The friezes inside the seventeen

A wallpaper pattern has translations in two directions. Choose one of them, take the line through the origin in that direction, and keep only those symmetries of the pattern that map that line onto itself. What is left is a group acting on a strip, so it must be one of the seven frieze groups — because the seven are all there are.

That is not a deep theorem — it follows from the frieze classification being complete — and the table it produces is more informative than the argument. Two plane groups can have the same symbol structure and different rows. Two rows of the same plane group can be different friezes. And the pair the whole notation exists to distinguish turns out to distinguish itself here too, in a way that has nothing to do with notation.

What “the group along a row” means, exactly #

The construction has to be stated carefully, because there are two plausible readings and only one of them gives a group on a strip.

An operation of the plane group either maps the chosen line onto itself, onto a line parallel to it, or onto a line in some other direction. The operations that matter are the first kind. The condition has two halves and both are needed.

The direction must survive. The linear part must send the direction vector to plus or minus itself. A fourfold rotation sends an axis to the other axis, so it drops out; a half turn sends every direction to its own reverse, so it always survives.

The line must not move sideways. An operation can preserve the direction and still slide the line across itself onto a parallel copy. A glide in pg does exactly that along one of its two directions, and the operations that do it are not symmetries of the strip — they are symmetries of the family of strips.

Both conditions are integer arithmetic in the lattice basis. Write the direction as a primitive lattice vector and complete it to a basis of the lattice; in that basis the linear part becomes upper triangular, and it reads off directly as a sign along the row, a sign across it, and an offset along it. Those three numbers are exactly the coordinates the frieze enumeration closes over, so the strip group is named by the same function that names the seven — a satisfying meeting in the middle, since one side comes from the wallpaper machinery and the other from a table on the other side of the same file.

What the table says #

Thirty-two rows, and every one of them lands on one of the seven, which is the assertion the figure makes while drawing. Three features are worth pulling out.

All seven appear. The seventeen plane groups between them realise every frieze group along a principal direction — a small completeness result, and not one that had to be true: nothing in the plane classification promises the strip classification will be exhausted by it.

p2 is the commonest, along nine rows. A half turn survives every direction, so any plane group containing one contributes p2 along any row where nothing else survives. That includes the rows of p4, p6 and pgg, all of which are richer groups than their rows suggest.

A group is not the sum of its rows. p4 has eight operations per cell and both of its rows are p2, the strip group with two. Everything that makes p4 a fourfold group tilts one axis into the other and therefore leaves no trace on either row. A reader reconstructing a plane group from its rows would build something much smaller.

The pair that swaps #

The most striking line in the table is the one the notation exists for.

p3m1 has p1m1 along the axes and p11m along the bisectors. p31m has exactly the reverse. Two groups with the same lattice, the same point group and the same number of operations, distinguished here by which of two frieze groups appears along which family of directions — measured from the operations, with no symbol involved.

This is worth dwelling on because the pair is usually presented as a subtlety of Hermann–Mauguin — a matter of which position the m sits in. It is not. The notation is reporting a difference that exists in the operations, and the row groups are an independent way of seeing the same thing: in one group a mirror runs along the lattice rows, in the other it runs across them, and no relabelling of axes turns one into the other because the lattice’s own threefold rotation does not carry one family onto the other.

The essay on the pair makes the argument from the mirror lines and the fundamental domains. This is the same fact reached from a third direction, and the agreement of three routes is worth more than any one of them.

The other pair, which does not swap #

p4m and p4g are the other awkward couple, and they behave differently.

p4m gives p2mm along both families. p4g gives p2 along the axes and p2mg along the diagonals. The asymmetry is real and it is not a mirror image of the p3m1 case: p4g is not “p4m with the families exchanged”, it is a group whose axial mirrors have been replaced by glides, and a glide along a row moves the row sideways, so it contributes nothing to the row group at all.

That is the general lesson of the table. A glide is invisible to the row it lies along, and richly visible to the row it lies across. It is the same asymmetry that makes glides awkward to spot by eye in a wallpaper pattern, expressed as a fact about subgroups.

Where the rows agree and the groups do not #

Sorting the seventeen by their pair of row groups is a cheap thing to do once the table exists, and the result says how coarse the invariant is.

Four groups share the pair (p2mm, p2mm): pmm, cmm, p4m and p6m. Their orders are four, four, eight and twelve, and their lattices are rectangular, centred, square and hexagonal — about as much variety as the classification contains — and along a principal row every one of them looks like the frieze with everything. Anything that distinguishes them acts between the rows rather than along them.

Three share (p2, p2): pgg, p4 and p6. Again the orders differ — four, four, six — and again the distinguishing operations are the ones that carry one direction into another, or, in pgg’s case, the glides that slide a row sideways.

Three share (p11m, p1m1): pm, cm and p31m. This one is the least expected of the three, because p31m is a hexagonal group with a threefold rotation and the other two have no rotation at all. Along a row, none of that is visible: what a row can see is a mirror across it and a mirror along it, and all three have exactly that.

So the row pair is a genuine invariant and a weak one. It separates the seventeen into ten classes rather than seventeen, and the classes it produces cut across lattice type, order and rotation. It is worth having for exactly the reason a weak invariant usually is — it is measurable on a fragment, where a strong one is not.

Reading a pattern along one line #

Turning that round gives a procedure, and the procedure is what somebody looking at a real surface actually does.

Pick a row of motifs. Ask three questions of it, each of which is a look rather than a construction: is there a mirror running along the row; is there a mirror running across it; is there a half turn taking the row to itself reversed? Those three answers name the row group, and the row group eliminates most of the seventeen.

Two cautions, both of which the table makes concrete. A single row is not enough, because the classes above are pairs and a pair needs both. And the order matters: p3m1 and p31m have the same two friezes and put them along different families, so the reader has to know which family was looked at first — which is the same information the notation’s positions carry, arrived at without the notation.

What the rows are good for #

Three uses, in increasing order of practicality.

Telling patterns apart. Two candidate identifications for a real pattern can often be separated by looking along a single row, which is a much smaller thing to check than a whole cell. If the row through a given point has a mirror across it, a great many groups are eliminated at once.

Reading a fragment. Ornament is frequently damaged, interrupted or bounded, and a strip of it may survive where a two-dimensional patch does not. The row group is what such a fragment can actually support, and the table says which plane groups remain possible given a row group — which is a much weaker conclusion than an identification, and an honest one.

Understanding the classification’s shape. Every plane group contains frieze groups, so the frieze classification sits inside the plane classification as a family of subgroups rather than as an easier warm-up exercise. The containment ordering among the seventeen has this as a companion: a plane group’s rows are a coarse invariant of it, and groups that share their rows are the ones a superficial look confuses.

What the round trip checked, and how #

Every row in the table is computed and then checked against something arrived at another way.

The computation collects the operations that fix a row, converts each to its class along the strip, closes the set under composition, renormalises onto the lattice the closure actually generates, and names it. That final naming is the same function the sixteen-candidate enumeration uses, so it applies the same renormalisation and inherits the same negative test: omit the rescaling and the frieze classification gives ten groups rather than seven.

Two assertions run while the figure is drawn. Every row group is one of the seven — a strip group that came out as anything else would mean the computation had left the classification, and there would be nowhere for the result to be. And p1’s rows are p1, which is the cheapest sanity check available: the plane group with no symmetry beyond translation cannot have a row with more.

The offsets are also checked rather than assumed. An operation preserving a row slides it along by a whole repeat or by half of one, and anything else — a third, a quarter — would mean the row is not a lattice direction at all. The computation asserts that every offset it meets is a whole or a half, which is the kind of assertion that never fires until a direction is chosen carelessly, and then fires immediately.

The rows of a frieze group #

The construction runs one dimension down as well, and there it degenerates in a way worth noting because it explains why the table above has two columns rather than one.

A frieze group already has a single translation direction, so “the group along a row” is the whole group. There is nothing to filter and nothing to choose. The exercise is therefore trivial for friezes and interesting for wallpaper groups, and the reason is precisely that a plane group has two directions to be asked about and a strip has one.

That is also why the two columns are not interchangeable. For the rectangular groups the two directions are genuinely different — pm has a mirror across one and nothing across the other — and swapping them changes the symbol without changing the group, which is the setting problem the notation index sets out. For the square and hexagonal groups the two families are related by the rotation, so the columns are two genuinely different questions rather than one question asked twice.

Where the exactness stops #

Two directions, not all of them. The table takes each lattice’s principal families — the axes and the diagonals, or the axes and the bisectors — because those are the directions the classification is organised around. Every other primitive lattice direction also has a row group, and for most groups it is the trivial one. A complete answer would be a function on all primitive directions rather than a table with two columns, and the table is the part worth drawing.

A row group is a subgroup, not a pattern. What has been computed is which symmetries of the plane pattern preserve a line. It is not a claim that the points along that line form a frieze pattern with that group: the row of a pattern may contain no motif at all, and a line through empty space still has a row group.

It is two-dimensional, like everything the machinery here decides. The three-dimensional analogue is the rod groups, of which there are infinitely many families, and nothing on this page bears on them.

Where the ladder goes next #

The classification these sit inside is the seventeen, and the argument that produces it one branch at a time is the classification proof.

The pair that swaps its rows is p3m1 and p31m, where the same distinction is made from the mirror lines instead.

The strip’s own classification, and the renormalisation that makes it come out at seven, is why sixteen become seven. The full table of which group is drawn where, including every row group above, is on the group index.

What the pictures here cannot show. The table is the argument and it is a table: the claim that a particular frieze group lives along a particular row is a claim about which operations survive a filter, and no drawing of a pattern displays a filter. The two wallpaper figures show where the mirrors and glides fall, which is the evidence for the claim rather than the claim itself — and a reader who wanted to verify a row would have to run the same filter by hand along it.

The same construction cuts a ribbon #

The rows in this table are lines drawn inside an infinite pattern, and the construction has an immediate physical reading: cut the pattern along one of them and keep a strip of finite width. What survives is the same group.

A ribbon has a frieze group, and it is the row group of its edge direction. The operations that survive a cut are exactly those that preserve the edge’s direction and do not move it sideways — the two conditions this essay states, arriving from a knife rather than from arithmetic.

Two cuts through the same pattern can leave different groups. A hexagonal pattern cut along a lattice row gives one frieze group; cut along a direction bisecting two rows it gives another. The material is identical, the width can be identical, and the strips are not the same object.

Graphene is the case everybody has met. A sheet of it has the symmetry of a hexagonal pattern with mirrors, and the two natural cuts through it — the one that leaves a row of atoms alternating up and down, and the one that leaves paired atoms along the edge — have different frieze groups. The two edges are known by the shapes they present, and their electronic behaviour differs sharply: one of them carries states localised at the edge that the other does not.

Symmetry does not by itself produce those states, and it does decide which are permitted. The row group is what a calculation on a ribbon has to respect, it fixes which wavevectors along the ribbon are special, and it says which edge modes can be degenerate. A ribbon of a given width is a one-dimensional crystal, and this table is the list of one-dimensional crystals a two-dimensional one can be cut into.

Rods and layers, one dimension up #

The construction generalises, and the generalisation is where the objects with unfamiliar names come from.

Take a space group and a line. Keep the operations preserving the line’s direction and not moving it sideways, and the result is a group with translations in one direction and finite extent across — a rod group. There are seventy-five of them, and they classify the symmetry of anything long and thin in a crystal: a chain of atoms, a dislocation line, a nanotube.

Take a space group and a plane. Keep the operations preserving the plane, and the result has translations in two directions and finite extent across — a layer group, of which there are eighty. That is what a cleave leaves, and the essay computing sections is this construction with the line replaced by a plane.

All three are the same arithmetic. An object of lower dimension sits inside a periodic one, and its group is the stabiliser of that object in the containing group. Seven friezes inside seventeen plane groups; seventy-five rod groups and eighty layer groups inside two hundred and thirty space groups.

And all three lists are complete for the same reason. The stabiliser is a subgroup of the containing group, it has translations only along the object, and the classification of such groups is finite — so the answer must be one of the listed groups whatever the containing group and whatever the object. The table on this page checks that in the smallest case, where every row can be computed and read.