Why it is a group and not a list

Suppose a designer sets out to make a pattern with exactly two symmetries in it: a mirror running north–south, and another running east–west. Nothing else. A perfectly reasonable design brief, and it is impossible.

Reflecting across the first mirror and then across the second turns the plane through a half turn about the point where they cross. A pattern invariant under both reflections is therefore invariant under that half turn, whether the designer wanted it or not. The two symmetries in the brief have silently become three, and the third was never chosen.

This is the property that makes the subject a subject.

Closure, stated plainly #

If $a$ and $b$ are both symmetries of a pattern, then $a$ followed by $b$ is a symmetry of that pattern.

The proof is one sentence: $a$ leaves the pattern indistinguishable from itself, and so does $b$, so doing both leaves it indistinguishable from itself. There is nothing more to it, and everything in the classification follows from it.

The consequence is that the symmetries of a pattern are not a menu from which items may be selected. They are a closed system. Put two in and the third appears; put the third together with a fourth and a fifth appears; and the process either terminates in a finite consistent collection or it runs away and contradicts something.

That is the sense in which the classification is a theorem rather than a survey. Nobody has to search the world’s ornament looking for a pattern nobody has drawn yet. The question is which closed systems are possible, and closed systems can be enumerated at a desk.

The other two conditions #

Closure is the interesting one. Two more conditions come almost free, and stating them completes the definition of a group.

There is an identity. Doing nothing is a symmetry of everything. It reads like a technicality and it is structurally necessary: composition needs a neutral element or the arithmetic does not work.

Every symmetry has an inverse. Every rigid motion can be undone by another rigid motion — turn back, slide back, flip back — and the undoing is itself a symmetry of the pattern, since it maps the pattern onto itself just as surely as the original did. Composing an operation with its inverse gives the identity, which is where the identity’s necessity becomes obvious.

Associativity holds automatically, because composing motions is composing functions and function composition is associative. So the symmetries of a pattern satisfy the four group axioms without anybody having to arrange it, and the word “group” in this subject is a description rather than a definition imposed from outside.

What forcing looks like in practice #

The abstract statement is easy to nod at. The concrete consequences are the reason the subject has any content.

Three compositions do most of the work in the plane. Two reflections in lines crossing at $\theta$ compose to a rotation through $2\theta$ about the crossing point. Two reflections in parallel lines a distance $d$ apart compose to a translation by $2d$ perpendicular to them. A rotation followed by a reflection whose mirror passes through the rotation centre is another reflection, in a line at half the rotation angle from the first.

Each of these turns a pair of chosen symmetries into a third that was not chosen. Two mirrors at 45° force a quarter turn. A mirror and a translation along it force a glide at half the translation. A half turn and a mirror force either a second mirror or a glide, depending on whether the twofold centre sits on the mirror line.

A worked case: the brief that cannot be met #

Return to the designer at the top of this essay, who wanted two perpendicular mirrors and nothing else, and follow what happens.

The two mirrors force a half turn about their crossing point. That is three operations, plus the identity: four. But a wallpaper pattern also has translations, and each mirror composed with a translation along itself gives a glide with half that translation attached — so glide axes appear, parallel to both mirrors, running between them. And each mirror composed with a translation across itself gives a second mirror, parallel to the first and half a repeat away.

So the brief’s two mirrors have become, per unit cell, four mirrors, several glide axes, and twofold centres wherever mirrors cross. The group is pmm, it has four operations modulo translations, and the designer’s original request — exactly two symmetries — was not a design that would be hard to draw. It was a description of nothing.

The general moral is worth stating in its strong form: in this subject nobody chooses a set of symmetries. A designer chooses one or two, and the algebra supplies the rest. What is actually being chosen, whether the designer knows it or not, is one of seventeen closed systems.

Generators: saying less and meaning more #

Because everything is forced, a group can be specified by a small number of its members rather than by the whole collection.

Two operations generate p4m’s eight — a quarter turn and one mirror. One operation generates p4’s four. The group p1 has no generators at all beyond the translations that every pattern has. This is not merely economical: it is the form in which every group on this site is defined, and it is the form in which the classification is proved. A branch of the argument says “suppose the highest rotation order is four and there is a mirror through the fourfold centre”, and that hypothesis is a generating set.

The compact notation for the seventeen is exactly a notation for generators, which is why Hermann–Mauguin symbols look like instructions rather than names.

Subgroups: what a pattern also is #

Closure works downward as well as upward. Inside every wallpaper group sit smaller closed systems — subgroups — and they are the reason a single pattern can honestly be described in more than one way.

Take a pattern in p4m and ignore its mirrors. What is left is closed: quarter turns compose to quarter turns and half turns, and everything has an inverse. That is p4, a subgroup of p4m of index two. Ignore the quarter turns as well and p2 remains, then p1 below that. Every wallpaper group sits at the top of a small tower of subgroups, and every group is a subgroup of something except the maximal ones.

This is not idle bookkeeping. A crystal that undergoes a phase transition typically loses symmetry rather than exchanging it arbitrarily: the low-temperature structure’s group is a subgroup of the high-temperature one’s, and which subgroup it is constrains what the transition can do to the physical properties. The same relation governs the appearance of twinning, where a crystal grows in several orientations related by an operation the low-symmetry structure has lost.

There is a converse worth noting. A pattern whose group is p1 has no symmetry beyond translation — but it is still, in every sense, a wallpaper pattern, and p1 is a perfectly respectable group. It is the one that the round trip finds hardest to certify, because almost any simple motif accidentally has more symmetry than none at all.

Where the finiteness comes from #

The classification is finite because forcing cuts both ways: it manufactures operations, and it also manufactures contradictions.

The argument runs roughly as follows. A wallpaper pattern has a lattice of translations. Any rotation in the group must map that lattice onto itself, which restricts the rotation order to one, two, three, four or six — the crystallographic restriction, and the point at which the finiteness enters. That leaves five cases for the highest rotation order present. Within each case, the possible arrangements of mirrors and glides relative to the rotation centres are enumerated, and forcing removes most candidates by showing that a proposed arrangement generates an operation inconsistent with the lattice, or generates an arrangement already counted.

Five rotation cases, a handful of mirror configurations in each, and forcing to prune: seventeen survive. The essay on the seventeen walks the branches.

It is worth pausing on how unusual that outcome is. A classification problem in mathematics has no general right to terminate. The finite simple groups were classified only after several decades and some ten thousand pages, and the answer includes twenty-six sporadic cases that fit no family. The plane groups, by contrast, fall out of an afternoon’s careful case analysis, and the answer is a number small enough to print on one page. The reason is the crystallographic restriction: it converts an infinite space of candidate rotations into five, and everything downstream inherits that finiteness.

The important structural point is that this argument could not be run at all if symmetries were a list. A list has no rules about which items may sit beside which. A group does, and the rules are what make an exhaustive search possible.

The point group, and what it forgets #

There is a second group hiding inside every wallpaper group, and it is often the more useful one.

Take every operation in the group and throw away its translation part, keeping only the rotation-or-reflection part. What remains is the point group: a finite group describing the pattern’s symmetry directions without reference to where anything sits. For p4, p4m and p4g the point groups are respectively the cyclic group of order four and the dihedral group of order eight, twice.

That immediately explains something that puzzles readers first meeting the seventeen: why p4m and p4g are different groups despite having the same operations available. The point groups are identical. What differs is where the mirrors sit relative to the fourfold centres — through them in p4m, between them in p4g — and that positional information is exactly what the point group discards.

Why the number is not thirty-two, or five #

A natural guess is that the number of wallpaper groups should equal the number of point groups compatible with a lattice, which in the plane is ten. It does not: it is seventeen. The gap is the placement information, plus the glides.

Ten point groups, each combined with the lattice types it is compatible with and each with its symmetry elements placed in every consistent way, gives thirteen groups whose operations are all symmorphic — meaning the group can be assembled by choosing an origin at which every rotation and reflection sits with no translation attached. The remaining four are non-symmorphic: pg, pgg, pmg and p4g, whose defining operations are glides that cannot be reduced to a mirror at any choice of origin.

Those four are the ones ornamentalists working by eye tended to conflate with their symmorphic neighbours, and they are the ones the round trip catches when a figure claims one and shows the other. The same distinction, scaled up, splits the two hundred and thirty space groups into seventy-three symmorphic ones and a hundred and fifty-seven that are not.

The group is not the pattern #

One clarification, because conflating these two things causes real errors.

A group is an abstract object: a set of operations with a composition rule. A pattern is a set of points. Many different patterns share a group — every pattern in p4m has the same eight operations per cell, whatever its motif looks like — and the relationship runs one way. The group determines what symmetries a pattern has; it does not determine the pattern.

This is why generating a pattern from its group requires a choice of motif, and why that choice is not innocent. Feed the group a motif with symmetry of its own and the resulting pattern has more symmetry than the group used to make it. The group was applied faithfully; the picture is nonetheless of something else.

Who noticed, and when #

The word “group” arrived in mathematics through Évariste Galois in the 1830s, in the entirely different context of the solvability of polynomial equations. Its application to symmetry came later and from several directions at once: Camille Jordan’s 1869 attempt at classifying the motions of space, Felix Klein’s Erlangen programme of 1872 proposing that every geometry is the study of what a group of transformations leaves unchanged, and Evgraf Fedorov’s derivation of the plane and space groups in 1891.

Klein’s formulation is the one that reframed everything. Geometry had been the study of figures; after 1872 it was the study of invariants under a group, and symmetry stopped being a decorative property of certain figures and became the organising principle of the whole field. Crystallography arrived at the same conclusion from the other end, having started with real minerals and worked backwards to the algebra that constrains them.

The two traditions barely spoke to one another for decades, which produced the subject’s most irritating legacy: two complete and incompatible notations for the same groups. Crystallographers use Hermann–Mauguin symbols, which describe generators; chemists and spectroscopists use Schoenflies symbols, which describe abstract group types; and orbifold notation, introduced by John Conway in the 1980s, describes the quotient surface and is by some distance the easiest of the three to learn. All three name the same seventeen objects. A reader meeting the subject through more than one source will meet all three, usually without being warned that this is what is happening.

Where the ladder goes next #

The immediate next step is to use the group rather than describe it: a pattern is the orbit of a motif, which is what makes every figure on this site something grown rather than drawn.

After that, the constraint that supplies the finiteness. Groups of plane symmetries are limited by the lattice they must be compatible with, lattices come in five kinds, and the rotations a lattice tolerates are five in number by an argument one line long.

What the pictures here cannot show. Composition is a claim about operations, and an operation is not visible — only its effect on a motif is. The figures on this page show the effect and mark the element; the claim that the composed operation was already a member of the group is checked by comparing operations, not by comparing pictures.

Which briefs can be met #

The designer’s brief fails, and it is worth saying exactly which briefs succeed, because the answer is a complete one and it is short.

A brief is a request for a set of symmetries. It can be met exactly when that set is closed under composition and inverses — when it is already a group. Otherwise the pattern will have the requested symmetries and more, and the extras are not negotiable.

So “two perpendicular mirrors and nothing else” fails, since the composite is a half turn and the half turn is not on the list. “One mirror and nothing else” succeeds: the composite of the mirror with itself is the identity, which is present, and there is nothing further to force.

“Two parallel mirrors and nothing else” fails differently. Their composite is a translation by twice the distance between them, and composing that translation with itself gives another, so the brief forces infinitely many operations rather than one. A designer asking for two parallel mirrors has asked for a frieze.

The general answer is therefore a list this collection already has. The sets a plane pattern’s symmetries can form are the seventeen plane groups, the seven friezes, the finite groups of a rosette, and their subgroups — and any brief that is not one of those is not a brief about a pattern. Checking a brief is checking closure, which takes a few compositions.

Nearly a symmetry is not a symmetry #

There is a reason to insist on the exactness of the closure argument, and it appears in the one part of this subject where the exactness is unavailable.

Closure does not survive approximation. Suppose two operations each carry the pattern onto itself to within some small error. Compose them and the errors add: the composite is a symmetry to within twice the error. Compose again and it is three times, and a group built from generators of a fixed accuracy has elements of unbounded inaccuracy.

So “approximately symmetric” describes nothing that behaves like a group. There is no closed set, no forcing, no classification — and in particular no argument of the kind that gives seventeen, because that argument runs by composing operations repeatedly and reading what is forced.

This is why an aperiodic pattern needed a new framework rather than a tolerance. A quasicrystal is not a crystal whose translations are slightly wrong; it has no translations at all, and describing it as nearly periodic makes every argument in this essay unavailable. What replaced them was a construction in which exactness is restored one dimension up — a cut through a lattice in a higher-dimensional space, where the symmetries really do compose.

The moral generalises past this subject. A property preserved under composition can be reasoned about by composing; a property preserved only approximately cannot, however small the approximation. Closure is not a technical condition attached to the definition of a group. It is the whole of what makes the definition worth having.