Reading the symbols

Hermann–Mauguin symbols are the standard way to name a crystallographic group, and they are opaque on first meeting in a way that puts readers off the whole subject. They are also not names. A symbol is a positional description: each character says what exists in a particular direction, which is why p3m1 and p31m are different groups rather than typographic variants of one, and why pm is really a four-character symbol with two positions left out for saying nothing.

Which of those positions the mirror occupies is a matter of which axis is called first, and the derivation below is honest about it: it reports pm as p11m because that is where the mirror sits in this site's tables, where the literature more often writes p1m1. The two are the same group under a relabelling of the axes, and the short symbol is identical either way — the position carrying the choice is exactly the position the short form drops, which is the notation quietly declining to record a decision that has no content.

Every decoding on this page is computed. The group's operations are generated from its generators, each reflection is sorted into a family of directions by the direction it reverses, and the letter for that family is read off what is present — a mirror if there is one, otherwise a glide, otherwise nothing. The symbol assembled that way is then required to equal the group's standard name. Seventeen derivations, seventeen agreements; a wrong convention anywhere would surface as a symbol that failed to match, and the build would stop.

The long-form argument for why the notation is built this way, what it costs and what it cannot say is the essay on reading Hermann–Mauguin. This page is the reference that essay implies: every symbol the site uses, decoded in one place.

The three conventions everything below rests on

A letter names the direction it is perpendicular to. An m in a position means a mirror across that family of directions, so the mirror line runs at right angles to the direction the position names. This is the convention that makes the symbol useful to an experimenter, because directions are what a diffraction pattern is indexed by.

A position is a family, not a single direction. The four mirrors of p4m fall into two families of two: the fourfold rotation carries one axis onto the other, so both share a position. Each family below is generated by taking one direction and carrying it round by the lattice's own rotations, rather than by listing what is in it.

A mirror outranks a glide in the same family. Every mirror composes with a lattice translation to give a glide parallel to it, so a family containing a mirror always contains glides as well. The symbol names the mirror — which is why p3m1 is not written "p3g1" even though its glide lines are perfectly visible in any drawing of it.

The seventeen, decoded

Each row gives the short symbol as it is normally written, the full symbol with every position restored, and what each character claims — followed by the operations the derivation actually found. The mirror and glide counts are per cell.

The seven friezes

A strip has one translation direction and two others that matter — across the strip and along it — and neither can be carried onto the other by anything in the group. So a frieze symbol drops nothing except when there is nothing to say: p1m1 keeps its trailing 1, where p4mm is allowed to become p4m.

What a symbol does not fix

The origin. The symbol says which operations exist and in which directions, not where the origin sits, which is why the International Tables list more than one origin choice for many groups.

The setting. Which axis is called first is a choice, and the short symbols are deliberately insensitive to it — p1m1 and p11m both shorten to pm because the position carrying the choice is exactly the position that gets dropped. Where the two families are genuinely inequivalent, nothing may be dropped, and the symbol distinguishes two real arrangements: that is the hexagonal pair, and it is the whole reason the notation is positional rather than a list of names.

The pattern. A symbol names a group, and a group is a set of motions. Two patterns with nothing visually in common can carry the same symbol, and a drawing can have more symmetry than its recipe asked for — which is what the round trip behind every figure here is for.