A blind cell costs half the cloths at any real repeat

A blind set hides less the thinner it is spread took every shape a set of blind intersections can have at four by four — every union of rectangles a pair of colour orders can make, in any number of colours — and counted how many of the 22,874 four-by-four cloths each shape keeps apart. The rule it found was that spreading beats gathering: four blind cells scattered one to a thread keep 3,632 surfaces, the same four gathered into a square keep 2,402.

It asked the question that decides whether the rule is worth a designer’s attention. At eight by eight a blind count can be spread much further. Does the separation keep rising as it is spread over sixteen threads instead of eight, or does it level off? The catalogue at eight by eight is too large to take whole, and the answer turns out not to need it. It needs one factor, and the factor makes the rule nearly vanish.

What a blind set’s surfaces are #

A blind intersection is one where the warp and the weft crossing there are the same colour, so the cell looks the same whichever is on top. Two drafts look alike under a blind set exactly when they agree on every cell the set does not cover. So the distinct surfaces are the patterns on the uncovered cells that some draft actually has.

If every pattern on the uncovered cells came from some draft, there would be $2^{n^2 - b}$ of them for $b$ blind cells in an $n$-by-$n$ repeat, and the arrangement could not matter at all. The arrangement matters only because not every pattern comes from a draft. A draft must interlace: every end and every pick must be on each face somewhere in the repeat, or it is not a cloth, as the catalogue has required from the start. So a pattern on the uncovered cells is a surface only if the covered cells can be filled in to give every thread both faces.

That splits the count in two:

$$\text{distinct surfaces} = f(B)\cdot 2^{\,n^2 - b}.$$

The count sets the bound. The arrangement sets $f(B)$: the share of patterns on the uncovered cells that some interlacing draft completes. It is at most one, and the rule the four-by-four census found is a statement about $f$.

The four-by-four census is worth seeing again before the factor is found, because its shape is what has to be explained.

Within each blind count the surfaces rise with the threads touched, and they rise steeply: the four-cell line runs from 2,402 at four threads to 3,632 at eight, half as much again. The spread is the variable, and the question is whether its effect is a property of colour-and-weave or of the catalogue it is measured on. The rise is too regular to be about the look of any particular cloth: every added thread buys roughly the same factor, whatever the shape. That regularity is the clue that it comes from something simpler than the weave.

The threads a blind set misses #

When does a pattern fail to complete? Take a thread the blind set touches. If its uncovered cells happen to lie all on one face, a covered cell can be filled with the other face and the thread interlaces. A thread the blind set does not touch has no such help: its cells are all uncovered, and if they happen to lie all on one face, nothing can rescue it.

A thread of $n$ cells lies all on one face in two of its $2^n$ patterns, a chance of $2^{1-n}$. If the threads the set misses fail independently, the share that survives is

$$f(B) \approx \left(1 - 2^{1-n}\right)^{2n - t(B)},$$

where $t(B)$ is the number of picks and ends the blind set touches, out of $2n$. The arrangement enters only through the threads it touches, and each thread it leaves alone costs a factor of $1 - 2^{1-n}$.

At four threads that factor is seven eighths, and a square touching four threads of eight leaves four untouched: $(7/8)^4 = 0.586$, which is exactly the census’s 2,402 over 4,096. At eight threads the factor is 127/128, and each untouched thread costs under one per cent.

Sampled, and set against the closed form #

The census at eight by eight is out of reach, but $f$ can be sampled: draw patterns for the uncovered cells at random and ask whether some filling of the covered cells makes every thread interlace. A pattern in which no thread is constant is accepted at once; the rest need a small search over the covered cells of the threads that need help. Forty thousand patterns a shape put $f$ to within about a tenth of a per cent.

The sampling reproduces the four-by-four census exactly: 0.884 for four scattered singles against the census’s 3,632 over 4,096, and 0.585 for the square against 2,402. It is then run at every repeat from four to twelve.

Every curve climbs toward one. At four threads a pattern of sixteen cells interlaces in every thread only a third of the time; at eight, 88 per cent of the time; at twelve, 99 per cent. A long thread is rarely all on one face, and the interlacing requirement that shaped the four-by-four census stops binding. The gaps between the shapes close as the curves climb, because every shape’s curve is climbing toward the same ceiling.

The four scattered singles are the exception that proves the form. At four and five threads they touch almost every thread, so the untouched-thread factor is nearly one, and what limits them is the touched threads failing — which at a short repeat they still can, when a thread’s one blind cell is also needed by its crossing thread. That is why their share dips from four threads to five, and why the closed form overstates the ratio in the hero figure at the two shortest repeats. From six threads up the form and the sampling agree to within the sampling.

At eight by eight, only the threads touched matter #

The answer to the question as asked is in this figure. At eight by eight, every shape of eight blind cells that touches the same number of threads keeps the same share of its bound, to within the sampling, whatever rectangles it is made of. Four dominoes and four singles with a square both touch twelve threads and both keep 96.9 per cent. The row of eight touches nine and keeps 94.6; two squares touch eight and keep 94.0; the two-by-four block touches six and keeps 92.5. Eight single cells, one on every thread, touch all sixteen and keep 99.9 per cent — within a thousandth of the bound.

So the separation does keep rising as the set is spread, all the way to sixteen threads, and it rises by the same step for every thread added: about 0.75 per cent, which is one in 128. It does not level off at one blind cell per thread; it arrives at the bound there, because once every thread is touched there is no thread left to fail.

At four by four, the census found that shapes touching the same number of threads could still differ, and that the ties were broken by how the rectangles sat. At eight by eight that second-order effect has gone below the sampling. The arrangement is a count of threads touched, and nothing more.

Every blind cell costs very nearly half #

That puts the four-by-four census’s most striking number in its place. One blind cell took the four-by-four catalogue from 22,874 surfaces to 14,414, and each further scattered cell took about 37 per cent of what was left — a factor of 0.63 a cell, much gentler than the factor of a half a hidden bit should cost.

The factor was gentle because the four-by-four catalogue is so small a share of all patterns. A cell hidden in a small catalogue often hides a difference with a pattern that could not interlace anyway, so it hides nothing. At eight by eight almost every pattern interlaces, so almost every hidden cell hides a real difference. Eight blind cells spread one to a thread cost a factor of 226, where eight independent halvings cost 256 — a factor of 0.508 a cell.

The six shapes drawn span the whole range, and the range is seven and a half per cent. Between the most gathered and the most spread way of making eight cells blind, the catalogue keeps 92.5 or 99.9 per cent of what the count allows. Against that, each blind cell added or removed changes the catalogue by a factor of two. The count is the lever; the arrangement is a trim.

What this means for a colour order #

A designer choosing a colour order is choosing a blind set, and the colour order that hides least showed that with two colours the blind count is $ab + (n-a)(n-b)$ for $a$ dark ends and $b$ dark picks. That count is now the whole story at any repeat a colour-and-weave design is usually drawn at.

At eight ends, a two-colour order’s blind set touches every thread whenever both colours appear in both warp and weft, because its two rectangles, one per colour, then together cover every pick and every end. So two-colour designs at eight by eight are already at the bound, and each one hides exactly as many distinctions as its count says. The shape question the four-by-four census raised belongs to three and more colours, and there it is worth a few per cent at most.

That answers the practical form of the question. At a real repeat, a designer who wants the weave to show through should minimise the blind count and stop there. Arranging the blind cells to touch more threads buys at most the few per cent the hero figure shows at eight, and nothing at twelve. The four-by-four rule was true and it was a small-repeat rule.

Three colours at eight ends, worked #

A three-colour order is where the shape question survives, so it is worth one worked pair. Take eight ends coloured three, three and two in colours one, two and three, and eight picks coloured the same way. Every colour is shared, so the blind set is three squares, three by three, three by three and two by two, twenty-two blind cells touching every thread: its surfaces are the bound $2^{42}$ to within a thousandth.

Now keep the ends as they are and colour the picks four of colour two and four of colour three. Only colours two and three are shared: a three-by-four rectangle and a two-by-four, twenty blind cells touching thirteen threads. The three untouched threads are the three ends of colour one. They cost $(127/128)^3$, a little over two per cent, and the blind count, two fewer, is worth a factor of four in the other direction.

That is the trade at any real repeat: two blind cells are worth four times the catalogue, and three untouched threads are worth two per cent of it. A designer arranging stripes and checks — a check is two stripes and a tartan is one counts how the orders combine — needs only the blind count to know how much of the weave the colours will hide.

Why small repeats exaggerate #

This is not the first time a count on a four-by-four catalogue has overstated what happens at a real repeat. Silence lives in the lifting plan found that four-end threadings are silent by coincidence more often than eight-end ones, because a small repeat has symmetries by accident; the complete window asked whether a census of small patches still separates at larger repeats. The shared mechanism is the catalogue’s own constraints, interlacing and symmetry, which bind a four-by-four draft hard and an eight-by-eight draft hardly at all.

The practical rule for reading any four-by-four count is to ask which part of it comes from the constraint. Here it could be separated exactly, because the constraint entered as one factor per untouched thread. The count $2^{n^2-b}$ was never in doubt; the four-by-four census measured it through a lens that made the arrangement look important, and the lens was the requirement that every thread interlace.

How the sampling was taken #

The blind sets are unions of rectangles on separate picks and separate ends, which is the only kind a pair of colour orders can make; every such set of four cells and of eight that fits an eight-by-eight repeat is enumerated, six and thirty-one of them, and the scattered set and the block are taken at every repeat from four to twelve.

For each, forty thousand patterns on the uncovered cells are drawn from a seeded random sequence. A pattern survives if the covered cells can be filled so that every end and every pick has both faces: a thread not constant on its uncovered cells is satisfied at once; a constant one needs a covered cell of the other face, and a thread wholly covered needs two covered cells that differ; the fills are searched only over the covered cells those threads need.

What is required of it. The sampled share must match the four-by-four census’s exact counts for the scattered set and the square within three standard errors; a gathered set must keep a larger share at every longer repeat; spreading must never lose; and at eight by eight the set with one blind cell on every thread must keep more than 99 per cent of its bound.

What the sampling cannot say #

Which of the surfaces are cloths anybody weaves. The catalogue at every size is every matrix in which each thread interlaces, and most of those have floats no weaver would accept. Designing to a float limit restricts a catalogue to drafts whose floats are short, and a float limit is a much stronger constraint than interlacing; under it, the threads a blind set misses would fail far more often, and the arrangement could matter again.

And whether an eye tells the surfaces apart. Two surfaces that differ in one uncovered cell are counted as two. Whether a viewer sees a single-cell difference in a colour-and-weave effect at arm’s length is a question about vision, and a shading’s centre and the moiré essays show how much depends on scale.

Who worked out which part #

Colour-and-weave is old weavers’ knowledge, the blind intersection its central idea. The census of blind shapes at four by four, and the rule that spreading beats gathering, are the earlier essay’s.

What is added here is the split of a blind set’s separation into the count’s bound and an arrangement factor; the closed form for that factor as one term per untouched thread; the sampled confirmation at every repeat from four to twelve; and the finding that at eight by eight the arrangement matters only through the threads touched and by a few per cent at most.

Still open: the same question under a float limit #

Everything here counts every interlacing matrix as a cloth. A weaver’s catalogue is narrower: no float longer than some limit, often three or four. Under a float limit, a thread the blind set misses must have no run longer than the limit on its uncovered cells, and the chance of that failing is not $2^{1-n}$ but the chance a random thread has a long run, which at eight threads and a limit of three is well over a half.

So under a float limit the untouched-thread factor stays large at every repeat, and the arrangement of a blind set may matter as much at eight by eight as interlacing made it matter at four. The sampling above runs unchanged with the limit added to its test, and whether the four-by-four rule returns in force for cloths a weaver would actually make is what it would say.