One class, two names

Reading a class off its axes derives a Hermann–Mauguin symbol from a group’s own matrices: the symbol is a report on three families of directions, read in an order the lattice system fixes, and every letter of it can be computed. This essay derives the other notation from the same matrices.

The interesting thing is not that both can be done. It is that they are answers to different questions, so the two derivations share no step — and a reader who has one symbol cannot obtain the other without going back to the group.

Two questions, and what each notation answers #

Hermann–Mauguin names directions. 4/mmm says: along the principal direction there is a fourfold axis with a mirror perpendicular to it; along the second family of directions there are mirrors; along the third family there are mirrors. Which directions those families are is decided by the crystal system, and that is the whole reason the notation is crystallographic — it presupposes a lattice, because without one there is no canonical set of direction families to report on.

Schoenflies names a construction. D4h says: take a fourfold axis, add four twofold axes across it, then add a horizontal mirror. It says nothing about which directions those are, and it does not need to. That is why it travels perfectly between a molecule and a crystal, why spectroscopy and chemistry use it, and why crystallography does not.

The consequence is that neither symbol is a function of the other. 3m and C3v are the same class; so are 32 and D3; and knowing that the first symbol has an m in its second position tells nothing about whether the corresponding Schoenflies letter is v or d without asking where the mirrors sit relative to the twofold axes.

Six questions, asked in an order that matters #

The derivation is a decision procedure and its steps have to be taken in the right order. Two of them are where a naive implementation goes wrong.

First: is there more than one axis of order three or more? If so the class is one of the five cubic ones — T, Th, Td, O, Oh — and there is no principal axis to find. Asking for a principal axis first fails here, because the four threefold axes of a cubic class are equivalent and no rule picks one.

Second: which axis is principal? The obvious answer — the axis of the highest-order rotation — is wrong, and the class that shows it is 4̅2m. That class has three twofold axes and its principal one carries a fourfold rotoinversion; taking whichever twofold comes first names the class D2, which exists, has a different order, and is a different class. The principal axis is chosen here by how much of the group acts about it: four of the eight operations of 4̅2m are about the rotoinversion axis and only two about each of the others.

Then the rest. Are there n twofold axes across the principal one — D family or C family? Is there a mirror perpendicular to it — the h suffix? Are there mirrors containing it — v, or d when they bisect the twofolds? Is there a rotoreflection of order 2n and nothing else — the S family?

Reading the families off the table #

Sorted by system, the Schoenflies column falls into families that the Hermann–Mauguin column scatters, and the comparison is the quickest way to see what each notation is organised by.

The C family — C1, C2, C3, C4, C6 — is the cyclic groups: one axis, nothing else. In Hermann–Mauguin they are 1, 2, 3, 4, 6, so here the two notations agree almost perfectly, because a bare axis is both a construction and a direction.

The D family — D2, D3, D4, D6 — adds n twofolds across the principal axis, and Hermann–Mauguin writes them 222, 32, 422, 622: one numeral per direction family, which is a very different-looking string for a very simple construction.

The suffixes h, v and d are where the notations diverge most. C4v and C4h are both “a fourfold axis with mirrors”, differing in whether the mirrors contain the axis or cross it; Hermann–Mauguin writes them 4mm and 4/m, where the slash is the whole of the distinction. A reader who has not learned that the slash means perpendicular to the preceding direction cannot decode 4/m at all, and a reader who has can decode any symbol in the system.

The S family — S4 and S6 — is the one that exists only because rotoreflections are a separate kind of operation in the Schoenflies scheme. Hermann–Mauguin writes them 4̅ and 3̅, as rotoinversions, and the two descriptions of the same operations differ by a relabelling that depends on the parity of the order. That mismatch is the single most confusing thing about holding both notations at once, and it is the reason S6 and 3̅ look like different orders of the same idea and are the same group.

Perpendicular is not a property of a matrix #

The derivation needs a metric, and this is the one place it reaches beyond integer arithmetic.

“Four twofold axes perpendicular to the principal axis” is a geometric condition. The axes are integer vectors in a lattice basis, and whether two of them are at right angles depends on the metric of that basis — the same metric tensor the fourteen Bravais lattices are classified by. In a cubic basis the vectors [100] and [010] are perpendicular; in a hexagonal basis they are at 120°.

So the derivation asks the question against the metric of the class’s own basis, exactly, and never in Cartesian coordinates. That keeps it in the same arithmetic as the rest of this site: the matrices are integral, the metric is a small table of integers, and a dot product is a sum of products.

The alternative — orthogonalising the basis and asking in Cartesian coordinates — introduces irrational numbers into a decision that has no irrationality in it, and puts a tolerance where none is needed. A character does not know its basis makes the same point about a physical property computed in a lattice basis, and it applies here word for word.

Two classes with two names each #

Two of the thirty-two are written two ways in the Schoenflies literature and both are correct, which is worth stating because a derivation that produces one of them looks wrong to a reader expecting the other.

The group generated by a threefold rotoinversion is S6 to anyone thinking of it as a rotoreflection of order six, and C3i to anyone thinking of it as a threefold with a centre. Its Hermann–Mauguin symbol is 3̅ and is unambiguous. Similarly the group containing only the identity and the inversion is Ci or S2, and its Hermann–Mauguin symbol is 1̅.

The derivation here produces S6 and Ci, and the check against the standard list carries both spellings of each pair rather than declaring a house preference. That is not fastidiousness: a check that accepted only one spelling would fail a correct derivation, and a check written to accept whatever the derivation happened to produce would not be a check at all.

Where Schoenflies runs out, and where Hermann–Mauguin does #

Each notation fails where the other is at home, and the failures are informative.

Schoenflies cannot name a space group. Schoenflies did assign symbols to all two hundred and thirty — D2h¹ through D2h²⁸ and so on — but the superscript is an index into a list rather than a description: nothing about D2h¹⁶ says which glides it has or where. Hermann–Mauguin’s Pbcn says exactly which operation lies along each direction, which is why the Tables use it and why reading Hermann–Mauguin is a skill worth acquiring.

Hermann–Mauguin cannot name a molecule’s group without an arbitrary choice. A free molecule has no lattice, so there is no canonical family of directions to report on; naming ferrocene’s symmetry in Hermann–Mauguin requires inventing axes for it. Schoenflies says D5d and needs nothing else — and the group is not a crystallographic one in any case, which Hermann–Mauguin has no way to express.

Neither handles a magnetic group without an extension. The magnetic classes need a prime on the operations that reverse time, and both notations have been extended to carry it — Shubnikov’s notation is Hermann–Mauguin with primes, and the Schoenflies extension is less standard. Which magnetism a class permits uses the primed Hermann–Mauguin form.

The trap in the principal axis, in full #

The 4̅2m case is worth working through, because it is the whole reason the second question in the procedure is phrased as it is.

The class has eight operations: the identity, three twofold rotations, two fourfold rotoinversions, and two mirrors. Ask for the highest-order proper rotation and the answer is two, achieved by three different axes. Pick any of them and count the twofolds perpendicular to it: there are two, which is n for n = 2, so the class comes out as a D family; look for a horizontal mirror and there is none; look for vertical mirrors and — depending on which axis was picked — there may or may not be one. So the answer is D2 or D2d according to an arbitrary choice, and D2 is a real class of order four, which this one is not.

Choosing the axis by how much of the group acts about it settles it without a special case. About the rotoinversion axis act the identity, the twofold along it, and the two rotoinversions: four operations. About each of the other twofolds act the identity and that twofold: two. The rotoinversion axis wins, the two remaining twofolds are perpendicular to it, the mirrors contain it and bisect them, and the answer is D2d.

The general rule that emerges is that a principal axis is a property of the group’s action rather than of any single operation. That is the same lesson what a trace decides draws about naming an operation from its matrix, applied one level up: an individual matrix does not know what the group around it is doing.

Both notations name the same thirty-two objects, and neither was consulted in making them. The classes themselves come from an enumeration that knows nothing about symbols at all: the integer matrices a lattice permits are closed into groups, the groups that are conjugate in the holohedry are identified, and what is left is thirty-two. Every symbol on this page — in either notation — is then derived from one of those groups. That ordering is the reason the agreement between the two columns means anything: if the classes had been produced from a list of symbols, the symbols reproducing the list would be no evidence about anything.

What the derivation is checked against #

The thirty-two Schoenflies names are the one list from a book that appears in this computation, and they appear as the thing the derivation is checked against rather than as its source. That is the same arrangement the Hermann–Mauguin derivation uses, and the reason is worth restating: a derivation that consults a table is a lookup, and a derivation checked against a table is an argument with a test.

Three things are asserted while the table draws.

Thirty-two distinct names. A derivation that collapsed two classes onto one symbol — which is exactly what happens when 4̅2m is named D2 — produces thirty-one distinct strings, and the count is checked before anything is drawn.

Every derived name is one of the standard ones. Missing names and extra names are both reported, and both are empty.

The pairing is with the Hermann–Mauguin symbols the site already derives. The two columns of the table come from the same crystalClasses() enumeration, so a class cannot appear with one notation’s symbol and another class’s other symbol.

Which notation a reader meets first, and why it matters #

There is a practical consequence of the two-notation situation that goes beyond bookkeeping, and it is the reason this rung exists rather than a footnote.

A student of chemistry meets Schoenflies, in the context of molecular symmetry and vibrational spectroscopy, and typically never meets Hermann–Mauguin at all. A student of crystallography meets Hermann–Mauguin, in the context of space groups, and meets Schoenflies only as a curiosity. So two people can know the thirty-two classes thoroughly and be unable to talk about them, which is a real and common failure of communication in materials work.

The translation table is therefore a genuinely useful object rather than a display of thoroughness, and the fact that both halves of it are derived rather than copied means it can be extended: a class not on any table — a magnetic class, a class in another dimension — gets both names by running the same two procedures rather than by consulting anyone.

There is a second consequence in reading the literature. A paper reporting a piezoelectric measurement will name the class in Hermann–Mauguin and cite tensor components indexed by the crystal system; a paper reporting a Raman spectrum on the same material will name it in Schoenflies and label the modes by irreducible representations. They are describing the same crystal, and a reader who cannot move between the two symbols will not notice that the classes with a property in the first paper are the ones with a particular representation in the second.

What the two notations agree about, and what that says #

For all their differences, the two symbols carry the same information, and the sense in which that is true is precise: both are complete invariants of the class, so each determines the other through the class. Two classes with the same Hermann–Mauguin symbol are the same class and therefore have the same Schoenflies symbol.

What differs is which facts are readable. From 4/mmm a crystallographer reads off immediately that there are mirrors perpendicular to both the second and third direction families, which is what decides several physical properties. From D4h a spectroscopist reads off immediately that the group is the dihedral group of order eight with a horizontal mirror, which is what decides the character table and hence the selection rules.

So the notations are two projections of one object, chosen for two trades, and the reason both survive is that each is short in the cases its trade meets most. That is a better account of why a field has two notations than the usual one, which is historical accident — Schoenflies was writing about the finite groups in 1891 and Hermann and Mauguin about crystals in 1928, and both notations were fitted to what their authors needed to say.

Where the ladder goes next #

The crystal-classes anchor now has five rungs: the enumeration, the fingerprint that was not checked, what a trace decides, reading a class off its axes, and the second notation.

The rung above is character tables, which is where the Schoenflies symbol earns its keep: the irreducible representations of a class are labelled by symbols that are Schoenflies-flavoured, and the selection rules of vibrational spectroscopy are read off them. This site has stayed out of representation theory deliberately — the claim registry records a boundary with a neighbouring site about exactly this — and crossing it would need that boundary redrawn rather than merely stepped over.

The nearer rung is the space-group symbols in Schoenflies, which would show why an index into a list is not a name: the twenty-eight groups sharing the symbol D2h differ in ways nothing about the symbol expresses, and putting them beside their Hermann–Mauguin symbols is the clearest available argument for the latter.

Short symbols, and what they leave out #

The Hermann–Mauguin column holds abbreviations, and knowing what has been abbreviated is what makes the symbols readable rather than memorable.

The full symbol names every direction. For the class written 4/mmm the full form is 4/m 2/m 2/m: a fourfold axis with a mirror across it, then a twofold axis with a mirror across it in the second direction, and the same in the third.

The short form drops what is implied. A mirror containing the principal axis, composed with the fourfold rotation, forces the twofold axes across it, so writing them is redundant — and the short symbol keeps the mirrors and drops the axes. Likewise mmm is short for 2/m 2/m 2/m, where the three twofolds are forced by the three perpendicular mirrors.

Which means the short symbol can mislead about what is present. mmm names three mirrors and the class contains three twofold axes and a centre as well, none of them written. Nothing has been omitted from the group; the notation is naming a generating set rather than an inventory.

And the position of each character is the direction it refers to. That is the whole design: 4mm and 4/mmm and 422 differ in what sits in each slot, and the slots mean fixed directions decided by the crystal system. It is why a symbol has to be read against a system rather than as a word.

Where one notation stops working #

The two schemes are of comparable usefulness for the thirty-two classes, and they part company completely one level up.

Hermann–Mauguin extends to space groups directly. Put a lattice letter in front and replace each mirror by the glide it actually is, each axis by the screw it actually is, and the symbol still names directions: Pnma says a primitive lattice with an $n$-glide, a mirror and an $a$-glide in the three directions.

Schoenflies does not extend at all. Its space-group symbols are the class’s symbol with a superscript — the sixteenth group of class $D_{2h}$, written $D_{2h}^{16}$, which happens to be Pnma. The superscript is a serial number in a list, carrying no information about what the group contains.

So the choice is settled by the level of description. A molecule has a point group and no lattice, and the Schoenflies symbol names its construction compactly. A crystal has translations, and the only symbol that can describe them is the one built from directions in the first place.