The four plane groups a molecule packs in

The densest lattice packing in the plane is a question about discs, and discs are the easy case: every direction is alike, so the answer is a property of the lattice and the search is over a two-parameter region with a corner in it.

A molecule is not a disc. It has protrusions and hollows, and packing molecules tightly is not a matter of choosing a lattice — it is a matter of getting one molecule’s protrusion into another’s hollow. That is a question about the operation relating the two, and not about the lattice at all.

Kitaigorodskii asked it in the 1940s and answered it with a rule short enough to state in a line: a mirror puts a bump against a bump.

The claim is not that symmetry causes dense packing. It is that a molecule packing densely must be related to its neighbours in particular ways, and that most of the seventeen relate them in a way that wastes space. The direction of the argument runs from geometry to a permission and stops there; what turns the permission into a prediction is the separate assumption that a crystal takes the densest arrangement available, which is chemistry rather than counting.

Four rows, pushed together #

The rule is a claim about geometry and can be measured. Take a shape with a protrusion on one edge and an indentation on the other — a caricature of a molecule, deliberately neither round nor symmetric — and lay a row of it. Then lay a second row on top, related to the first by each of the four operations in turn, and push the second row down until it touches.

The three operations that keep the shape the same way round come within two per cent of one another. The mirror is twenty-two per cent worse, and the picture shows why: every protrusion of the lower row meets a protrusion of the upper one, and the two rows are held apart by both of them together.

The four numbers are worth reading off rather than summarising, because the shape of the result is in the gaps between them. Translation gives 0.736, the half turn 0.734, the glide 0.723 and the mirror 0.575. The first three span 1.7 per cent — less than the difference between two determinations of the same crystal’s density — and the fourth is not near them at all. The measurement is not a ranking with the mirror at the bottom; it is a cluster and an outlier, and that is what makes the argument a filter rather than a preference. A rule that put the four operations in order would predict that structures drift towards the best of them; a rule that separates three from one predicts that the fourth is absent, which is what the survey below actually shows.

That twenty-two per cent needs saying carefully, because there are two of it. The mirror row reaches 78 per cent of the density the translation row does, so it is 22 per cent less dense; the translation row is 28 per cent denser than the mirror row. Both are the same pair of numbers against different denominators, and the figure’s own description now gives both rather than one, because a percentage with an unstated base is the commonest way a measurement of this kind is misquoted.

The rise is the more physical statement of the same thing. A molecule here is 0.90 units tall. Translation, half turn and glide all lift the second row by about 1.22 — a third more than one molecule, because a lumpy row is not flat on top — and the mirror lifts it by 1.56. The extra 0.34 is exactly the height of the bump, which is what “a protrusion against a protrusion” means in a number: the two rows are separated by both bumps in series instead of by one bump nesting into a hollow.

And the measurement has a control. Take the bump off the shape, leaving a plain rectangle, and the four operations give identical densities to four decimal places — in fact to eight, at 0.99740843 apiece. The preference is about interlocking and about nothing else, and a control that came back with any spread at all would mean the rows were being built differently rather than packed differently. That is the check that keeps the result from being an artefact of the procedure: the same bisection, the same four operations, one detail of the shape removed, and the whole effect goes with it.

Why a half turn works and a three-fold does not #

The half turn deserves a paragraph on its own, because it is the case where the answer is least obvious.

A half-turned neighbour is not the same shape the same way round in the sense a translated one is — it has been turned upside down. What matters is that it has not been reflected: the surface it presents across a contact is the original surface rotated, not mirrored. A protrusion at one place on the boundary maps to a protrusion at the diametrically opposite place, and since a general molecule’s bump is not diametrically opposite itself, the two miss each other and interlock.

That is why the argument needs the bump to be off centre, and the shape built for the measurement above has it off centre for exactly that reason. A shape with its bump exactly in the middle of an edge would meet its own half turn bump-to-bump, and would be a special case rather than a general molecule — which is the same reason this collection’s motif is a comma rather than a dot.

One consequence of that is worth stating because it decides how far the argument reaches. The half turn’s success is a fact about a general shape and not about every shape. Give the molecule a two-fold axis of its own perpendicular to the layer and the half turn becomes the identity on it, so the operation relates the molecule to a copy indistinguishable from a translated one and buys nothing; give it a bump exactly at the centre of an edge and the half turn puts that bump against itself. Neither case is exotic — a great many molecules are centrosymmetric — and in both the packing argument still permits the group while the interlocking it was based on has quietly stopped happening. What survives in those cases is the weaker statement that the operation does not actively hurt, which is why the rule is stated as a permission throughout.

A three-fold centre fails for a different reason and one no choice of shape repairs. Three molecules around a point at 120° leave a triangular void in the middle unless the molecule’s own outline happens to close it, and a general outline does not. The failure is not at a contact but at the centre, which is why a rosette is a worse packing than a row however the shape is chosen.

Filter the seventeen #

A plane group is a set of operations, and a molecule sitting in one is related to its neighbours by them. If any of those operations is of the kind that puts a bump against a bump, the packing pays for it.

Four of the seventeen. p1 has nothing but translations; p2 adds half turns; pg has glides; pgg has both. Every other group is refused, and for one of two reasons.

A mirror is the first reason and covers most of the refusals — pm, cm, pmm, pmg, cmm, p4m, p4g, p3m1, p31m, p6m.

A rotation of order three, four or six is the second, and it is a different failure. Such a centre arranges molecules in a rosette, and a rosette of a general shape leaves a hole in the middle of it that nothing fills. A half turn does not: two molecules related by one sit head-to-tail, which is the tightest arrangement two lumpy shapes have.

That filter has the same form as several others in this collection — walk the seventeen, look at each operation, keep the groups whose operations all pass. Two of them fold into a surface because their operations all move every point; four of them permit a close packing because their operations all interlock. Different criteria, same shape of argument, and the two lists overlap in p1 and pg without either containing the other.

Coordination six, which is what “close” means #

There is a second half to Kitaigorodskii’s argument, and it is the one that turns a preference into a classification.

A dense packing of a two-dimensional shape has each molecule touching six others — the same coordination number a packing of discs has, and for the same reason: six is what a plane allows when every neighbour contributes a contact rather than a corner. His result is that a layer with coordination six can be built for a molecule of general shape only in the four groups above, and that in the others the coordination falls to four or fewer.

That is why the filter is a filter rather than a ranking. A group with a mirror in it does not merely pack a few per cent worse; it cannot reach the coordination a close packing has, because the mirror-related contacts are the ones that fail. And a group with a three-fold centre loses contacts at the centre rather than at a face, which is a different arithmetic with the same conclusion.

One consequence is worth extracting because it runs against intuition. Symmetry is a hindrance to packing, not a help. The more operations a group has, the more of them are of the kind that puts a bump against a bump, and the worse the packing gets. p1 — the group with no symmetry whatever beyond its translations — is on the permitted list, and p6m, the most symmetric of the seventeen, is as far from it as a group can be.

That is the reverse of the relationship symmetry has to almost everything else in this collection. A more symmetric crystal has fewer independent elastic constants, fewer unique reflections, a smaller asymmetric unit and a shorter list of permitted properties; symmetry simplifies. Here it costs, and it costs because the packing question is about surfaces meeting rather than about quantities being constrained.

The same filter in space #

In three dimensions the permitted list gains the inversion — a molecule and its point reflection meet head-to-tail, exactly as two molecules related by a half turn do in the plane — and the half turn becomes the two-fold screw. What is refused is what was refused before: mirrors, and rotations with a fixed axis.

The list of four is not the list of five parallelohedra, and the two are worth separating. The shapes that fill the plane by translation alone are a classification of tiles; this is a classification of groups, and the object being packed is not required to tile anything. A molecular crystal has gaps in it — a packing fraction of about seventy per cent is normal — and the argument is about which arrangement wastes least, not about which wastes nothing.

What the literature does #

Here is where the argument earns the essay, because there is a survey to check it against.

About a third of all published organic crystal structures are in P2₁/c. A quarter are in P1̅. Then P2₁2₁2₁, C2/c and P2₁, and those five together are more than eighty per cent of everything. There are two hundred and thirty groups available and five of them do nearly all the work.

Four of the five pass the filter outright. Their operations are exactly the permitted ones: P2₁/c has a screw, a glide and an inversion and nothing else; P1̅ has the inversion alone; P2₁2₁2₁ has three screws; P2₁ has one.

C2/c is the interesting one, because it does not pass the filter as stated — it has a two-fold rotation axis in it. What it also has is a layer whose own symmetry is one of the permitted four, and the rotation relates molecules in different layers rather than neighbours in contact. That is the rule in its proper form: Kitaigorodskii’s argument is about the layer, and a three-dimensional group qualifies when it contains a permitted layer, not when every one of its operations is permitted. The filter above is the strict version, and C2/c is the case that shows the difference.

The coefficient the rule was invented to explain #

The filter is the memorable half of Kitaigorodskii’s argument and it is not the half he started from. What he started from was a number, and the number is worth having because it is what makes “close packing” a measurement rather than a description.

The packing coefficient is the fraction of the cell that the molecules themselves occupy: the volume of one molecule, times how many are in the cell, divided by the cell’s volume. For organic crystals it lands between about 0.65 and 0.77, and the upper end of that range is close packing of spheres at 0.74 — which is the observation the whole argument rests on. Molecules of every shape, with no relationship to one another, fill space about as efficiently as identical spheres do.

That is not obvious and it is not required. A collection of awkwardly shaped objects could easily pack at 0.5, and a few do. The claim that they generally do not is what needs explaining, and the explanation is the interlocking above: a bump in a hollow, achieved by translation, half turn, glide or inversion, and not achieved by a mirror.

The coefficient also makes the filter testable in a way the group census cannot. A structure in a permitted group with a coefficient of 0.6 is packing badly despite having the right operations available, which says the rule is a permission and not a mechanism — the operations make dense packing possible and the molecule’s own shape decides whether it happens.

Wallach’s rule, which is this argument’s oldest consequence #

There is a prediction hiding in the filter that was made fifty years before the filter, and setting the two beside each other is the cleanest test the argument has.

A racemic crystal contains both hands of a molecule; an enantiopure one contains only one. The filter says the four best packing operations are translation, half turn, glide and inversion — and a crystal of one hand cannot use the glide or the inversion, because either would produce the other hand. So an enantiopure crystal is packing with two of the four operations available and a racemic one with all four.

A racemate should therefore pack slightly better on average, and that is Wallach’s rule, stated from measured densities in 1895 and unexplained until Kitaigorodskii. The prediction is about an average and it is weak: modern surveys find racemates denser than their enantiopure counterparts in rather more than half of the pairs where both are known, which is a real effect and not a law. That is what a permission-based argument should predict — a shifted distribution rather than an inequality — and claiming more than that from it is the standard way this kind of reasoning goes wrong.

Where the argument came from #

Kitaigorodskii published this in 1945 and expanded it into a book in 1955, and the circumstances are worth a paragraph because they are unusual for a result this collection would otherwise treat as pure geometry.

He had no database. What he had was a few hundred determined structures and a set of wooden and plaster models of molecules, which he pushed together on a table to find out which arrangements were dense. The rule above is what that exercise produced, and the survey it now agrees with — a million structures, of which four fifths sit in five groups — did not exist for another forty years.

That sequence is the opposite of the usual one in this collection. Every other count here is a theorem checked against the Tables; this one is an observation made on models, turned into a geometric argument, and then confirmed by a body of data collected for entirely different reasons. The confirmation is the strongest evidence any claim on this site has, and it is evidence rather than proof.

What the rule is and is not #

It is a permission, and the survey is a measurement. Nothing here predicts that a molecule will crystallise in P2₁/c. What is derived is that certain operations allow a dense packing and others do not; that dense packings are energetically favoured is a separate claim, about forces this collection does not compute, and the agreement between the permission and the survey is evidence for that separate claim rather than a consequence of the arithmetic.

That is the same division every symmetry permission on this site observes, and it is worth being strict about here because the agreement is so good that the temptation to call it a derivation is real.

A chiral molecule cannot use half the list anyway. A molecule with a hand cannot sit in a group containing a mirror, a glide or an inversion, because those operations would produce its mirror image — which is a different compound. So a single enantiomer is restricted to the groups with only proper operations before the packing argument is applied at all, and P2₁2₁2₁ and P2₁ are the two commonest of those. The two constraints are independent and both are visible in the survey.

The measurement here is two-dimensional and the survey is not. Everything pushed together above is a row of a plane shape, and the space groups checked against it are three-dimensional. The step between them — that a three-dimensional packing is a stack of close-packed layers, so that the layer’s group decides — is Kitaigorodskii’s and is not derived here. It is stated, and the C2/c case above is what it costs to state it loosely.

The rule is about shape, not about chemistry. Hydrogen bonds, stacking of flat aromatic rings, ions of opposite charge: each imposes its own preferences and each can overrule a packing argument that treats a molecule as a lump. Kitaigorodskii’s rule works as well as it does because for most molecules the lumpiness dominates, and it works least well for exactly the molecules a chemist finds most interesting.

Nor does it say anything about how many molecules are in the cell. A structure with two molecules in the asymmetric unit and one with a half — a molecule sitting on an inversion centre — are both common, and both are consistent with the rule. Whether a molecule sits on a special position is a question about the molecule’s own symmetry against the group’s, which is Wyckoff’s arithmetic rather than Kitaigorodskii’s, and the two constraints act at the same time on the same structure.

And the same constant appears elsewhere in this collection. Kitaigorodskii’s other measurement — that molecular crystals hold about one non-hydrogen atom per eighteen cubic ångströms whatever the molecule — is the packing argument’s numerical shadow, and it is what makes the count of unknowns against observations computable before any data exist. A rule about which operations interlock and a constant about how densely matter packs are the same observation measured two ways.