How chiral, as a number

The groups a single hand may sit in answers the question a group can answer. A set of points either admits an improper operation — a reflection, an inversion, a rotoinversion — or it does not, and if it does not it is chiral. That is a bit, and a bit is the most any classification can ever return.

For most of the questions chirality is asked about, a bit is not enough. Two molecules are both chiral; one is a nearly symmetric thing with a substituent out of place and the other is a helix. Whether a racemate can be separated by crystallisation, how strongly a substance rotates light, whether a synthesis has produced anything worth measuring — all of these are questions about how much, and a classification that returns the same word for both cases has not begun to answer them.

So: a number. And the useful thing about asking for one on this site is that it can be computed exactly rather than sampled, and that its answer can be checked against the classification it is meant to refine.

The distance to the nearest symmetric set #

The construction is Avnir’s continuous symmetry measure, and its idea takes one sentence. Take a set of k points, centre it, and ask for the nearest set that does have a mirror:

S = ( Σ |P_i − Q_i|² ) / ( Σ |P_i|² ), minimised over every mirror-symmetric Q.

The denominator makes it scale-free, the centring makes it translation-free, and minimising over the mirror’s orientation makes it rotation-free. It is zero exactly when the set already has a mirror. And it is a distance, so unlike a verdict it can be compared, ordered and plotted.

The definition is easy. Computing it looks as though it should not be, because it is a minimisation over all symmetric configurations — an infinite set — and that is where most treatments reach for an optimiser.

Why there is nothing to optimise #

Three observations collapse the whole thing to a finite list of sums, and it is worth going through them because each one removes a search that a general-purpose optimiser would have had to do badly.

Fix the mirror and the pairing, and the nearest symmetric set is written down. A mirror sends each point to some point, so it induces a pairing; given that pairing and that mirror line, the closest symmetric arrangement replaces each point by its average with the reflection of its partner. There is nothing to search.

The pairing must be an involution. A reflection is its own inverse, so the permutation it induces on the labels must be too: every point is either fixed by the mirror or swapped with exactly one other. This is a constraint rather than an optimisation, and dropping it is the standard mistake — minimising over all k! permutations returns a smaller number that is the distance to nothing at all. For four points, fourteen of the twenty-four permutations are not involutions, and the best of them gives 0.18 where the measure is 0.005.

And the best line has a closed form. Write the points as complex numbers. Reflection in the line at angle θ is z ↦ e^2iθ z̄, so the residual for a pairing π is

R(θ) = ¼ Σ |p_i − e^2iθ p̄_π(i)|² = ½ ( D − Re( e^2iθ C̄ ) )

with D = Σ|p_i|² and C = Σ p_i p_π(i). Only one term depends on the angle and it is a cosine, so the minimum is at 2θ = arg C and its value is (D − |C|)/2 — the angle is the argument of a complex number, not the end of a sweep.

What is left is

S = min over involutions π of ½ ( 1 − |Σ p_i p_π(i)| / Σ |p_i|² )

a minimum over a finite enumerable list, each entry one sum. There is no grid anywhere in it, which is why a set with a mirror comes back at exactly zero rather than at the smallest number a sweep happened to reach.

The closed form also hands over a bound nobody was looking for. |C| ≥ 0, so S ≤ ½ for every set whatever, with equality only when the best pairing makes the sum vanish exactly. A measure with a ceiling is a measure whose values can be compared across sets of different sizes without further normalisation.

The check the classification supplies #

A new quantity claiming to refine an old one has to agree with the old one where the old one speaks, and here the check is unusually clean because the two computations share nothing.

Take one asymmetric motif of three points and apply each plane point group to it. The classification says the orbit is chiral exactly when the group contains no reflection. The measure knows nothing about groups — only about distances between arrangements — and it must return zero for the groups with a mirror and something positive for the rest.

It does, and the zeros are exact — a difference of identical numbers rather than a small residue.

The motif has to have three points and the reason is this site’s own. Any two points have a mirror: the line through them, and their perpendicular bisector. So a two-point motif makes every orbit achiral and turns the whole check vacuous. That is the motif must be a comma again, one dimension down — the same failure, where a probe too symmetric to distinguish the cases reports success on all of them. The first version of this check used two points, reported zero for every group, and would have passed a test that asked only whether the mirror groups gave zero.

What the measure adds, and where it stops agreeing #

Beyond the zero the two quantities part company, and the parting is the point.

Every motif in that table is chiral. A classification says so about all six and says nothing more. The measure spreads them from 0.042 down to 0.00002 — a factor of two thousand — and the ordering is not arbitrary: the top of the table is a triangle with no two sides nearly equal, and the bottom is a quadrilateral that is very nearly a parallelogram.

Chirality has no sign, and handedness does #

The commonest error about all of this is to treat “how chiral” and “how strongly left-handed” as the same quantity, and the measure refuses to support that reading in the plainest possible way: a set and its own mirror image have the same measure. Both are equally far from the nearest symmetric set, because the nearest symmetric set to one is the mirror image of the nearest symmetric set to the other. The measure cannot tell the hands apart and is not supposed to.

Handedness is the other quantity and it does have a sign. Computed here as the oriented area of the polygon through the points, normalised by the spread, it changes sign under reflection and is zero only for a degenerate set. It is the same quantity one hand only colours its orbits by, and the same one an experiment cannot see without an anomalous signal.

The table above has a case that separates them about as sharply as anything could. The irregular quadrilateral has the largest handedness in the table and the smallest measure. It is as one-handed as anything there and almost not chiral at all — nearly a parallelogram, so nearly symmetric, but with a definite sense of circulation that is large because the points are spread and ordered.

That is not a curiosity about a contrived example. It is the reason optical rotation and chirality are not in proportion: a molecule can be strongly dissymmetric in the sense that decides the sign of a rotation while being close to a symmetric structure in the sense that decides its size, and the two are different measurements of different things.

What the ladder shows about the two ends #

The sliding triangle is worth one more look, because both of its zeros are exact and they are not the same zero.

At the left end the third vertex is directly above the midpoint of the other two, and the mirror is the vertical line. At the right end the third vertex is as far from the right-hand one as the two fixed vertices are from each other, so the triangle is isoceles about a line at an angle — a different mirror, in a different place. The measure is exactly zero at both, and the mirror line jumps between them: it is not one symmetry weakening and recovering, it is one symmetry lost and another gained.

That is why the family has to pass through chirality to get from one end to the other. A continuous path between two mirror-symmetric arrangements whose mirrors are in different places cannot stay symmetric throughout, because the symmetry it would have to keep is not the same symmetry at the two ends. The peak in the middle is where the two candidate mirrors are equally bad, and the measure at that point — 0.042 — is the smallest distance to any symmetric arrangement, not to either of the two the ends supply.

The handedness over the same family behaves entirely differently, and the comparison is drawn faintly on the same axes. It does not vanish at either end. An isoceles triangle traversed in a fixed order has a perfectly good oriented area; it simply also has a mirror, which is a statement about the set rather than about the ordering. A quantity that depends on how the points are listed and a quantity that does not are different kinds of object, and the second is the one a symmetry argument can ever be about.

Where the exactness stops #

Computed here: for each set of points, the centroid, the involutions of its labels, the complex sum for each, the closed-form angle and residual, and the minimum over the list. Also the oriented area, as the separate signed quantity.

The plane, and only the plane. Everything above is two-dimensional, and chirality is a statement about a set and the space it sits in. 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. So none of these numbers transfers to a molecule without redoing the construction in three dimensions, where the closed form for the best mirror is a rotation problem rather than an argument of a complex number — the same idea, more arithmetic, and this site’s machinery does not do it.

The number of points is what bounds it. Involutions of n labels grow very fast: twelve points give a hundred and forty thousand pairings, sixteen give forty-six million. Every orbit drawn above is twelve points or fewer, and that bound decides which groups appear rather than any argument about them. A measure for a patch of a wallpaper pattern is out of reach by this route, and getting there needs a different algorithm rather than a faster machine.

And it measures distance to a mirror, not to symmetry in general. Achirality in the plane means having some improper operation, and in the plane the improper operations are the reflections and the glides — but a finite set has no translations, so its improper operations are reflections and nothing else. That makes the restriction exact here and it would not be in a periodic setting, where a set can be carried onto itself by a glide with no mirror anywhere.

What a measure is for #

There is a general point underneath, and this collection has met it before from the other side.

Near-symmetry, and the tolerance that is not here argues that this site’s detector is exact and needs no tolerance, because a pattern either has a symmetry or it does not, and that introducing a tolerance would replace a decidable question by an arbitrary one. That argument stands. What this essay adds is that there is a second question, not a softened version of the first: not does it have a mirror, to within ε but how far is it from having one, which is a well-posed question with an exact answer and no parameter in it.

The two are different in a way worth being precise about. A tolerance turns a yes-or-no question into a yes-or-no question with a knob on it, and the knob decides the answer. A distance replaces the question with a different one whose answer is a number, and there is no knob. The first is a compromise; the second is a measurement.

Fifteen may rotate light is where the distinction earns its keep: eleven of the thirty-two classes are chiral and fifteen may rotate the plane of polarisation, and neither count says anything about how much a given crystal rotates it. Symmetry decides which effects are permitted; it never decides their size. A continuous measure is what sits in the gap, and it is worth having precisely because it does not pretend to be a symmetry argument.

Who worked it out #

The measure is David Avnir’s, with Hagit Zabrodsky and Shmuel Peleg, from the early 1990s, and the folding–unfolding algorithm they gave for it is the closed form above written as a geometric recipe. The chirality measure is the case where the symmetry asked for is a reflection, and the same construction with a rotation asked for gives a continuous measure of any symmetry at all.

The idea that chirality should be a matter of degree is older and was resisted for a long time, on the reasonable ground that chirality is a group-theoretic property and group-theoretic properties do not come in degrees. Kelvin’s definition of 1904 — a figure is chiral if it cannot be brought into coincidence with its mirror image — is a bit, and it is the right bit. What changed is the recognition that the question asked about a molecule in a flask is usually not Kelvin’s.

The mathematics has one wrinkle worth recording. There is no canonical continuous chirality measure: the one above depends on choosing a distance between point sets, and other choices give other measures that order the same sets differently. What every reasonable choice agrees on is the zero, which is exactly the part the classification already knew.

Where the ladder goes next #

Back, to the classification a measure refines rather than replaces: the groups a single hand may sit in, and the sixty-five space groups a chiral molecule can crystallise in.

Sideways, to the motif this site draws everything with and the reason it has three points: the motif must be a comma, where a probe too symmetric to distinguish the cases reports success on all of them.

And to what symmetry does and does not decide about a measurement: fifteen may rotate light, and eleven are chiral, where the classification says which crystals may rotate the plane of polarisation and has nothing whatever to say about by how much.

The same measure in three dimensions #

The construction here is two-dimensional, and the case a chemist wants is three — so it is worth saying what carries over, because the answer is that everything does except the closed form’s simplicity.

The definition is unchanged. The measure is still the squared distance to the nearest set having some improper operation, normalised by the size of the set, minimised over every such operation and over every pairing.

The enumeration is larger in two ways. The improper operations of space are not only the reflections: there are the rotoinversions as well, so achirality means having any one of them and the minimisation runs over a bigger family. And the operation carries a direction as well as a type, so the search is over a sphere of orientations rather than a circle of angles.

A closed form survives, and the trick is the one that survives in every dimension. For a fixed operation and a fixed pairing, the nearest symmetric set is obtained by folding — apply the operation to each point, average each point with its partner’s image, and unfold by applying the operation back. That average is the nearest symmetric configuration, exactly, and the residual is one sum.

So the computation stays a minimisation over a finite list of pairings, with the orientation handled by a small eigenvalue problem rather than by a search. What grows is the list, since the involutions of n labels grow faster than anything, and the practical limit is a molecule of a couple of dozen atoms rather than a couple of hundred.

And the three-dimensional measure is the one with a literature. It is used to say how far a coordination polyhedron is from being an ideal octahedron, how far a molecule is from being planar, and how far a protein’s fold departs from a symmetry it nearly has — every one of them a question a group answers with a bit and a distance answers with a number.