A hand made of pieces that have none

The groups a single hand may sit in sorts the plane groups by one question: does the group contain an operation that reverses orientation? Five do not, and a pattern made from a single-handed piece can only have one of those five. The criterion is about the group, and it is exact.

What it is not about is the pieces. Two hundred and thirty or two hundred and nineteen says so in a paragraph: “a crystal of a non-chiral molecule can grow in a Sohncke group — the arrangement carrying a handedness the contents do not.” Quartz is silicon dioxide, whose tetrahedra are superposable on their mirror images, and a quartz crystal is left-handed or right-handed. Sodium chlorate does the same with ions that each have a mirror.

That paragraph establishes that it happens and stops there. It does not say when an arrangement of achiral pieces comes out chiral, and when it keeps the mirror its pieces have. The answer is exact, and it is a statement about the group’s normaliser.

The piece’s mirror is the only candidate #

Take a motif m with a mirror σ and no other symmetry — three points, two of them mirror images and the third on the mirror line — and repeat it by a group G with no operation that reverses orientation. The pattern is the orbit of m under G: every copy g·m for g in G.

Suppose the pattern has some operation τ that reverses orientation. It carries the original motif onto one of the copies, say g·m, so the operation g⁻¹τ carries the motif onto itself. The only symmetries of the motif are the identity and its mirror, and g⁻¹τ reverses orientation, so g⁻¹τ is the motif’s mirror σ. Every reversing symmetry the pattern could have is a copy of the motif’s own mirror, moved by G.

So the question is whether σ itself is a symmetry of the pattern. It carries a copy g·m to σg·m, which can be written as (σgσ⁻¹)·σm, and σm is m. So σ maps the orbit to the orbit exactly when σgσ⁻¹ belongs to G for every g — exactly when σ conjugates G onto itself. That is the definition of the normaliser.

The pattern is achiral exactly when the motif’s mirror belongs to the normaliser of G. Otherwise it is chiral, however symmetric each piece is. Nothing about the argument uses a picture; it uses that the motif has exactly one non-trivial symmetry and that the pattern is an orbit, which is the orbit as the pattern read in reverse.

Most directions lose the mirror at once #

The normaliser is a small set, and the first constraint on it comes from the lattice.

A symmetry of a periodic pattern carries its lattice of translations onto itself, so its linear part is one of the lattice’s own symmetries — the holohedry is the ceiling on what any symmetry of the pattern can be. A mirror whose direction is not a mirror direction of the lattice cannot be in the normaliser, and so a motif turned to a generic angle always makes a chiral pattern. There is no measurement to make: the mirror is not a symmetry of anything periodic built on that lattice.

The most extreme case is the oblique lattice, which has no mirror direction at all. Put an achiral motif into p1 or p2 on an oblique lattice, at any angle and any position, and the pattern is chiral. The only achiral patterns available are on lattices that have mirrors to offer.

That is already a strong statement about crystals, because most orientations of a molecule are not special. A crystal grown from achiral molecules, in a group without mirrors, keeps a mirror only if the molecule’s mirror happens to line up with one of the lattice’s, and whether a crystal makes that alignment is a question of packing rather than of symmetry.

Along the lattice’s mirrors, only certain lines #

A motif whose mirror does point along a mirror direction of the lattice still has to be in the right place. The normaliser of G contains a given mirror direction only along certain lines, and the rest of the lines in that direction lose the mirror as surely as a generic angle does.

The lines were found by testing rather than by reading them off a table. For each group and lattice, every mirror line of the lattice passing through a point of a grid of twelfths was tried: a motif with exactly that mirror was placed, its orbit generated in exact rational coordinates, and the symmetry of the resulting points detected from scratch.

In p4 on the square lattice, thirty-six distinct lines were tried and six keep the mirror. Four of them pass through the four-fold rotation centres, and the pattern becomes p4m. Two run halfway between them, and the pattern becomes p4g. The same motif, moved by a quarter of a cell without being turned, changes which achiral group it makes — or makes a chiral one.

In p6 only six of seventy-two lines keep the mirror, and every one gives p6m. In p3 twelve of seventy-two do, and they split between p31m, nine lines, and p3m1, three: the difference between those two groups is whether the mirrors pass through every three-fold centre or only through one kind, and which kind of line a motif’s mirror lies on decides it. In p2 on a rectangular lattice four of twelve lines keep the mirror, two through the half-turn centres giving pmm and two between them giving pmg; on a square lattice the diagonal lines are available too and add four more giving cmm.

And in p1, which constrains nothing, every line in a mirror direction of the lattice keeps the mirror. A pattern with no symmetry but translation has no rotation centre for a mirror line to miss, so the only condition is the direction.

Which groups a rescued pattern can be #

The groups the census finds are not a miscellany, and the list for each starting group can be predicted before anything is drawn.

A pattern that keeps the motif’s mirror has symmetry G together with σG — the original operations and the original operations followed by the mirror — so its group contains G at index two. That makes it one of the groups that a symmetry can climb back to in one step, read from below: an index-two supergroup of G. And it contains a genuine mirror, because σ is one. So the possible rescued groups are the index-two supergroups of G that contain a mirror, and no others.

Checked against the census, that rule accounts for every name in it and for every name missing from it. Above p1 the index-two supergroups with a reversing operation are pm, pg and cm; the census finds pm and cm and never pg, because pg’s reversing operations are all glides and a motif’s mirror is not a glide. Above p2 they are pmm, pmg, pgg and cmm; the census finds pmm, pmg and cmm and never pgg, for the same reason. Above p4 they are p4m and p4g, both found; above p3, p3m1 and p31m, both found; above p6, only p6m. A pattern built from achiral pieces in a group with no mirror can become every achiral group one step up except the ones made entirely of glides.

That puts a sharp line through the four plane groups a lumpy molecule packs in. Two of the four, pg and pgg, are achiral, and neither can be reached by an achiral molecule keeping its mirror. A molecule’s mirror can never become a glide of the pattern — a glide moves every point, and the molecule’s mirror fixes a line through it — so the achiral groups a crystal reaches by keeping its molecules’ mirrors are the ones that have mirrors, and a crystal whose group is pg or pgg has kept no molecular mirror at all. Its achirality comes from pairs of molecules related by glides, not from any symmetry of the molecules, and it would be achiral in exactly the same way if the molecules had a hand.

Checked two ways #

The rule has two sides that share nothing but the group, and they were compared at every line.

One side is detection. The pattern’s points are handed to the same detector that verifies every pattern on these pages, which knows nothing about the motif or the group that produced it, and it reports whether any operation reversing orientation carries the points onto themselves. The other side is the normaliser test: conjugate the group by the motif’s mirror and ask whether the group comes back. Across thirteen pairs of group and lattice and four hundred and sixty-eight mirror lines, the two agree every time.

Where the pattern is achiral, its group is named by construction rather than by resemblance: a change of basis and origin is searched for that carries one of the seventeen standard groups exactly onto the operations detected. Every achiral pattern found this way is one of the seventeen, and none of them is ever the group the motif was placed in — which would mean a mirror had appeared from nowhere.

What the normaliser is doing here #

The normaliser has appeared on these pages as a bookkeeping object. The same pattern, described twice introduces it as the set of motions that turn one description of a structure into another description of the same structure. Here it is doing something with physical consequences: its mirrors are the only places an achiral molecule can put its mirror and keep it.

The connection is direct once stated. A mirror in the normaliser of G is a motion carrying the pattern’s description to another description of the same arrangement; a mirror not in it produces a different arrangement, the mirror image of the pattern. For a pattern built from an achiral motif, that mirror image is either the same pattern — if the motif’s mirror is in the normaliser — or the pattern’s enantiomorph, and then the crystal has a hand.

The metric enters exactly as it does in the normaliser’s dependence on the cell. p2 on an oblique cell has no mirror in its normaliser at all; p2 on a rectangular cell has four lines of them, on a square cell eight, and on a hexagonal cell twelve. The group has not changed. A molecule crystallising in p2 has more places to keep its mirror the more symmetric the cell accidentally is, which is the specialised-metric result read as a statement about chirality.

Quartz, sodium chlorate, and the plane’s version of them #

The argument does not use the dimension. In space a molecule with an improper symmetry, placed in a Sohncke space group, makes an achiral crystal only if that improper operation, moved into position, normalises the space group; and in space the improper operations include the inversion and rotoinversions as well as mirrors. Nothing above computes that census. What the plane shows is its shape.

Quartz and sodium chlorate are the standard crystals of this kind. The silicon-oxygen tetrahedra of quartz have mirrors, and they are joined corner to corner into helices round screw axes of one sense; the tetrahedra’s mirrors lie nowhere the screw group can accept them, so the crystal is chiral and comes in two hands. The chlorate ions of sodium chlorate each have a three-fold axis and three mirrors, and they sit in a cubic group of rotations and screws with none of those mirrors available. Both are quoted rather than computed here, and both are the three-dimensional form of the first panel of the figure at the top.

The practical consequence is that a chiral crystal can grow from a solution that has no hand. A solution of sodium chlorate is achiral; its crystals are left-handed or right-handed, and which hand a given crystal takes is decided at nucleation. Stirring the solution while crystals form can make almost all of them one hand, because fragments of the first crystal seed the rest — an observation from 1990 that turned a symmetry argument about arrangements into a demonstration of how a hand can be selected where none was supplied.

That is also why optical rotation in a crystal says nothing about the molecules. A crystal of an achiral substance in a chiral group can rotate the plane of polarisation, and its solution cannot.

What the census does not reach #

Motifs with more than one symmetry. The argument assumed the motif’s only non-trivial symmetry is its mirror. A motif with two mirrors, or a mirror and a rotation, has more candidates for the pattern’s reversing symmetries, and the rule becomes a condition on the whole of the motif’s group: the pattern is achiral when the motif’s group meets the normaliser in some reversing operation. The tests here use motifs with exactly one mirror, and the richer case is not computed.

Position within the line. A motif whose mirror lies on a rescuing line can still sit anywhere along that line, and that freedom does not affect the verdict. What it affects is whether the motif lands on a point of higher symmetry — a rotation centre on the line — and then the pattern has more symmetry still, which is the reason a motif must be a comma in the first place. The motifs here were placed at generic points of their lines.

Whether a crystal chooses a rescuing line. Symmetry lists the lines on which an achiral molecule keeps its mirror; packing decides whether the molecule sits on one. A mirror puts a bump against a bump, which is why close-packed molecular crystals avoid mirrors, and why achiral molecules crystallise in chiral groups far more often than an argument from symmetry alone would suggest.

How close to a rescuing line a real molecule sits. The verdict above is a bit: on the line the pattern is achiral, a hair off it the pattern is chiral. A real molecule is not placed exactly; its mirror lies at a measured angle and position with an error bar, and a molecule whose mirror is a small fraction of a degree off a rescuing line gives a crystal that is chiral in the strict sense and nearly achiral in every sense a measurement can reach. That is the case how chiral, as a number is for — not whether the arrangement has a mirror but how far it is from one — and it is also the case near-symmetry and the tolerance warns about from the other side. A structure refined with a tolerance will report the rescued group, with its mirror, for any molecule close enough to the line, and the tolerance, not the crystal, will have decided whether the crystal has a hand.

The two errors are not symmetric in consequence. Reporting a nearly-rescued arrangement as achiral puts a mirror into a structure that has none, which merges two enantiomorphic crystals into one description and hides the fact that the substance can crystallise in two hands. Reporting a genuinely rescued arrangement as chiral does the opposite and invents a hand. Neither is visible in the arithmetic of the group, which is exact on both sides of the line; both are visible only in how far the molecule actually sits from it.

What was computed, and how. For each of p1, p2, p3, p4 and p6 on each lattice it can sit on, every mirror of the lattice’s holohedry through every point of a grid of twelfths, deduplicated to one per line; a motif of three rational points with exactly that mirror; its orbit under the group; the symmetry of the orbit, detected; the normaliser test on the mirror; and, for the achiral patterns, the standard group carried onto the detected operations by a change of basis.

Who noticed the arrangement could be handed #

Pasteur separated the two hands of tartrate crystals by hand in 1848, and the lesson taken from it for a century was that crystal handedness reports molecular handedness. Quartz was the standing counter-example throughout: its optical activity had been measured by Arago and Biot before Pasteur began, and it vanished when the crystal was melted, because the hand was in the arrangement. The distinction between a chiral crystal and a crystal of chiral molecules was drawn cleanly in the twentieth century, once structures could be solved and the tetrahedra could be seen to have no hand.

The group-theoretic form — that the arrangement is achiral exactly when a symmetry of the piece normalises the group of the arrangement — is the kind of statement the normaliser tables of the International Tables make possible, and it is implicit in how those tables are used to decide when two descriptions of a structure are enantiomorphs rather than the same structure.

Still open: the census in space #

The plane has five groups without mirrors and a handful of lines in each. Space has sixty-five Sohncke groups, and a molecule’s improper symmetries include inversions and rotoinversions, each of which a space group’s normaliser may or may not contain along given lines and at given points. Which molecular symmetries can survive in which Sohncke groups, and at how many positions, is a finite table that the plane’s version above says exists and does not compute.