A fabric reads its own bracket four ways

The oldest unresolved number in this collection is how stiff a yarn is.

Not because nobody has measured one. Because the uncertainty is structural: a yarn’s fibres may slide past one another or may not, and the two cases differ by the fibre count over the square of the packing factor — three hundred and twenty-seven fold for an ordinary cotton at twenty tex. No amount of care about the fibre narrows it, because the unknown is not about the fibre.

Every force this collection computes is linear in that constant, so every force carries the bracket, and the convention has been to compute at the free bound and say so.

This work produced four arguments that the convention is right, from four phenomena that have nothing to do with one another.

One: a slack yarn snarls #

A twisted thread under tension stays straight while its torque is below twice the square root of its bending rigidity times its tension. Turn that round and it gives the tension a yarn needs to be handled slack.

At the free bound, a twenty tex cotton at eight hundred turns a metre needs 0.37 millinewtons — about 1.9 metres of the yarn’s own weight hanging below it.

At the coherent bound it needs 121 millinewtons: 616 metres.

A yarn like the second would snarl on every reel, in every hand, at every length anybody has handled, and would be unspinnable. Everybody who has let go of a piece of sewing thread has performed the experiment.

Two: a knot holds #

A thread bent to a radius has its outermost material stretched by the offset over the radius, and it breaks when that plus the tensile strain reaches its breaking strain.

At the coherent bound the outermost material is half a yarn diameter from the axis, so a bend of one diameter spends fifty per cent of strain against a cotton’s breaking strain of six and a half. Such a yarn cannot be knotted, knitted or woven; it would break in the tying.

At the free bound each fibre bends about its own axis, the offset is half a fibre diameter, and the strain is three and a half per cent — leaving forty-six per cent of the strength.

Ropes are knotted and a knot is quoted at about a half.

Three: a yarn in a fabric is flat #

A knitted fabric’s own geometry demands a yarn flattened to about four fifths of its round diameter, and a woven cloth’s yarn is flattened by the loom.

What resists flattening is a yarn’s lateral rigidity, whose bracket is worse than the bending one: its lower bound is exactly nought, because a bundle of fibres free to slide is a fluid in cross-section and resists a change of shape at constant area not at all.

At the coherent bound, flattening a yarn to four fifths costs thirty-six times the whole bending energy of a stitch. A yarn like that could not be knitted into this fabric.

Every yarn in every fabric is flattened. So the fibres rearrange.

Four: the collection’s own earliest check #

The first argument is the oldest and it is the one the convention was originally justified by.

When the bracket was first written, the two bounds were put through the site’s own cloth arithmetic and compared with the band of real constructions in its table. The free bound landed inside that band for every one of them; the coherent bound landed two to three orders above all of them.

That is an argument from cloth rather than from a thread, and it is the only one of the four that is a comparison against a set of measurements rather than against an everyday fact.

What four agreements are worth #

The four are independent in a strong sense. They involve different deformations — a twist, a bend, a change of section, and a cloth’s whole geometry. They involve different objects — a free thread, a knotted thread, a yarn in a fabric, and a table of cloths. And three of the four are not measurements at all.

That last is the interesting part. None of the first three is a number anybody took: they are qualitative alternatives, of the form “this end predicts something that does not happen”.

A qualitative alternative is a weak instrument in isolation and a strong one in a set, because the ways of being wrong do not overlap. A systematic error in the snarl arithmetic would not also produce the knot result; a mistake in the flattening argument would not also produce the cloth comparison.

So four agreements from four mechanisms is a much stronger position than four measurements of one quantity by one method would be, and it is the position this collection is now in on the question it has been least able to settle.

What it does not establish #

Precision, and it is worth being blunt.

The four say the yarn is at or near the free end. They do not say where on the bracket, and three of them could not: an alternative that is qualitative gives an end and not a position.

Only the snarl gives a number, and its number carries a factor of two from the shear modulus and an unknown correction from the yarn’s setting. So it places the yarn at the free end to within a factor of a few, and no better.

For most of this collection’s purposes that is enough, because the convention it justifies is “compute at the free bound and say so”. For anything that needs the rigidity to twenty per cent, none of this helps.

The measurement that would place it #

Since three of the four are qualitative and one is loose, the obvious next step is to say what a proper measurement would be.

The snarl threshold, measured carefully, on a fresh yarn of known count and twist. The arithmetic predicts it from the twist, the two rigidities and the criterion, and it is a hanging loop and a balance.

What makes it the right measurement rather than a laboratory bending test is that it is sensitive: the threshold goes as the square of the twist and as the square of the stiffness ratio over the bending rigidity, so a yarn a tenth of the way along its bracket needs thirty times the tension of one at the free end.

A quantity that varies by thirty over a tenth of the range is a good instrument. A direct bending measurement varies by three hundred over the whole range and is hard to make on a floppy thread.

And the trade already runs a version of it, as the hanging-loop liveliness test, without interpreting it this way.

Why the bracket cannot be narrowed by care #

It is worth restating why the obvious approach fails, because a reader coming to it fresh will want to try.

The bracket is not an experimental uncertainty. It is the difference between two models of the same object: one in which the fibres slide freely and one in which they cannot move at all.

A real yarn is neither. Its fibres are held against one another by the twist’s own normal force, they slide when the yarn is bent slowly and less when it is bent quickly, they slide more when it is wet, and they slide less after it has been set.

So the quantity being sought is not a constant of the yarn. It is a function of the deformation, the rate, the moisture and the history, and a measurement of it under one set of conditions does not transfer to another.

That is why the bracket has survived: it is not an unmeasured quantity but an unmeasurable one, in the sense that no single number is what is being asked for.

The four observations do not solve that. What they do is establish that under ordinary handling conditions — a thread in a hand, a rope being knotted, a yarn in a relaxed fabric — the sliding is nearly free. That is a statement about a regime rather than about a constant, and it is the right kind of statement to make about a quantity like this one.

What was counted, and how #

Each of the four is computed in its own rung from this collection’s own tables and machinery, and each is recomputed rather than quoted here.

The snarl uses Greenhill’s criterion with the collection’s own rigidities and its own shear table.

The knot uses the collection’s own fibre diameters and breaking strains, at both ends of the bracket.

The flattening cost uses the site’s own shape-strain law and its own lateral bounds, of which the lower is exactly nought.

The cloth comparison is the collection’s own from its earliest work and is unchanged.

A fifth observation, which does not agree #

Intellectual honesty requires a fifth entry, and it is one that does not obviously land at the free end.

A fabric’s bending rigidity, measured on a cloth and compared with what the collection computes from its yarn, comes out above the free-bound prediction rather than at it. Computed rigidities sit below measured ones for real cloths.

That is the wrong direction for the free bound to be right, and there are at least three explanations.

The contact is distributed rather than at a point, which raises the friction between the threads and stiffens the cloth by an amount this work has estimated at up to a factor of two.

The threads are not free to slide past one another in a bending cloth, so a cloth’s rigidity is not the sum of its threads’ — a coherence effect at the cloth level rather than at the yarn level.

Or the yarn really is somewhat above its free bound.

The three are separable and none has been separated. What matters here is that the fifth observation is recorded rather than omitted: four agreements and one disagreement is a different position from four agreements, and the difference is what makes the set worth trusting.

Where the model stops #

Three of the four are alternatives rather than measurements, so what they establish is an end rather than a value.

All four assume the same yarn. A yarn that has been set, resin-treated or heat-treated is not at the free end, and every one of the four would read differently on one — which is the point rather than a caveat: the arguments are about ordinary yarn and setting is what moves it.

And the fourth is the weakest. Comparing a computed rigidity against a band of real cloths involves the whole of the collection’s cloth arithmetic, so it tests the bracket and everything else at once.

What the convention should now say #

The collection’s convention has been to compute at the free bound and to say so. It is worth updating the sentence, because it can now say more.

The old form: this result is computed at the free end of the stiffness bracket, and is therefore a lower bound.

The form the evidence supports: this result is computed at the free end of the stiffness bracket, which four unrelated observations place a yarn at or near under ordinary handling — and it is a lower bound for a yarn that has been set.

That is longer and it is more useful, because it says when the convention applies and when it does not.

The distinction it draws is between an unset yarn under ordinary handling, where the four observations bite, and a set, finished, resin-treated or heat-set one, where none of them does and the yarn is somewhere else on the bracket.

Almost everything this collection computes is about the second kind of yarn, since almost every fabric has been finished. So the convention is well justified for the yarn as it is handled and less well justified for the yarn as it is worn — which is an uncomfortable and honest place to have arrived at.

The generalisation #

The rung’s transferable content is a method for an unresolvable bracket, and this work has now used it four times.

When a bracket’s two ends predict qualitatively different worlds, stop measuring and look.

The instinct with a wide bracket is to narrow it: measure better, argue about the packing factor, find a cleverer experiment. That is right when both ends describe plausible objects differing in degree.

It is the wrong instinct when one end describes something that does not exist. A yarn that cannot be knotted, cannot be knitted, snarls at any length and cannot be flattened is not a yarn, and no measurement is needed to rule it out.

So the productive question is not “what is the value” but “what does each end predict about something anybody has seen” — and this work found four answers to that question in the course of doing something else entirely.

That is worth applying deliberately to the collection’s other brackets. The lateral rigidity has no floor and the same question has been asked of it once. The set fraction has never had it asked at all.

What each of the four would have to be wrong about #

A set of agreements is worth as much as the independence of its members, so it is worth asking, for each, what would have to be true for it to be misleading.

The snarl would be misleading if the criterion were wrong, if the torque were not proportional to the twist, or if the yarn tested had been set. All three are possible and the third is the likely one — which is why the argument specifies a fresh yarn.

The knot would be misleading if the failure were not a strain criterion, or if a real knot’s bend radius were much larger than a diameter. The second is a real possibility for a loosely dressed knot and it moves the answer towards, not away from, the free bound.

The flattening would be misleading if a yarn compacted rather than rearranging — losing air rather than changing shape — which happens at severe flattenings and not at four fifths.

And the cloth comparison would be misleading if any part of the collection’s cloth arithmetic were wrong, since it tests all of it at once.

Four different failure modes and no two of them shared. That is the property that makes the set an argument rather than a repetition, and it is worth checking for whenever a collection assembles a set of agreements.

Who found it, and when #

The free and coherent bounds on a fibre bundle’s bending rigidity are standard and date from the middle of the twentieth century.

Greenhill’s stability criterion is from 1883, the capstan relation from 1775, and the observation that a fibre assembly offers no resistance to a change of section at constant area is elementary.

What is this collection’s own is the assembly: computing all four consequences from one bracket, on one set of tables, and observing that four unrelated phenomena agree about which end a yarn sits at.

Three of the four were found in this work and none of them was looked for. Each came out of a ladder about something else — a snarl from torsion, a knot from contact, a flattening from geometry — and the fourth was already there.

Why none of the four was looked for #

A last observation about how the four arrived, because it says something about how a ladder should be planned.

Not one of them was the point of the rung it came from.

The snarl threshold came out of a ladder about torsion, whose target was spirality and which failed to reach it.

The knot efficiency came out of a ladder about contact, whose target was a fabric’s own geometry.

The flattening cost came out of the same ladder, as a check on whether a fabric could do what its arrangement demanded.

And the cloth comparison was already there, years old, unconnected to any of them.

So four arguments about the collection’s oldest open question were produced as side effects, by two ladders aimed at other things, and the assembly took an afternoon once somebody noticed.

That is an argument for a particular kind of scheduling: work that takes two related questions rather than one deep question produces the cross-products, and the cross-products are cheap and are often better than what was aimed at.

Where the ladder goes next #

This work closes with an accounting. Two ladders, two new libraries, a torsional rigidity and a contact measurement, and a fair statement of what this collection’s model of a thread can now do and where it still stops.

Where this collection’s thread model now stands.