Weaves as plane patterns

A draft is a pattern of filled and empty squares that tiles the plane. That makes it a periodic plane pattern, and periodic plane patterns are among the most thoroughly classified objects in mathematics.

The complication — and it turns out to be the interesting part — is that a draft’s two states are not two colours in the ordinary sense. Filled means the warp is on the face and empty means the weft is, and those are physically different things about the same cloth.

Two kinds of symmetry #

An operation that maps a draft onto itself can do one of two things.

It can preserve the states: every filled square goes to a filled square. Slide a plain weave by two ends, and that is what happens.

Or it can exchange them: every filled square goes to an empty one and every empty one to a filled square. Slide a plain weave by one end and that is what happens — the draft becomes its own complement, which is the same cloth seen from the other side.

Both are symmetries and calling them the same kind loses the whole distinction. The second says something physical: the cloth is the same from the front and from the back.

Balance decides which exist #

Here is a result that falls straight out of counting.

A draft with more filled squares than empty ones can have no state-exchanging symmetry, because such an operation would have to map a larger set onto a smaller one. So a warp-faced weave — a 3/1 twill, a satin — has none, exactly and provably.

A balanced weave, with as much warp on the face as weft, can have them and generally does. Plain weave and a 2/2 twill both do.

That is a physical statement recovered from arithmetic. A balanced cloth looks the same from both faces because its draft is its own complement under some operation; a warp-faced cloth does not, because no operation can do that job. The figure above computes it for each weave by trying every candidate operation against the repeat.

What the operations are #

The candidates are the symmetries a rectangular repeat can have, combined with translations.

Translations by any number of ends and picks, mod the repeat. The half turn about a point. Mirrors along and across. And, when the repeat is square, the transpose and the quarter turns.

The transpose is worth singling out. It exchanges the roles of ends and picks — which physically means looking at the cloth with warp and weft interchanged, as if it had been turned ninety degrees and rewoven. For a plain weave that is a state-exchanging symmetry; for a 3/1 twill it is not a symmetry at all, since the twill has three warp on the face and one weft and the transposed version has the reverse.

Where the wallpaper groups come in #

The classification of periodic plane patterns gives seventeen wallpaper groups, and allowing operations to exchange two colours gives forty-six two-colour groups. A draft is exactly such an object, so the machinery applies.

Two cautions, and they are the reason this site does not simply hand the question over.

The two states are not interchangeable colours. A pattern coloured red and blue is unchanged in meaning if the two are swapped; a draft is not, because warp-up and weft-up describe different physical situations. The mathematics of two-colour groups applies to the pattern; the interpretation of a colour swap as “the other face of the cloth” is extra.

And the interesting constraints on a draft are not symmetry constraints. What makes a weave a weave — that it hangs together, that its floats are short enough, that it can be made on a given number of shafts — has nothing to do with its symmetry group. Two drafts with the same group can be a good cloth and an impossible one.

So the symmetry is a real property, computable, and it is not the property that decides whether the draft is any use.

Running the test #

The counts in these figures are produced by trying every candidate operation against the repeat, which is worth describing because it is short and exhaustive.

For each of the eight symmetries of a square — identity, half turn, two mirrors, transpose, anti-transpose and two quarter turns — and for each translation within the repeat, apply the operation to the draft and compare the result with the original. If every cell matches, the operation is a state-preserving symmetry. If every cell is the opposite, it is a state-exchanging one. If neither, it is not a symmetry at all.

The quarter turns and the transposes are only tried when the repeat is square, since otherwise they map an $n \times m$ block onto an $m \times n$ one.

The exhaustiveness matters. A symmetry claimed by inspection is a claim about a search, and a search is exactly the kind of thing an eye does badly and a loop does well — which is the same argument this site makes about integrity and about float length.

What symmetry does decide #

Three things it genuinely controls.

Whether the cloth is reversible. A state-exchanging symmetry means the two faces are identical, which decides whether a cloth can be used either way up.

Whether the pattern has a direction. A twill has a diagonal, which is a symmetry statement: it has no mirror across the diagonal direction, so it is chiral in the plane and a garment cut from it has a right and a wrong way round. The diagonal is not a thread, and it is a symmetry property.

How the repeat can be laid out. A drafting convention has to say where the repeat starts, and a weave with a large symmetry group has many equivalent choices — which is why two sources can print different-looking drafts of the same cloth.

The satin, which is nearly symmetric #

The satin condition looks like a symmetry statement and is not quite one, which is worth untangling.

A satin’s interlacings are placed by an arithmetic progression, which gives the pattern a large translation group: sliding by the move takes each interlacing onto the next. That is a symmetry, and it is why a satin looks uniform.

But the condition that makes a satin a satin — that no diagonal forms — is not a symmetry condition. A twill has just as much translational symmetry and does form a diagonal. The distinction is about how the interlacings are distributed rather than about what maps the set onto itself, and it is closer to a question about even spacing than about groups.

That is a fair summary of the whole relationship. Symmetry describes a draft accurately and does not capture what a weaver is choosing between, and the satin theorem is an arithmetic result rather than a symmetry one.

Why a satin has so many symmetries #

One number in the figure is worth explaining because it looks anomalous.

A five-end satin comes out with more state-preserving symmetries than a plain weave has, which seems wrong for a weave with an irregular-looking scatter of marks.

The reason is the arithmetic progression. Sliding a satin by its move takes every interlacing onto the next one, so the move is a symmetry — and so is every multiple of it, which is every translation in the repeat. A satin’s interlacing set is a single orbit under translation, which is as much translational symmetry as a repeat of that size can have.

What a satin lacks is the point symmetries — the mirrors and rotations — because the progression has a direction. It is highly symmetric under translation and barely symmetric under anything else, which is a different profile from plain weave’s and produces a similar total by a different route.

That is a good illustration of why a bare count of symmetries is a poor summary. Two weaves with the same total can have completely different groups, and the classification — which is about the arithmetic of the move rather than the symmetry — is the more informative question for a satin.

The two counts are equal or one of them is zero #

There is a pattern in the figure’s numbers that the figure does not comment on, and it is not a coincidence: a plain weave has sixteen state-preserving symmetries and sixteen state-exchanging ones. Exactly the same number.

It has to be, and the argument is three lines.

Take all the operations that map the draft onto itself or onto its complement, and call that set G. It is a group: composing two of them maps the draft onto itself or its complement again, and inverses are in it. The state-preserving ones form a subgroup H, because composing two preserving operations preserves and composing two exchanging ones also preserves.

Now suppose there is at least one exchanging operation x. Then composing x with every element of H gives an exchanging operation, and every exchanging operation arises that way exactly once — because composing two exchanging operations lands in H. So the exchanging ones are a coset of H, and a coset has exactly as many elements as the subgroup.

So the split is either fifty-fifty or a hundred to nothing, and nothing else is possible. A draft has as many state-exchanging symmetries as state-preserving ones, or it has none at all. There is no draft with twelve of one and four of the other.

That is worth stating for three reasons.

It is a check on the search. The figure’s counts come from trying every candidate operation against the repeat, and a count of twelve against four would be an arithmetic impossibility rather than an unusual weave — so the two columns checking against each other is a live test of the enumeration, free, on every draft it is run on.

It sharpens the balance result. Balance is necessary for an exchanging symmetry, because such an operation maps the filled squares onto the empty ones and the two sets must be the same size. It is not sufficient — being the same size does not make one set carry to the other — so the honest statement is that a warp-faced weave has none, provably, and a balanced weave has either as many as it has preserving ones or none. Which of the two it is has to be searched for, and the ninety balanced drafts in the four-by-four census are the place to look.

And it makes reversibility a yes-or-no rather than a degree. A cloth is not slightly reversible. Either some operation carries its face to its back, in which case as many do as leave it alone, or none does. That matches the physical fact rather better than a count would: a fabric either looks the same from both sides or it does not, and there is no partial case.

Where the analogy is genuinely useful #

Two places it earns its keep.

Colour. Once coloured threads are threaded in an order, the visible pattern is a genuine two-colour plane pattern in the ordinary sense, and the machinery applies without caveat. Colour and weave takes that up, and houndstooth from a plain twill is the standard demonstration.

Design. A designer laying out a large figured repeat is doing exactly what an ornamentalist does — arranging motifs so that the repeat is not obtrusive — and the symmetry vocabulary is the right one for that job. The half-drop and brick repeats of textile design are glide symmetries by another name.

The colour orders make it literal #

The two-colour machinery applies with caveats to a draft and without any to the visible face, and the difference is worth restating.

A draft’s two states are physically different: warp on the face and weft on the face. Exchanging them is a real operation — it is the cloth seen from behind — but it is not a relabelling of arbitrary colours.

Once ends and picks are threaded in a colour order, the visible face is a two-coloured pattern in the ordinary sense. Its two colours are interchangeable by definition, its symmetry group is one of the two-colour plane groups without qualification, and the effects that result are exactly the ornamental patterns the classification was invented for.

So there are two symmetry questions about any coloured cloth and they have different answers. The draft’s symmetry says whether the fabric is reversible; the face’s symmetry says what the pattern looks like. Conflating them is easy and produces confident wrong statements about both.

What this account leaves out #

Two limits.

The symmetry is of the draft, not the cloth. A real fabric has yarn irregularity, a selvedge, and a direction imposed by weaving; none of that is periodic and none of it is in the group.

Nothing here classifies. The figure counts operations for particular weaves. Working out which of the seventeen groups a given draft belongs to, and which drafts realise which groups, is a genuine question that this site does not answer at foundation.

The three-way comparison #

Setting three weaves side by side makes the balance result concrete.

Plain weave has the most of both kinds: sixteen state-preserving operations and sixteen state-exchanging ones on its two-by-two repeat. It is balanced, so exchange is available; it is the simplest repeat, so almost everything is a symmetry.

A 2/2 twill is balanced too, and has state-exchanging symmetries — but fewer relative to its repeat size, because the diagonal breaks the mirrors that a plain weave has.

A 3/1 twill and a satin have none at all, exactly, because they are warp-faced and no operation can map a larger set of filled squares onto a smaller set of empty ones.

The last of those is the useful one to carry away. Reversibility is decidable from the draft, by counting, and it is the same fact as the visible difference between a cloth’s two faces. A great many properties in this subject turn out to be countable in that way, which is why the matrix formulation earns its place.

What a repeat is, as a lattice #

One more piece of vocabulary, because it connects the subject to the wider classification cleanly.

A draft’s repeat defines a lattice: the set of translations that map the pattern onto itself. In a rectangular repeat those translations are generated by two perpendicular vectors, which makes it the simplest kind of lattice there is.

But drafts are not obliged to have rectangular repeats. A satin’s natural repeat is arguably oblique — the move vector and the pick vector generate the same lattice as the square repeat does, and describing a satin by its move is describing it in that basis rather than the conventional one.

That is a familiar situation wherever periodic patterns are studied: the same lattice has many bases, the conventional one is chosen for clarity rather than being canonical, and quantities that matter are the ones invariant under the change. On this site the invariant quantities are float length, interlacing count and layer count — none of which depends on how the repeat was drawn — which is why they can be quoted without a convention attached.

Which groups actually occur #

The classification says seventeen groups exist. It does not say a draft can have all of them, and the question can be settled by enumeration.

Twelve. And the five missing are exactly p3, p3m1, p31m, p6 and p6m — every one of them needing a three-fold rotation.

The reason is not about four-by-four repeats. A symmetry of a draft has to permute the intersections of warp and weft, so its linear part maps the square grid to itself, and the isometries doing that are the eight symmetries of the square. None of them has order three, so no draft of any size can have three-fold symmetry.

That is the crystallographic restriction in an unusually concrete setting: the lattice is not something the pattern chooses, it is the warp and the weft, and it is square. The seventeen groups a draft can have does the classification properly, including the distinctions — mirror against glide, primitive against centred — that no drawing settles.

Where the ladder goes next #

The place the two-colour idea becomes literal is colour and weave, where the threading order produces patterns the draft gives no hint of.

The directional consequence is the diagonal, which is the most misdescribed feature in the subject.

What the pictures here cannot show. A symmetry is a statement about operations, and an operation is not visible. The counts in these figures are the output of trying every candidate against the repeat, and a reader comparing the bars is comparing computed numbers rather than anything in the drawings.