An eight-shaft error hides in the row's mirror

A proof plan has no slide in it took every threading of eight ends on four shafts, found every double error a lifting plan hides, and re-wove each under eight other plans. The hidden errors nested: whatever a 1/3 twill, a point twill or a broken twill hides, a 2/2 twill hides too. Its explanation was symmetry — a plan with more ways to carry one writing of its cloth onto another has more ways to absorb a mistake — and its last paragraph named the place that explanation would be tested.

An eight-shaft loom is pegged with satins as well as twills. A twill’s rows are one row slid a shaft a pick; a satin’s are one row slid three or five. Those are different slides and neither contains the other, so there was every reason to expect that a satin and a twill would hide different errors, that neither would hide the other’s, and that no single plan would sit above them all. A mill changing between a satin and a twill on one warp would then need a check for each.

The expectation is wrong twice over.

One end on each of eight shafts #

The four-shaft census put eight ends on four shafts, two to a shaft. On an eight-shaft loom the corresponding repeat puts one end on each shaft — the fewest ends that need all eight shafts — so a threading is an order of the eight shafts, and the straight draw, the skip draws and the satin draws are all of this kind. There are 40,320 such orders, and a threading started at another end is the same threading, so the census takes the 5,040 that begin on the first shaft and lets them stand for the rest.

A double error moves two ends each to a wrong shaft. Twenty-eight pairs of ends, seven wrong shafts for each, give 1,372 double errors a threading, and seven million in all. Twenty-eight of each threading’s errors are exchanges, in which two ends take each other’s shafts; the rest leave one shaft with two ends and another with none.

Each error is woven under each plan and sorted the way the four-shaft census sorted them: caught if some end or pick no longer interlaces, silent if the drawdown is a writing of the meant cloth — the same cloth slid, turned half round or turned over, which how many cloths are there counts as one cloth — and different otherwise.

Twelve ways to peg eight shafts #

Seven are twills, chosen to span what a twill’s row can be. The 1/7, 2/6 and 3/5 are the plain warp-faced twills. The 4/4 is the balanced one. The 1/2/3/2 is a fancy twill whose row reads the same backwards; the 3/1/1/3 and 2/1/1/4 are fancy twills whose rows do not. Two are satins, the only regular eight-end satins there are, of move three and move five — the count that leaves no six-end satin leaves exactly these two at eight. Two are interrupted twills, broken and point, made from the 4/4 by turning its second half. And one has no symmetry at all: eight fixed rows, every shaft and pick lifted and lowered, no row a slide of another.

A plan that lifts its shafts in identical pairs — a 2/2 twill pegged on eight shafts lifts shaft one with shaft five — is a four-shaft cloth on an eight-shaft loom and hides every exchange between paired shafts trivially. It is left out for the reason plain weave and hopsack were left out at four shafts.

Only an exchange can hide #

The first result is a simplification, and it comes before any plan is compared with another.

Every silent error under every plan is an exchange. An error that puts two ends on one shaft and leaves another shaft empty is never silent here: the drawdown then has two identical columns and is missing a third, and a writing of a cloth whose eight columns all differ cannot have two the same. Several plans go further and catch every such error outright. Under the 1/7 twill and both satins each pick lifts exactly one shaft, so an empty shaft leaves one pick lifting nothing at all, and all 6.8 million non-exchange errors fall out of the cloth before anybody needs to look at it.

The plans differ sharply in how they show the rest. The 2/6 twill lifts two neighbouring shafts each pick, so a pick goes bare only when an error empties both of them; that happens in 21 per cent of the non-exchange errors, which it catches outright, and it weaves the rest as different cloths. The 3/5 and 4/4 twills, and every fancy twill, catch none: with three or more shafts lifted every pick, an empty shaft never leaves a pick bare, and every such error is woven, as a cloth that is merely wrong. A check under a plan that lifts one shaft a pick fails loudly; one under a plan that lifts half the shafts fails quietly, by weaving a cloth that has to be compared with the one intended.

So the silent errors are a small set — at most 1,888 of 141,120 exchanges, one in seventy-five — and about a quarter of all threadings carry one under the plans that hide most. The question of nesting is a question about that small set.

A satin is a twill with its shafts renumbered #

The first half of the expectation fails on a fact about the satin’s plan rather than about any threading.

The satin of move three lifts shaft one on the first pick, shaft four on the second, shaft seven on the third, and so on round. Read its shafts in that order and it is the 1/7 twill. So any threading woven under the satin weaves exactly what the renumbered threading — every shaft number multiplied by three, counted round eight — weaves under the twill. The satin’s slide by three is the twill’s slide by one seen through a renumbering of the shafts.

That alone would only say a satin hides the same number of errors as a twill, at renumbered threadings. The census says more: the satins and the 1/7 twill hide the same errors at the same threadings, all 1,536, error for error. So do the 2/6, the 3/5 and the 1/2/3/2 twills. The reason is in what a silent error turns out to be.

Every silent error is a copy #

The example is typical. Exchanging the first and eighth ends of 1 2 3 4 6 7 8 5 gives 5 2 3 4 6 7 8 1. Read the meant threading backwards — 5 8 7 6 4 3 2 1 — and replace every shaft number n by 10 − n, counted round eight, and the result is 5 2 3 4 6 7 8 1: the mistaken threading. The error has turned the threading into a copy of itself, read the other way along the ends with its shaft numbers reflected.

Every silent exchange in the census is a copy of one of four kinds. A threading can be read from another end or backwards along the ends, and its shaft numbers can be shifted by a constant or reflected:

• slid, shafts shifted — 256 exchanges in the census make this kind;
• reversed, shafts reflected — 1,536, and every one of the 256 is among them;
• reversed, shafts shifted — 192;
• slid, shafts reflected — 192.

That makes 1,888 exchanges that copy the threading, of 141,120. And the census checks the other direction for every plan: no exchange that is not a copy is silent under any plan. A double error is hidden only when it has turned the threading into a relabelled or reversed version of itself, which the cloth then shows as a writing of itself.

Renumbering the shafts by three or five carries a copy of each kind to a copy of the same kind — a shift stays a shift, a reflection a reflection — which is why the satins hide the same exchanges as the twill and not merely the same number of them.

The plan’s row decides which copies it hides #

Which copies a plan hides depends on its row.

Every plan that is one row slid each pick hides whole kinds, never part of one. Which kinds is read straight off the row, whose own symmetries are the one-dimensional case of the seventeen groups a draft can have:

• Any slid row hides the slid-and-shifted copies. Shifting every shaft number by one is sliding the plan’s cloth by a pick, and that is a writing. The 2/1/1/4 twill has nothing more than this and hides 256.
• A row that reads the same backwards — 1/7, 2/6, 3/5, the 1/2/3/2, and the satins’ single lifted shaft — also hides the reversed-and-reflected copies, because reflecting the shaft numbers and reading the ends backwards turns the cloth half round. That adds 1,280 and makes 1,536.
• A row whose reverse is its counterchange, as the 3/1/1/3’s is, hides the slid-and-reflected copies instead, because its cloth reflected is its cloth turned over. That adds 192 and makes 448.
• The 4/4 twill’s row has every one of these properties. It reads the same backwards; slid four places it is its own counterchange; reversed it is its counterchange slid. It hides all four kinds, all 1,888 copies there are.

The interrupted twills behave differently, as interrupting a slide should. The broken and point twills hide the reversed-and-shifted copies whole and only parts of the other kinds — 384 and 576 errors — because each copy is absorbed in some of their picks and not in others.

The order has a top, and it is the balanced twill #

That accounts for the nesting diagram completely. The order is not total: the 3/1/1/3 hides 192 slid-and-reflected copies the satins do not, and the satins hide 1,280 reversed-and-reflected copies the 3/1/1/3 does not. The broken and point twills sit apart from the slid plans, the broken inside the point. But the order has a top, because one plan hides every copy that exists, and every error any plan hides is a copy. Whatever is silent anywhere is silent under the 4/4 twill.

At four shafts the top was the 2/2 twill, the balanced twill of that loom; at eight it is the 4/4, the balanced twill of this one. In both, the plan at the top is the one whose row is half up and half down, because only such a row can be its own counterchange slid, and that is the symmetry that adds the last two kinds.

The slide is the smaller hiding place #

The four-shaft account explained a silent error by the twill’s slide: an end moved to the next shaft, the cloth slid a pick, and nothing visible. At eight shafts that explanation covers a sixth of what happens.

Under the 1/7 twill, 256 silent errors are hidden by a slide and 1,280 by a half turn. The plan’s row reading the same backwards hides five times as many mistakes as its slide does. The difference is in what each copy asks of the threading. A slid-and-shifted copy asks that exchanging two ends reproduce the whole threading with every shaft number moved along by one constant and every end moved along by another — a coincidence along all eight ends at once. A reversed-and-reflected copy asks only that the threading, read backwards with its numbers reflected, differ from itself in two places: that it be one exchange away from a mirror symmetry. The census finds five threadings-and-exchanges of the second kind for every one of the first.

That is where an eight-shaft error hides: in the row’s mirror. The practical consequence is that a check woven under a plan whose row reads the same backwards — every plain twill, both satins, the symmetric fancy twills — is blind to every exchange that turns the threading into its own mirror image.

What a check under each plan proves #

The census reads as a guide to checking a warp before a cloth is woven.

A warp checked under the 4/4 twill has passed the weakest check there is among these plans: every exchange any of them would hide, it hides. A warp checked under a satin has passed exactly the check a 1/7 twill gives, and re-pegging from one to the other shows nothing new — so a mill weaving satin and plain twill from one warp gains no assurance from having seen both.

A warp checked under the 2/1/1/4 twill has passed a much stronger check, hiding only the 256 slid copies — and a different cloth from the one intended still has to be noticed, which which weave hides a fault found the eye does unevenly; and one checked under the unrelated plan has passed the strongest: no double error is silent under it. The proof plan of the four-shaft account carries over unchanged, with one difference. At four shafts 864 pairs survived every plan because they were the threading started at another end, not errors at all. With one end on each shaft no exchange can start a threading elsewhere — that would change every digit — so at eight shafts the unrelated plan exposes every exchange without exception.

And because the hidden errors are copies, a weaver does not need the census to find them. An exchanged pair is silent under a symmetric twill or a satin exactly when the threading, read backwards with its shaft numbers reflected, is the threading as entered. That is a check on the threading itself, made on paper, before a pick is woven — the twin of the shortest notation has the longest mistakes, where one digit of a threading decides a whole column.

How the census was taken #

Every threading that begins on the first shaft and uses each of eight shafts once — 5,040 — and every double error in it, 1,372 each, woven under each of twelve stated plans. A drawdown is eight bytes; an error changes two columns, which is two bits a byte. A silent error is one whose drawdown is among the 256 writings of the meant drawdown — 64 slides each read as it is, turned half round, turned over, or turned along the picks and counterchanged — built once a threading and looked up. A cheap test passes first: a writing has the same counts of lifted shafts in its rows as the meant cloth, or their complements, and an error whose counts differ is not looked up.

Each silent error is labelled with the first kind of writing that absorbs it, and each exchange is tested against the four kinds of copy — the threading read from each of eight ends forwards or backwards, with every shaft number shifted by each of eight constants or reflected about each.

Required of it: the cheap test never discards a silent error, checked against a full walk without it on the first two hundred threadings under two plans; every plan’s silent errors are copies; every slid plan hides whole kinds and the interrupted twills one kind whole; the satins and the four symmetric twills hide identical sets; the 4/4 twill’s set contains every other; the unrelated plan’s is empty; and every error is classed once.

What the census cannot say #

One end on each shaft only. A repeat of sixteen ends on eight shafts, two to a shaft, has the four-shaft census’s richer structure — exchanges between two ends on one shaft, threadings that are their own copies started elsewhere — and its counts will be larger. Whether the 4/4 still sits on top there is a further census; the argument that the top is the plan with every row symmetry suggests it should.

Twelve plans. They span the row symmetries a slid plan can have and include the two common interruptions, but there are many more eight-shaft plans than twelve, and how many twills a repeat admits counts how many the twills alone come to. A plan whose row has no symmetry beyond its slide behaves like the 2/1/1/4; one with every symmetry like the 4/4; the census does not walk the rest.

Double errors. A single wrong digit cannot be silent with one end on each shaft — it leaves a shaft empty — and triple errors are not walked. The copy argument applies to them unchanged: a triple error is silent only if it makes a copy, and the plan’s row decides which copies it hides.

And it says which plans hide an error, not where the error is. That a drawdown has shown an error says the threading is wrong; three mistakes and the shape each one leaves is about reading where.

Who worked out which part #

Checking a threading by weaving it is as old as the loom, and that a satin is a twill with its shafts reordered is in effect how weavers have always drafted satins — by counting a move across the shafts. The census of eight-shaft silent errors, the finding that every silent error is a copy of the threading, the sorting of copies into four kinds by the plan’s row, and the nesting with the 4/4 twill at its top were done here, extending the four-shaft crossing in the proof-plan account and the one-plan census of silence lives in the lifting plan.

Still open: whether the balanced twill stays on top at two ends a shaft #

The census here walked the eight-shaft repeat with one end to a shaft. The repeat a mill most often enters on an eight-shaft loom is sixteen ends, two to a shaft, and there an exchange between the two ends on one shaft changes nothing, a threading can be its own copy started half a repeat along, and the counts are of a different order: 16!/(2!)^8 orders before any symmetry is divided out. Whether every silent error there is still a copy of the threading — and so whether the 4/4 twill, with every row symmetry, still hides everything any other plan hides — would say whether one proof plan, checked once, certifies the threading a mill actually enters.