Wetting moves a cloth to another locus

This collection has a set of states a cloth can reach without any yarn changing length: the constant-thread-length locus, one-dimensional, parameterised by how the two systems divide the cloth’s thickness between them. Everything about extension, crimp interchange and Poisson behaviour is a statement about moving along it.

The locus is built at a fixed thickness D, which is the sum of the two thread diameters. Wetting changes D. So a wetted cloth is not at a different point of its own locus. It is on a different locus, and the two are not connected by any deformation the cloth can perform.

That is the whole structure of the question, and it decides what may be compared with what.

What may be compared with what #

An earlier rung was written to settle whether bending energy could supply the crimp ratio that Peirce’s geometry leaves undetermined, and it found that it cannot — but it also found what the energy does determine, and being careful about that is what makes this rung possible.

The least-energy point of a constant-thread-length locus is a state at a different sett from the one the cloth was quoted at. That is why writing its crimp ratio onto a figure of the quoted construction would put the ratio of one state on a drawing of another, and it is why that correction was refused.

But shrinkage is the change of sett. So the object that was the wrong answer to the crimp-ratio question is exactly the right answer to this one: the state a cloth relaxes to is a real thing about a real fabric, and comparing two of them — one dry, one wet — is a comparison of two objects of the same kind.

The rule this rung obeys is therefore: a locus minimum may be compared with another locus minimum and never with a table row. Both states here are computed the same way, by the same scan of the same energy over the same machinery, with one input changed.

What the numbers are #

Of the eight cloths in this collection’s table, five have a wet state at all. The other three are the subject of the swelling a cloth cannot take.

The batiste closes up by 2.54 per cent across and 2.56 along. The muslin by 3.09 and 3.14. The poplin by 3.67 and 1.82 — the only markedly anisotropic one, and it is anisotropic because it is the only markedly unbalanced construction. The voile by 0.93 and 0.94.

And the cheesecloth grows, by half a per cent in each direction.

Those sizes are the right order for real cotton. Reported wash shrinkages for woven cottons run from one to five per cent for the first wash of an unfinished cloth, and a mercerised, pre-shrunk shirting is specified at under two. This model has no plasticity, no felting and no residual loom tension in it, and it lands in the range anyway — which is worth noticing but not worth leaning on, because three mechanisms this collection can compute and two it cannot all point the same way and the agreement could be luck.

The scrim that gets bigger #

The cheesecloth is the interesting one and the reason is the column of the swelling table that looks like a rounding error.

There are two competing effects. The thicker partner costs extra crimp: at constant thread length a thread that must go round a fatter obstacle has less length left over to span the cloth with, so the spacing closes. And the fibre is longer: cotton gains one and two tenths per cent of length in water, which is extra thread to span with.

The first term goes roughly as the square of the thickness over the span, so it is large in a close cloth and small in an open one. The second is flat — a hundredth is a hundredth wherever it is.

So there is a construction where they balance, and below it the second wins. Suppress the axial swelling in the arithmetic and every cloth shrinks without exception: the cheesecloth goes from opening by 0.50 per cent to closing by 0.76. Put it back and it reverses.

An open scrim comes out of a wash wider than it went in, and the reason is that a cotton fibre is one per cent longer wet.

The cover factor where it changes hands #

The crossing between the two terms is not a property of a particular cloth. Sweep the sett of a balanced cotton cloth and the width shrinkage crosses zero at one place, and the number worth having is the cover factor there rather than the sett.

It comes out at 0.229904, and it is the same to six figures at 15, 20, 30, 45, 60 and 100 tex.

That is not a coincidence and it is not a fit. Both competing terms scale with the yarn diameter — the geometric term through the thickness over the span, the axial term through the extra thread length, which is a fraction of a length that itself scales with the diameter at fixed cover — so where they cross is a ratio and the count divides out. The corresponding sett runs from 15.9 ends per centimetre at 15 tex down to 6.2 at 100, and the cover at every one of them is the same number.

A cotton cloth set below a cover of 0.23 gets bigger in the wash, whatever it is made of and whatever count it is woven in. That is a statement with no free parameter in it and it is the cleanest thing the water ladder produced.

It is a different number for every fibre #

The crossing cover is set by the ratio of a fibre’s axial swelling to its transverse one and by nothing else, which the ordering across fibres says plainly.

Viscose, whose swelling is the least anisotropic of the swelling fibres at 8.3 to one, crosses at 0.306. Silk at 12.9 to one crosses at 0.262. Wool at 13.3 crosses at 0.259. Cotton at 16.7 crosses at 0.230.

The direction is the mechanism restated: the axial term is what opens a cloth out, so a fibre with proportionally less of it needs a more open cloth before it can win.

Two fibres have no crossing at all and both are informative. Flax swells a thousandth along its length, because its length is fixed by a crystalline structure water does not get into, so the axial term is essentially absent and a linen cloth only ever shrinks — which is exactly the reputation linen has, and is one of the two places water separates two fibres this site’s geometry treats as one. Nylon swells 2.3 per cent across and 1.9 along, which is nearly isotropic, so the axial term wins everywhere and a nylon cloth of any construction gets very slightly bigger.

Where the crimp goes #

The shrinkage is the visible half of what happens; the crimp is the other half and it moves much more.

Wetting the muslin takes its warp crimp from 9.45 per cent to 14.36 and its weft crimp from 6.41 to 11.12. The batiste goes from 8.27 and 6.03 to 12.45 and 10.11. The poplin’s weft crimp more than quadruples, from 1.57 per cent to 6.70.

So a cloth that shrinks three per cent has gained five points of crimp in each system. The crimp change is nearly twice the dimensional change, and the reason is that most of the extra thread path goes into going round a thicker obstacle rather than into pulling the cloth in.

That has a practical consequence the trade knows in a different vocabulary. A cloth’s thickness rises far more than its dimensions fall, and its weight per unit area rises by more than the areal shrinkage, because the same thread mass is in a smaller area and the crimp has taken up more of it. That is the same arithmetic relaxation does dry, with a larger driver, and it is why a cloth gains weight by losing size.

What was counted, and how #

Four assertions, and the interesting one is a negative.

The two forms of the closure condition agree. The solver works with a margin in millimetres and the essays argue with a condition in cosines, and the two are the same statement divided by the thickness. They are checked against each other at every cloth and five swellings, to 1e-12, because a change to one that did not reach the other would leave the figures and the prose disagreeing with nothing to say so.

The wet minimum is geometric for a balanced cloth. The rigidities used here are dry ones and a wet fibre is not as stiff as a dry one. What rescues the result is that six of the eight cloths have equal counts in warp and weft, so the two rigidities are equal whatever they are and the minimum cannot depend on them. Run at both ends of the bending bracket and required to agree to the locus’s own sampling resolution — which is the check that says the unknown does not matter rather than the assumption that it does not.

Swelling closes a cloth and the axial term opens it. Asserted in two halves: with the axial swelling suppressed every cloth must shrink, and with it restored at least one must grow. A result of the opposite sign in either half would mean the swelling or the locus had been applied backwards.

And the crossing cover is count-free. Asserted to one part in a hundred thousand across counts spanning a factor of seven, which is the tolerance an identity is entitled to and not the tolerance an approximate agreement passes at.

A defect this rung found in its own arithmetic #

Two, and both were in the sweep rather than in the model.

The first was a cache keyed on a name. Every memo in the wet machinery was keyed on the cloth’s name, which is unique across the eight rows of a table and is not unique across the constructions a sweep invents. A sweep of forty setts under one name returned the first answer forty times, to the last decimal place, and the flatness was the only thing that gave it away. A name identifies a row; a construction is its four numbers.

The second was an operator precedence error that made the shrinkage exactly zero whenever the dry relaxed spacing happened to be exactly one millimetre. That is a sett of ten per centimetre, which is a perfectly ordinary construction, and it showed as two suspicious zeroes in a sweep of forty points.

Neither would have been caught by any assertion in this file, because both produced numbers that were the right shape and the right magnitude. What caught them was plotting the sweep and looking at it, which is not a check and is not automatable and is why a figure is drawn from the same arithmetic the essay quotes.

Where the model stops #

The bending rigidities are dry. The two unbalanced cloths’ answers therefore carry an unquantified error, and they are the poplin and the filter cloth. The six balanced ones do not, and the assertion says so.

There is no plasticity. Dry this model’s cloth and it returns exactly where it started. A real cotton does not: the first wash takes most of the shrinkage and later washes take much less, which means something has been spent permanently. This model can say nothing about that at all, and it is the largest thing standing between this rung and a shrinkage a specification could quote.

The relaxed state is a least-energy state, which assumes the cloth gets there. Friction holds a cloth’s construction where the loom left it, and a soaking without agitation relaxes far less than a tumble. So this computes the state a cloth would reach with enough help, and what agitation buys applies here unchanged.

And the swelling is at saturation. A cloth in humid air is somewhere between dry and this, and the whole ladder scales with the swelling rather than switching on at it.

What this does not say about a wash #

A number of the right size is a temptation and it is worth putting the limits in one place before anybody quotes these figures against a label.

These are hygral shrinkages, not wash shrinkages. A wash also relaxes the cloth, and that is a separate mechanism with a separate size that this arithmetic knows nothing about. A label’s number is at least the sum of the two, and probably of three — which is the distinction two shrinkages and one tape measure is about.

They are computed at saturation. A cloth in a wash is saturated; a cloth in a humid room is not, and the swelling — and therefore all of this — scales roughly with the moisture the fibre has taken up.

And they are reversible. Every figure here comes back on drying. So the honest reading of “a muslin shrinks 3.1 per cent” is that a muslin is 3.1 per cent smaller while it is wet, and comes back. That is a real and observable thing — it is why a shirt is tight straight out of a machine — and it is not what a label means by shrinkage.

Stating all three is not hedging. A figure of the right order that measures a different quantity from the one it will be compared against is the most misleading kind of agreement there is, and this collection has been caught by one before.

The generalisation #

Changing a parameter that a solution set was built from does not move a system within that set; it replaces the set.

That sounds obvious written down and it is the mistake this ladder was closest to making. The locus is such a familiar object here that the natural first move was to ask where on it a wetted cloth sits, and the answer is nowhere: the swollen cloth’s states are not states of the dry cloth at all. Every quantity the locus computes — the extension available, the exchange rate, the Poisson ratio — has to be recomputed rather than re-read.

The wider version is worth stating because this collection has now met it twice. An energy closes a kinematic model only over the set it is minimised on, and naming that set is the whole of the work. Here the set changed, and noticing that it had changed is what made the comparison legitimate.

Who found it, and when #

That a cotton cloth shrinks when it is wetted and that the shrinkage is mostly recovered on stretching has been known since cloth was washed. Peirce’s own 1937 paper gives the geometrical explanation in one line — the yarns swell, so the crimp increases, so the cloth contracts — and does not compute it.

Computing it is not hard once a solver exists, and the reason it is rarely done is that the useful quantity in the trade is the residual shrinkage after finishing, which is dominated by mechanisms this model does not have. So the geometric part is known, agreed and not much used.

What is this collection’s is the crossing cover, the observation that it is independent of the yarn count, and the finding that it is the axial swelling — the column everybody rounds to zero — that puts it where it is.

Where the ladder goes next #

Downward, to the three cloths that are missing from every table in this essay. They have no wet state at constant thread length at all, and what a cloth does when its geometry has run out is the largest force this collection has computed.