Crystallography & Symmetry
patterns derived from their groups
Into space

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

Assumes Sixteen candidates, ten groups and The half of a translation that is not a choice.

15 min read 7 figures Exactly this manyNo origin removes itSymmetry is decidable

Quartz comes in two forms. They have the same chemistry, the same density, the same cell dimensions and the same diffraction pattern to within the precision of an ordinary measurement. One rotates the plane of polarised light to the left and the other to the right, and no amount of turning a crystal of one will make it into the other.

The symmetry groups of the two are called P3₁21 and P3₂21. They are counted as two of the two hundred and thirty, and as one of the two hundred and nineteen.

2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.
Fig. 1 The whole of the difference. One threefold screw advances a third of a cell per turn and the other advances two thirds, which is the same helix wound the other way. There is no rotation of space that turns the left picture into the right one, and there is a reflection that does.

The test, and what it has to be careful about

A group is chiral — a word this site has already used of a motif, meaning the same thing one dimension down. The crystallographic word is Sohncke, after Leonhard Sohncke who first listed the sixty-five of them — when every one of its operations preserves handedness. In matrix terms every linear part has determinant +1: rotations and screws and translations, and no mirror, glide, inversion or rotoinversion anywhere.

Only a chiral group can be one of a pair, because a group containing a mirror contains its own reflection already. This is conjugation asking its usual question — when are two subgroups the same subgroup seen differently — with the transformations restricted to those a lattice permits.

Being chiral is not sufficient. Most chiral groups are their own mirror image: reflect P2₁ and what comes back is P2₁, because a 2₁ screw wound the other way is still a 2₁ screw — half of the way up is half of the way up whichever direction the turn goes. The pairs are the chiral groups where the reflection produces something genuinely different.

The chiral groups among 22, by determinant. 22 space groups, each asked whether any of its operations reverses a hand — which is whether any linear part has determinant −1. 19 of them are chiral, the property crystallographers call Sohncke, and they are the only groups an enantiomorphic pair can be drawn from, since a group containing a mirror contains its own reflection already. Being chiral is not enough: the right-hand column applies the second condition, that the group carry one of the eight screws whose rise is not one half. 7 of the groups here pass both, and the screws each group contains are read off its own operations rather than off its symbol.
Fig. 2 The population the question is asked of, decided rather than listed. Each group’s linear parts are multiplied out and their determinants taken: all +1 and the group is chiral, one −1 anywhere and it is not. The right-hand column applies the second condition, which is that the group carry one of the eight screws whose rise is not one half — read off its own operations, not its symbol. A group failing either test cannot be half of a pair, and the two tests fail for quite different reasons.

Computing the reflection needs one piece of care, and getting it wrong is easy in a specific way this site got wrong first.

The obvious matrix to reflect with is diag(−1, 1, 1) — flip the a axis, leave the others. It is improper, its determinant is −1, and it is the first thing anybody writes.

It is not a symmetry of a hexagonal lattice, and it is not a symmetry of a monoclinic one. A hexagonal cell has its a and b axes at 120°; negate one and the angle becomes 60°, which is a different cell. So conjugating a hexagonal group by that matrix produces a group described in a cell that is not a cell of the lattice, and comparing it with the original compares two different things. Used as the mirror here, it reported P6₃ as half of an enantiomorphic pair. It is not: 6₃ is its own mirror image, for the same reason 2₁ is — three sixths one way is three sixths the other.

The matrix that always works is inversion. Its determinant in three dimensions is (−1)³ = −1, so it is improper, and it belongs to every one of the seven holohedries, so it is a symmetry of every lattice. It sends an operation (M, t) to (M, −t), which flips the sign of every rise and leaves everything else alone — exactly the geometric operation of winding every helix the other way.

Running it

Conjugate every operation of a group by the inversion, then ask whether the result is the same group at some origin. The origin has to be searched over, because the mirror image may be the original group written from somewhere else, and a comparison that assumes the origin is fixed would call a self-mirror group a pair.

And the inversion is not the only improper transformation available. A group is its own mirror image if any improper symmetry of its lattice maps it back to itself, so all of them are tried and the group is declared one of a pair only when none does. That is the difference between a search that has finished and a search that stopped at the first thing it tried.

Among the thirty-seven groups this site defines, the answer is four groups in two pairs: P3₁ with P3₂, and P4₁ with P4₃. Everything else chiral — P1, P2, P2₁, C2, P222, P2₁2₁2₁, P4, P4₂, P3, R3, P6, P6₃, P23, P2₁3, I4₁ — comes back to itself.

3 screw axes. 3 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁, 4₂, 4₃. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 1 of those drawn is its own mirror image; the rest come in left- and right-handed pairs.
Fig. 3 Why 4₂ is not in a pair and its neighbours are. Reflecting 4₁ gives a helix advancing three quarters per turn, which is 4₃; reflecting 4₃ gives 4₁. Reflecting 4₂ gives a helix advancing two quarters per turn the other way, which is the same helix, because a half turn per half cell has no handedness to reverse.

That is the general rule and it falls out of the arithmetic rather than being imposed: the reflection of n_m is n_(n−m), so a screw is its own mirror image exactly when m = n − m, which is exactly when m is half of n. The three screws satisfying that are 2₁, 4₂ and 6₃, and this site’s enumeration of the eleven reports them as the self-paired ones.

The other way to say it

There is a second description of the same fact that some readers will find more natural, and it is worth having both because they fail in different places.

A chiral group has no operation that reverses handedness. Its normaliser in the full group of rigid motions — the transformations that map the group to itself — therefore either contains an improper element or does not. If it does, the group is its own mirror image; if it does not, applying an improper transformation lands on a different group, and the two are a pair.

That formulation makes the eleven look like a statement about normalisers, which they are, and it makes the computation look harder than it is. The version this site runs is the concrete one: conjugate by each improper symmetry of the lattice, compare against the original at every origin, and report a pair when nothing matches. Both answers agree, and the concrete one has the advantage that its scope is visible — it says out loud that the transformations tried are the lattice’s own.

P3₁, in the two diagrams the Tables print. Space group P3₁, number 144, projected down c on a primitive hexagonal cell. The symmetry elements drawn: 4 3₁ screw axes, 1 3₂ screw axes. 3 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.
Fig. 4 One of the pair, in the Tables’ projection. Three general positions at heights zero, a third and two thirds, and threefold screws marked at the three positions a hexagonal cell puts them. Its partner’s diagram is this one with the heights read as their complements, which is a small enough change on paper to explain why the distinction was missed the first time anybody tried to make the list.

Where eleven comes from

The full list of pairs across the 230 is P3₁/P3₂, P3₁12/P3₂12, P3₁21/P3₂21, P4₁/P4₃, P4₁22/P4₃22, P4₁2₁2/P4₃2₁2, P6₁/P6₅, P6₂/P6₄, P6₁22/P6₅22, P6₂22/P6₄22, and P4₁32/P4₃32. Eleven pairs, twenty-two groups.

Every one is built on a screw axis whose m is not half of n: a 3₁ or 3₂, a 4₁ or 4₃, a 6₁, 6₂, 6₄ or 6₅. That is the whole source. Eight of the eleven screws are chiral and the pairs are the groups those eight can generate — subject to the group having nothing improper in it to spoil the effect.

Subtract eleven from 230 and the answer is 219, which is the number of space groups up to affine equivalence: up to any invertible linear change of coordinates, including the ones that reverse handedness. That is the number an algebraist gets asking about abstract isomorphism classes of these groups, and it is the right answer to that question. P3₁ and P3₂ are isomorphic as abstract groups; there is a bijection preserving composition, and it is the reflection.

Two hundred and thirty is the answer to a question about crystals. A crystal is a physical object with a handedness, and a grown quartz crystal is one of the two and not the other. The classification that counts them separately is the one that describes what is on the bench.

11 screws, each with its own reflection. 11 screw axes, each drawn twice: once as it is, and once with its rise negated, which is what reflecting it does. The reflection is taken by the inversion rather than by negating a cell axis, because negating a turns a hexagonal cell's 120° into 60° and is not a symmetry of that lattice at all — and used as the mirror it reports 6₃ as half of an enantiomorphic pair. The inversion sends an operation's translation to minus itself and belongs to every holohedry, so the rise m/n becomes (n − m)/n. 3 of the screws drawn come back to themselves, and those are exactly the ones whose rise is one half; the rest are left- and right-handed twins, and the computed partner is checked against the enumeration's own.
Fig. 5 The source, drawn. Each of the eleven screws appears twice in its own panel: once as it is, and once with its rise negated, which is what reflecting it does. The reflection is taken by the inversion rather than by negating a cell axis — negate a and a hexagonal cell’s 120° becomes 60°, which is not a symmetry of that lattice at all, and used as the mirror it reports 6₃ as half of a pair. Eight of the eleven come back as a different screw and three come back as themselves, and the three are exactly those whose rise is one half. The partner each panel computes is checked against the enumeration’s own pairing, so two routes have to agree before the figure is drawn.

Where it stops being about symmetry

The reason anybody outside crystallography cares is that a chiral space group is the only kind a single-handed molecule can crystallise in, and that has consequences with a body count.

A molecule with a handedness — a sugar, an amino acid, most drugs — cannot be superposed on its mirror image. If a crystal contains only one hand, then the crystal has no operation that reverses handedness, because such an operation would have to produce the other hand and there is none present. So a single-enantiomer crystal is in one of the sixty-five Sohncke groups, always, without exception, and that is a hard constraint rather than a tendency.

P2₁2₁2₁, in the two diagrams the Tables print. Space group P2₁2₁2₁, number 19, projected down c on a primitive orthorhombic cell. The symmetry elements drawn: 8 2₁ screw axes. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.
Fig. 6 The commonest of the sixty-five and the commonest space group for a chiral molecule: three mutually perpendicular 2₁ screws, no mirror, no glide, no inversion. Every operation preserves handedness, which is exactly what a crystal made of one enantiomer requires. It is also its own mirror image, so the choice of hand is entirely in the molecule and not at all in the group.

The converse gets used constantly and is worth stating precisely, because it is nearly true rather than true. Determining a structure and finding it in a Sohncke group establishes that the crystal is chiral. It does not by itself establish which hand the molecule is — the diffraction from a left-handed crystal and a right-handed one are almost identical, and separating them needs the small deviation from Friedel’s law that anomalous scattering provides. That measurement is the standard way absolute configuration is assigned, and it works because the failure of a symmetry is measurable when the symmetry is nearly exact.

Where the exactness stops

Three limits, and the middle one is a real gap in what this site computes.

The pairs here are found among thirty-seven groups, not two hundred and thirty. The same caution the opening essay gives about the 230 applies here. Two of the eleven pairs live in the table this site defines and the other nine do not, because the groups they belong to are not among the ones the essays argue about. The number eleven in this essay is from the literature; what is computed is that P3₁/P3₂ and P4₁/P4₃ are pairs and that fifteen other chiral groups are not.

The search is over the lattice’s own improper symmetries. A transformation that changed the shape of the cell — stretched one axis, sheared another — is not considered. For these systems there is no improper transformation of that kind which is not already in the holohedry, so nothing is missed here; but the scope is a choice rather than a theorem and it is stated rather than assumed.

A group being chiral is not the same as a crystal being chiral in the way a chemist means. A crystal of a non-chiral molecule can grow in a Sohncke group — the arrangement carrying a handedness the contents do not, which is the accidental symmetry question with its sign reversed — quartz is silicon dioxide, which has no handedness at all as a molecule, and the handedness is entirely in how the tetrahedra are arranged. Sodium chlorate does the same. So “chiral space group” is a statement about the arrangement and only sometimes about the contents, and the two get conflated routinely.

What the diagram cannot show, and what can

The two members of a pair have diagrams that are almost the same picture. In projection down the axis, P3₁ and P3₂ differ in whether the general positions climb by a third or by two thirds as they go round — which is three numbers printed beside three dots, and reading them the wrong way round is a mistake anybody could make once.

The axonometric view helps more than usual here, because it is the one case where the third dimension carries the entire content.

P3₁ and P3₂: one turn, two rises. One general point carried round by the screw of P3₁ and by the screw of P3₂, 2 cells of climb in each, with the turn between successive images drawn rather than the chord — a straight line between two images of a three-fold hides which way round the point went. The first climbs 1/3 of a cell per turn and the second 2/3, and both return the point to its own column after 3 turns, which is checked here rather than drawn. Both orbits occupy the same 3 heights within a cell — 0, .33, .67 — so what separates the groups is not the set of heights but which height goes with which turn: the same linear part carries a rise of .33 in one and .67 in the other, and negating every rise carries one group exactly onto the other.
Fig. 7 One general point carried round by each group’s own screw, two cells of climb apiece, with the turn between successive images drawn as an arc in the cell’s own coordinates rather than as a chord — a straight line between two images of a threefold hides which way round the point went. Both panels turn the same way, because both use the same linear part; what differs is the climb, a third of a cell per turn against two thirds. The two orbits occupy the same three heights, which is the trap: what separates the groups is not the set of heights but which height goes with which turn, and negating every rise carries one group exactly onto the other.

What no diagram of either group shows is which one a particular crystal is, because that is a question about the sample rather than about the group. Both diagrams are correct pictures of real crystals; the assignment of a diagram to a sample is a measurement.

Sixty-five, and why that number is the useful one

The eleven pairs are the headline, and for anybody working with crystals the more useful number in this essay is sixty-five.

Sixty-five of the two hundred and thirty space groups are chiral. That is the complete list of arrangements available to a crystal made of a single-handed molecule, and it is a little over a quarter of them — which means that determining a structure and finding it in one of the other hundred and sixty-five is, on its own, proof that the sample is not enantiopure.

The distribution within the sixty-five is extremely uneven, and the reason is packing rather than symmetry. P2₁2₁2₁ and P2₁ between them account for the large majority of chiral organic structures, and the essay’s own diagram of the first shows why: three perpendicular screws give four general positions with no special positions at all, which is exactly what a molecule of no particular shape wants. A group with special positions is only useful to a molecule that has the corresponding symmetry itself, and most molecules have none.

That last observation is the Wyckoff argument doing work outside its own essay: the availability of special positions constrains what can crystallise where, and a group that acts freely — with no special position anywhere — imposes nothing. The two plane groups that act freely are p1 and pg; the space groups that do are more numerous and more useful, and P2₁2₁2₁ is the most useful of them.

Choosing the wrong member of a pair

For an enantiomorphic pair the choice between the two is not a choice of hand within one group — it is a choice of group, and getting it wrong has a different signature from the ordinary absolute-structure problem.

Refine a structure in P3₁21 when the crystal is P3₂21 and the model is the mirror image of the truth. Every intensity the model predicts, in the absence of anomalous scattering, is correct: Friedel’s law makes a structure and its mirror image indistinguishable in the magnitudes, so the refinement converges, the residual is fine, and nothing complains.

What is wrong is every coordinate and the symbol above them. A published structure in the wrong member of a pair describes a substance that is the mirror image of the one measured — which for a chiral compound is the other enantiomer, a different substance with different properties.

The remedy is the one the sixty-five need generally: include the anomalous scattering, refine a parameter measuring how much of the inverted structure is present, and read it. A value near one says invert the coordinates and change the symbol, since the two go together for a pair and only the coordinates change for the other fifty-four Sohncke groups.

That is the practical content of the eleven. For the fifty-four, the group is the same either way and the hand is a property of the structure. For the eleven, the hand is in the symbol.

Proper affine, which is the equivalence 230 uses

The two numbers are described above as counts under two equivalences, and it is worth naming the first one precisely, because affine is doing more work in one of them than a reader might expect.

Two hundred and nineteen counts up to affine equivalence: any invertible linear map plus a translation. That permits stretching and shearing as well as reflecting, so a hexagonal group and a stretched copy of it are one — which is right, because the classification is of groups and not of the metrics they act on.

Two hundred and thirty counts up to affine equivalence with the determinant required to be positive. Everything else is permitted, so the difference between the two counts is exactly the maps that reverse handedness, and nothing else.

The stretching is common to both, which is worth saying because it explains why the difference is eleven and not more. A group and its mirror image always have the same lattice shape — reflection does not stretch anything — so the two counts differ only where the reflection cannot be undone by a proper map, and that happens only where a screw’s sign is at stake.

So the second number is not a coarser classification in general; it is the same classification with one bit removed, and the bit is the sign in the subscript.

Fedorov’s 229

The historical note is the best argument for taking the distinction seriously, because the first person to complete this classification got it wrong in exactly this place.

Evgraf Fedorov’s 1891 enumeration produced 229 groups. What he had done was identify one enantiomorphic pair as a single group — which is a perfectly defensible position, and it is the position that gives 219 if applied consistently. Applied once, to one pair among eleven, it gives 229, which is not the answer to any question.

Arthur Schoenflies, working independently and from a different direction, had 230 with errors of his own. The two corresponded, each found the other’s mistakes, and both published corrections. William Barlow’s independent third derivation in 1894, arriving from sphere packing rather than from algebra, is why the number has not been seriously doubted since.

The lesson is not that Fedorov was careless. It is that “the same group” is a question with more than one correct answer, and that a count is meaningless without the equivalence attached — which is the same point the enumeration of one class makes about sixteen and ten, and the same point this site’s two-colour count makes about seventy-four and forty-six. Three different places in this collection where the number depends on the question, and in every one of them the honest thing is to give both numbers and say which is which.

Where this goes next

The space groups in class mm2 (P), counted. Every way of attaching translations to the generators of mm2 (P): 64 assignments close into a group of the right size, 16 survive moving the origin, and 10 survive relabelling the axes — which is the number the International Tables record for this class. the ten primitive orthorhombic groups with a polar axis. Sixteen candidates, ten groups One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact. Why p4 cannot be drawn with dots. The same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption. The motif must be a comma A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption. A conjugacy class of p4m. One conjugacy class of p4m drawn in place: every copy of the same symmetry that the group can carry onto every other. Conjugation was applied to each of the 8 operations by each of them in turn, and the kind and order of the result was checked to match every time. The same symmetry, somewhere else Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 39 that link here.

The objects this essay names

Each one links to every other essay that touches it.

Affine equivalenceChiralityEnantiomorphHandednessScrew axisSohncke groupSpace group