The swelling a cloth cannot take

Peirce’s geometry ties a cloth’s two thread systems together with one equation: the crimp height taken by the warp and the crimp height taken by the weft must add to the cloth’s thickness. Thickness one system takes is thickness the other cannot have.

Each system can supply only so much height. A thread of length l wrapping a partner of thickness D rises highest when its straight portion has just vanished, at an angle of l/D, and the height there is D(1 − cos(l/D)) — which is closed form, with no search in it. It is Peirce’s closure condition asked for its own limit rather than for a solution.

So there is a condition for a cloth to exist at all:

$$h_1^{\max} + h_2^{\max} \;\ge\; D$$

and swelling raises the right-hand side in exact proportion while raising the left-hand side much more slowly, because a fixed thread length wraps a thicker partner through a smaller angle.

For three of the eight cloths here, twenty per cent of swelling breaks it.

The condition has no sett in it #

Substitute the closed forms and the thickness cancels:

$$\cos\!\left(\frac{l_1}{D}\right) + \cos\!\left(\frac{l_2}{D}\right) \;\le\; 1$$

That is worth writing out, because of what is not in it. There is no spacing. Whether a wet cloth exists is a statement about two thread lengths and a thickness, and the sett has said everything it has to say once it has fixed those.

The two forms — the margin as a length, and the cosine condition as a pure number — are the same statement divided by D, and they are checked against each other at every cloth and five swellings. Doing that is not redundancy: the closed form is what this essay argues with and the margin is what the solver uses, and a change to one that did not reach the other would leave the argument and the figures disagreeing with nothing to say so.

The critical swelling, cloth by cloth #

Bisect on the margin — which is monotone, because the demand rises linearly and the supply sublinearly — and each cloth has a swelling at which it loses its last state.

The sheeting loses it at 9.29 per cent, less than half of cotton’s twenty. The filter cloth at 17.18. The duck at 19.85.

The other five survive: the poplin has headroom to 31.79 per cent, the muslin to 37.39, the batiste to 43.49, the voile to 75.82, and the cheesecloth has no critical swelling anywhere below one and a half, because an open scrim has thread to spare.

The duck is the case worth pausing on. Its critical swelling is 19.85 per cent against a fibre swelling of 20.0, so it fails by fifteen hundredths of a point — and cotton’s swelling is a measurement with a reported range of fourteen to twenty-three per cent. So the duck’s answer is not merely close, it is undecidable at the precision of the constant: at the bottom of the range the duck is comfortably fine and at the top it is comfortably not.

That is the honest report and it is why the assertion on this arithmetic is about a boundary rather than about a set of cloths. Just below its critical swelling a cloth must have a state and just above it none, which is what makes the number a boundary rather than the end of a search; and the eight must not all share a value, which is what makes it a property of the cloth.

What the boundary depends on, and what it does not #

Since the condition has no spacing in it, it is fair to ask what a cloth’s critical swelling is a function of.

Two thread lengths and a thickness — but the thread lengths came from the sett, through the Peirce solution at the quoted construction, so the sett is in the answer after all, once removed. It arrives the same way a diameter arrives from a count. What has been separated out is that the sett enters only through the thread lengths, which is a much stronger statement than “the sett matters”.

The practical form of it is a ratio: l/D, the thread length per crossing divided by the cloth’s thickness. Write the condition in those terms and it is a statement about two pure numbers, one per system. A cloth whose threads have a lot of length per unit of thickness is safe and one whose threads have little is not, and everything about counts, setts and covers is upstream of that single ratio.

The eight cloths order themselves accordingly. The cheesecloth has 30 tex yarns at 10 ends per centimetre — an enormous span per crossing against a modest thickness — and is untouchable. The sheeting has 25 tex at 28, which is a short span against a thickness two thirds as large, and it is the first to fail. The duck’s 60 tex at 16 is a large thickness and a large span together, which is why it lands almost exactly on the boundary.

So the quantity that decides is not density and not fineness but their combination, and the reason the cover factor does not predict it is that cover is d/p while this is l/D — a length along the thread against a length through the cloth, rather than two lengths in the plane.

Something has to give #

Past the critical swelling there is no state at constant thread length, so the fabric cannot take the swelling as a shape change. Something must change length or change volume, and there are exactly two candidates.

Compact the yarn. Squeeze the swollen yarn back to a diameter the geometry can hold. The fibre volume fraction rises above its free-swollen value and van Wyk’s cube law prices it — the same law this collection uses for a preform under pressure and a wool bale in a press.

Stretch the thread. Leave the yarn alone and lengthen the threads until they can reach round the swollen partner. The yarn’s tensile stiffness prices it.

The cloth takes whichever is cheaper, so the answer is a minimum over two routes rather than a calculation, and the comparison is the result.

The stretching route does not exist #

For the sheeting, stretching the threads far enough to wrap a fifth-thicker partner needs 9.1 per cent of strain.

Cotton’s breaking strain, computed from the two constants this collection already carries — a tenacity of 0.35 newtons per tex divided by a specific modulus of 5.26 newtons per tex — is 6.65 per cent.

So the thread would break before it was long enough. There is one route and not two, and the comparison is settled without any arithmetic about which is dearer.

That breaking strain is a derived number rather than a new row in a table, and it is worth saying how it is derived because it is a straight-line extrapolation of a curve that is not straight. A tenacity is a breaking specific stress and a specific modulus is a specific stress per unit strain, so their quotient is the strain at break on a linear stress-strain line. A real cotton yarn’s curve is concave, so it breaks at a larger strain than a linear extrapolation gives — around seven to eight per cent, against this 6.65. That makes the conclusion safer rather than shakier: the true breaking strain is nearer to 9.1 than the computed one, and still below it.

For the filter cloth the stretch route needs 2.24 per cent of strain, which is survivable, and there the comparison has to be made on price: 178.9 megapascals of stress against 1.59 of compaction pressure. Compaction wins by two orders.

The pressure #

At cotton’s own swelling the sheeting’s yarn must come back from a free-swollen diameter to the one its critical swelling allows, which raises its fibre volume fraction from 0.600 to 0.723 — and van Wyk gives 7.8 megapascals.

That is three orders of magnitude above anything else this collection computes. The contact force at a crossing of a relaxed sheeting, which had to be read backwards out of a measured fabric thickness, is 0.42 newtons; spread over the projected area of one crossing that is well under a megapascal.

The filter cloth comes out at 1.59 megapascals and the duck at 0.078, and the ordering is exactly the ordering of how far each is past its own critical swelling. That monotonicity is what is asserted rather than the value, because the value carries van Wyk’s constant and that has a reported spread of nearly four — 0.003 to 0.011 — while the shape of the answer has none.

So the height of this curve is known to a factor of four and its shape is not. A megapascal is the honest statement; 7.803 is what the arithmetic prints.

Why megapascals is not absurd #

A number three orders above everything around it deserves a sanity check, and there are two.

Osmotic swelling pressures really are of this order. The pressure a swelling gel can generate against a constraint is tens of megapascals, and a fibre’s water uptake is a swelling of the same kind. Nothing here computes that pressure independently — the number above is what it costs to compact the yarn, not what the water can supply — but the two being the same order is what makes the compaction route physically available at all. If the swelling pressure available were kilopascals, the fibres simply would not swell in a constrained cloth.

And the consequences are visible. A closely woven cotton that has been wetted and dried is measurably harder, stiffer and denser than one that has not, and a cloth that has been through a wash-and-dry cycle several times is denser again. Something is being pressed permanently, and a pressure of the order of a fibre’s own transverse modulus is what would do it.

That second observation is also this rung’s largest limitation, and it is worth being blunt about: this model is entirely recoverable and the real process is not. Dry the cloth here and the yarn springs back to 0.600 packing exactly. A real cotton does not, which is why the first wash shrinks a garment and the tenth barely moves it.

The three cloths and what they have in common #

It is worth asking whether the three that fail have anything in common besides failing, because if they do then the result generalises beyond this table.

They are the sheeting, the filter cloth and the duck. By cover they are the first, fourth and third densest of the eight. By yarn count they are 25, 40 and 60 tex — the three coarsest but one. By sett they are 28, 20 and 16 ends per centimetre — which is to say the sheeting is among the densest and the duck among the openest.

Neither cover nor count nor sett sorts them out, and that is the finding restated. The sheeting fails because it is closely set for its count; the duck fails because it is very thick for its span; the filter cloth is between the two. Three different routes to the same failure, which is what a condition on a combination looks like from outside.

What they do share is being the three heaviest cloths in the table by weight per unit area, and that is not an accident either: weight per unit area is thread mass per unit area, and thread mass goes with diameter squared while area goes with spacing squared, so a heavy cloth is one with a large diameter for its spacing — which is the same combination once more, in a third dress.

What was counted, and how #

The boundary is a boundary. Just below each critical swelling a state must exist and just above it none, checked at a step of one part in a million. That is what distinguishes a boundary from the last value a search happened to reach, and this collection has shipped a solver that could not tell the difference twice.

The pressure is monotone in the excess. Swept on the sheeting from its critical swelling to its fibre’s, in eight steps, and required to rise at every one. That is the statement with no measurement in it: a cloth pushed further past the point where its geometry ran out must press harder.

And the scale is a fabric scale. The cloth furthest past its own critical swelling must exceed one megapascal, which is a floor set far below the answer because what is being asserted is an order and not a value. Applying the same floor to every cloth would have failed on the duck, and failed for the right reason wearing the wrong clothes.

Compaction is cheaper, and for one cloth the alternative does not exist. Asserted both ways: every compacted cloth must prefer compaction, and at least one must have a stretch route beyond its own breaking strain.

Where the model stops #

The compaction is uniform along the thread. A real yarn is squeezed only near its crossings and lies free between them — which is the same non-uniformity that makes a flattened thread a record of a force — so the pressure computed here is an average charged over the whole length. The peak at a crossing is higher and this collection cannot say by how much, because that needs a contact mechanics it does not have.

Van Wyk’s law is for a random fibre assembly. A twisted yarn is not random: its fibres are helices under tension from the twist, and compacting it is not the same problem as compacting a batt. The law is the right family and the constant is being asked to do work it was not fitted for.

There is no water in the mechanics. Water plasticises cellulose, so a wet fibre’s transverse modulus is lower than a dry one’s and the pressure to compact a wet yarn is lower than this. The direction is known and the size is not, and it would reduce the headline number.

And the swelling is treated as a rigid demand. In fact a fibre in a constrained space swells less, because the constraint opposes it; the equilibrium is between the osmotic pressure and the mechanical one. This model applies the free swelling and then asks what it costs, which overstates both.

The generalisation #

A geometric model that refuses is naming where the physics has to enter.

The refusal here is not a bug and it is not a limitation of the solver. Peirce’s geometry has no state past a certain swelling because there genuinely is none: no arrangement of two inextensible threads of those lengths around obstacles of that thickness satisfies the closure condition. The geometry has therefore located, exactly, the point at which something outside it must take over, and it has done so without knowing anything about what.

That is a more useful failure than a wrong answer. A model that had quietly returned a state past the boundary — which is what a bisection that treats every failure the same way does, and what this collection’s own solvers did twice — would have hidden the most interesting thing about the problem.

The habit worth carrying is to make a model’s refusals explicit and named, and then to read them as findings.

Who found it, and when #

That a wetted cotton yarn swells and that a densely woven cotton becomes nearly impermeable when wet is old and practical knowledge; Ventile and Grenfell cloth are both built on it.

The geometric limit does not appear in that literature as far as this collection can tell. Peirce’s own papers give the closure condition and use it in the direction it is usually needed — solve for a state — and do not ask when there is none.

Van Wyk’s law is from 1946 and is the standard account of compressing a fibre assembly.

What is this collection’s is the critical swelling, the observation that the condition has no sett in it, and the finding that the alternative route is closed by the fibre’s own breaking strain rather than by its price.

Where the ladder goes next #

Sideways, into the criterion this whole collection is built on. A cloth that presses on itself has a contact force, and a contact force is exactly what the criterion has always needed and never had without a measurement.