The colour order that hides least

Colour and weave as a two-colour problem found the loss. Where the two threads crossing are the same colour the intersection looks identical whichever is on top, so with a balanced order half of them are blind and the 22,874 four-by-four drafts collapse onto 256 surfaces — eighty-nine weaves apiece, with no way to tell them apart.

Its last figure carried a caption with a question in it: “The blind fraction is not symmetric in the two orders — a warp of long runs against a weft of short ones hides less than either would against itself, which is a lever nobody uses and the account’s next question.”

The lever is real and the caption named the wrong variable.

The arrangement never matters #

There are sixteen two-colour orders at four ends and 256 ordered pairs of them. Running the surface census on every pair gives three numbers each: the blind intersections, the distinct surfaces, and the largest class of drafts a single surface fails to separate.

Every pair with the same colour counts gives the identical three numbers. A warp of 0001 and a warp of 0100 behave exactly alike against any weft, because both carry one dark end of four. A warp of 0011 — two dark ends in a run — behaves exactly like 0101, which alternates them.

The sweep collapses 256 pairs onto 25 cells, one per pair of counts, and the largest of those cells stands for thirty-six distinct colour orders that cannot be told apart by any of the three measurements. That is required rather than observed: the census requires each cell to hold exactly one outcome, so an order that broke the rule would fail the check rather than quietly widen a sentence.

So run length is not the variable. Houndstooth’s alternating orders and a two-and-two stripe’s runs make exactly the same intersections blind and confuse exactly the same drafts. The pattern on the cloth is wholly different; the information the surface destroys is identical.

And the blind count has a closed form #

Once the arrangement is out of the way the count falls out in one line. With a dark ends of n in the warp and b in the weft, an intersection is blind exactly when its two threads match, so

$$\text{blind} = ab + (n-a)(n-b),$$

which at four ends is checked against the census at all twenty-five cells:

The corner is the whole answer to the caption’s question. A solid warp of one colour against a solid weft of the other has no blind intersections at all, and the surface separates every draft in the catalogue — all 22,874, each with its own appearance.

That is not a subtle effect. It is the difference between a colour order that throws away 99 per cent of the structural information and one that throws away none.

What a mixed pair saves, exactly #

The caption asked whether a mixed pair hides less than the two orders paired with themselves. It does, and the amount is an identity rather than a trend.

Writing the self-pairings’ mean less the cross-pairing:

$$\tfrac{1}{2}\big[\text{blind}(a,a) + \text{blind}(b,b)\big] - \text{blind}(a,b) = (a-b)^2.$$

One apart saves one blind intersection; two apart saves four; four apart saves sixteen, which is all of them. The lever is a difference of colour proportion between the two systems, and its value is the square of that difference.

It is worth seeing why the identity holds, because it is short. Writing x = a/n and y = b/n, the blind fraction is xy + (1−x)(1−y); the mean of the two self-fractions less that is exactly ½[(x−y)² + (x−y)²] = (x−y)². The two terms are the two colours, and each contributes the same squared difference — which is the algebraic form of the observation that making the warp darker and the weft lighter helps twice, once in each colour.

The blind count is not the statistic #

Having a closed form for the blind count invites reading it as the quantity that matters, and the sweep says it is not.

Eight blind intersections gives 196, 254 or 256 surfaces depending on which pair of counts produced them, and the largest confused class is 196 or 256. So a designer who counts blind intersections and stops has a number that does not settle the question the number was computed for.

The reason is that blindness is a property of positions and the separation is a property of the map. Two colour orders can leave the same eight intersections invisible and leave them in different places — and where they are decides which drafts become indistinguishable, because a draft is distinguished by the intersections that remain sighted and which of those are informative depends on the geometry.

(2,2) leaves the most surfaces and the largest classes at once, which sounds contradictory and is not: it has 256 classes averaging 89 drafts and a largest of 256, so its classes are very unequal. (0,2) has 196 classes averaging 117 and a largest of 196, so its classes are nearly uniform. One of them confuses badly in a few places and the other confuses moderately everywhere, at the same eight blind intersections.

Which makes a chambray the one cloth that can be read from a photograph #

The corner has a consequence for a different question entirely, and it is the sharpest practical thing on this page.

Analysis — taking a cloth apart to recover what it is — is the inverse of the notations this account has been measuring, and the hardest part of it is the weave. Three notations for a repeat each write a draft and none of them is recoverable from a finished fabric without unpicking it, because the face shows which thread is on top and a cloth has two faces and a thickness.

A two-colour cloth changes that, because the surface is the draft rendered in colour. And the sweep says exactly when the rendering is faithful: at no blind intersections, the surface determines the draft uniquely.

So a cloth with a dark warp and a light weft can be analysed from its face alone — every intersection announces which system is on top — while the same cloth in one colour cannot be analysed at all without a microscope and a needle, and the same cloth in a balanced two-colour order is one of eighty-nine that look identical.

That is a statement about a photograph. A chambray photographed at thread resolution carries its complete drawdown; a balanced colour-and-weave cloth photographed at the same resolution carries 1/89th of one. Nothing about the cloths differs except which ends were dyed.

It also says what a cloth costs to specify. A specification names a cloth by its yarn, its setts and its weave and the weave field is the one nobody can check against a sample — unless the sample is coloured in the corner of this grid, in which case the check is a photograph and an enumeration.

The opposite arrangement is a tone #

It is worth setting this beside the only other way this account makes a two-valued surface, because the two are exact opposites and they are usually confused.

A colour-and-weave surface is made by dyeing the threads. The draft is fixed and the colours vary, so the surface is a function of the draft and the colour order, and blindness is information the colours destroy.

A shaded surface is made by moving the draft. A shading changes two things at once: the threads are all one colour and the tone comes from the fraction of warp on the face, so the surface is a function of the draft alone and nothing is blind — but the surface carries only the fraction, which is one number a tone rather than sixteen.

So the two make two-valued cloth from opposite ends, and they lose information in opposite ways. Colour and weave keeps the positions and loses the identities; shading keeps the identities and loses the positions. A colour-and-weave surface at the corner of this grid loses nothing at all, which is why it is the only one of the two that can be inverted.

And that explains a practical asymmetry the trade has and does not explain. A damask is designed on point paper and specified by its weave, because nothing about its surface recovers the weave. A colour-and-weave check is specified by its colour order and its weave together, and a mill checking one against a sample is checking the colour order — which is the field that, at a balanced order, is the only one the sample can verify.

Why nobody uses the lever #

The corner of the grid — a solid warp of one colour against a solid weft of another — is not an exotic construction. It is the commonest colour arrangement in weaving: a cloth with a dark warp and a light weft, which is what a chambray is, what denim is, and what most shot cloths are.

So the lever is used constantly and it is not used for this. A chambray is woven that way for its colour, and the fact that its surface is a faithful record of its interlacing is a side effect nobody has had a reason to name.

What is genuinely unused is the middle of the grid. A designer wanting the pattern that a mixed colour order gives — the colour-and-weave effects that are nowhere in the draft — reaches for a balanced order, because that is what every pattern book shows, and a balanced order is the worst cell in the table. A one-and-three order against a three-and-one gives 1,022 surfaces where a two-and-two gives 256: four times as many cloths distinguishable, with a largest confused class a fifth the size, and it is still a two-colour cloth with a colour-and-weave pattern in it.

That is the lever stated usefully. If a designer wants both an effect and a legible structure, the two orders should differ in proportion rather than in arrangement — and the arithmetic says by how much it is worth differing, which is as much as the design will bear.

What the grid says about the catalogue’s own shape #

The separations in the table are not arbitrary numbers and two of them are recognisable.

22,874 at the corner is the whole sweep — every four-by-four draft in which each end and each pick reaches both faces — so the corner’s surface map is a bijection and the catalogue is seen entire.

256 at the balanced pair is $2^8$, and the eight is the sighted intersections. With eight of sixteen blind, the surface can encode at most $2^8$ patterns, and the census finds every one of them occupied. So the balanced order is not merely lossy; it is maximally lossy in the precise sense that its image fills its own ceiling, which the census required rather than assumes.

That ceiling is what makes the other cells interesting. (1,2) also has eight blind and reaches only 254 of the 256 available, and (0,2) reaches 196. Two surfaces the eight sighted intersections could have encoded are never produced, because the blind positions in those cells are arranged so that some sighted patterns are unreachable — which is a statement about the interlacing condition rather than about colour. A draft must have every end and every pick reaching both faces, and that constraint removes exactly those two.

So the grid’s cells are doing two things at once: throwing away information through blindness, and inheriting the catalogue’s own constraints through what is left. The catalogue is 426 cloths and 22,874 drafts, and the difference between those two numbers is a different quotient again — a colour order confuses drafts, and the cloth relation confuses drafts, and the two quotients are not nested in either direction.

What was counted, and how #

The census is this account’s own surface census, run at every one of the 256 ordered pairs rather than at the handful an essay would quote. It builds each draft’s surface as a string of colours and counts the classes, so “distinct” means distinct in appearance under that colour order and nothing weaker.

The collapse onto counts is required at every cell, not observed on a few: each of the twenty-five cells is required to hold exactly one triple of measurements, so a pair of orders that behaved differently from another with the same counts would fail the census rather than be missed.

The closed form is required at every cell too, against the enumeration, rather than derived and trusted. It is one line of algebra and it is the sort of line that is wrong by a factor of two.

And the identity is checked at all twenty-five pairs rather than at the two the argument was noticed on.

The refutation the census carries is the one that earns its place: it requires that some blind count correspond to more than one separation, so a sweep in which the blind count did decide the answer would fail — which is the claim this essay is denying, made falsifiable.

What the sweep cannot say #

It is two colours and four ends. At more colours the blindness condition is the same — two threads of the same colour — but the counting is over a partition rather than over a pair of proportions, and the identity above is a two-colour identity. At more ends the closed form is unchanged, because it never used n = 4; the separations are a census over a larger catalogue and nothing here predicts them.

“Distinct surface” is not “visibly different”. The census separates two drafts if any intersection differs in colour, which a photograph would separate and an eye across a room would not. A float’s length decides what the eye reads, and a surface differing in one intersection of sixteen is a surface nobody would call a different cloth. So the separations counted here are an upper bound on what a viewer distinguishes, and the ordering between colour orders is what survives.

And the largest-class figure is about the worst case, not the typical one. A pair whose largest class is 256 has one surface standing for 256 drafts and most of its other surfaces standing for far fewer. Quoting the largest alone would make (2,2) look uniformly catastrophic, which the mean of 89 says it is not.

Who found it, and when #

Colour and weave is old — the effects are in every nineteenth-century pattern book and houndstooth is older than the books — and the arrangements are always given as recipes: four-and-four, two-and-two, one-and-one, with the pattern drawn beside them.

What is absent from all of them is the observation that the recipe does not matter. A pattern book’s whole organisation is by arrangement, because the arrangement is what the reader sees, and the quantity measured here is invisible to a reader: it is how much of the cloth’s construction the colouring conceals.

This account’s contribution is the sweep, and the result it produced is a correction to its own caption — which is the honest reason to record it here rather than quietly. The caption guessed that run length was the variable because run length is what a colour order looks like, and the census says the variable is a count. The two coincide often enough in the examples anybody draws that the guess was reasonable and it was still wrong.

Still open: what a third colour does to the corner #

The corner of this grid — no blind intersections, every draft distinct — is reached by a solid warp against a solid weft, and it is reached at two colours. Whether it stays reachable as colours are added is not obvious in either direction.

More colours should help, because a blind intersection needs a match and matches get rarer as the palette grows. More colours should hurt, because a cloth of many colours in both systems has a surface so busy that the eye reads nothing — which the census cannot see, since it separates on any difference at all.

The first half is a count and can be done: the blind fraction with warp proportions $p$ and weft proportions $q$ over $m$ colours is $\sum_i p_i q_i$, which is minimised when the two distributions have disjoint supports and is zero exactly then. So the corner generalises: a palette split between the two systems, with no colour used in both, is blind nowhere at any number of colours.

The second half is not a count and this account has no instrument for it. What it would need is a model of how much difference the eye resolves at a thread spacing, which is a question about vision rather than about cloth, and which every result on this account has so far been able to avoid.