Colour and weave as a two-colour problem

Colour and weave established that the pattern a reader sees on a cloth is not in the draft: thread the same weave with a different colour order and the surface changes completely, with no intersection having moved.

This essay asks the question that follows and is much sharper. Given the surface, how much of the weave can be recovered?

The answer is: much less than anybody would guess, and the amount can be counted exactly.

Blind intersections #

The whole argument turns on one observation, and it is immediate once stated.

At an intersection, the colour seen is the warp’s where the warp is on the face and the weft’s where it is not. So if the two threads crossing there happen to be the same colour, the intersection looks identical either way up. The weave at that intersection leaves no trace on the surface at all.

Call such an intersection blind. The number of them is a property of the colour order alone, and it is arithmetic:

$$\text{blind} = |\{(i,j) : a_j = b_i\}|$$

for warp colour order $a$ and weft colour order $b$. With two colours, if the warp has $n_0$ threads of the first colour and $n_1$ of the second, and the weft likewise, the count is $n_0 m_0 + n_1 m_1$.

For the classic one-and-one order over a four-by-four repeat — dark, light, dark, light in both directions — that is $2\times2 + 2\times2 = 8$. Half the intersections are blind.

What that costs #

If eight intersections carry no information, the surface is determined by the other eight, and there can be at most $2^8 = 256$ distinct surfaces.

Run the enumeration and every one of them occurs. So the 22,874 four-by-four drafts collapse onto 256 surfaces, at an average of eighty-nine drafts apiece, with the largest class holding 256.

That is a catastrophic loss of information and it is exact. Eighty-nine different weaves — different float lengths, different interlacing counts, some of them describing more than one cloth — produce cloth nobody can distinguish by looking, given that colour order.

The one-and-one plain weave is the sharpest illustration. Threaded one way it produces horizontal stripes; shifted by a single end it produces a checkerboard. Two cloths that could not look more different, from the same weave and the same two colours, differing only in where the colour order starts.

The order that hides least still hides half #

A natural next question is whether a cleverer colour order recovers more of the weave. It does not.

The blind count is $n_0 m_0 + n_1 m_1$ with $n_0 + n_1 = m_0 + m_1 = 4$. Minimising it over the possible splits gives the answer immediately: the minimum is at $n_0 = n_1 = m_0 = m_1 = 2$, and the minimum value is eight.

No two-colour order on a four-by-four repeat blinds fewer than half the intersections. A balanced order is the best available and it is still half.

Going the other way makes it worse fast. A three-and-one order blinds ten of sixteen, leaving only sixty-two distinct surfaces in the whole census and a largest class of 480.

Adding colours helps, and it helps in exactly the way the arithmetic predicts. With every thread a different colour, no two crossing threads match, nothing is blind, and the surface determines the weave completely. That is why a warp-printed or ikat cloth, where the colour varies continuously, gives away its structure — and why the traditional colour-and-weave cloths, which are two-colour, do not.

What this says about the named cloths #

The colour-and-weave tradition names its fabrics after their appearance, and the enumeration explains why it has to.

Houndstooth is a two-two twill with four dark and four light in both directions. It is named for what it looks like, and there is no other option: several other weaves in the same colour order give the same figure, so naming it after its construction would name the wrong thing.

Shepherd’s check, dogtooth, birdseye, hairline and the rest are all named the same way, and the names do not partition constructions. Two mills can sell the same-looking cloth made two different ways, and neither is wrong.

That is a systematic feature rather than an accident of naming. The trade names appearances because appearances are what customers buy, and the enumeration says appearances are a lossy function of the construction — so the naming could not have gone the other way.

The information that survives #

Not everything is lost, and it is worth being precise about what gets through.

The surface does determine which colour each intersection shows, obviously. What it does not determine is which thread is on top.

So a property of the weave survives exactly when it is a function of the sighted intersections alone. Balance does not survive: two drafts in a collision class can have quite different warp-face fractions, because the blind intersections contribute to balance and not to appearance. Float length does not survive either, for the same reason. Nor does the layer count — some members of a collision class hold together and others do not, so a cloth that looks perfectly sound may be two cloths.

That last one is not hypothetical. It means a designer working from a coloured rendering of a cloth is working from an image that cannot tell them whether the cloth exists — and the check that could tell them operates on the draft, which the image does not determine.

What a designer does about it #

The loss is real and designers work in spite of it, so it is worth setting out how.

Design in the surface, then choose any weave in the class. If eighty-nine drafts give the same appearance, the designer is free to pick whichever of them has the properties they want — the shortest float, the firmest interlacing, the fewest shafts. The collision is a constraint satisfied for free rather than an obstacle, once the direction of the argument is turned round. A designer choosing a houndstooth is choosing an appearance and is at liberty to choose the construction independently.

Use the blind intersections for the back. The surface cannot see them, so what happens there is available for something else: stitching a backed cloth, placing a second colour on the reverse, or simply keeping the float short where the front does not care. This is the most useful consequence in practice and it follows immediately from the count.

And test the construction, not the picture. Since the appearance does not determine whether the cloth holds together, the integrity check has to be run on the draft. A design process that ends at a coloured rendering has skipped it.

That is the reversal worth taking from the whole essay. The blindness looks like an obstacle when the question is “what weave is this cloth”, and it is a degree of freedom when the question is “what cloth shall this weave be”.

The same problem, elsewhere #

The structure of this argument turns up wherever a visible output is a lossy function of a hidden state, and two neighbouring cases are worth naming because both are met in the same trade.

Printing. A printed check and a woven one can be indistinguishable at a distance, and the printed one has no structure at all behind the pattern. The information loss is total rather than partial: nothing about a printed surface constrains the weave beneath it. The trade’s usual test is to look at the reverse, which is a way of getting a second observation of the same hidden state — and it works precisely because a printed cloth is white on the back while a woven one is not.

Jacquard. A large figured cloth is designed as an image and converted to a draft, and the conversion is exactly the inverse problem this essay says is ill-posed. What makes it tractable is that the designer supplies the missing information by choosing the weaves: a jacquard design specifies which of several weaves fills each region of the image, and the image alone would not determine it. So the practice already contains the answer to the theoretical problem, arrived at because the alternative did not work.

Both cases share the shape of the argument. The surface is a projection; a projection loses a dimension; and recovering the lost dimension needs either a second observation or a decision.

Where the effect was noticed #

Colour and weave is old as a practice and young as a subject.

The cloths are traditional — shepherd’s check is Border weaving, houndstooth is Scottish, and both are considerably older than any written account of why they look as they do. What distinguishes them from most traditional patterns is that they are not patterns in the weave: the same cloth in one colour is a plain twill and nothing more, which means the effect was discovered by somebody threading two colours and being surprised.

The systematic treatment belongs to the twentieth-century textile design literature, where colour-and-weave effects are usually presented as a catalogue — this colour order on that weave gives this figure, tabulated. That is a useful reference and it is the opposite of an argument: a table says what happens without saying why, and a table cannot say what does not happen.

The framing here — as an information problem, with a count of blind intersections and an enumeration of collision classes — is this site’s own, and it is the natural thing to do once the weave is treated as a matrix. What it adds to the catalogue is the negative result, which no catalogue could contain: that the entries are not unique, that the non-uniqueness is exactly measurable, and that it is worst precisely where the tradition concentrates.

Against the plane-group question #

Setting this beside the plane-group classification sharpens both.

That essay classified the uncoloured pattern of a draft — filled and empty squares — and found twelve of the seventeen groups occurring. This one is about a genuinely two-coloured pattern, and the relationship between them is not the one it looks like.

The plane group of a draft is a property of the interlacement. The plane group of the coloured surface is a property of the colour order and the interlacement together, and the two need not agree at all. A p1 draft can have a highly symmetric surface, if the colour order supplies the symmetry; a p4m draft can have an asymmetric surface, if the colour order breaks it.

So the visible symmetry of a colour-and-weave cloth is telling a viewer about the colouring rather than about the weave, which is one more thing on the list of what the surface fails to report.

The reverse side, which is a second observation #

There is one more piece of information available on a real cloth and it is worth counting too, because it changes the answer.

The back of the fabric shows the complement: where the warp is on the face, the weft is on the back. So a blind intersection on the front is blind on the back as well — the two threads are the same colour, and swapping which is visible changes nothing. Looking at the reverse gives no extra information about the blind intersections whatever.

What it does give is a check on the sighted ones, which were already determined. So the reverse of a colour-and-weave cloth is exactly as uninformative as the front, and the collision classes are unchanged by turning the cloth over.

That is a genuinely surprising result and it runs against the trade’s usual instinct, which is that the back of a cloth reveals what the front is hiding. For a printed cloth it does; for a colour-and-weave cloth it does not, and the reason is that blindness is a property of the colours meeting, which is the same on both sides.

The blind fraction, for any number of colours #

The count of blind intersections is worked out above for two colours and the general form is one line, because the count is a probability of agreement.

If the warp uses colours in proportions pᵢ and the weft in proportions qᵢ, the blind fraction is Σ pᵢqᵢ — the chance that a random warp thread and a random weft thread happen to match. Two consequences follow immediately and both generalise the essay’s findings.

Balance is always best. For a fixed set of colours used the same way in both systems, Σpᵢ² is minimised when the proportions are equal — so a balanced order blinds least at any number of colours, not merely at two. Going away from balance always costs information, and the three-and-one order’s ten blind intersections are one case of a general inequality.

And at c colours used evenly, the blind fraction is exactly 1/c.

The third colour buys more than the fourth, and the returns diminish as 1/c: the fraction of intersections that are legible goes 50, 67, 75, 80 per cent, so the first extra colour recovers a third more of the weave and the next recovers an eighth more. That is why an ikat or a warp-print gives its structure away and a two-colour check does not, and it now has a rate rather than an observation.

The cloth that hides nothing needs only two colours #

The stronger statement is not about how many colours there are but about whether they are shared, and it falls straight out of the same expression.

If the warp’s colours and the weft’s are disjoint sets — no colour appears in both — then every term of Σpᵢqᵢ has a zero in it and the blind fraction is

exactly zero,

whatever the proportions and however few colours there are. One dark warp against one light weft is completely legible: every intersection shows which thread is on top, the surface determines the draft, and there is no collision class at all.

That is not a curiosity. It is denim — an indigo warp and an undyed weft — and it is why denim’s twill line is the most conspicuous weave structure in ordinary clothing while a shepherd’s check made of the same two colours shows nothing about its weave. The difference between the two cloths is not the number of colours or the weave; it is whether each system keeps its colours to itself.

It is also how every weave diagram in this collection is drawn, and how a weaver samples a new draft: one dark warp, one light weft, because that is the colouring under which the draft is visible. The practice is universal and the reason for it is this identity.

So the design space divides cleanly and the dividing line is a set intersection rather than a count:

Share the colours between the systems and the weave disappears — half the intersections blind at two colours, and the classic checks are what that looks like.

Separate them and the weave is fully exposed — no intersections blind at any number of colours, and the twill lines, the satins and the drafting diagrams are what that looks like.

Everything between the two is the partial case, blinded in proportion to how much the two palettes overlap. And a designer wanting a cloth whose structure reads at a distance has a one-line rule that needs no enumeration: do not put the same colour in both directions.

What the arithmetic assumes #

Three simplifications, and the second is the one that matters most in practice.

The intersection is drawn as a square of one colour. In real cloth the threads are round, the one on top spreads over its neighbours, and the visible colour at an intersection is a mixture weighted towards whichever thread is raised. That means a real cloth leaks a little information the model says is unavailable — a blind intersection is not perfectly blind, because the raised thread’s colour dominates slightly even when both are the same hue.

Only two colours. The whole argument is about matching colours, so it is the number of distinct colours that decides everything. Three colours in the warp and three in the weft, chosen so no warp colour matches a weft colour, blinds nothing at all.

And the repeat is four by four. The blind fraction is a property of the colour order and generalises directly; the collision-class sizes are not, because a bigger repeat has more intersections and the counts grow exponentially. What holds at any size is the fraction: a balanced two-colour order blinds half the intersections whatever the repeat.

One more count, for the surface itself #

A last enumeration, because it answers the question a designer would actually ask.

If a two-colour order on a four-by-four repeat admits 256 surfaces, how many of those are distinct patterns rather than the same pattern shifted or turned? Reducing by the symmetries — translation, rotation, reflection — collapses them a good deal further, and the number of genuinely different colour-and-weave figures available at this repeat is small enough to be listed on a page.

That is why the tradition has perhaps a dozen named checks rather than hundreds. The design space is not being under-explored: it is small. Houndstooth, shepherd’s check, birdseye and their relatives are not a selection from an abundance, they are most of what exists at that scale.

Going to a larger repeat opens it up quickly, and the price is a loom with more shafts and a cloth with a larger pattern. That trade — pattern scale against shaft count — is the constraint every shaft weaver works under, and it is the reason large figures need a jacquard.

Where the ladder goes next #

This is the top of the pattern ladder as it stands. Below it are the plane groups, colour and weave where the effect is introduced, and weaves as plane patterns at the base.

The nearest thing elsewhere is the diagonal is not a thread, which is the same complaint about a different visual feature: the eye reads a structure that is not there. And the general form of the complaint — that the properties easy to see are not the properties that matter — is the argument the integrity check was built to make.