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 block onto an 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.
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.
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.