A mispick is one row in the wrong place

A loom makes its cloth one pick at a time, and each pick is laid into whatever shed the harness has opened, then beaten up against the fell. If the harness opens the wrong shed, the pick is still laid, still beaten up, still in its proper place. Nothing has broken and nothing is missing. The cloth is simply not the cloth that was designed.

That makes a mispick a very clean object to reason about: it is one row of the matrix replaced by another row, with everything else untouched. Which means it can be applied to every matrix, exhaustively, and the results counted.

Every four-by-four draft in which each end and each pick interlaces — 22,874 of them — with each of its four picks replaced by each of the sixteen possible sheds, the correct one excluded. 1,372,440 substitutions.

The claim #

Of every way a four-by-four cloth can be given one wrong pick, 63.2 per cent leave a cloth that still hangs together, 36.6 per cent leave a thread with nothing holding it, and 0.25 per cent split the cloth into layers.

The last of those is the interesting one, because a missing end never splits a four-by-four cloth at all. The smaller fault is the one that can break the cloth, and the reason is exact.

The argument for the inversion #

The criterion this collection uses reads every crossing as a relation — this thread passes above that one — and asks whether the resulting digraph is strongly connected. A cloth is one cloth exactly when every thread can be reached from every other by going upwards.

Deleting a thread deletes relations. Every remaining relation still points where it pointed, so a path between two surviving threads either survives or loses a step it did not need. Nothing is turned round, and at this repeat nothing is stranded.

Substituting a pick reverses relations. Wherever the wrong shed disagrees with the right one, a crossing that had the warp above now has it below, and the arrow in the digraph turns round. A reversed arrow is exactly what a separation is made of: it is how a set of threads comes to be above everything it crosses.

So the two faults are not larger and smaller versions of the same thing. They are different operations on a directed graph, and only one of them can partition it.

What was counted, and how #

Sixteen rows are generated, each pick of each draft is replaced by each of them, the one that matches the original is skipped, and every result is put through the same three questions: is everything still interlaced, is what remains one cloth, and what is its longest float.

Three things are asserted while it runs.

• The outcomes sum to the trials, which is the census times four picks times fifteen wrong sheds.
• Some substitutions separate the cloth, so the outcome is reachable — and this is the assertion that carries the essay’s claim, since it is what distinguishes a substitution from a deletion.
• A float grows somewhere, which catches an arithmetic slip in the run-length measurement.

The whole sweep is about five seconds of work, which is the same order as the collection’s other censuses and is memoised so that every figure drawn from it reads the same numbers.

The same wrong shed is harmless here and fatal there #

The average over a million substitutions is not the useful statistic. What matters is that the outcome depends almost entirely on which draft the mistake was made in, and hardly at all on how wrong the shed was.

A shed differing from the correct one in a single end can split a cloth. A shed differing in three of four can leave everything sound. Nothing about the size of the mistake predicts the size of the fault — which is the same lesson the three places a mistake can be made delivers at the scale of the piece, arriving here at the scale of the repeat.

The practical form of that is uncomfortable for anyone hoping to reason about faults by inspection: a weaver looking at a mispick can see how far the pattern is disturbed and cannot see whether the cloth has come apart, and the two are not related.

The single reversed intersection #

The finest-grained version of a mispick is a single wrong lift: one end that should have risen and did not, in one pick. Over the whole census that is 365,984 reversals, and the counts are their own small essay.

Two hundred and sixty-nine thousand five hundred and sixty-eight leave a sound cloth. Ninety-five thousand leave a thread loose. One thousand one hundred and fifty-two split the cloth. Ninety drafts are immune to every one of their sixteen possible reversals.

The float arithmetic is the sharper result. A reversal in the middle of a float shortens it; a reversal at a binding point joins the two floats it was separating, so a 3/1 twill whose longest float is three can be given one of seven by a single wrong intersection. That is the mechanism behind the census result the last essay in this ladder rests on, and it is why the worst case for a weave is so much worse than its average case.

What it does to the float #

The structural outcomes are one half of what a mispick is; the visible mark is the other, and the quantity behind the mark is the float.

Of the substitutions that leave a sound cloth, a great many lengthen the longest float — and the mechanism is the one the single-reversal figure shows. A reversal does not stretch a float. It joins two of them. An intersection that was binding a thread down was separating two runs of that thread on the surface, and turning it round removes the separation, so the two runs become one and the new float is the sum of them plus the intersection itself.

That is why the worst case for a weave is so much worse than its typical case. A 3/1 twill’s longest float is three; the worst single reversal in it produces one of seven, because it joins two threes across the single binding point between them. A weave with long floats separated by single binding points is maximally exposed to this, and that describes almost every weave anybody uses.

Why a loom makes this mistake at all #

The structural argument is indifferent to how the wrong shed came about, but the ways it can are worth naming because they decide what the fault looks like down the piece.

A single failure repeats. A dobby peg missing from one row of the chain, a card punched wrongly, a lag set incorrectly: each of those produces the wrong shed every time that row comes round, so the fault is not one pick but every fourth pick, or every eighth, for as long as the chain runs. That is a periodic fault, and a periodic fault is worth an order of magnitude in visibility over a random one.

A momentary failure does not. A heald that sticks, a shed that fails to clear, a pick laid while the harness was still moving: those give one wrong pick and then correct themselves, and what results is a single bar across the cloth.

So the same structural fault — one row of the matrix wrong — arrives in two forms that differ enormously in what they cost. The first is a design fault that will run the length of the piece and can be corrected by re-pegging; the second is an accident that costs the width of the cloth times a pick, which is almost nothing.

Which sheds are dangerous, which is the opposite of the guess #

The census reports outcomes over every draft and every wrong shed at once, and the average hides a distribution that runs against every expectation about it.

Sort the substitutions by how wrong the shed was — how many of the four intersections it got the wrong way round:

A pick laid in the exact complement of its own shed — every intersection reversed, which is the largest mistake a single pick can be given — never splits the cloth. A shed wrong in one end alone splits it three times in a thousand.

The reason is the same one that made a missing end harmless. Reversing every relation of one thread keeps that thread consistently related to everything it crosses: it is now under what it was over and over what it was under, which is a thread that has changed sides rather than a thread that disagrees with itself. Reversing some of them is what leaves a thread above part of the cloth and below the rest, which is precisely the condition a separation is built from.

So the ordering of danger is: a nearly-correct shed is the worst, a wholly reversed one is safe, and the loose-thread outcome runs the other way entirely — the more wrong the shed, the likelier the pick it lays is on the same side all the way across and held by nothing.

One coincidence in the table is not explained here. The complement row’s three counts — 50,736 sound, 40,760 loose, none separated — are the counts of the missing-end census exactly, to the last unit, over the same number of trials. The two operations are different and the mechanisms producing their loose threads are different: a complemented pick strands an end, and a deleted end strands a pick. That they produce identical totals looks like a bijection through the census’s own complementation symmetry rather than an accident, and nothing here has proved it.

Why the nearly-correct shed is the dangerous one #

The table’s ordering is the essay’s most counter-intuitive result and it is worth putting the mechanism in one sentence, because once it is said the ordering stops being surprising and starts being obvious.

A separation needs a set of threads that lies consistently above everything outside it. Consistency is the whole requirement: a thread that is above some of its crossings and below others cannot be part of a layer, because it is tied to both sides.

A complemented pick is consistent. Every one of its crossings has been reversed, so the pick that was under all four ends is now over all four, and it has simply changed sides. It is still related to every end in the same direction as before, with the sign flipped, and a uniform flip of one row’s relations cannot partition a graph that was connected.

A pick wrong in one place is inconsistent. Three of its relations agree with the design and one does not, and that single disagreement is what can leave an end above the picks on one side of it and below on the other. The disagreement is the raw material of a layer, and one is enough.

So the danger is not proportional to the error’s size; it is proportional to the error’s heterogeneity. A wholly wrong pick and a wholly correct one are both consistent and both safe; everything between them carries some inconsistency, and the row with one intersection wrong carries the most concentrated form of it.

That also explains the second column running the other way. A loose thread is a thread nothing passes over, which is a statement about a whole row or column being on one side — so a pick laid in a shed that lifts every end is a pick under everything, held by nothing, and the more uniformly wrong a shed is the likelier that becomes. The two failure modes have opposite relationships to uniformity, which is why no single measure of how wrong a shed is predicts either of them.

The practical form is a warning about inspection. A weaver can see how far a shed departed from the design, and that is the quantity the table says is least informative. What matters is whether the departure was uniform across the width, which is not a thing anybody looks at and which is the difference between a bar in the cloth and a cloth in two pieces.

Where the model stops #

A substituted pick is treated as a permanent change to the repeat. It is not: a single mispick disturbs one row and the draft returns to normal afterwards, so what the census examines is a cloth in which every repeat has that pick wrong — which is the periodic case above, not the accidental one. For the accidental case the right object is a tiling with one row disturbed, which is what the single-reversal figures use and which the census does not.

A missing pick and a doubled pick are not covered. Both are ordinary weaving faults, both change the number of rows rather than their contents, and both need the spacing arithmetic that a matrix does not carry. A doubled pick is two threads in one shed, which the encoding cannot express at all.

The criterion remains topological. It says a pick held by one crossing is attached and does not ask how firmly, so a substitution that leaves a pick held by a single crossing is reported as sound. In a slippery yarn that pick will work its way out, which is a question about friction rather than about the matrix.

And the enumeration is at four by four, which is small enough that a wrong shed differs from the right one in at most four places. At a larger repeat there are far more nearly-correct sheds and the distribution of outcomes must shift towards the harmless end; nothing here has measured how fast.

The generalisation #

In any system built out of directed relations, removing an element degrades and reversing one can partition. That is the transferable result, and its practical form is a priority: look for the components that are working backwards before looking for the ones that are missing, because a missing component announces itself and a reversed one does not.

The asymmetry has a second half worth carrying. A deletion can be detected by counting; a reversal cannot. Anything that inventories a system’s parts will find the missing thread, and nothing that inventories will find the crossing that goes the wrong way — every part is present and correct, and only the relation is wrong. That is why the check this collection makes is a check on connectivity rather than on contents.

Who found it, and when #

Mispicks are as old as weaving and their appearance is entirely familiar. The counting is not, and could not have been done before the criterion existed: what a wrong shed does to the integrity of a cloth is a question that needs a decidable test, and the test is recent.

The inversion — that a substitution can break a cloth and a deletion cannot — appears to be new here, and it is the sort of result that is obvious once stated and unavailable until the enumeration is run. It also settles something about how faults ought to be ranked: the trade ranks them by how much cloth they condemn, which puts a broken end far above a mispick, and the structural ranking is the other way round.

Where the ladder goes next #

The scale changes next: from what a mistake does to a repeat, to where in the loom the mistake was made and what shape it leaves on the piece. A threading error, a lifting error and a tie-up error are the same size of mistake and differ by four orders of magnitude in how much cloth they touch.

Sideways, the float a reversal creates is the quantity that decides whether anybody sees the fault, and asking every draft which of them keeps that float shortest gives an answer that disagrees with the trade’s rule.