Reading a class off its own axes

4/mmm looks like a name. It is a sentence, and it has three clauses.

Each crystal system comes with an ordered list of symmetry directions — a first, a second and a third — and the symbol reports, for each in turn, two things: the highest-order axis lying along that direction, and whether a mirror is perpendicular to it. Nothing else. A direction carrying an axis and a perpendicular mirror is written n/m; a direction carrying only a mirror is written m; a direction carrying nothing is written 1.

Two dimensions has the same arrangement, and this site already derives all seventeen plane-group symbols that way. What three dimensions adds is a third position, the rotoinversions, and one convention that has to be stated rather than computed.

Finding the axis on a direction #

A proper rotation fixes its own axis pointwise: M d = d, where d is the direction as an integer vector in the lattice basis. So the test is one matrix–vector product and an equality of integers.

The improper operations need care, and getting them wrong loses whole positions from whole symbols. A rotoinversion is −1 times a rotation, so it reverses its own axis: 4̅ about c sends c to −c. Testing only for a fixed vector finds no axis on the primary direction of 4̅, 3̅ or 6̅, and every symbol built on one of them comes out beginning with 1.

So the test for an improper operation’s axis is M d = d or M d = −d. And that same second condition, applied to a mirror, is how the mirror is found — a mirror perpendicular to d is exactly a mirror whose normal is d, which is exactly a mirror reversing d.

Which leaves an ambiguity that the ranking resolves. On a direction carrying a two-fold rotation and a perpendicular mirror, both operations reverse or fix d, and the symbol wants the axis to be the two-fold and the mirror to be the mirror. Determinant separates them: the axis is chosen among determinant +1 operations and among the improper ones that are not mirrors, and the mirror is asked about separately.

The ranking, and what it is for #

Several operations can sit on one direction. The class 4/mmm has, along c, a four-fold rotation, its square (a two-fold), its cube, a 4̅, a 3̅… no — it has the four-fold, the two-fold, the 4̅ and the inversion’s product with them. The symbol reports the highest-ranking one, and the ranking is:

$$6 \;>\; \bar 6 \;>\; \bar 3 = 4 \;>\; 3 = \bar 4 \;>\; 2 \;>\; m$$

with the ties broken by preferring the proper axis. That ranking is not arbitrary: it orders by how much the operation constrains, and it is what makes 4/m come out of a group that also contains a two-fold on the same axis. Report the two-fold instead and the symbol is 2/m, which is a different class of half the size.

The one place the tie matters is 4 against 4̅. A group with a four-fold rotation and a perpendicular mirror contains 4̅ as their product, and the symbol is 4/m; a group with 4̅ and no four-fold rotation is 4̅, and it contains no mirror at all. Preferring the proper axis is what keeps those two apart.

The directions are the system’s, not the group’s #

The list of directions belongs to the crystal system. That is why an empty position is written 1 rather than dropped: 4/m and 4/mmm differ in what their second and third positions hold, and a notation that omitted empty positions could not say so.

Two things in that table are worth stopping on. The cubic system’s primary direction is not its highest axis — ⟨100⟩ carries the four-folds where there are any, and the three-folds that actually make the system cubic are in the second position, which is why m3̅m has its 3̅ in the middle. And the trigonal and hexagonal systems share a direction list, which is what makes the trigonal classes describable in hexagonal axes and is the setting every table prints them in.

Trailing 1s are dropped once the positions are read, which is the short form: 321 becomes 32, 3m1 becomes 3m. That is done last, so the long form is available to anything that needs to know which position was empty.

What the notation buys, and what it costs #

It is worth being explicit about why crystallography writes symbols this way rather than naming the groups after people or numbering them, because the answer is the same one that makes the derivation possible.

A Hermann–Mauguin symbol is a specification. Handed 4/mmm and the tetragonal direction list, the group can be rebuilt: put a four-fold on c, a mirror across it, two-folds on ⟨100⟩ and ⟨110⟩ with mirrors across those, and close. The symbol is the generating set. Schoenflies notation — D₄ₕ for the same class — names the abstract group and its type, which is more useful for representation theory and tells nobody where anything is.

That is why the symbol can be derived at all. A name cannot be computed from a group; a specification can be read off one.

The cost is that the symbol is opaque until the direction list is known, and the direction list is different for every system. 3m and m3̅ are not related; the first is a three-fold with mirrors through it, the second is cubic and has its mirror on ⟨100⟩ and its 3̅ on ⟨111⟩. Reading the second as “m, then 3̅” in the way the first is read as “3, then m” gets the wrong group. This is the notation drill the site’s plan names as one of its three standing risks, and the way past it is to show the derivation rather than the table.

A symbol needs a setting, and the setting is a choice #

Here is the part that cannot be computed, and saying so is the honest half of the derivation.

Turn 4̅2m an eighth of a turn about its own axis. The two-folds land where the mirrors were and the mirrors where the two-folds were. It is the same group — the same abstract group, the same crystal class, related to itself by a rotation — and it now derives as 4̅m2.

Both are real. The Tables print 4̅2m and also print 4̅m2 as a setting of it. Nothing in the group prefers one.

So the derivation has to choose, and the rule it uses is stated rather than smuggled: fill the earliest position with the highest-ranking element. Compare positions in order; within a position, an axis outranks a bare mirror and a higher order outranks a lower one. On 4̅2m that puts the two-fold second, because a two-fold outranks a mirror.

The exception, and why it is orthorhombic #

That rule gives the Tables’ symbol for thirty-one of the thirty-two. The one it does not is mm2, which it derives as 2mm.

2mm is a real setting of a real group and is not what anybody writes. The reason is that the orthorhombic system is the one with no principal axis: its three directions are a, b and c, which are interchangeable, and nothing in the group prefers one of them. So the convention cannot be “highest first”, because there is no first. The convention crystallography actually uses is that the odd direction goes last — the polar axis is c — and the class is mm2.

The exception is stated as a clause with its own name rather than worked around, and the difference matters more than it looks. A rule with a named exception is a rule; a rule quietly bent until it fits is a table with extra steps. Written the second way, nothing would record that the orthorhombic system is different, and the next person to extend the derivation would rediscover it.

It is worth being exact about which half of the convention is arbitrary. That the three orthorhombic directions carry different things is a fact about the group, and the table above computes it. Which of them is written first is not, and no reading of the matrices will produce it. A community that ordered the positions the other way would derive 2mm throughout, would have a self-consistent notation, and would disagree with every table printed since 1935. The derivation on this page is therefore two things stacked: a computation that is forced, and a single stated preference sitting on top of it, and the value of separating them is that only the second can ever be a matter of opinion.

The six classes that have a setting at all #

Only six of the thirty-two can be written more than one way, and which six is itself a small result rather than a list.

A setting question arises exactly when the secondary and tertiary positions hold different things and the group has an operation that swaps those two families of directions. If both positions hold the same token, swapping them changes nothing: 4/mmm has 2/m in both and 422 has 2 in both, so neither has an alternative. If a position is empty and the other is not, swapping produces 1-then-something, which the trailing-1 rule cannot recover.

The six are mm2 (also m2m), 4̅2m (4̅m2), 32 (312), 3m (31m), 3̅m (3̅1m) and 6̅2m (6̅m2). Four of the six are trigonal or hexagonal, and that is not a coincidence: the hexagonal direction families ⟨100⟩ and ⟨11̅0⟩ are thirty degrees apart, and a three-fold axis does not carry one family onto the other while a six-fold does. So a class built on 3 has two inequivalent ways to place its secondary elements and a class built on 6 has one.

That is the same sentence, one dimension up, as the one this site’s p3m1 and p31m essay is about — and in the plane the two are different groups rather than settings of one. The next essay is about why the answer changes.

What the derivation is checked against #

Thirty-two symbols come out. Three things are then required, and the enumeration throws if any fails:

• the thirty-two are distinct. A classification that had split a class produces two identical symbols, and one that had merged two produces thirty-one.
• the seven holohedries are among them — 1̅, 2/m, mmm, 4/mmm, 3̅m, 6/mmm, m3̅m. This is the containment the whole enumeration assumes and would otherwise never test.
• they are the thirty-two the International Tables record. That list appears in the code once, is read by nothing that produces an answer, and exists only to be compared against — the same arrangement the arithmetic-class counts use.

The third is the strongest single check in the field, because a symbol encodes where each element sits and not merely how many there are. A classification with a wrong class in it does not produce a wrong-looking symbol; it produces a symbol that is not on the list.

The third check is worth reading as a statement about the whole collection rather than about this derivation. Thirty-two symbols come out of the operations, and the thirty-two the Tables record are written down once, read by nothing that produces an answer, and compared against at the end. Every column of every plate on this site — the classes sorted into systems, the properties marked on them, the Laue classes — is built from the derived list and not from the recorded one, so an error anywhere in the derivation reaches the comparison before it reaches a reader.

The failure this check is built to catch #

The derivation is also the field’s most sensitive test, and the sensitivity is worth being concrete about, because it is what makes it worth doing at all rather than attaching names from a table.

Suppose the enumeration had a genuine error — a subgroup missed by the cyclic extension, or a merge made that should not have been. Every count downstream would still be an integer, every stereogram would still draw, every property table would still fill. The classification would be wrong and nothing about it would look wrong.

But the symbol would break, because a symbol is not a label on a class — it is a report on where that class’s elements sit. A merge that should not have happened leaves thirty-one classes and thirty-one symbols, and the missing one is a holohedry or is not; a split leaves thirty-three, two of which derive to the same string. Either way the assertion that the derived set equals the recorded set fails and names the discrepancy.

That is the same property the notation index has in two dimensions, and the gate there carries the refusal explicitly: handed the hexagonal direction families in the wrong order, p3m1 must derive as p31m and back. The three-dimensional version of that refusal is the one this field’s own gate carries — read the tetragonal secondary and tertiary in the wrong order and 4̅2m must come out as 4̅m2.

The other notation, and what it refuses to say #

There is a second naming system for these same thirty-two groups, it is in wide use, and setting the two side by side says exactly what a Hermann–Mauguin symbol is for.

Schoenflies notation names a group by its abstract structure and its generators: C₄ᵥ is a four-fold axis with vertical mirrors, D₄ₕ a four-fold axis with two-fold axes across it and a horizontal mirror, T_d the tetrahedral group with mirrors. It is older than Hermann–Mauguin, it is shorter, and it is what spectroscopists and chemists use almost universally.

What it does not carry is a direction. C₄ᵥ says there are vertical mirrors and does not say which vertical directions they sit on, because the notation has no positions in it to say so. That is not an oversight; it is the point. A molecule floating in solution has no axes to refer to, so a name that specifies orientations would be specifying something that does not exist, and the useful name is the one that identifies the group up to any rotation whatever.

The cost appears exactly where this essay’s derivation earns its keep. 3m1 and 31m are both C₃ᵥ. 4̅2m and 4̅m2 are both D_2d. The six classes with a setting question collapse in Schoenflies to one symbol apiece, because the question a setting answers — which of two inequivalent direction families holds the mirrors — is one the notation cannot phrase. In a molecule there is nothing to answer; in a crystal there is a lattice, and the two arrangements are distinguishable, which is why crystallography needed a notation with positions in it and molecular spectroscopy did not.

So the two notations are not competing spellings of one idea. Schoenflies names a group; Hermann–Mauguin names a group in a basis. One is an invariant and the other is a description, and the derivation on this page is possible only for the second — there is nothing to derive from operations if the answer is not allowed to mention where they point.

That has a practical edge worth stating for anybody reading across the two literatures. A conversion table between them exists and is one-to-many in the direction that matters: going from Hermann–Mauguin to Schoenflies throws information away and always succeeds, while going the other way requires a choice for six of the thirty-two and is therefore not a conversion at all. A paper reporting C₃ᵥ for a crystalline material has not reported which of the two arrangements it found, and the difference is measurable.

The same asymmetry runs one level up. The two hundred and thirty space groups have Schoenflies symbols too — D_2h^16 and the like — but there the notation has degenerated into a group symbol with a serial number attached, since the abstract structure no longer determines the group. That is worth noticing as a limit rather than a criticism: a naming scheme built to be basis-free stops being informative exactly when the translations start to matter, and the crystallographic notation, which was built around a basis from the start, carries straight through.

Where the exactness stops #

The derivation names classes, not space groups. A space-group symbol carries the lattice letter in front and screws and glides in the positions — P2₁/c rather than 2/m — and those come from the translations, which a point group does not have.

The setting rule is a convention and is labelled as one. It reproduces the Tables and it is not derived from anything; a different community could reasonably print 4̅m2 throughout and nothing would be wrong.

Some symbols in the wild are neither the long nor the short form. m3̅ and m3̅m drop a numeral the tetragonal analogues keep, for a reason that is real — in a cubic group the four-fold is implied by the three-folds and the mirror, and in a tetragonal one it is not — and that irregularity is written into the shortening rule as its own clause.

Where this goes #

The setting question has one case where it stops being a convention and becomes a fact about the lattice. In the plane, p3m1 and p31m are two different groups, because the lattice comes with them. As point groups, 3m1 and 31m are one class — and the two subgroups are each normal in 6/mmm, so no operation of the hexagonal holohedry carries either onto the other. What relates them is a rotation of thirty degrees, which is a symmetry of space and not of any hexagonal lattice.