Which weave hides a fault

Every weaving text says it and every weaver believes it: a busy weave hides a fault. Choose a twill or a crepe for a cloth that has to be made in a hurry, on old machinery, from indifferent yarn, and the small mistakes will disappear into the pattern.

The claim is about the eye, and this collection does not have one. What it has is the float — the length of thread lying on the surface between one binding point and the next — which is the quantity behind lustre, snagging, abrasion and everything else that makes a surface look and behave as it does. So there is a defensible proxy: a fault is visible when it puts a length of thread on the surface that the weave does not otherwise produce.

Measured that way, the trade’s rule is wrong, and it is wrong in an unusually complete manner.

The claim #

Ninety-nine point six per cent of all weaves hand a single wrong intersection a float of seven ends. The ninety that do not are the plain weave and the ribs — the least busy weaves there are.

The distribution has exactly two values in it. Not a spread, not a tail: two. And the weaves in the good column are precisely the ones nobody recommends for hiding a fault.

A two-valued distribution over twenty-two thousand drafts is the part that should be surprising, because nothing about the question suggests it. The float a wrong intersection produces looks like the sort of quantity that would take every value from one to seven depending on what the neighbouring ends were doing, and it does not: it takes seven, or it takes one, and the census is over ninety-nine per cent the first. Whatever is going on is a property of nearly every draft rather than a property of particular ones, which is a different kind of finding from a ranking and is why the essay can state it as a claim rather than as a table.

And the direction of the answer is the reverse of the trade’s. A busy weave is recommended for hiding faults because a busy surface hides everything, which is true of dirt and of small irregularities and is not true of this. A wrong intersection in a busy weave breaks a float that was there to be broken; a wrong intersection in a plain weave has nowhere to go, because a plain weave has no float longer than one and a wrong crossing is simply a crossing the other way. So the weave with nothing to disturb is the weave that a disturbance cannot enlarge.

The argument, which is about joining rather than stretching #

The reason is one sentence long once it is seen.

A reversal does not lengthen a float. It joins two of them.

An intersection that binds a thread down is separating two runs of that thread on the surface. Turn it round and the separation is gone, so the two runs become one — and the new float is the first run, plus the second, plus the intersection that used to divide them.

A 3/1 twill has floats of three, separated by single binding points. Reverse one of those binding points and the result is 3 + 1 + 3 = seven. The weave’s own longest float was three; one wrong lift has more than doubled it.

That is the mechanism, and it explains why the census has only two answers. Almost every draft has runs separated by single binding intersections somewhere, because that is what a float-bearing weave is; and wherever that pattern occurs, the worst reversal produces the sum of the two runs it divides.

Which weaves escape, and why #

The ninety that hold the fault to a float of three are the drafts whose surface runs are short and evenly spaced, so that no single reversal has two long runs to join.

• The plain weave and its complement — two drafts. Every intersection binds, no run is longer than one, and a reversal joins 1 + 1 + 1 = 3.
• Eighty-eight rib and block drafts. Warp ribs, weft ribs, baskets and the uneven partitions between them. Their binding regions are two intersections wide rather than one, so reversing a single intersection does not remove a boundary — it narrows it, and the two runs stay separate. The float grows by one, from two to three, and no further.

That second reason is the general one and it is worth stating on its own: a weave escapes the join when no boundary in it is only one intersection wide. A single-intersection boundary is a boundary a single mistake can delete; a two-intersection boundary is not. Everything about which weaves are exposed follows from that, and it has nothing to do with how busy the pattern is.

Read by the relative measure — the float the fault makes against the float the weave already has — the ordering changes and the plain weave becomes the worst of all: its longest float trebles, from one to three, which is the largest proportional disturbance any weave suffers. The eight-end satin is the best on that measure, at 1.14, because a float of eight against one of seven is a difference nothing can see.

So the two measures disagree, and the disagreement is not a defect in either of them. They are answers to different questions:

• Will the fault produce a length of loose thread the cloth cannot carry? — the absolute measure, and here the plain weave and the ribs win.
• Will the fault look different from its surroundings? — the relative measure, and here the long-float weaves win.

Which one matters depends on what the fault is for. A float of seven in a satin is invisible and will snag exactly as the rest of the satin does. A float of three in a plain weave is conspicuous and structurally trivial.

What the trade is actually right about #

The rule survives, but it is a rule about something else, and this collection has the instrument to say what.

A busy weave hides a fault because of what it already occupies, not because of what the fault becomes. A patterned cloth has energy of its own at coarse spatial frequencies; a plain weave has none — all of its structure is at the finest period a cloth can have. So a mark of any kind on a plain ground stands alone in an empty spectrum, and the same mark in a twill or a crepe is competing with lines the cloth already has.

That is the coherence argument applied to the cloth rather than to the fault, and it is a completely different mechanism from the float. It also predicts something the float measure cannot: that the hiding works best when the fault’s own period is close to the weave’s, and fails when it is not.

So the honest verdict on the trade’s rule is: right about the outcome, wrong about the cause, and consequently wrong at the edges. A busy weave does hide small marks. It does not reduce what a fault does to the structure — it makes it worse, by handing a single wrong lift the sum of two floats — and it stops helping as soon as the fault is large enough to be read as a length of thread rather than as a shading.

What was counted, and how #

Each draft is laid out as cloth before being damaged, and that is the step that decides whether the census means anything. A reversal made inside a four-by-four repeat is a fault in every repeat of the piece, which is a mistake in the design; a fault at the loom happens once. So every draft is tiled to eight by eight, the reversals are applied to the cells of one repeat, and the longest run is measured over the tiling.

Getting that wrong is not subtle in its consequences and is very easy to do: the first version of this census flipped cells inside the two-by-two repeat of a plain weave, where a single reversal leaves a pick on one side of every remaining end, and reported the plain weave as having no float at all rather than a float of three.

Four things are asserted while the sweep runs.

• No draft is improved by a fault in it, over the whole catalogue. A ratio below one would mean the run measurement was finding a shorter float after a reversal than before, which is arithmetically impossible and is what a boundary error in the tiling produces.
• Some drafts hold the fault to half again their own float, which is the good column being non-empty.
• The best ground by the relative measure is a short-float weave, asserted rather than observed, because it is the finding that contradicts the rule.
• And nine drafts in ten share one ratio, asserted as a share above ninety per cent — the signature of a single mechanism producing the whole distribution rather than a spread of causes.

How often the bad reversal happens is the other system’s share of the face #

The census counts the worst reversal a weave can suffer, which is the right question for a specification and the wrong one for a piece of cloth. A fault at the loom does not choose the worst intersection; it lands where it lands. So the third measure is the expected outcome, and it reconciles the two the essay has so far found in conflict.

Which reversals are dangerous is decidable in one line. Reversing an intersection that is inside a float creates a binding point and shortens the float. Only reversing a binding intersection joins two runs — so for warp floats, the dangerous positions are exactly the cells where the weft is up, and their share of the repeat is the weft’s share of the face.

That share runs the opposite way from the severity, and it runs the opposite way by construction: a weave with long floats has few binders, and few binders means few places a mistake can do the damaging thing.

The last column is the share times the damage, and reading it settles the argument the two earlier measures started.

On the expected measure the extremes are safe and the middle is not, which is the same verdict the absolute and relative measures reached from opposite directions and could not agree on. A 2/2 twill is safe because the damage is small; an eight-end satin is safe because the damage almost never happens; and the three weaves between them sit at exactly one extra pick of float per fault, for three different reasons that happen to multiply to the same number.

And the coincidence in that column is not a coincidence. The expected increase is the weft’s share times the join’s excess over the weave’s own float, and for a weave built by a shift rule both factors are functions of the same run length — one falling and one rising. Their product is nearly flat over the whole middle of the range, which is why the trade never developed a rule about it: there was nothing to notice.

The eight-end satin’s 0.13 is the one row that is not the product of a genuine trade-off, and the essay above says why. Its damage figure is capped by the repeat rather than earned, so the number is small because a longer float would not fit rather than because anything is protecting it. A cloth whose fault outcome is limited by the size of its own repeat is a cloth that has run out of room to get worse, which is a different kind of safety and one that disappears the moment the repeat grows.

Designing against it #

If the exposure comes from single-intersection boundaries, a designer can do something about it, and the something is cheap.

Widen the binding. A weave whose floats are separated by two intersections rather than one cannot have two of its floats joined by a single mistake. That is what a basket weave is, and what a hopsack is, and what every construction that doubles its ends or its picks amounts to. It costs interlacing — and interlacing is firmness, so the cloth is stiffer and sets more openly — but it removes a whole class of fault outcome.

Or accept the join and make it invisible. A weave whose float is already long enough that seven is unremarkable does not need protection, which is the satin’s position: it is maximally exposed by the absolute measure and maximally protected by the relative one.

What does not work is the middle. A weave with moderate floats separated by single binding points — the twills, which is to say most of what is woven — is exposed by both measures at once: its worst case is more than double its own float, and its own float is short enough that seven ends of thread on the surface look like something. The census says so plainly, since that middle is where the 22,640 sit.

The split is not between plain weaves and busy ones. It is between balanced weaves and unbalanced ones. A 2/2 twill is as safe as a basket, because its floats are separated by two intersections; a 3/1 twill of the same repeat is the worst weave in the table, because they are separated by one. Two weaves a weaver would put in the same family, differing by a factor of more than two in what a single wrong lift does to them.

The satin’s 8 is capped rather than earned. Joining a float of seven to another across a single binder would give fifteen, and the repeat is eight — so what the reversal actually produces is a thread on one side of the cloth for the whole repeat, which is the loose-thread outcome rather than a long float. The satin is not protected by anything; it has simply run out of room.

Where the model stops #

A float is a proxy for visibility and is not visibility. Whether a length of thread on the surface is noticed depends on the yarn’s lustre, the light, the colour and the finish, none of which this collection has. What is claimed is a structural quantity that correlates with being noticed, and the correlation is an assumption.

Only single reversals are counted. A real mispick reverses a whole row, and a tie-up error reverses a lattice; both are compositions of the operation counted here, and the composition need not behave like the sum of its parts.

The tiling is eight by eight and the fault repeats at that period. So the census is really about a fault every eight ends and eight picks rather than about an isolated one, which is close enough for the local join arithmetic and is not the same thing.

And the enumeration is at four by four. The catalogue’s satins are larger and are handled directly, but the census’s two-valued answer is a fact about drafts on this repeat, and there is no reason to expect exactly two values at six by six.

The generalisation #

When a fault’s cost comes from joining two existing features rather than from creating a new one, the systems most exposed are the ones with the largest features and the thinnest separations between them. That inverts the usual reasoning, which is that a system with big components is robust because a small perturbation is small relative to them.

The diagnostic question: does this failure make something bigger, or does it remove a boundary? The first scales with the size of the failure and the second scales with the size of the things on either side of it — so a small failure in a well-separated system is not a small failure at all.

And the second lesson is about proxies that disagree. Two defensible measures of the same intuitive quantity — how much a fault shows — gave opposite orderings here, and the right response is not to choose between them but to notice that the intuition was two questions wearing one word. Where a rule of thumb has survived for a century, it is usually right about something; the work is finding out what, and the answer is often not what its holders would say.

Who found it, and when #

The rule that a patterned cloth hides a fault is trade knowledge of indefinite age and appears in every practical text. The float arithmetic here is this collection’s own and follows from machinery built for entirely different purposes — the census of every four-by-four draft, and the run-length measurement that has been used for lustre, abrasion and float limits since the beginning.

The join mechanism appears not to be written down, though it is obvious once stated, and it has a practical consequence worth carrying beyond faults: any operation that removes a binding intersection from a float-bearing weave produces the sum of the floats on either side. That applies to a mistake, to a deliberate simplification of a draft, and to any process that reduces interlacing — and it says that such changes are not gradual.

Where the ladder goes next #

This closes the ladder about faults, which has run from what a fault costs, through what it does to the structure, to whether it shows. The float that decides the last of those is the same quantity that decides lustre, snagging and how close a cloth can be set, so a weave chosen to make a fault inconspicuous is a weave chosen against several other things at once.

Sideways, the reversal whose consequences are counted here is the elementary operation behind every fault in this ladder: a mispick is a row of them, and a tie-up error is a lattice of them.