A cloth cannot shrink past its own crimp

Every other number in this field is a number at a stated loom tension, a stated crimp ratio, a stated swelling. This one is not. It has no fitted parameter, no measured coefficient and no model in it beyond the definition of crimp, and it puts a hard bound on the worst a fabric can do.

A cloth relaxing at constant thread length cannot lose more than c/(1 + c) of a dimension, where c is the crimp it ends up with.

The derivation, which is three lines #

Let a thread of length l lie in a cloth at crimp c. By the definition of crimp it spans

p = l / (1 + c)

The most a relaxation can do is take the cloth from a completely straight thread — crimp zero, spanning the full l — to the crimped state spanning l/(1 + c). That is the largest possible change, because a thread cannot be straighter than straight.

The fractional loss is therefore

(l − l/(1 + c)) / l = 1 − 1/(1 + c) = c / (1 + c)

and no arrangement of loom tensions, temples, wetting or tumbling can exceed it. It is a consequence of the thread having a fixed length and of nothing else.

Why it is c/(1 + c) and not c #

The difference between these two looks like pedantry and is not, because the ratio between them is the size of the error people make.

The confusion comes from which length the fraction is of. A crimp of twelve per cent means the thread is twelve per cent longer than the cloth. Shrinkage is quoted as a fraction of the cloth’s original length, which in the extreme case is the straightened thread. Twelve per cent of the smaller quantity is not twelve per cent of the larger.

Concretely: a cloth whose relaxed crimp is 12.79 per cent has a ceiling of 11.34 per cent, not 12.79. At a crimp of 50 per cent the ceiling is 33.3 per cent, not 50. At a crimp of 100 per cent — a thread twice as long as its cloth, which happens in a terry pile — the ceiling is 50 per cent.

The gap grows with the crimp, which means the mistake is smallest exactly where it matters least. For a shirting the two figures differ by a point and a half. For a heavily crimped construction they differ by a third.

What the bound is worth #

A bound that is never approached is a curiosity. This one is approached, and knowing it changes what a designer looks at.

The ceiling is a property of the finished cloth’s crimp, and the crimp is decided by two things this site has computed since the foundation: the number of interlacings and the sett. So the ceiling is decided at the drawing board.

• A plain weave interlaces at every intersection. It has the most crimp of any weave at a given sett and therefore the highest ceiling — the most it can possibly move.
• A 2/2 twill interlaces half as often. Its threads bend half as many times per unit length, so its crimp is markedly lower, so its ceiling is lower.
• A satin interlaces least of all, and a five-end satin’s threads run nearly straight between widely spaced bindings.

This is the float doing its usual work. Float length sits behind lustre, drape, abrasion, tear strength and setting, and here it is again behind dimensional stability, in the same direction it always runs: fewer interlacings mean a limper, glossier, more slippery, more stable cloth, and a weaker and more snag-prone one.

The sett does the same thing in the opposite direction #

Crimp rises with closeness, which this site derived from Peirce’s closure condition at the foundation: the crimp height each system must supply is set by the yarn’s thickness and does not change, while the horizontal room to supply it in shrinks as the threads crowd. Same rise over a shorter run is a steeper angle and more length spent.

So a densely set cloth has more crimp than an open one in the same yarn, and a higher ceiling. A tightly woven fabric is potentially less dimensionally stable than a loose one, which is the opposite of the intuition that tight means stable.

The intuition is not stupid, and it is worth saying what it is right about. A tight cloth resists distortion in use: it shears less, it bags less, it recovers better. What it does not do is resist relaxation, because relaxation is not a distortion — it is the fabric going where it wanted to go all along, and a tight cloth wants to go further.

Where the ceiling itself moves #

The ceiling is c/(1 + c) for the crimp the cloth ends up with, and that is the subtlety that makes the bound useful rather than merely true.

Wetting swells the fibre. A larger D means the two systems must fill a greater thickness between them, so both relaxed crimps rise — and the ceiling rises with them. At a twenty per cent swelling the relaxed crimp of the standard cloth here goes from 12.79 to 20.26 per cent and the ceiling from 11.34 to 16.85.

That is the honest statement of why a fabric can shrink further than a dry-relaxation figure suggested it could. Nothing has broken the bound. The bound was computed for the dry cloth and the cloth being measured is the wet one.

What the bound looks like across the standard weaves #

The ceiling is c/(1 + c) for whatever crimp the finished cloth has, so putting numbers on it means putting numbers on the crimp — which this site’s solver does for a plain weave and does not, directly, for anything else.

What can be said exactly is the ordering, and it comes from the interlacing count. A thread’s crimp is accumulated one bend at a time: each interlacing is a place where the thread rises over or dips under the thickness of the crossing system and spends length doing it. Halve the number of bends per unit length and, to a first approximation, halve the length spent.

The approximation is worth being careful about, because it is not exact and the reason is interesting. A thread with fewer, more widely spaced bindings does not simply bend half as often — it also has more room between bindings, so each bend can be gentler, and a gentler bend spends less length than a sharp one. So the crimp falls faster than the interlacing count does, and a satin’s advantage is larger than the table’s ratios suggest.

That is the same mechanism as the weave angle falling as a cloth opens out: more room means a shallower turn.

The bound in the other direction #

A bound on shrinkage is also a bound on extension, and the symmetry is exact.

A cloth pulled in a thread direction extends by having crimp pulled out of it, and it cannot extend by more than the crimp holds — the same c/(1 + c), read as a gain rather than a loss. Beyond that point the yarns themselves must extend, which for cotton means a few per cent more before something breaks.

So a woven fabric’s extensibility in its own directions and its relaxation shrinkage are the same quantity, bounded by the same number, differing only in which state is taken as the reference. A cloth that is very stable is a cloth that does not extend, and a cloth with a lot of give is a cloth with a lot to give back.

That equivalence is worth having because extensibility is easy to measure non-destructively and crimp is not. Pulling a specimen to the point where its load–extension curve turns sharply upward — the point where the crimp has run out and the yarns take over — gives the crimp directly, and therefore the ceiling.

What was counted, and how #

The ceiling is asserted rather than displayed, which is the point of having it.

relaxationRange computes it once per family as c/(1 + c) from the relaxed crimp, and then checks every row against it. A row exceeding the ceiling would mean the shrinkage arithmetic had produced a length from nowhere, which is the one class of error this whole field could make and not notice, because a plausible-looking percentage is a plausible-looking percentage.

The check has bite in exactly one place: at a loom tension of 1 — every last bit of crimp removed — the computed shrinkage equals the ceiling, to the last digit. That is the equality case of the bound and it is where a sign error or an off-by-one in the crimp bookkeeping would show as an overshoot.

The relaxed crimp itself comes from peirce() and is fed back through assertPeirceConsistent, so the number the ceiling is computed from is one the model’s own equations have re-derived forwards.

Where the model stops #

The bound assumes the thread length is fixed, and that is the assumption everything here rests on. A yarn that consolidates under repeated wetting — twist settling, fibres migrating and packing — really does shorten a little on its own account, and a cloth made from it can pass the ceiling. That is not a violation of anything; it is a different mechanism arriving. The bound is a bound on crimp recovery, and it says so.

It is a bound per direction, not on area. A cloth at the ceiling in both directions loses 1 − (1 − 0.1134)² ≈ 21 per cent of its area, and area shrinkage is not the sum of the two.

And it says nothing about how much of the ceiling a given fabric will use. The bound is the worst case. What actually happens depends on the loom tension, the finishing route and how much mechanical action the fabric receives, none of which is derivable from geometry.

The bound read as a specification test #

The most useful thing about a parameter-free bound is that it can be applied before a fabric exists, and the application is one comparison.

A customer states a residual shrinkage. A designer has a construction in mind and can compute its finished crimp. If c/(1 + c) is below the specification, the fabric cannot fail it however it is finished; if it is above, the fabric may or may not, and the answer depends on the whole finishing route.

That divides specifications into two kinds and the division is worth making early. A specification below the ceiling is a geometry problem: choose the weave and the sett, and the fabric is safe by construction. A specification above it is a process problem: no construction settles it, and the answer is a compressive shrinking range or a resin finish, which are capital decisions rather than design ones.

Ordinary numbers put most apparel specifications in the second class. A two per cent residual needs a finished crimp under 2.04 per cent, which is a satin at an open sett — not a shirting, not a sheeting, not anything a garment is usually made of. So the trade’s reliance on pre-shrinking is not a preference; it is what the ceiling leaves.

The other direction is where the bound earns something a mill would not otherwise have. A fabric that measures a residual above its own ceiling has broken the assumption rather than the arithmetic, and the assumption is that the thread length is fixed. So an overshoot is a positive finding: it says the yarn itself has consolidated, which is a fibre and twist question rather than a construction one, and it points the investigation at the spinner rather than at the finisher.

A bound that can be exceeded only by a named mechanism is a diagnostic, and that is worth more than a bound that is merely never approached.

It is also the only test in this field that can be run on a fabric nobody has woven yet, which is what makes it worth putting on a drawing board rather than in a laboratory.

Why a result with no free parameter is worth more #

This is the second result in this collection with that shape — the first was the leno’s grip, where a friction coefficient cancelled out of a comparison — and the reason for pointing at the shape rather than the number is the same.

Most claims in this field take the form: at a friction coefficient of μ, or a loom tension fraction of t, quantity A exceeds quantity B by so much. Those are honest and they come with an obligation to quote the parameter and say what survives its range. A claim of that kind fails if the parameter turns out to be different, or to vary between the two cases being compared.

This claim has no such exposure. It says: a dimension cannot fall by more than c/(1 + c), and the only input is the crimp, which is measurable directly by unravelling a thread and straightening it against the cloth it came from. There is nothing to be wrong about except the definition of crimp.

The value of a parameter-free bound is not that it is more accurate. It is that it fails differently. A fitted result fails quietly, by drifting as conditions change. A derived bound fails loudly or not at all, and when a measurement exceeds it the right conclusion is that a different mechanism is present — which is information rather than error.

The bound is per direction and the area is not #

One arithmetic point worth being explicit about, since area is what a purchaser notices.

A cloth at its ceiling in both directions loses 1 − (1 − 0.1134)² of its area, which is 21.4 per cent, not 22.7. Area shrinkages compound rather than add, in the same way and for the same reason the pre-shrinking arithmetic does not subtract.

The compounding runs the helpful way here — the area loss is always less than the sum of the two — and it is small at small shrinkages and grows. At two per cent each way the sum overstates by 0.04 points; at eleven per cent each way it overstates by 1.3.

So the sum is a usable approximation for a stable cloth and not for an unstable one, which is the wrong way round for a rule of thumb: the approximation is worst exactly where the number matters most.

Who found it, and when #

The arithmetic is not attributable to anyone and is implicit in every textbook definition of crimp. What is unusual is stating it as a bound and putting it on a figure, and the reason it is rarely done that way is probably that the trade approaches shrinkage from the measurement end. A mill measures what its fabrics do and tabulates it; a bound on what they could do is not a number anyone needs to run a finishing works.

It becomes useful in the other direction — designing a fabric to a stability specification rather than measuring one that exists. There the ceiling says immediately whether the specification is reachable at all: if a customer wants under two per cent residual and the construction has a ceiling of eleven, the fabric will need pre-shrinking rather than careful weaving, and that is a decision about equipment rather than about cloth.

Where the ladder goes next #

If a cloth is going to lose eight per cent of its length and the customer will not accept it, there are two options and only one of them is practical. The fabric can be designed to want less — fewer interlacings, opener sett, and everything that goes with them — or the shrinking can be done before the cloth is sold.

The industry does the second, and the next rung is about what the label then means, which is not what it appears to mean.