A float limit gives a blind set its arrangement back

A blind cell costs half the cloths at any real repeat ended a question and reopened it in one paragraph. It had found that the rule the four-by-four census was so clear about — spread a blind set thinly and it hides less — fades to nothing as a repeat grows. Four cells spread over gathered keep half as much again at four threads, three per cent more at eight, a fifth of a per cent more at twelve. A designer at a real repeat, it concluded, should minimise the blind count and stop.

Its catalogue was every matrix in which each thread interlaces. That is not what anybody weaves. A weaver’s catalogue has a float limit, and the fading depended on a fact a limit reverses: that a long thread is rarely in trouble.

The one number that changed direction #

The earlier account rested on one factor per thread. A thread the blind set does not touch has no help: if its cells happen to lie all on one face, nothing can rescue it, and a surface with such a thread is not a cloth. That happens to a share $2^{1-n}$ of a thread’s patterns, and at eight threads that is one in 128. Untouched threads cost almost nothing at a real repeat, so touching them buys almost nothing.

Put in a limit and the question for an untouched thread changes. It no longer has to avoid lying on one face all the way round. It has to avoid lying on one face for any stretch longer than the limit, counted round the repeat, and a thread has as many places to do that as it has cells.

The two curves run in opposite directions, and everything below follows from that. Only interlacing, the share allowed at eight threads is 99.2 per cent and rising. Under floats of at most five it is 86.7 and falling; at most four, 74.2; at most three, 52.3 — and at sixteen threads those are 76.0, 55.4 and 26.2. Designing to a float limit measured this fall as the cost of the limit to a catalogue of twills. Here it has a second consequence. An untouched thread, under a limit, is a thread in real danger, at every repeat, and increasingly so.

Spreading keeps paying #

So the sampling is run again with the limit added to its test. A pattern on the uncovered cells counts as a surface only if some filling of the blind cells gives a draft whose every float, along every end and every pick, is inside the limit.

The answer is the first figure. At eight threads, with no limit, four spread cells keep 1.03 times what the square keeps. Under floats of at most four they keep 1.42 times as much, under three, 1.44, and under five, 1.29. The first of those figures runs out; the three others do not. At twelve threads the unlimited ratio is 1.002 and the limit of four still holds it at 1.32, and the limit of five at 1.26.

The ratios are not perfectly smooth. They move by a few hundredths with the repeat, and the limit of three drops to 1.34 at seven threads between 1.53 at six and 1.44 at eight. Some of that is the sampling, whose error at these counts is up to two hundredths, and the rest looks like a thread’s own arithmetic: which patterns fit a limit depends on how the repeat divides, and the share falls in steps rather than smoothly. What does not appear is any trend down. The four-by-four rule did not get weaker with the repeat. It stayed near the strength it had at four threads, where the unlimited catalogue’s ratio was 1.51.

The reason the plateau is a plateau and not a climb is worth stating, because the falling curves might suggest the advantage should grow. What a touched thread gains is one free cell, and what one free cell is worth to a thread is a ratio: the share of patterns it can rescue over the share that needed no rescue. Computed exactly, that ratio stays between 1.31 and 1.33 under floats of at most three at every repeat from eight threads to fourteen; between 1.17 and 1.18 under four; between 1.09 and 1.10 under five. A single cell can repair one bad stretch and a long thread with one bad stretch is about as common relative to the good ones at twelve threads as at eight. The value of a free cell does not depend on how long the thread is, so spreading does not fade and does not grow.

A form built on that ratio, with each thread surviving on its own, gets the plateau’s level wrong while getting its existence right: it predicts 2.1 under floats of three where 1.44 is sampled. One blind cell has to serve both the pick and the end through it, and it can only take one value, so a cell that rescues its pick by lying on the face may be no use to its end. The threads a set touches are not independent, and without a limit that hardly mattered, because they were almost never in trouble.

A blind cell is cheaper than a half #

The earlier essay’s largest claim was about the count, not about the arrangement: at a real repeat a blind cell costs very nearly half the catalogue, because almost every hidden cell hides a real difference. Eight scattered cells cost a factor of 226 against the 256 eight halvings would cost.

That is the second thing a limit undoes.

Under floats of at most three, a blind cell costs 0.39 of a halving, a factor of 1.31 and not 2. Under floats of at most four it costs 0.60, a factor of 1.52. Each cell leaves 0.66 of what was there, which is back beside the 0.63 the four-by-four catalogue’s scattered cells left — the very figure the earlier essay explained away as a property of a small catalogue. It was not about small catalogues. It was about constrained ones.

The mechanism is the one that explanation named. Hiding a cell loses nothing if the two drafts it would merge were not both in the catalogue, and in a tightly constrained catalogue they very often are not: flip one cell of a draft inside a float limit, and the result has a good chance of breaking the limit. At floats of at most three a little over half of a thread’s patterns are allowed, and a cell lies on two threads. A cell’s hidden bit was, much of the time, not free to begin with.

So the two factors have traded places. Without a limit, the count is the lever and the arrangement is a trim worth three per cent. Under a tight limit the count is a weaker lever and the arrangement a stronger one, and they come within reach of each other.

Five scattered beat four gathered #

Within reach is measurable. Under floats of at most three, one more blind cell costs a factor of about 1.31, and spreading four gathered cells gains a factor of about 1.43, and 1.43 is larger.

Five blind cells, scattered, keep 1.09 times the surfaces that four gathered do. Three million patterns a set hold the error near one and a half per cent, so the ordering is real. A colour order that makes one more cell blind can show the weave better than the order it replaced, provided the extra cell comes with a spread. Six cells fall back below, at 0.84, so the crossing happens once and is narrow: under this limit an arrangement is worth about one blind cell and a little more.

The crossing does not survive a looser limit. Under floats of at most four, a cell costs a factor of 1.52 and spreading gains 1.42, and four gathered keep 1.08 times what five scattered do. The rule the earlier essay gave is not simply wrong; it is a rule for loose catalogues, and somewhere between a limit of four and a limit of three at eight threads it stops being the whole answer. Minimising the count is still most of the job until there.

Shapes that tied no longer tie #

The earlier result that looked most final was a tie. At eight by eight, without a limit, every set of eight blind cells touching the same number of threads kept the same share of its bound, to within the sampling. The arrangement was a count of threads touched and nothing more.

They do not tie now. Shapes touching twelve threads run from 5.9 to 6.4 per cent, a spread of about a twelfth, well outside the sampling. How many cells fall on one thread now matters, since two free cells on a thread are worth less than two times one, and adjacent free cells less than separated ones, as a free cell in a bad stretch can only mend that stretch: under floats of three, two free cells side by side leave 75 per cent of a thread’s patterns alive and two set four apart leave 97. All of it sits far below the bound: the whole range runs from 3.5 per cent for a two-by-four block to 8.4 for eight single cells, and the block is left with about one in twenty-nine of the patterns its count would allow.

That difference between shapes means the number the earlier essay found decisive is now only the largest term in a sum, which is what it looked like at four by four. The four-by-four census had the structure right and the reason wrong, and so did the essay explaining it: both blamed the size of the catalogue, and what mattered was how hard the catalogue pressed.

Colour orders that were equals #

The most striking part is about two colours, where nothing was supposed to happen at all.

The colour order that hides least proved, across all sixteen four-end orders on each side, that for two colours only the counts matter: the blind count is $ab + (n-a)(n-b)$, and every arrangement with those counts separates exactly as many drafts as every other. An exhaustive census confirmed it at every pair. A blind cell costs half the cloths at any real repeat extended the thought: two-colour designs at eight threads touch every thread and are already at their bound, so each one hides exactly as many distinctions as its count says.

Under a float limit that fails, and by a large factor.

A four-and-four stripe keeps 12 per cent of its bound under floats of at most three; a two-and-two and a one-and-one keep all of it, with no failure among eight hundred samples. All five have thirty-two blind cells and sixteen threads touched. Nothing about them that the earlier accounts measured differs at all.

The four-and-four case can be done by hand. A blind cell is where a warp and a weft of one colour cross, so along a dark pick the blind cells are at the four dark ends and the free cells are at the four light ones, and they sit together. A stripe of four is a stretch of four cells on every crossing thread that no blind cell can reach. Under a limit of three those four cells must not all lie on one face, and that fails in two patterns of sixteen, one in eight. Sixteen threads each survive with seven chances in eight, and $(7/8)^{16}$ is 11.8 per cent, which is the sampled figure.

A stripe wider than the float limit takes surfaces away by a closed form, in this regular case, and no other property of the order enters it. So a stripe’s width has a meaning here that it does not have in a check of two stripes: measured against the limit, it says how much of the weave’s variety the colours have closed off.

Fitting is necessary and not sufficient. The order 00101101 has a widest stripe of two and keeps only 68 per cent, and 00010111, with a stripe of three that exactly fits, keeps 57. In both, no thread is doomed by a stretch alone. The failures there are the other mechanism, a blind cell asked to serve its pick and its end with one value. A regular stripe reaches its bound where an irregular one with stripes no wider does not. The likely reason is that a regular order’s blind cells fall where its picks and ends do not compete for them, but only the five orders drawn were sampled and that reason has not been proved.

That connects two things the earlier accounts had treated as unrelated. A designer who picks a stripe by what the pattern shows, in colour and weave as two colours, is also picking how many of a shop’s drafts can be told apart in that order. A coarse stripe does not merely hide cells, which is what a count sees. It removes the drafts in which those cells would have been free to differ.

What this changes for a designer #

The earlier essay’s trade was that two blind cells are worth four times the catalogue and three untouched threads are worth two per cent. Under floats of at most four at eight threads, two cells are worth a factor of about 2.3 and four touched threads, gained by spreading a square, about 1.4; under floats of at most three they are 1.7 and 1.4. A designer working to a real limit has two levers of comparable size, not one lever and a trim, and with two colours has a third the count never mentions: keep every stripe inside the limit and prefer the regular ones.

None of this reaches the case of a loose limit. Floats of seven at eight threads is the unlimited catalogue, and the sampler returns it. The float decides most of what a cloth will wear like, and floats and abrasion give the reason a limit is usually short: it is at the tight end, the ordinary end for a hard-wearing cloth, that a colour order has this second job.

How the catalogue was walked and sampled #

The catalogue is the one every count here has used, cut by the limit. The sets are the ones a pair of colour orders can make: unions of rectangles on separate picks and ends, with the two-colour orders given as orders and every cell where they agree blind. A float is the longest run of one value along a thread, counted cyclically around the repeat, so a limit of $n - 1$ requires only that a thread interlace.

Random patterns would almost never survive at these limits, since a cloth with sixteen threads each near a half is one pattern in many thousands. So a pick the set does not touch is drawn only from the patterns inside the limit, and the estimate is multiplied by the share those are. The ends are then tested, and if they pass, the blind cells are tried. Hundreds of thousands of patterns a point, three million for the crossing, and 150,000 a shape for the thirty-one sets. For the two-colour orders, thirty-two blind cells are too many to try every filling of, so a search abandons a filling as soon as any thread’s known cells break the limit, over eight hundred patterns an order.

The share of one thread’s patterns is counted, not sampled. Required of all this: that the sampler agree with walking all $2^{16}$ drafts at four by four under limits of two and three; that a limit of $n - 1$ return the unlimited sampler; that spreading keep its advantage at ten threads as at eight under a limit of four; and that at one blind count the four-and-four stripe stay under a fifth of its bound while the two-and-two reaches it.

What the arithmetic leaves out #

A float limit is a straight line drawn through a judgement. A cloth with one float of four and the rest of two is rejected by a limit of three, and a cloth full of floats of three is accepted, and a weaver would not treat them that way. The limit is applied to both warp and weft floats at one value, and the figures would move if a warp-faced cloth’s two systems had different limits.

Most catalogue members are still not cloths anybody weaves. Inside a limit of three there are drafts no one would choose on any other ground, and the sampling counts each equally. A weaver’s real catalogue is narrower still, and may have its own structure that the factor found here does not see.

Counting is not seeing. Two surfaces differing in one cell count as two, as they did when four-end threadings came out silent by coincidence and a small repeat exaggerated. Whether a four-and-four stripe’s lost variety is variety a viewer would miss, or whether the eye reads an eight-end shading by its coarser marks and would never have told those drafts apart, is the question that would say how much of this is a designer’s business. The count says what is lost and the picture cannot say whether it mattered.

And the repeat stops at twelve. At floats of three, nine threads is the longest measured, because even with picks drawn only from the allowed patterns, too few samples survive past it. The plateau beyond is argued from the free cell’s value, which is exact, and not from a sample.

Who worked out which part #

Colour and weave, blind intersections and float limits are all old working knowledge; the limit in particular is ordinary mill practice. The four-by-four census, the closed form for the unlimited factor and the finding that the rule fades are the earlier essays’ own.

What is added here is the same count run inside a float limit, and three results: that the arrangement’s advantage holds level with the repeat instead of fading, because a free cell’s value is independent of the thread’s length; that a blind cell costs well under a halving, and that this, not the small catalogue, was the true reading of the four-by-four figure; and that two-colour orders of one count, which are exactly equal without a limit, are not equal inside one, by a closed form when a stripe outruns the limit.

Still open: what a limit on one system does #

Everything here applies one limit to warp and weft alike, and few cloths are specified that way. A warp-faced sateen has floats of four in one system and of one in the other, and a designer choosing a colour order for it faces a different catalogue in each direction.

An asymmetric limit has one thing to say that a symmetric one cannot: whether a stripe in the warp costs as much as the same stripe in the weft. By the four-and-four argument, a warp stripe is a stretch of free cells on picks, governed by the limit on picks, and the weft stripe the same for ends. So a warp-faced cloth might carry coarse stripes one way and not the other, and if the sampling agrees, it would say which direction of a check a designer can coarsen for free.