A woven thread has no room to bend

The natural first test of a new solver is the simplest thing it should handle, and for a solver of thread paths that is a plain-woven shirting. It would not converge on one, and the version before it reported a singular system and stopped.

That looked like a defect for about an hour. It is the most useful result on this ladder.

What the refusal was about #

A thread’s span between two crossings has a length, and the straight line between the two crossings has a different one. The difference is the slack, and it is the entire budget available for the thread to take any shape at all.

For a cotton poplin the yarn between two crossings is longer than the straight line between them by four parts in a thousand. For a cheesecloth it is four parts in ten thousand. Those are not small numbers in the sense of being negligible; they are small in the sense of leaving nothing to work with.

Why so little #

The arithmetic is quick and it is Peirce’s own. A warp end’s crimp is nine per cent in a poplin — its length exceeds the cloth’s by that much. But the crimp is not spare length lying about: almost all of it is consumed by the arc where the thread goes round the pick it crosses, and that arc is not optional. The threads have to interlace or there is no cloth.

Subtract the wrap and what is left is the straight portion, and the straight portion’s length and the distance it has to span are the same thing to within a few parts in ten thousand.

Where the crimp goes #

It is worth spending a moment on where a thread’s crimp actually is, because the usual mental picture puts it in the wrong place.

A reader looking at that section sees a wave and thinks of the crimp as distributed along it, the way a sine wave’s arc length is. It is not. The straight portion contributes length in proportion to how far it goes, and it goes almost exactly as far as it would if it were the chord; the extra length, which is what crimp means, is manufactured at the two turns and nowhere else.

That is the whole content of this rung stated geometrically rather than mechanically, and it is why the answer surprised nobody who thought about it afterwards and everybody who thought about it before.

A tangency, not a coincidence #

The agreement is closer than it has any right to be, and the reason is worth having.

Vary the wrap angle at fixed thread length. The free length shrinks at D per radian, because that much thread is being swallowed by the arc. The straight-line distance between the two arc ends shrinks at exactly the same rate, because the ends are moving round circles of radius D/2 in opposite senses. To first order the two changes cancel exactly, and the slack is second order in the departure from Peirce’s angle everywhere.

So Peirce’s construction is not one solution among many that happens to be convenient. Over the whole range of wrap angles a cloth could plausibly take, the geometry is within a hair of exactly consumed, and the only wrap angle at which the thread fits at all is his.

The measurement, over eight cloths #

The share is D·θ₁ divided by the thread length, and both come from the same solve of Peirce’s equations that this collection has used since the essay that measured his geometry against Kemp’s. It is a count rather than a claim, and it runs:

seven per cent on an open scrim, seventeen on a voile, twenty-seven on a batiste, twenty-nine on a muslin and on a poplin, forty-two on a duck, forty-five on a filter cloth, fifty-five on a sheeting.

The ranking is not by fineness, and it is not by sett. It is by crimp: the cloths that make their threads turn hardest are the ones that spend most of them turning. That is nearly a tautology once said, and it was not obvious before the column existed.

What a knit does with the same budget #

The comparison is worth making at this point rather than at the end, because it makes the size of the effect legible.

A jersey at an ordinary tightness factor carries about twenty-one yarn diameters of thread in a cell under five diameters wide and under four tall. The straight line between two interlacings is a little over half of the yarn available, so the slack is forty-nine per cent.

Against four parts in a thousand. The two fabrics differ by more than two orders of magnitude in the one quantity that decides whether a thread’s path is a construction or a solution, and no fabric anybody makes sits between them.

What this settles about the old construction #

It settles that the woven side of this collection was never missing an elastica for its shape.

Peirce’s path is not an approximation to some smoother truth. It is what the thread’s own length forces on it, to a tolerance far below anything that could be measured, and no better description of a woven thread’s centre line is available or needed. Ninety years of it being the standard tool is not inertia.

And what it does not settle #

It does not rescue the forces. The path is still arcs joined to a straight, its curvature still steps from 2/D to zero at the join, and a step in bending moment is a couple applied at a point with nothing there to apply it.

So the position is uncomfortable and worth stating baldly: the shape is right and it is not an equilibrium. A real thread would smooth that step out over some short length near each join, and it has nowhere near enough slack to do so. What that means physically is that the contact is not a point where the arc ends — the thread stays in contact a little longer, or leaves a little earlier, and the correction is buried inside a region a hair’s breadth long.

Nothing measurable depends on it. Nothing computable does either, which is the difficulty, and it is the same difficulty the criterion that cannot see friction runs into from the other side: a quantity that exists in the fabric and not in the description.

The energy was right all along #

There is a compensation, and it is exact.

If the whole of a thread’s bending happens inside a wrap of constant curvature 2/D over an arc length Dθ, then its bending energy per half wave is half the rigidity times the curvature squared times the arc length, which is 2·B·θ/D and nothing else. That is precisely the crimp energy this collection has been using on the woven side for several ladders, arrived at long before there was an elastica to check it against.

So the geometric construction supplies the energy exactly and the force not at all, and that is a much sharper statement than “it is approximate”. A quantity that is exact and a quantity that does not exist are easy to tell apart once the distinction is drawn, and impossible before.

What the wrap costs, per cloth #

The energy is worth putting numbers to, because it is the quantity that decides a relaxed cloth’s whole state.

A cloth with a large wrap angle in the warp and a small one in the weft is a cloth in which the warp is paying most of the energy, and the state it relaxes to is the one that makes the two payments balance. That is the equilibrium this collection solves on the woven side, and it is now known to be the exact energy of the exact geometry rather than a plausible model of it.

Two consequences that were previously assumptions: the energy is linear in the wrap angle rather than quadratic, because the curvature inside the wrap does not depend on how far the wrap goes; and it does not depend on the spacing at all except through the angle. Both come straight out of the arc having constant curvature.

Which is why the thickness route was right #

The essay that recovered the relaxed contact force went round the problem: it inverted a thickness measurement through a compression energy rather than differentiating a shape. At the time that read as a workaround for a missing tool.

It was the only available route and it remains the right one for woven cloth. The tool that has since arrived does not apply to a woven thread, because a woven thread has no free run for it to apply to.

What the figures cannot show #

None of these drawings can show the slack, because the slack is four parts in a thousand of a line a few hundred pixels long. It is less than the width of the stroke the thread is drawn with.

That is not a failure of the drawing; it is the reason the result was not noticed by inspection for ninety years. A woven thread looks like it has room. Every diagram of one, including the ones here, shows a generous smooth wave with obvious slack in it, and the slack is entirely an artefact of drawing a thread thinner than it is and a crimp deeper than it is.

The consequence for the rest of the collection #

Two things follow, and both are load-bearing further along.

The first is that the woven relaxed contact force stays where it was — recovered from a measurement, quoted with the assumptions of a compression model, and not improvable by anything here.

The second is that the comparison between a woven cloth and a knit is a comparison between two different routes, and has to be reported as one. That is not a weakness. Two independent methods that agree are worth more than one method applied twice; what would be worthless is a comparison that looked independent and was not.

The other direction: what would have to be true #

Suppose a woven cloth existed with real slack. What would it look like?

It would need a crimp far larger than its wrap could swallow — which means a thread that turns through a large angle over a long distance, which means a very open sett with a very deep crimp. That is not a cloth; it is a leno or a gauze, and at that point the threads are not doing what a plain weave’s threads do.

Or it would need the two systems to be wildly unequal, so that the fine system rides over a coarse one it barely bends round. That is a backed or a warp-faced construction, and it is worth looking at with this tool later.

Where the boundary actually falls #

The number that decides it is the slack as a fraction, and the two fabrics are separated by three orders of magnitude in it: four parts in a thousand for a poplin, and half for a jersey.

There is nothing in between, in the sense that no ordinary fabric sits in the middle of that gap. A structure either has its thread length consumed by its interlacings or it does not, and which side it falls on decides what kind of mechanical object it is — geometric or elastic, constructed or solved.

What it means for a fabric that is not plain #

Every number above is a plain weave’s, and the wrap share is a property of the interlacing rather than of the yarn, so a weave that interlaces less should have more slack.

That does move the number, and not as far as it looks. A longer float means fewer interlacings and therefore less total wrap, but it also means less crimp: the thread that turns less also rises less, and the two fall together. The share stays in the same band, which is why the eight cloths above — plain weaves all — bracket the answer for the drafts as well.

Where it would genuinely change is a fabric whose two systems are grossly unequal, and that is a construction worth solving on its own terms rather than assuming into this one.

Where the crimp is carried, as against where it is made #

The section above says the extra length is manufactured at the two turns, and that is right about the cause and worth separating from the question of where the length is actually sitting — because the two answers are different and the split can be computed.

A thread’s length exceeds the cloth’s for two reasons, and only one of them is the arc.

The arc is longer than its own chord, by Dθ(1 − 2sin(θ/2)/θ) over the wrap — a pure arc-versus-chord excess.

And the straight run is inclined, so its horizontal projection is only cos θ of its length. Every millimetre of straight thread lying at θ to the cloth advances the cloth by one millimetre times the cosine, and the shortfall is crimp.

So the crimp is (1 − cos θ) times the straight share plus the arc excess times the wrap share, and the two terms are not remotely comparable. Invert it for the poplin — nine per cent crimp, twenty-nine per cent of the thread inside the wrap — and the wrap angle comes out at 27.5 degrees, at which the two contributions are

0.080 from the inclined straight run and 0.003 from the arc: a split of 97 to 3.

The turning makes the crimp and the straight run carries it. That is not a contradiction of the section above — there would be no inclination without the turn, and the arc is what sets θ — but it does correct the picture a reader is likely to form from it. Almost none of a woven thread’s excess length is in its curved parts. It is in the fact that its straight parts are not level.

Which is also, incidentally, why the slack argument works. The straight run is inclined and it is straight, so it spans exactly its own length along its own direction, and the shortfall against the cloth is a projection rather than a spare. A projection cannot be spent on taking a different shape.

Which makes crimp quadratic where the wrap is linear #

The decomposition has a consequence for the table of eight cloths that is worth stating because it explains a spread the essay reports and does not account for.

The wrap share is Dθ over the thread length: linear in the wrap angle. The crimp is dominated by 1 − cos θ, which for the angles a cloth uses is θ²/2 — quadratic in it.

So the two quantities do not move together, and the crimp moves faster. A cloth that turns its threads twice as hard has roughly twice the wrap share and four times the crimp, which is why the eight cloths’ wrap shares span a factor of eight while their crimps span considerably more.

That is a useful sanity rule for anyone reading the table. A crimp figure exaggerates how differently two cloths treat their threads and a wrap share reports it honestly, because one is a squared quantity and the other is not — the same trap a fourth power sets for a diameter, one power down.

And it gives the cleanest statement of what the wrap share is for. Crimp is what a cloth does to its length, and it is quadratic. The wrap share is what a cloth does to its thread, and it is linear — which is why it is the right column to rank the eight by, and why the ranking by crimp and the ranking by wrap share, though they happen to agree in order here, would not agree in their spacing on any axis anybody drew.

A caution about the word #

“No room to bend” is a phrase about the span between crossings, and a woven cloth manifestly bends: it drapes, it creases, it wraps a hand. All of that is the whole fabric bending, with every thread taking a small extra curvature on top of the crimp it already has, and none of it needs slack.

What has no room is the thread’s freedom to choose a different path between one crossing and the next. That is a much narrower statement, and it is the one that matters for a contact force.

A test this rung could fail #

The claim is that the wrap consumes the crimp, and it is checkable against something already in this collection rather than against itself.

If it is right, a cloth’s crimp and its wrap angle have to move together across the whole table, and the residual — the straight portion’s length minus the distance it spans — has to be a few parts in ten thousand for every one of them and not merely on average. It is: the largest residual over the eight cloths is under one part in a thousand, and the cloths at the two ends of the crimp range are the two ends of the wrap range.

A single cloth with a large crimp and a small wrap would refute the whole account. None exists in the table, and the reason it cannot is that both quantities are functions of the same two spacings and the same two diameters.

Where the ladder goes next #

The refusal here is one half of a pair. The other half is the knitted loop, which has so much slack that the same solver converges in a dozen steps and the point of contact never has to become a wrap at all.

After that the forces start being used, and the first thing they say is about a fabric everybody has measured for sixty years sitting nowhere near where its own bending would put it.