A thread's hand is not a choice

Chiral in the plane is not chiral in the room settles the question for a sheet, and the answer is that the two chiralities are independent. A wallpaper pattern covered in mirrors can be a sheet with a hand, because a mirror line becomes a half-turn about that line once the sheet is allowed to have two sides and the operation is allowed to turn it over. Over the seventeen plane groups there are sixty-three sheets, and exactly seventeen of them are chiral in space — one over each plane group, whatever that group contains.

That essay ends on a thread, which repeats in one direction rather than two. Its operations carry signs for the two directions across it as well as for the one along it, and the natural guess is that the same independence appears: that some achiral thing wound into a thread comes out with a hand, as an achiral wallpaper wound into a sheet does.

It does not, and the reason is that a thread has no sign left to choose. Of the seventy-five rod groups, thirty-two are chiral, they lie over nine of the twenty-seven axial crystal classes, and which nine is decided by the class’s linear parts alone. Nothing about how a thread is built, wound or handed can move a rod group across that line.

Its direction is another matter. Whether anything exchanges the two ends of a thread depends on how the class lies along it, and three classes can lie two ways, so for those three the answer is not fixed by the class. What a thread can choose is its direction. Its hand it cannot.

Two signs, and what each of them is about #

Every operation of a rod group carries the axis of the rod onto itself — that is what makes it a rod group rather than a group of something else — so it does two independent things.

It sends the axis direction to itself or reverses it, by a sign $\sigma$, and it acts on the plane across the axis by a $2 \times 2$ orthogonal matrix $A$. Its determinant in space is $\sigma \det A$, so

• a turn or screw about the axis keeps both, and is proper;
• a half-turn crossing the thread reverses both, and is proper;
• a mirror across the thread, or a rotoreflection, reverses the direction and keeps the plane’s hand, and is improper;
• a mirror or glide containing the axis keeps the direction and reverses the plane’s hand, and is improper.

A rod group is chiral when every one of its operations sits on the proper diagonal. The four-fold rotoinversion is the clearest case of an operation that looks proper and is not: it turns the thread through a quarter and reverses the axis, so $\sigma = -1$; and the quarter-turn it performs in the plane across the axis has determinant $+1$; and the product is $-1$. A structure with that operation contains both hands, however helical it looks in a drawing that shows only part of it. And a second question uses the same square: a group is polar along its axis when every operation has $\sigma = +1$, so that nothing exchanges the two ends of the thread. That is a real property of a real object. A protein chain runs from its amino end to its carboxyl end and the two ends are chemically different; a symmetry reversing the chain would have to carry one onto the other, and there is nothing for it to carry.

The two conditions pick out a diagonal and a row of the same four cells, so neither contains the other, and the seventy-five sort into four boxes rather than two.

Nine classes, and a hand that follows from them #

Both signs are conditions on the linear parts of the operations and on nothing else, and a rod group’s linear parts are its crystal class, placed on the rod. Chirality belongs to the class: an improper operation is improper however the class is turned, so a class is chiral or achiral for all the rod groups built over it at once.

Polarity belongs to the placing. A monoclinic two-fold may lie along the rod, where it turns the thread and keeps its ends apart, or across it, where it turns the thread end for end. Class 2 is polar the first way and not the second. The same holds for m, whose mirror may contain the rod or cut it, and for mm2, whose two-fold may be the rod or cross it. 2/m can lie either way too, but its centre reverses the rod in both. So polarity is a property of a class together with the way it lies on the rod, and those three classes appear on both sides.

Nine of the twenty-seven classes have no improper operation: the five pure rotation axes 1, 2, 3, 4 and 6, and the four with half-turns added across them, 222, 32, 422 and 622. The derivation of the seventy-five attaches translations along the axis to each class and counts what survives an origin shift, and the counts over those nine are 1, 3, 3, 4, 6, 2, 3, 4 and 6 — thirty-two in all. Class 2 carries three because its two-fold may lie along the rod, as a rotation or a screw, or across it, where no screw is possible.

The arithmetic is worth doing twice, because a count taken one way and a count taken another way disagreeing is how an error of this shape shows itself. Counted operation by operation over the groups, thirty-two have no operation of determinant $-1$; counted by summing the per-class totals over the nine classes with no improper linear part, thirty-two again. The two are the same number reached from opposite ends, one from the groups and one from the classes.

Both counts once agreed on thirty-one, and both were wrong. They were taken over a list of rod groups that put each class on the rod one way only, and so lacked p211, the two-fold across the rod, which is chiral. Two counts over one list can only check each other’s arithmetic, never the list itself. The derivation of the seventy-five sets out how that list came to have the right total anyway, and the census here is now taken over the corrected one, which agrees with the Tables class by class.

Thirty-two of seventy-five is 43%, which is a larger share than the corresponding one for the groups of space — the classification of which groups a single hand may sit in leaves a smaller fraction, because space has cubic classes and inversion centres and rods have neither. A rod is a thinner object and correspondingly harder to make achiral.

There is a second reason the share is large, and it is about what the translations can carry. A rod group’s translation lattice is a single direction, so the only intrinsic translation an operation can have is along the axis — a component across the axis is either removed by moving the origin off the axis, or would make the operation’s square a translation across the axis, which a one-dimensional lattice does not contain. That is the fact the enumeration of the seventy-five runs on, and it works the same way here: a class with $n$ pitches available along the rod carries $n$ rod groups that way, and when the class is chiral all $n$ of them are. So the chiral classes are not merely nine of twenty-seven; they are nine that between them carry the largest pitch counts, because the pure axes are exactly the classes with nothing to remove a pitch.

Four boxes, all of them occupied #

Sixteen rod groups are chiral and polar. Their classes are the five pure axes laid along the rod, so these are the groups of a single-stranded helix: a turn with a climb, and nothing else. Sixteen are chiral and not polar, and every one of them has a half-turn crossing the thread: the fifteen over 222, 32, 422 and 622, and p211, which is class 2 with its only two-fold lying across the rod. The half-turn is exactly what costs the thread its direction.

Thirteen more are polar and achiral, over m, mm2, 3m, 4mm and 6mm with a mirror or glide containing the axis and nothing reversing it. The same m with its mirror cutting the rod, and the same mm2 with its two-fold across it, are not polar. And thirty are neither — every group with an inversion centre, a mirror across the thread or a rotoinversion, which is where most of the twenty-seven classes sit once the axis is allowed anything at all.

The thirteen polar achiral groups are the ones a reader is least likely to have a picture of, and they have a simple one. Their mirrors or glides contain the axis and nothing reverses it, so the object is a thread with flat sides: a ribbon, a strip, a column with a mirror plane running down it. It has a direction, because nothing turns it end for end, and it has no hand, because a reflection down its length carries it onto itself. An idealised instance is a chain of glycine, which has no stereocentre, drawn fully extended and flat. The plane of the chain is a mirror, a two-fold screw along the chain carries each residue onto the next, and the chain still runs from its amino end to its carboxyl end. The obvious instance fails. A planar zig-zag of identical atoms has a mirror in its own plane and also one across the chain through every atom, so its ends are alike and it is not polar at all.

The pair that is worth carrying is the first two, because it is the difference between one strand and two.

A single helix has no operation crossing its axis: everything that carries it to itself turns it and slides it along, and the arrow along the chain is preserved. Put a second copy of the same helix beside it running the other way, and a half-turn about an axis at right angles to the thread carries each strand onto the other. That half-turn reverses the direction along the thread and reverses the hand of the plane across it, so its determinant is $+1$ and the pair is still chiral — but the two ends of the thread have become alike.

That is why a double helix of two antiparallel strands has dyad axes perpendicular to its own axis and a single strand does not, and it is a symmetry statement rather than a chemical one: the dyads are forced by the antiparallel arrangement and would be there in any two-stranded thread built the same way.

Reflected, and sometimes a different group #

A chiral group has a mirror image, and the mirror image is a group of the same kind. Whether it is the same group is a further question, and it is the question that decides whether a material can be described by one entry of the list or needs two.

Reflecting in a plane containing the axis leaves every climb alone and reverses every turn, so a screw of pitch $k/n$ comes back as one of pitch $(n-k)/n$. A pitch of zero and a pitch of exactly one half are unchanged; everything else moves. Over the thirty-two chiral rod groups that gives eight enantiomorphic pairs, accounting for sixteen groups, and sixteen groups that are their own mirror image.

The pairs sit over five of the nine chiral classes: one each over 3, 4, 32 and 422, and two each over 6 and 622, since an axis of order six has two pitches away from zero and a half on each side. The distribution across the orders is worth reading directly. An axis of order $n$ has $n$ pitches, of which the fixed ones under reflection are $k = 0$ always and $k = n/2$ when $n$ is even; the rest pair off. So order 1 has one pitch and no pair, order 2 has two pitches and both are fixed, order 3 has one pair, order 4 has one pair, and order 6 has two. Summed over the five pure axes laid along the rod that is four pairs and eight self-mirror groups, and p211 adds a ninth self-mirror group, having no pitch at all. The classes with half-turns across the axis give the other four pairs and seven more self-mirror groups.

This is the same arithmetic that makes the eleven screw axes split into pairs and singletons, and it arrives here without the space group’s complication that a change of cell can turn one enantiomorph into the other. A rod group has one axis and one lattice direction, so there is no alternative cell to choose and no way for a reflected group to come back looking like the original under a relabelling.

An achiral rod group never moves under reflection, and the reason is a one-line argument rather than a computed fact: the group contains an improper operation, so the reflection it would be carried by is already realised inside it, and conjugating by an element of a group carries the group to itself. Checked over all forty-three achiral rod groups, none moves.

Why a sheet gets a choice and a thread does not #

The two answers look different and the arithmetic is identical. In both cases an operation is proper in space exactly when what it does to the directions of periodicity and what it does to the rest have determinants of the same sign.

The difference is what the second factor is. A plane group is a group of motions of the plane; making a sheet out of it means deciding, for each operation, whether the sheet is turned over or not, and that is a homomorphism of the point group onto $\{\pm1\}$ with as many choices as the group has index-two subgroups. Sixty-three sheets over seventeen plane groups, and exactly one chiral sheet over each — including the twelve plane groups that are covered in mirrors, where the chiral sheet is the one turned over by every operation that reverses orientation in the plane.

A rod group is already a group of motions of space. Its operations come with their action on all three directions, and no sign is free. So the hand is settled at the level of the class and there is no construction, however ingenious, that produces a chiral thread over 4mm.

One freedom is left to a thread, and it is not the hand. A class that singles out more than one direction can be laid on the rod along either, and for 2, m and mm2 that choice decides whether the thread has a direction. It never decides the hand, because turning a set of operations to lie a different way changes none of their determinants. The sheet’s free sign lands on chirality and the thread’s free choice lands on polarity, and the same product of two signs accounts for both.

The contrast also explains a fact about rolling that would otherwise look like a coincidence. Rolling a plane pattern into a tube sends each operation of the sheet to a definite operation of the tube, and the assignment is forced by the rolling rather than chosen: a translation becomes a screw, a half-turn becomes a half-turn crossing the tube, and a mirror or glide becomes a mirror or a rotoreflection. Proper goes to proper and improper to improper, every time. So a rolled pattern is chiral exactly when the frieze it wraps is chiral as a plane pattern, and the independence that holds for sheets fails for tubes — for the same reason it fails for rod groups in general, which is that the rolling has used up the freedom.

The seven families of groups with a single axis, which rolling the friezes round a cylinder produces, sort the same way: two of them are chiral and five are not.

What the sorting is, and what it is not #

Three limits are worth stating plainly.

Chirality of the group is not chirality of the structure. A rod group with no improper operation permits a structure with a hand; it does not require one. A structure whose atoms happen to sit at positions with extra symmetry can be achiral while its group is chiral, in exactly the way a hand made of pieces that have none traces the opposite case, where achiral pieces make a handed arrangement. The group is a permission, and the standing gap between permission and prediction runs through here as it does through the classes with a direction of their own.

Polar along the axis is not the same as polar in the ordinary crystallographic sense. A crystal class is called polar when it leaves some direction invariant and unreversed; a rod group is called polar here when it leaves its own axis unreversed, which is a question about one particular direction rather than about the existence of any. The thirteen polar achiral rod groups have mirrors containing the axis, so they are not what a physicist would call a polar group in every respect, and the word is doing a narrower job.

The sorting says nothing about how much of a hand there is. Two structures with the same chiral rod group can be very differently handed — a tight helix and one so slack that a small displacement would make it planar — and the group answers one bit for both. That is the standing limitation of any classification by symmetry, and it is the gap a continuous measure of chirality exists to fill; the measure applies here unchanged, taken over one repeat of the thread rather than over a finite cluster.

And the census is over groups rather than over structures. How many of the thirty-two chiral rod groups are actually realised by known fibres, polymers and nanotubes is a different and much harder question, and one where the answer would be a survey rather than a derivation. What a thread scatters is where the measurement enters, and it reads a helix’s turn and climb rather than its group.

Still open: the same two signs for a layer’s edge #

A sheet has an edge, and the edge is a thread. A layer group restricted to the operations that carry a boundary line of the sheet to itself is a rod group, and the restriction is a homomorphism whose image is one of the seventy-five — which raises the question the two censuses here do not answer together. The eighty layer groups are built from the same twenty-seven axial classes as the rod groups, and each of them has edges in as many directions as its lattice has. Which rod groups appear as the edge of which layer, and whether a chiral layer can have an achiral edge or the reverse, is a table with two enumerations already on this page and a third computation between them.

The other question left is about the sixteen. Sixteen chiral polar rod groups is the same number as the sixteen screws a crystallographic axis admits, and it is the same sixteen: a chiral polar rod group is nothing but an axis with a pitch. The coincidence is not a coincidence, but stating why in the right way — that the polar chiral classes are exactly the cyclic ones, so the group is its own axis — makes it a definition rather than a theorem, and a definition dressed as a result is worth catching.