A cloth extends by moving its crimp

Take a strip of ordinary cotton sheeting and pull it along the warp. It gives a little, and then it stops giving, and the stopping is abrupt enough that anybody who has done it once can feel where the end of the travel is.

The standard account of that end of travel is a single sentence, and every textbook and every trade note gives it: the extension available is the warp crimp, because once the warp is straight there is nothing left to take out. It is a clean argument, it is obviously correct, and on most cloth it is wrong by a factor of between two and four.

What the two rungs below leave open #

Crimp interchange is the mechanism: pull along the warp, the warp straightens, the weft has to take up the bending it stops doing, and the cloth narrows. Nothing stretches and everything moves.

What that conserves is the arithmetic, and it stops at an honest limitation. Inextensibility fixes the new warp crimp once the new length is named, and then leaves one equation in two unknowns for the weft: any width and crimp whose product is the weft’s thread length will do. The trade closes the system with a bookkeeping rule — the crimp the warp gives up is added to the weft in percentage points — which is a convention rather than a consequence and which knows nothing about either of the stops.

Both rungs name the second stop and neither computes it. That is what this one is for.

Peirce’s equations, read forwards #

Peirce’s 1937 geometry gives a plain weave a thread path made of circular arcs joined by straight runs. Around each crossing the thread turns through an arc of radius D/2, where D is the sum of the two yarn diameters; between crossings it runs straight. Two equations follow for each thread system,

$$p = (l - D\theta)\cos\theta + D\sin\theta, \qquad h = (l - D\theta)\sin\theta + D(1 - \cos\theta),$$

with l the thread length per crossing, θ the weave angle and h the crimp height. And one condition ties the two systems together: the two crimp heights must fill the thickness of the cloth, so $h_1 + h_2 = D$.

Given the two spacings that is a solved system, which is how the site has used it since its earliest essays. Given the two thread lengths it is something else entirely.

Nothing stretching means exactly one thing in these equations: l₁ and l₂ are constants. Fix them, and the closure condition $h_1 + h_2 = D$ becomes one equation in the two weave angles.

One equation, two unknowns. So the states a cloth can reach form a one-parameter family — a curve in the plane of its own two dimensions, and not a region. Pick any warp angle in range and the weft angle is decided; pick a cloth length and the width is decided with no freedom left over.

That is the missing statement the rung below could not supply, and it did not have to be borrowed from anywhere. It was in the geometry the whole time, and reading Peirce’s equations forwards rather than backwards is the whole of the trick: solved from a pair of spacings they give a state, and solved from a pair of thread lengths they give a locus.

The curve is monotone, which is worth stating because it is the interchange restated: along its whole length the cloth cannot get longer without getting narrower, and cannot get narrower without getting longer. There is no state anywhere on it in which both dimensions have risen.

The bound everybody quotes #

The end of the curve on the extending side is what a reader wants, and there are two candidates for it.

The first is the one in the books. A thread cannot be straighter than straight, so when the warp’s weave angle reaches nought the warp has given everything it has, and the extension is exactly the warp crimp. A cloth with eight per cent of warp crimp gives eight per cent. This is a genuine bound and it can never be exceeded.

What it is not is the bound that stops the cloth.

The bound that actually binds #

All the crimp the warp gives up has to go somewhere, and there is only one place for it. The weft takes it, its weave angle rises, and its straight runs between crossings get shorter. When those straight runs reach nothing the weft is wrapped as tightly round its neighbours as the geometry permits — it has jammed — and the curve stops there whatever the warp still had in hand.

This is the same jam that caps the sett, reached from the other side. There the cloth is squeezed until its threads touch; here the thread length is held and the weave angle is driven up to meet it. The condition is identical: the straight run l − Dθ goes to zero.

So the extension available along the warp is decided in the weft, and the warp’s own crimp has almost nothing to do with it.

What was counted, and how #

Eight plain weaves, quoted as nominal constructions rather than as measurements of particular fabrics — a count and a sett of the kind a mill states, spanning from a scrim to a close sheeting. Diameters come from the counts by conservation of volume at a packing factor of 0.6, which is the site’s standing route from a yarn count to a diameter. Every row is a plain weave, because Peirce’s geometry is a plain-weave geometry.

For each, the reference state is solved from the two spacings and checked by running Peirce’s own equations forwards on the answer. The two thread lengths are then held and the curve is sampled at 241 states, at each of which both thread lengths are reconstructed from the crimps and the spacings and compared with the originals. The worst departure over the whole table is one part in ten thousand million million, which is arithmetic noise and nothing else.

The check runs at every sampled point rather than at the ends, and that is deliberate. A deformation that conserves length at its two endpoints and drifts in between is exactly the failure a two-state figure cannot see, and this site has been drawing two-state crimp figures since its earliest essays.

Six of the eight are stopped by the weft. The two that are not are the two openest — a cheesecloth at 20.5 per cent warp cover and a voile at 31.1 — and on both the two bounds are the same number to every decimal place, because the warp really does reach straight before anything else happens.

The changeover, and it is the same on every yarn #

The two openest cloths behave one way and the six closer ones the other, which invites the obvious question: where exactly does it change over, and what decides it?

The condition is exact and it is a single inequality. The warp reaches straight when the weft’s crimp height h₂ has grown to the whole thickness D of the cloth. The largest height a weft can supply is at its own jam, where θ₂ = l₂/D and the height is D(1 − cos(l₂/D)). Setting the second at least equal to the first gives cos(l₂/D) ≤ 0, which is

$$l_2 \ge \frac{\pi D}{2}.$$

The weft must carry at least a quarter of the circumference of the circle it bends around. Below that it jams before the warp can straighten, and the crimp bound is simply unreachable. It is a statement about two lengths, and there is nothing else in it.

Every length in the model scales with the yarn diameter, so the changeover is a ratio of spacing to diameter and not a construction. Solved for a square plain weave it comes out at a spacing of 2.9638 diameters, which is a cover of 0.3374 and a crimp of 5.9998 per cent — and the same six figures come back for a 5 tex yarn, a 20 tex yarn and a 100 tex yarn, agreeing to the last place the arithmetic carries. That the crimp lands on six per cent to four significant figures is a coincidence of the numbers and not a fact about anything.

The practical form of it is short. A cloth with more than about a third of its surface covered by warp is stopped by its weft. Almost every cloth anybody wears is on that side of the line: the cover factor of a shirting is nearer three quarters.

The other direction is much longer #

One thing falls out of the arithmetic that nobody appears to have said, and it is visible in every curve above.

The curve is far longer on the shortening side than on the extending one. A batiste reaches 6.56 per cent of extension and 29.23 per cent of contraction; a muslin 6.59 and 26.87; even the close sheeting, which extends 4.03 per cent, can be shortened by 10.19. A cloth at constant thread length can be made a great deal shorter than it can be made longer, by a factor of four or more.

The reason is symmetric with everything above. Shortening the cloth along the warp lets the warp crimp grow, and it can grow until the warp jams — which is a long way, because a thread wrapping further round its neighbours has more travel in it than a thread flattening out. Meanwhile the weft straightens rather than jamming, and a straightening thread is never the thing that stops.

That is the geometry behind relaxation shrinkage, and it says something the shrinkage essays could not. A cloth cannot shrink past its own crimp is the standing statement; what the curve adds is that the room available in that direction is much larger than the room available in the other, so a finishing process that leaves a cloth stretched has put it somewhere it can travel a long way back from.

Four kinds of cloth, decided by two covers #

The changeover condition is a statement about one direction, and the same inequality holds in the other with the systems exchanged. Writing both down sorts every plain weave there is into four classes, and the boundary in each direction is the same number.

Pulled along the warp, the cloth stops when the weft jams unless the weft’s own span is long enough — l₂ ≥ πD/2, which for a square cloth is a warp cover under 0.3374. Pulled along the weft, the roles reverse: the warp jams unless l₁ ≥ πD/2, which is a weft cover under the same figure.

So the classification is a pair of yes-or-no answers on two numbers a specification already carries:

The bottom row is nearly every cloth anybody wears, because a shirting is around three quarters covered in each direction and a sheeting more. For those the standard account is wrong in both directions at once, and the two errors are independent — a cloth can be overstated by a factor of two one way and four the other.

The middle two rows are the interesting ones and they are not rare. An unbalanced cloth can sit on opposite sides of the line in its two directions, which means the textbook rule is right about it one way round and wrong the other. A warp-faced cloth set close in the warp and open in the weft — a poplin, a rep, a warp-faced twill — has its warpwise extension decided by the weft’s jam and its weftwise extension decided by the warp’s own crimp, so a laboratory testing it in both directions gets one number that agrees with the standard account and one that does not. That is exactly the shape of evidence that gets read as a bad specimen.

And the top row explains why the rule survived. An openly set cloth is the case the account is right about, and an openly set cloth is what somebody demonstrating crimp interchange by hand would reach for, because the mechanism is visible in it. The rule was formed on the cloths that show the mechanism and applied to the cloths that do not.

Two cautions on the table. The threshold is the same number in both columns only because D is the same for both systems — it is the sum of the two diameters, which is symmetric — so the asymmetry between the rows comes entirely from the two spacings. And 0.3374 is a square cloth’s figure; for unequal counts the condition is still l ≥ πD/2 exactly, and converting it to a cover needs the two diameters separately.

Where the model stops #

Peirce’s circular section is an idealisation and a generous one. A real yarn flattens where it is gripped, and the racetrack section gives a flattened yarn less crimp for the same cloth and a different jam. Every number here would move under that model. The arguments would not: the jam is still a straight run going to zero and the changeover is still a comparison of two lengths.

The circular model refuses the densest real cloths outright. A 44-end poplin in 15 tex cotton has no solution at all in it — the section asks for more room than the sett leaves — so the poplin row in the table is set at 32 ends per centimetre. That is a real limitation and it is the same disagreement, seen as a refusal rather than as a number.

There is no force anywhere in this essay, and there cannot be. The site records “no force in the arithmetic” as an open shortfall and this rung does not close it. What is computed here is which states a cloth can reach; a load appears only as the thing that decides which of them it sits at. So the essay can say the sheeting stops at 4.03 per cent and cannot say what it takes to get there — and the rung below is right that the load rises steeply long before the geometry runs out, which means the achievable extension is smaller again than the reachable one.

And the model is a plain weave. A twill’s threads pass over more than one crossing before they turn, so the arc-and-straight construction is a different one. The direction of every argument survives and none of the numbers do.

Who found it, and when #

Peirce published the geometry in 1937 and stated the jamming condition in it. The constant-thread-length family is not his — he was solving for the state of a cloth at given spacings, which is the question a fabric analysis asks.

The biaxial deformation of a woven cloth with both threads inextensible belongs to the fabric-mechanics literature that grew out of him, and it is a substantial calculation rather than a formula. What is odd is that the bound has not travelled back into the general account. Every source that mentions the limit gives the warp’s crimp; the ones that mention the weft’s jam mention it as a caveat, without a number, and none this site has found says which of the two binds on an ordinary cloth.

The trade does know it, in the way trades know things. A weaver will say a close cloth “has no give in it”, which is a statement about the second bound and not the first, and will say so about a fabric whose measured crimp is substantial. The measurement and the behaviour disagree, and the disagreement is the whole of this essay.

Where the ladder goes next #

The curve computed here has a shape as well as two ends, and the shape is the next rung. Because there is only one degree of freedom, a cloth pulled both ways can only move along it — which turns out to mean that equal extension in both directions is not available at all, at any size, on any cloth.

Beyond that, the slope of the curve has a name everywhere else in mechanics, and it breaks every rule that name comes with.

Beside this rung sits the other mechanism, which is an order of magnitude larger and available only at an angle to the threads.

What the pictures here cannot show. Every curve on this page is a set of reachable states and not a path through them. Nothing here says how a cloth gets from one point to another, how long it takes, or whether it comes back the way it went — and it does not, because a real crossing has friction in it. The curve is the map, and the map has no journey drawn on it.