, and the reflection is performed in the plane by folding the paper over the line — which turns it over, and which in space is a rotation by half a turn about the line. The paper’s pattern has been reflected; the paper, as a solid object, has only been rotated. Whether that was a reflection or a rotation depends on whether the room has a third dimension.

Why a fold is a rotation #

The paper-folding picture has an exact version, and it connects this to a count made elsewhere.

Every motion of the plane is a product of at most three reflections, and whether a motion reverses handedness is the parity of how many it takes: one reflection for a mirror line, two for a rotation. Space has the same rule with planes in place of lines. A mirror line drawn on the sheet is one reflection in the plane; in space the motion that realises it without tearing the sheet is the product of two reflections in planes — the plane standing across the sheet through the line, and the plane of the sheet itself. Two reflections is an even number, so the fold is proper, and two planes meeting along a line compose to a rotation about that line through twice the angle between them. The planes are perpendicular, so the rotation is a half-turn.

That is also a special case of every proper motion of space being a screw: a rotation about an axis combined with a slide along it. A mirror line that turns the sheet over is a screw with no slide, a half-turn axis; a glide line that turns the sheet over is a screw with a slide of half a cell, a two-fold screw axis lying in the sheet. Neither is an operation the plane can express, and both are what the plane’s reflections become when the room has a third direction.

The parity count also says why the inversion appears where it does. A half-turn of the pattern is two reflections in the plane; turning the sheet over adds the reflection in the sheet’s own plane, which makes three reflections in space, an odd number, and the three perpendicular planes through a point compose to the inversion through it. The whole table of what each operation becomes is a count of mirrors, taken once in the plane and once in the room.

Which signs are allowed #

The signs ε cannot be chosen operation by operation. Composing two operations composes what they do to the normal, so ε is a homomorphism from G onto {+1, −1}, and a translation cannot turn the sheet over, because turning over is a matter of direction and a translation changes no direction. The construction of sheets over a plane group counts exactly these choices: the sheet with no operation turning it over, the sheet with a mirror in its own plane, and one sheet for each homomorphism of G’s point group onto {+1, −1}. Over the seventeen there are sixty-three.

Now ask which of them are chiral in space. A sheet is chiral in space when every operation has δε = +1, which is to say ε = δ for every operation. The determinant δ is itself a homomorphism onto {+1, −1}, and it vanishes on translations. So there is always exactly one choice of signs that makes the sheet chiral: turn the sheet over at precisely the operations that reverse orientation in the plane, and at no others.

That choice is one of the sixty-three every time, and it is a different one depending on the pattern.

If the pattern is chiral in the plane, δ is +1 for every operation, so ε = δ means nothing turns the sheet over. The chiral sheet is the one-sided sheet: top and bottom different, nothing exchanging them. That is five sheets, over p1, p2, p3, p4 and p6.

If the pattern is achiral in the plane, δ is −1 somewhere, so ε = δ turns the sheet over at every mirror line and every glide line and leaves it the right way up under every rotation. The chiral sheet is a sheet turned over by exactly the operations that reversed orientation in the plane, and every mirror line of the pattern has become a half-turn axis lying in the sheet. That is twelve more, one over each achiral plane group.

The count was run rather than argued: every one of the sixty-three sheets was built, the determinant of every one of its operations as a motion of space computed, and exactly seventeen came out with every determinant +1 — one over each plane group, and in each case the one whose signs equal the plane determinants operation by operation.

A pattern with mirrors, a sheet with a hand #

The surprising half is the second one, and a picture makes it concrete.

The left-hand drawing is a pattern any wallpaper printer would recognise as full of mirrors. The right-hand drawing has the same points in the same places, and it is a chiral object: the copies that were mirror images of each other in the plane are now related by half-turns about lines in the sheet, which moves them from above the sheet to below it and keeps their handedness in space. Seen from directly above, the right-hand sheet projects exactly onto the left-hand pattern. The projection has mirrors; the sheet has none.

This is not a curiosity of drawings. It is what a projection does to handedness, and it means that an image taken straight down a sheet cannot decide whether the sheet is chiral. A sheet of atoms whose top view has mirror lines may have them as mirror planes standing across the sheet, in which case it is achiral, or as half-turn axes lying in the sheet, in which case it is chiral, and the two top views are identical. What separates them is which atoms sit above the mid-plane and which below — the side of the sheet, which is exactly the information a projection discards.

The physical case people meet is a pair of stacked sheets turned against each other. Each sheet of graphene is achiral, with mirrors in the plane and a mirror in its own plane; two sheets stacked with a small twist between them lose every mirror and keep the half-turn axes lying between them. Seen from above, the moiré pattern has six-fold rotations and nothing obviously handed about it; in space the bilayer is chiral, and the sense of the twist is its hand.

A pattern with a hand, a sheet without one #

The other half is the flat-figure remark again, with more options than one.

Over p4, a chiral pattern, there are three sheets. The one-sided sheet is chiral in space, as established. The sheet with a mirror in its own plane is achiral: its top and bottom are the same, and that mirror reverses handedness in space whatever the pattern does. And the sheet turned over by the quarter-turns — where a quarter-turn of the pattern also flips the sheet — is achiral too: a quarter-turn that turns the sheet over is a rotoinversion, improper in space. Two of the three sheets over a chiral pattern have no hand.

The table shows how far apart the two properties are. Of the sixty-three sheets, five are chiral in both senses and thirty-eight in neither; eight are chiral in the plane and not in space, and twelve are chiral in space and not in the plane. Handedness in the plane neither implies nor is implied by handedness in space. A pattern tells how the sheet looks from above; the chirality of the sheet is a statement about the sheet.

What this does to the sixty-five #

The consequence for crystals is a warning about reading chirality off a projection or off a layer.

A crystal of a single hand must have a Sohncke space group, and in the plane the corresponding count is five of the seventeen. It is tempting to think of a layered crystal as a stack of layers with plane groups and to ask whether the layers are chiral. The arithmetic above says the question is badly posed. A layer whose plane group has mirrors can be a chiral layer, and stacking chiral layers of that kind gives a chiral crystal whose every projection down the stacking direction has mirrors. Conversely, a layer whose plane group is chiral can sit in a centrosymmetric crystal if the layer is turned over by an inversion.

So the five of the seventeen is a count about patterns, and it is the right count for a layer confined to a surface, where nothing can turn over. For a free-standing sheet the right count is one sheet over each of the seventeen, and which sheet is decided by what happens at the mirrors.

A substrate brings the two together. A crystal surface has vacuum on one side and solid on the other, so nothing can exchange the faces, and every operation that turned the sheet over is lost. The chiral sheet over p4m keeps only its rotations: it becomes the one-sided sheet over p4, which is still chiral. A substrate can remove the half-turn axes that made an achiral pattern into a chiral sheet, but what is left is chiral anyway, because removing operations can never create a mirror.

What an experiment on a sheet can see of its hand #

The projection argument above has a counterpart in diffraction, and the two limitations add.

A diffraction pattern of a single sheet measured with the beam perpendicular to it records the sheet’s structure projected onto its own plane — the same projection a top view gives — so it inherits the blindness of the top view: a sheet chiral by virtue of which atoms sit above and which below can give the same pattern as its achiral sibling with mirror planes across it. And even a measurement that does see the third direction faces the law that hides handedness: without anomalous scattering, intensities are unchanged by inverting the structure, so a chiral sheet and its mirror image scatter identically. The first limitation hides whether there is a hand; the second hides which hand it is.

What does see chirality directly is a measurement that couples to the sense of rotation — the difference in absorption of left and right circularly polarised light. Optical activity is permitted only in certain classes, and a sheet’s class is decided by its operations in space, not by its pattern in the plane. So the prediction of this essay is checkable in principle: of two sheets with identical top views, one with mirror planes across it and one with half-turn axes in it, only the second can show circular dichroism when light travels perpendicular to the sheet. That is precisely the difference the tables of layer groups record and the tables of plane groups cannot.

A surface cut from a crystal is the case where the question arrives already decided. What a cleave leaves is a one-sided layer, so the operations that turned the sheet over are gone, and the surface is chiral exactly when its pattern is chiral in the plane. For a surface, and only for a surface, the five of the seventeen is the right count.

What the construction does not cover #

Sixty-three sheets, not eighty layer groups. The sheets here are built from a plane group and a sign on the normal for each operation, which is the point-group half of the classification of layers. The eighty layer groups of the literature also distinguish translations and positions of the operations that turn the sheet over — a half-turn axis in the sheet and a screw along the same line are different layer groups over the same sheet here. The eighty are quoted, not derived, and the count of which of the eighty are chiral is not attempted.

A measure, not only a verdict. Everything above is a yes or no. A continuous measure of chirality is equally relative to its space: the distance to the nearest mirror-symmetric arrangement is a different number when the mirrors available are planes in space rather than lines in the plane, and a sheet that is nearly flat is nearly achiral in space whatever its pattern.

What was computed, and how. For each of the seventeen plane groups, every sheet over it as a set of operations of space, each lifted from an operation of the plane with its sign on the normal; the determinant of every such operation; which sheets have every determinant positive; and whether their signs equal the plane determinants. The drawing of the chiral sheet uses the operations of that sheet, applied to the site’s asymmetric motif.

Where the idea is written down #

Kelvin’s definition of chirality in 1904 — a figure is chiral if its image in a plane mirror cannot be brought to coincide with itself — mentions the mirror and not the room, and the room is implicit in it: a mirror of the plane is a line and a mirror of space is a plane, and the definition means whichever mirrors the space has. The relativity of chirality to dimension has been a standard remark ever since, usually made about triangles.

The layer groups were tabulated by Elizabeth Wood in 1964, in the form still used, and the operations that turn a layer over were part of the classification from the start. Applying the determinant rule to them is a one-line exercise; what makes it worth doing is the result that each plane group carries exactly one chiral sheet, and that the sheet over a pattern full of mirrors is chiral by turning every mirror into an axis.

Where this goes: the same arithmetic for a thread #

A rod — a helix, a fibre, a nanotube — repeats in one direction and lives in three, and its operations carry signs for the two directions across it as well as for the one along it. The same product of determinants decides whether it is chiral, and the answer should again be that a frieze pattern’s handedness in the plane and a rod’s handedness in space are independent. Which of the seventy-five rod groups are chiral, and how each of the seven friezes can be wound into a chiral rod, is not computed here.