A wet cloth is set closer than it was woven

The cover factor is the simplest number this collection carries and almost the most useful: a thread’s diameter divided by the distance to the next thread, which is the fraction of the cloth’s width that is thread rather than hole.

It is also the number water does the most obvious thing to. A cover is d over p. Wetting raises d and leaves p exactly where it was — a cloth put in water does not change size instantly; that takes time and needs the threads to move — so

$$\text{cover}_{\text{wet}} = \frac{(1+s)\,d}{p} = (1+s)\cdot\text{cover}_{\text{dry}}$$

and the sett at which threads touch side by side falls by the same factor.

That is an identity, not a result. It is asserted here to machine precision across every cloth and four swellings, because that is what an identity is entitled to, and because it is the one statement in this whole ladder a reader can check on the back of an envelope.

What the identity is worth #

A relation with no arithmetic in it can look like it is not saying anything. This one is saying two things and both matter.

It fixes what state the question is being asked in. A cloth’s cover changes when it is wetted and when it shrinks, and those are different changes at different times. Straight out of the water the spacings have not moved and the whole change is the diameter’s; a day later the cloth has relaxed to a new construction and the spacings have moved too. This essay is about the first moment, which is the one nobody computes because it looks trivial.

And it makes the comparison across cloths free. Since every cover is multiplied by the same factor, the ordering of the eight cloths by cover is unchanged and so is every ratio between them. Anything this collection has said about cover that was a comparison survives wetting untouched; anything that was an absolute number does not. That is the same shape as the float ladder’s finding about the capstan, and it is not a coincidence — a multiplicative correction always divides out of a ratio.

Which cloth comes closest to closing #

The eight cloths run from a cheesecloth at a warp cover of 0.205 to a sheeting at 0.523. Wet, they run from 0.246 to 0.628.

None of them reaches one, so no cloth in this table jams laterally when it is wetted. The closest is the sheeting, which arrives at 0.628 and still has more than a third of its width as hole. Even the poplin, which is the most warp-dense construction here at 32 ends per centimetre, goes only to 0.556.

That is worth stating because the expectation runs the other way. A close cotton shirting feels noticeably stiffer and less permeable wet, and the natural explanation is that the threads have closed up against each other. They have not. What has happened is in the thickness rather than in the width, and it is the swelling a cloth cannot take.

The jamming sett, and how far away it is #

The sett at which the swollen threads would touch is 10 divided by the wet diameter, in threads per centimetre. For the sheeting that is 44.6 against a quoted 28 — the cloth would have to be set sixty per cent closer than it is before its wet threads touched.

The headroom is smallest in the sheeting at 59 per cent, then the filter cloth at 76 and the poplin and the duck at 80. The cheesecloth has 307 per cent of headroom, which is another way of saying it is a scrim.

So the answer to “can a cotton cloth be woven so close that it jams when wet?” is yes in principle and no in this table. It would take a construction about sixty per cent denser than a sheeting, which is denser than anything woven in cotton at these counts — and by the time a cloth is that dense the through-thickness condition has failed several times over — which is why the cloth called impossible fails for a different reason again — so the lateral jam is not the binding constraint and never gets a chance to be.

The first contact a wetted cloth makes is with the thread it crosses, not with the thread beside it. That is the finding of this rung and it is the opposite of what the cover factor’s prominence would suggest.

Why the two constraints are not comparable #

It is worth being precise about why the lateral condition is so slack and the through-thickness one so tight, because the two look symmetrical and are not.

The lateral condition asks whether d has grown past p. Both are lengths in the plane, and in an ordinary cloth p is two to five times d, so there is a factor of two to five of room and a swelling of twenty per cent uses very little of it.

The through-thickness condition asks something else entirely. It asks whether the two thread systems can supply, between them, a crimp height equal to the cloth’s thickness — and the thickness is itself the sum of the two diameters. So the demand grows in exact proportion to the swelling while the supply, which is set by the thread lengths and does not grow at all except for the axial hundredth, does not.

One condition has a factor of several in hand and one has none. The through-thickness condition is not slack in a dense cloth even dry: the sheeting has 0.06 millimetres of margin on a thickness of 0.37, which is a sixth. Twenty per cent of swelling on a thickness of 0.37 is 0.075 millimetres of extra demand, and a sixth is not enough.

That is the whole of why the cover factor, which is the number a weaver reaches for and the number this collection reached for first, is the wrong number for this question.

The two covers a cloth has, and which one water moves #

A cloth has a warp cover and a weft cover and they are almost never equal, because almost no cloth is set the same way in both directions. The poplin is the extreme case in this table: 32 ends and 22 picks per centimetre in yarns of 15 and 20 tex, which gives covers of 0.463 and 0.368.

Wetting multiplies both by 1.2 and therefore leaves the ratio at 1.26 exactly where it was. So a warp-faced cloth is still warp-faced wet, by the same margin, and the whole vocabulary of balance and unbalance that this collection built on cover survives a wash unaltered.

That is not obvious in advance and it is not true of everything. The crimp ratio does not survive: wetting a poplin takes its warp crimp from 20.5 per cent to 24.2 and its weft crimp from 1.6 to 6.7, which is a change in the ratio from thirteen to less than four. So one of the two ways of describing how unbalanced a cloth is moves enormously in water and the other does not move at all.

Which one somebody means by “an unbalanced cloth” therefore decides whether the statement survives a wash. That is a fair warning about a piece of vocabulary this collection has used a great deal.

Why the spacings really do not move at first #

The claim that p is unchanged deserves more than an assertion, because a reader may reasonably think that a cloth put in water starts contracting immediately.

It does not, and the reason is that contracting requires the threads to slide against one another at their crossings. A cloth’s construction is held in place by friction: the pick spacing is where the beat-up put it, and moving it means every crossing along the thread letting go and re-gripping. That is what relaxation is about, it is why agitation helps, and it is why a fabric left to soak quietly relaxes far less than one that is tumbled.

Swelling needs none of that. Water enters the fibre and the fibre gets wider, at every point along the thread at once, with nothing having to slide. It is the fastest thing that happens to a wetted cloth and it is complete in seconds to minutes.

So there is a genuine separation of timescales, and it is what makes this essay’s state a real state rather than a convenient fiction: the swelling finishes before the relaxation starts.

What was counted, and how #

Cover scales by exactly the swelling ratio. Asserted across eight cloths and four swellings from five per cent to thirty-five, to within a tolerance of 1e-12. What it would catch is a later change that let a spacing move inside the cover calculation, which would make the number a different quantity wearing the same name — a wet cover of a shrunken cloth rather than a wet cover of the cloth as woven.

And the jamming sett is its reciprocal. Same assertion from the other side, because the two are the same statement and a version that computed them independently could disagree.

The headrooms are read off the same arithmetic and are not separately asserted, because there is nothing in them a check could catch that the identity does not already cover.

Where the model stops #

The spacings are held, and in a real wetting they are held only briefly. A cloth dropped in water swells in seconds and relaxes over minutes to hours, so the state computed here has a lifetime. It is the right state for the question — what does wetting do before the cloth has moved — and the wrong one for a garment coming out of a machine.

The swollen thread is treated as round. A thread in a close cloth is not round; it is flattened by the threads it crosses, and a flattened thread’s width is larger than its equivalent diameter. So the real wet cover is higher than the figure here and the real headroom smaller. The direction is known and the size is not, because the swelling changes the flattening too and this rung does not solve for that.

The cover is a single-system cover. A cloth’s warp cover and weft cover are computed separately and the fabric’s total cover is not their sum — the overlaps are counted twice, which is why the trade’s combined cover factor subtracts a product. That correction is unchanged by wetting in form and changed in size, and it is not carried here.

And nothing here is about permeability. Cover is often used as a proxy for how much air or water gets through, and it is a poor one even dry: the hole between four threads is not the complement of the cover, and it is a channel with a length rather than a rectangle. The wicking ladder computes that properly and would need its own wet treatment.

The number the trade actually uses, and what water does to it #

Weavers do not usually quote a cover factor as a bare fraction. The British system quotes it as the thread count divided by the square root of the cotton count, and the American as the thread count divided by the square root of the yarn number in its own units, and both come out as numbers in the twenties for an ordinary cloth rather than as fractions under one.

That is the same quantity with the diameter arithmetic folded in and a constant absorbed. Since a yarn diameter goes as the square root of its count, threads per inch over the square root of the count is threads per inch times a diameter times a constant — which is a cover.

The consequence for this essay is that the trade’s cover factor has no fibre in it. The constant that was absorbed contains the fibre density and the packing factor, and both were treated as fixed when the system was set up, which was reasonable when nearly everything was cotton. So a cover factor of 14 means one thing in cotton and a slightly different thing in wool, and it means a very different thing wet than dry — because the absorbed constant is exactly where the swelling would have gone.

This is a small instance of a pattern this collection keeps meeting: an index that works because everything it was calibrated on shared a property, used later on things that do not. The tightness factor a knitter uses has the same shape and the same blind spot.

What the same identity does to the hole #

The identity multiplies the cover and it does something much more violent to the space between the threads, which is the quantity the shower-proof cotton is actually about. That part can be taken one step here even though the flow through it belongs elsewhere, because the geometry is the same identity used twice.

The gap between two adjacent threads is p − d, which is p(1 − K) with K the cover. Wet, the diameter is 1.2d, so the gap is p(1 − 1.2K) and the hole between four threads — a square of that side — has an area in the ratio

[(1 − 1.2K) ÷ (1 − K)]².

The cover rises by twenty per cent and the hole falls by anything from a quarter to everything, depending entirely on where the cloth started. That is the amplification the cover factor’s own linearity conceals: a multiplicative change in the thread becomes a divergent change in the space between threads, exactly as the fourth-power rule describes on the dry side.

Which gives the shower-proof cotton a number #

The bottom row is the interesting one. A cloth at a dry cover of 0.833 has no hole at all when its threads swell by a fifth, because 1.2 × 0.833 is exactly one.

That is the Ventile condition, and it explains the cloth’s reputation as a threshold rather than a gradient. An ordinary sheeting at 0.52 loses a third of its openings and is merely damp; a dense cotton at 0.70 loses nearly three quarters; one at 0.80 loses ninety-six per cent and is, for practical purposes, closed.

And the last three points of cover do most of the work. Between 0.75 and 0.78 the residual opening halves. So the specification for such a cloth is a point or two of cover wide, which is why these fabrics are woven from long-staple cotton at very high setts on slow looms, sold on their construction, and difficult to make — and why nothing about the recipe is negotiable.

It also says why the effect cannot be had by any other route. The closure ratio depends on the cover and the swelling and nothing else — not the weave, not the yarn’s twist, not the finish. A cotton swells twenty per cent, so the threshold is 0.833 and there is one way to reach it. A polyester swells by a thousandth of a per cent, so no polyester cloth is shower-proof by this mechanism at any construction, which is exactly why waterproof synthetics are coated or laminated instead.

And it is a first-order figure, deliberately #

Two things make the table optimistic and both are worth naming so it is read as a bound.

A hole is a channel rather than a square, and its narrowest section is not its mouth — so a cloth whose projected opening has vanished may still have a passage through it, and the wet closure is less complete than the plan view says.

And the swollen thread is not round. A flattened thread is wider than its equivalent diameter, so the real cover is higher and the real hole smaller — which pulls the other way and moves the threshold below 0.833.

The two corrections are opposite in sign and neither is computed here. What survives both is the shape: a squared ratio with a zero in it, so the closure is negligible over most of the range and total over the last tenth of it. A mechanism with a threshold in it is what the trade observed, and the threshold is where a linear identity meets a difference of two nearly equal numbers.

The generalisation #

When two constraints look symmetrical, check which one has room in it before deciding which binds.

Lateral jamming and through-thickness jamming are both statements that a cloth cannot hold any more thread, and both are exact conditions on the same geometry. One of them is slack by a factor of several in every cloth and the other is tight in half of them. Nothing about the way they are usually written says so, and the cover factor’s prominence in the trade actively points at the wrong one — because cover is what a weaver can see, and the thickness condition is invisible.

The habit worth carrying is to write both conditions as ratios of the same two quantities where possible and then look at the ratios. Here that is d/p against D/(supply of crimp height), and the second is near one wherever the first is near a half.

Who found it, and when #

The cover factor is Peirce’s, and the observation that a wetted cloth becomes less permeable is as old as cloth. The trade’s rule of thumb — that a closely set cotton “closes up” in the wash and becomes shower-resistant — is the practical form of it, and it is the basis of Ventile and of the older Grenfell cloth: a very densely woven long-staple cotton that is nearly waterproof when wet precisely because it swells.

That effect is real and this rung says it is not lateral jamming. What it is instead is the pore between four threads closing, which is a much smaller hole than the space between two, and computing it properly belongs to the arithmetic of what a cloth lets through.

What is this collection’s is the arithmetic that says which condition binds, and the observation that the one the trade quotes has a factor of several in hand.

Where the ladder goes next #

To the condition that does bind. A thread of fixed length cannot wrap a partner that has grown by a fifth, and three of the eight cloths here have no wet state at all.