A proof plan has no slide in it

Silence lives in the lifting plan walked every threading of eight ends on four shafts and found that under a 2/2 twill’s lifting plan, nine threadings in ten can be mis-threaded by two digits without the cloth showing it. The cloth woven is another writing of the one meant: the same pattern slid along the picks. A mill checks a threading by weaving a sample, and the sample looks right.

That essay ended on the consequence that makes a silent error matter. The threading is still wrong, and the next lifting plan may show it. A warp stays in the loom while its plan is changed; that is how several cloths are woven from one warp. The question a mill would ask is which plans show the error, and the census had what it needed to answer: the silent pairs under one plan, and a drawdown for any other. This essay crosses the two.

Nine ways to peg the same four shafts #

A four-shaft loom can be pegged for many cloths over a repeat of eight picks. The census uses nine, chosen because they are the ones a four-shaft warp is ordinarily re-pegged between, and because between them they span the kinds of symmetry a plan can have.

Four are twills: the 2/2 twill in both directions, and the 1/3 and 3/1 twills, each row the row before slid one shaft along. Two are twills with the slide interrupted: the broken twill, which runs two rows forward and then two with the order turned, and the point twill, which runs its rows forward and back. Two lift the shafts in identical pairs: plain weave, shafts one and three together and two and four together, and the hopsack, one and two together and three and four. The last is a plan of eight unrelated rows, every pick and every shaft lifted and lowered and no row a slide of another — the plan the earlier census set against the twills and found almost silent-proof.

The crossing #

For each starting plan, the census takes every threading of eight ends on all four shafts that weaves a whole cloth under it, and every double error — two ends each moved to a wrong shaft — whose cloth is another writing of the one meant. Each such pair is woven again under the eight other plans and classed the way a second wrong digit classed errors: caught if some end or pick stops interlacing, different if the drawdown is another cloth, silent if it is still a writing of the same one.

Under the 2/2 twill there are 99,872 silent pairs on 36,320 threadings. Under the 1/3 twill, 56,352; under the broken twill, 29,888; under the point twill, 66,208. The reversed twill and the 3/1 are not taken as starting plans, for a reason the crossing itself supplies below. Every silent pair is classed under every other plan, and every classification adds up to the pair count it came from.

A mirror and a complement expose nothing #

The first thing the hero table says is where it is empty.

The 2/2 twill run the other way exposes none of the 2/2 twill’s silent errors, not one in 99,872. The reversed plan weaves the same cloth turned over, so a pair of drawdowns that are writings of one cloth under the first plan are writings of one cloth under the second: the relation how many cloths are there divides out includes the turn. The 3/1 twill exposes none of the 1/3’s, for the same reason in a different form: the 3/1 lifts exactly the shafts the 1/3 leaves down, so its cloth is the 1/3’s counterchanged, and a counterchange is a writing of the same cloth too.

Plain weave exposes none of anything. On four shafts, plain weave lifts one and three together and two and four together, so it is a two-shaft cloth with two shafts’ worth of choice in it. Every threading error a twill hides is an exchange among shafts, and plain weave cannot tell shaft one from shaft three. A warp that has been checked under a twill and is then woven plain has had its errors made harmless rather than found, which is a good outcome for the plain cloth and no information about the threading.

So three of the eight re-peggings are blind by construction. A mill that re-pegs a 2/2 twill warp for the same twill the other way, for a 3/1 or 1/3, or for plain weave, will not see a silent error it did not see before, and it should not expect to.

The broken twill exposes seven in ten #

The plans that do expose are the ones whose symmetries differ from the twill’s.

A broken twill exposes 70 per cent of the 2/2 twill’s silent errors. Its rows are the twill’s rows with the second pair reversed, so it has no single slide running through the whole repeat: the move that let a 2/2 twill absorb an error — “this end on the next shaft, with the whole cloth slid a pick” — works in half the broken twill’s picks and fails in the other half. The 1/3 twill exposes 44 per cent, the point twill 34 and the hopsack 34. The point twill is the weakest of the twill-like plans at exposing because it is a twill for most of its picks: it runs forward four rows and back four, and a slide that fits the forward run fits the backward run read the other way.

The example is a threading of the kind a four-shaft loom is entered with, every shaft used twice. Exchanging its first and fifth ends puts a shaft-two end where a shaft-three end should be and the reverse. Under the 2/2 twill plan the mistaken drawdown is the correct one slid along the picks, and the sample looks right. Under the broken twill plan the slide no longer carries one onto the other, and the cloth has a visible fault in two columns. The weaver who wove the sample under the twill and then re-pegged for a broken twill would find the fault in the first repeat of the new cloth.

The plans nest #

The table’s zeros are not scattered. They fall in a pattern, and the pattern is an ordering.

Every error silent under the 1/3 twill is silent under the 2/2 twill. So is every error silent under the point twill, and every error silent under the broken twill is silent under the point twill as well. The 2/2 twill sits at the top with 99,872 silent pairs, containing the point twill’s 66,208 and the 1/3’s 56,352, and the point twill contains the broken twill’s 29,888. The 1/3 and the point are not nested either way: each hides some pairs the other exposes.

The ordering has a simple reason. A threading error is silent when some symmetry of the plan’s cloth carries the mistaken drawdown onto the meant one, and a plan with more symmetry has more ways to do it. The 2/2 twill’s plan is the most symmetric of the four: it is a slide by one pick for each shaft, it is its own complement, and it is its own turn. The 1/3 has the slide and not the complement; the point twill has part of the slide and a reflection; the broken twill has less than either. Each step down loses symmetries and so loses silent errors, and none gains a symmetry the one above lacks.

The practical reading is the one that matters. A threading that has passed a woven check under a 2/2 twill has passed the weakest test there is among these plans. Any error it still carries will be hidden by every other twill a mill is likely to peg next, except the broken twill, the hopsack and the 1/3, and it is those three plans a mill is most likely to find it under.

Most errors are shown by three of the eight #

A silent error is not either exposed or hidden once and for all. It is exposed by some plans and not others, and counting how many of the eight other plans expose each pair says how exposed a typical error is.

Of the 2/2 twill’s silent errors, half are exposed by exactly three of the other plans, a quarter by four, and a tenth by five. None is exposed by more than five, because three of the eight are always blind to a twill’s errors. The 1/3 twill’s silent errors are exposed mostly by three; the broken twill’s by two or three. The distributions have a thin left tail — a few per cent of each plan’s silent pairs are exposed by one plan or by none.

The 864 that are not errors #

In every row of that figure the leftmost sliver is the same 864 pairs. They are silent under the 2/2 twill, the 1/3, the broken and the point, and under every plan they are re-woven under, including the plan of unrelated rows that exposes 99 per cent of everything else.

They are silent under every plan because they are not errors in the warp. Exchanging the first and fifth ends of 4 2 1 1 3 2 1 1 gives 3 2 1 1 4 2 1 1, which is the original threading read from its fifth end round to its fourth. Under any lifting plan whatever, the two threadings weave the same cloth started four ends apart, which is the same cloth. A census that counts drawdowns up to their writings counts them as errors because two digits changed; a weaver would not, because nothing about the warp is wrong.

So the plan of unrelated rows exposes every silent twill error that is really an error. It is a complete proof plan for four-shaft threadings of eight ends: a sample woven under it shows every double error that a 2/2 twill sample hid, except the ones that do not need showing.

A proof plan, and what it costs #

That suggests a practice the mill already half has. A warp is often woven for a few centimetres under a plain or twill plan to check it before the pattern starts. Weaving those centimetres under a plan with no slide in it instead — eight rows, each shaft lifted and lowered, no row a slide of another — would expose every silent double threading error a twill check misses.

The cost is one repeat of a cloth nobody wants, eight picks, which is what any check sample costs. The rule for choosing the plan is short: it must have no slide, no reflection and no complement among its own symmetries, so that no writing of its cloth other than the cloth itself can be reached by two wrong digits. The broken twill is the nearest ordinary plan to that and gets seven in ten; a deliberately unrelated plan gets all of them.

This is the practical close of the argument the shortest notation has the longest mistakes began. A threading digit is the most dangerous symbol in a draft because one digit changes a whole column. Three mistakes and the shape each one leaves followed the ones that show. What is left are the ones that hide, and they hide in a plan’s symmetries. The remedy is to weave under a plan that has none.

The same argument, turned through a right angle #

A draft has two halves, and everything above is about errors in one of them. A lifting plan says nothing without a threading: the cloth is the plan read through the threading, rows through columns, and the relation is symmetric. A threading error replaces a column; a mispick is one row in the wrong place replaces a row.

So the proof-plan argument has a twin. A mispick is silent when some symmetry of the cloth carries the mistaken drawdown onto the meant one, and the symmetries that do it are now slides along the ends, which a straight or regularly stepped threading supplies. A threading with no slide in it — entered so that no end is a shift of another’s pattern — is a proof threading for lifting-plan errors, as a plan with no slide is a proof plan for threading errors. The crossing here has not walked that twin; the argument transposes, and the counts would be the same census with rows and columns exchanged.

It also says why the two kinds of check are usually done in the order they are. How many shafts a draft needs is decided by the threading, and a threading is fixed for the life of the warp while the plan can be re-pegged in minutes. A proof plan is cheap because it is a plan. A proof threading would cost a re-entered warp, which is why nobody makes one, and why a mispick is caught by looking at the cloth — where which weave hides a fault says what the eye will and will not see — rather than by any designed check.

How the crossing was taken #

Nine plans, each stated. Every threading of eight ends that uses all four shafts is enumerated — 40,824 — and for each starting plan those that weave a whole cloth are walked, with every double error tried: each of the 28 pairs of ends moved to every pair of other shafts. An error is silent under a plan when its drawdown lies among the 256 writings of the meant drawdown, built once per threading and plan as eight-byte keys, exactly as the one-plan census built them.

Each silent pair is then woven under each other plan and classed as caught, different or silent, with the threading set aside for that plan if the correct threading does not weave a whole cloth under it. None was set aside: every threading that weaves under one of these plans weaves under all of them.

Four plans are started from. Plain weave and hopsack are left out as starting plans because they lift shafts in pairs, so over a million pairs of errors between paired shafts are trivially silent under them and say only that they are two-shaft cloths. The reversed twill and the 3/1 are left out because they are the 2/2 turned over and the 1/3 counterchanged, and the crossing confirms that neither exposes a single error of its partner’s.

What is required of it. The 2/2 twill’s silent pairs must be the one-plan census’s to the pair. A mirror and a complement must expose nothing. Every classification must add up. The same 864 pairs must survive every plan from every start. And the plan of unrelated rows must expose all but under two per cent of the 2/2 twill’s.

What the crossing cannot say #

It is four shafts over eight ends and eight picks. Longer repeats and more shafts will have plans with more and fewer symmetries, and the ordering found here — more symmetry, more silence — should carry over; the counts will not. A proof plan for eight shafts would be the same idea with a larger search for a plan with no symmetry at all, and how many twills a repeat admits suggests there are many such plans to choose from.

It is double errors only. A single wrong digit under a twill plan is silent one time in five hundred, and a triple error is a combination the census has not walked. The proof plan’s argument — no symmetry, so nothing to hide in — applies to them as well, but it is an argument and not a count.

And it says which plans expose an error, not where. A drawdown that differs from the meant one says that the threading is wrong, and a weaver then has to find which ends. Two columns differ in the example above, and in the census they are always the columns of the moved ends, but reading which shafts they should have been on is a second problem the crossing leaves alone.

Who worked out which part #

Checking a threading by weaving is as old as looms, and the risk that a sample looks right with a wrong entry is hand-weavers’ lore, usually told of point drafts. The census of silent errors under one plan is this account’s, from the four-by-four walk and its eight-end extension.

What is added here is the crossing: the matrix of which plans expose which plans’ silent errors, the nesting it reveals, the recognition that the errors surviving every plan are rotations of the threading and not errors, and the proof plan that follows — a check weave with no symmetry in it.

Still open: whether the order of symmetries is total at eight shafts #

At four shafts the plans nest by their symmetries, with the 2/2 twill at the top and the broken twill at the bottom. An eight-shaft loom has far more plans, among them twills of every step and satins whose rows are slides by a move of three or five, and the symmetries of those plans are not ordered simply: a satin’s slide and a twill’s slide are different slides, and neither contains the other.

Whether silence still nests at eight shafts — whether every plan’s silent errors sit inside some single most-forgiving plan’s, as they do inside the 2/2 twill’s here — would decide whether one check under one plan can ever certify an eight-shaft threading, or whether a mill that changes between a satin and a twill on one warp needs a proof plan for each.