p3m1 and p31m

Two of the seventeen have the same lattice, the same point group, the same number of operations, and the same list of what those operations are. They are nevertheless different groups, and no change of coordinates converts one into the other.

The difference is entirely positional, and this pair is the best argument in the subject for taking position seriously.

What is identical #

Being precise about the sameness makes the difference sharper.

Both groups sit on a hexagonal lattice, and neither could sit anywhere else, since threefold rotation requires it.

Both have six operations per cell: the identity, two threefold rotations, and three reflections.

Both have the same point group — the dihedral group of order six, written $3m$ in crystallographic notation. Strip the translation part from every operation in either group and the same abstract group of order six remains.

Both are symmorphic: there is a choice of origin at which every rotation and reflection sits with no translation attached.

Any classification that looks only at which operations exist will merge them. The seventeen are not classified that way, and the reason is this pair.

What differs #

The threefold centres of a hexagonal pattern sit at three inequivalent positions per cell: the corners, and the two points at one-third and two-thirds along the long diagonal. The mirrors, meanwhile, can run in two families of directions — along the lattice axes, or bisecting the angles between them.

In p3m1 the mirrors run in the family that passes through all three kinds of threefold centre. Every threefold centre lies on a mirror.

In p31m the mirrors run in the other family. One kind of threefold centre lies on a mirror; the other two do not.

That is the whole difference, and it is not reconcilable by relabelling. Rotating the coordinate system by $30°$ swaps the two families of directions, but it also rotates the lattice — which is not preserved by a $30°$ turn, since the hexagonal lattice’s holohedry contains rotations by multiples of $60°$ and not by $30°$. There is no operation available that exchanges the families while leaving the lattice alone.

The threefold centres, counted #

The argument depends on there being more than one kind of threefold centre, so it is worth establishing that carefully.

Place a hexagonal lattice with points at the cell corners. A threefold rotation about a corner maps the lattice onto itself, so every corner is a threefold centre. But so are two further points per cell: the positions one third and two thirds of the way along the long diagonal, at fractional coordinates $(1/3, 2/3)$ and $(2/3, 1/3)$. Rotating about either of those by $120°$ permutes the lattice points among themselves without moving that point.

So a p3 pattern has three inequivalent threefold centres per cell — inequivalent meaning that no operation of the group carries one onto another. That is not a fact about the drawing; it follows from the arithmetic of the hexagonal lattice, and it is why the classification has room for two distinct arrangements of mirrors.

A mirror family running along the axes passes through all three kinds. A mirror family running in the bisecting directions passes through one kind and misses the other two. There are exactly two families available on a hexagonal lattice, so there are exactly two groups, and the classification has no room for a third.

Site symmetry, which is the sharp way to say it #

The cleanest statement of the difference uses site symmetry: the group of operations that fix a given point.

In p3m1, a threefold centre has site symmetry $3m$ — the threefold rotation and the mirrors through it, six operations fixing that point.

In p31m, two of the three kinds of threefold centre have site symmetry $3$ — the rotation alone, three operations. Only one kind has the full $3m$.

Site symmetries are properties of the pattern that survive any change of coordinates, so a difference in site symmetry is a genuine difference between groups — in the same way that the choice of unit cell is a matter of description while the lattice type is not. That is the argument in its most portable form, and it generalises immediately: the way to tell two apparently identical groups apart is to compare the site symmetries of their special positions.

What the symbols are saying #

The Hermann–Mauguin symbols encode exactly this, which is what makes them worth learning to read.

Both symbols have three positions after the lattice letter: the rotation, then the first family of directions, then the second.

p3m1 is: primitive cell, threefold rotation, a mirror perpendicular to the first family, nothing in the second.

p31m is: primitive cell, threefold rotation, nothing in the first family, a mirror perpendicular to the second.

The digit 1 is not padding. It is the notation saying “nothing here”, and its position is the entire content of the distinction. A reader who mentally deletes the 1 as decoration has deleted the difference between two groups, which is a good demonstration that terse notation is not the same as arbitrary notation.

The same pair at fourfold #

The threefold case is not an isolated curiosity. The identical structure appears on the square lattice.

There is one asymmetry between the two pairs. p3m1 and p31m are both symmorphic; p4m is symmorphic and p4g is not, because in p4g no choice of origin removes the translation part from every operation simultaneously. So the fourfold pair differs in a second respect as well, which makes it slightly easier to tell apart and slightly less illustrative of the pure point being made here.

Two more ways to see the difference #

Neither of the following is a new fact. Both are the same distinction wearing different clothes, and different readers find different clothes convincing.

By fundamental domain. A fundamental domain is a patch of the plane whose images under the group tile the whole of it exactly once. For p3 the domain is a third of a cell. Adding mirrors halves it, so both p3m1 and p31m have a domain of one sixth of a cell — but the domains are different shapes, because the mirror lines that cut them run in different directions. Draw the two domains side by side and the groups are visibly distinct, without any need to talk about site symmetry.

By orbifold. Fold the plane up along its own symmetries and what remains is a small surface with marked points, and the two groups fold to different surfaces. In orbifold notation p3m1 is *333 and p31m is 3*3. The first says three mirror corners; the second says a threefold cone point sitting beside a mirror. Two different surfaces, so two different groups, and no calculation required beyond reading the symbols.

The orbifold reading is the one that most readers find clarifying, and it is a good illustration of a general point about notation: three schemes name these seventeen groups, and each makes a different fact obvious. Hermann–Mauguin makes the directions obvious. Schoenflies makes the abstract type obvious. Orbifold makes the distinctness obvious, which for this pair is exactly the thing in question.

Telling them apart in practice #

Confronted with an unlabelled pattern in one of these groups, the question to ask is not “are there mirrors” — both have them — but “do the mirrors pass through the rotation centres”.

That is a question about coincidence of position, and by eye it is genuinely hard. The threefold centres are not marked in a real pattern; they have to be located from the motif, and locating them requires already knowing which points the rotation fixes. A reader who guesses the centres slightly wrong will read the mirrors as passing through them when they do not.

The computational version of the question is trivial. Enumerate the operations, find the rotations, compute their fixed points, find the reflections, compute their axes, and ask whether any fixed point lies on any axis. Two lists of exact rationals and a comparison — no tolerance anywhere.

Why a classification bothers #

It is fair to ask whether a distinction this fine earns its place in a list of seventeen, and the answer is that it earns it physically rather than aesthetically.

Site symmetry determines what can sit where. An atom at a site with symmetry $3m$ must have a local environment invariant under those six operations; an atom at a site with symmetry $3$ need only be invariant under three. So the two groups permit different structures, and a material in one cannot be described by the other.

The consequences reach into properties. Site symmetry governs which vibrational modes are active in infrared and Raman spectra, whether an atom may carry an electric dipole, and which components of a tensor property are forced to vanish. Neumann’s principle — that the symmetry of any physical property must include the symmetry of the crystal — turns a positional distinction into a prediction about measurements.

A classification that merged p3m1 and p31m would predict the same spectrum for materials that give different ones. That is a sufficient reason to keep them apart.

Where the pair shows up in real materials #

The distinction is not confined to ornament. Its three-dimensional descendants govern real structures, and the difference is measurable.

The layered transition-metal dichalcogenides — molybdenum disulfide and its relatives — occur in polytypes whose stacking arrangements realise different site symmetries for the metal atom. Whether the metal sits at a site with a mirror through it or beside one determines which vibrational modes are Raman-active, and the resulting spectra differ visibly. A sample can be assigned to one polytype rather than another by counting peaks.

The general mechanism is Neumann’s principle: the symmetry group of any physical property of a crystal must contain the symmetry group of the crystal. A property that would be changed by an operation the crystal has cannot exist in that crystal. So the placement of a mirror relative to an atom is not an abstraction — it forbids or permits specific components of specific tensors, and the permissions are what an experiment measures.

Two of the more striking consequences: a crystal with an inversion centre cannot be piezoelectric, because inversion would reverse the polarisation that pressure induces while leaving the pressure alone. And a crystal whose group contains any improper operation cannot be optically active. Neither statement mentions the material’s composition. Both follow from placement alone.

The distinction in three dimensions #

The same phenomenon recurs among the space groups, and more often.

Several pairs of space groups have identical point groups and identical operation counts and differ only in placement. The two hundred and thirty include, for instance, both $P\bar{3}m1$ and $P\bar{3}1m$ — the direct three-dimensional descendants of this pair — and the naming convention is the same, with a 1 marking a family of directions that carries nothing.

The larger structural fact is that the space groups are grouped into thirty-two crystal classes by their point groups, and the classes have wildly different sizes. Some contain a single space group; the class $mmm$ contains twenty-eight. All twenty-eight have the same point group, and everything distinguishing them is placement and centring.

So the seventeen’s two awkward pairs are a small preview of the situation in space, where placement does most of the classifying work.

How the pair is verified here #

Both groups are generated and round-tripped on this site, and the pair is the one place where the round trip is doing the most delicate work it does.

Generating p3m1 and p31m from their standard generators gives two sets of six operations. Applying each to the three-point asymmetric motif gives two point sets. Handing each to the detector gives back two sets of six operations, and the requirement is that each detected set matches its own generating set exactly — not merely that both have six members.

The gate goes further: it asserts that the two groups are told apart, by requiring that the pattern generated from one is not certified as the other. Since the two have equal order, a check that compared only counts would pass while the figures were swapped, and the gate exists to make sure that cannot happen quietly.

Where the exactness stops #

The distinction drawn above is exact, and the exactness has a boundary worth marking precisely, because this pair is where it is easiest to overstate.

The decidability belongs to the point set, not to the material. Deciding whether a mirror passes through a threefold centre is a comparison of two exact rationals in the lattice basis, and it is exact because the positions were given as rationals. Atomic positions in a real crystal are refined from diffraction data and arrive with uncertainties, and thermal motion means an atom is not at a position at all but distributed around one. So the experimental question is never “does this mirror pass through that centre” but “is the departure from coincidence larger than the uncertainty in the coordinates”, and that is a statistical question with a threshold in it.

Assigning a group to measured data is a decision, not a computation. The standard procedure fits the data with the higher-symmetry group and asks whether the residual improves enough to justify the extra parameters the lower-symmetry group would need. Structures have been published in the wrong one of a pair like this, corrected later when better data arrived — the literature on missed symmetry is large enough to have its own detection software, and the commonest error is exactly the one the round trip here is built against: a structure refined in a group less symmetric than the crystal actually has.

And the pair is defined only up to the lattice. p3m1 and p31m are distinguishable because the hexagonal lattice’s holohedry contains no $30°$ rotation to exchange the two mirror families. Weaken the lattice — allow the axes to be inequivalent, as a strained thin film or a distorted layer would — and the distinction does not become hard to make, it stops existing, because the group that would have to be preserved is no longer there. What is being classified is a pattern together with its lattice, and the second half of that pair is as much an assumption as a measurement.

Where the ladder goes next #

The hazard that shadows this whole essay is the accidental symmetry a motif can introduce — and for this pair it is acute, since a dot at most positions turns either group into p6m.

The wider context is the classification itself, where this pair is one of five branches, and the notation, which was designed to be able to state exactly this distinction.

And the experimental route is systematic absences, where the question of where an element sits becomes a question about which reflections vanish.

What the pictures here cannot show. Two patterns on this page differ in the position of a mirror relative to a rotation centre, and neither the mirror nor the centre is a visible feature of a pattern — both are marks placed by the detector. A reader comparing the drawings is comparing annotations, and the annotations are the computation’s output rather than the drawing’s content.

A diffraction pattern does tell them apart #

The essay ends on how hard the two are to separate by eye. A diffraction experiment separates them easily, and the mechanism is worth stating because both groups are symmorphic and so neither produces a single systematic absence.

The distinction is in the intensities rather than in the absences. The mirrors of a group relate reflections, and the two groups’ mirrors run in different directions — so they relate different pairs of reflections, and the sets of reflections required to be equal are different.

That makes the two distinguishable by the merging test. Merge a data set in the arrangement one group implies and it merges cleanly; merge it in the other and reflections that are not equal are averaged together, and the residual jumps.

The Tables record the difference as two settings of one Laue class, written 3̄m1 and 3̄1m, and choosing between them is a routine step in a trigonal structure determination rather than a subtlety. The two settings are not two conventions: they are two different sets of equalities among the intensities, and the data decide which holds.

So the pair is hard to see and easy to measure, which is the reverse of the usual situation on this site and is worth noticing. What defeats the eye is that the difference is a coincidence of positions; what an experiment reads is not positions but which reflections agree.