Textile Structure
how cloth is put together
After the loom

A wrinkle cannot settle what a crease settles

The first rung of this ladder found that a pressed crease decides a question this collection had been unable to settle for two fields — whether a bent yarn's fibres slide or bend as one body — and it decides it by refusing: the coherent branch would strain the fibres by eighty-four per cent and cotton breaks at six. A wrinkle is the same fold at twenty times the radius, and there both branches are survivable. So the bracket that a crease collapses stays fourteen times wide at every radius anybody actually creases a cloth at accidentally.
15 min read 7 figures Measured, not claimedA mechanism, not a materialStructure before fibre

Worth reading first: A crease is a fold the crimp cannot supply · Which fibres crease, and why there are two answers · Recovery is measured and nothing predicts it.

The first rung of this ladder does something this collection rarely manages: it settles a question that two whole fields had left open.

The question is the bending bracket. When a yarn bends, do its fibres slide over one another and bend individually — in which case the strain is set by a fibre’s diameter — or does the yarn bend as one coherent rod, in which case it is set by the yarn’s? The two answers differ by a factor of fourteen at a twenty-tex cotton, and this collection has never had a way to choose between them.

A pressed crease chooses, by refusing. At a fold radius of a tenth of a millimetre the coherent branch strains the outer fibres by 84 per cent and cotton breaks at six to ten. So the coherent branch is not merely unlikely there, it is impossible, and the free branch at 6 per cent is the only survivor.

That is a genuine settlement and it is much narrower than it looks.

A cotton yarn at a fold, at both ends of its bending bracket. The same 20 tex cotton yarn bent to a radius of 0.084 mm under the two assumptions this site's bending bracket is drawn between. If the fibres slide freely past one another each bends about its own middle and the surface strain is 7.14%; if they are locked the bundle bends as a rod and the outermost fibre is strained 100%. The ratio is 14.0, which is the ratio of the two diameters and therefore the square root of the fibre count over the packing — so the bracket of 327 this site carries in a stiffness is a bracket of 14.0 in a strain. What the drawing cannot show is which of the two a real yarn does, and the answer is settled by a refusal: at the tightest fold a cloth can make, the locked bound asks for a strain of one, and a creased cloth does not fall apart.
Fig. 1 The bracket the crease collapses: two answers to what a bent yarn’s fibres do, a factor of fourteen apart. Only at the tightest folds does one of them become impossible.

The bracket is a constant ratio

The two branches have the same form. The strain at a bend of radius R in a body of diameter d is d/2R, so:

  • the free branch uses the fibre’s diameter — 11.9 µm for a twenty-tex cotton at a packing factor of 0.6;
  • the coherent branch uses the yarn’s — 167 µm.

The ratio of the two is the ratio of the diameters, which is √(fibres per yarn) — 14.0 for that yarn, at every radius.

fold radius free strain coherent strain
0.1 mm 5.97% 83.5%
0.2 mm 2.98% 41.8%
0.5 mm 1.19% 16.7%
1 mm 0.60% 8.4%
2 mm 0.30% 4.2%
5 mm 0.12% 1.7%

The bracket does not narrow. It is a factor of fourteen at every radius, and what changes with the radius is not how wide the bracket is but whether both of its ends are physically possible.

Which is why the crease settles it and nothing else does

Cotton’s breaking extension is 6 to 10 per cent. Read that against the table.

At 0.1 mm — a pressed crease — the coherent branch is 83.5 per cent and is out by an order of magnitude. Only the free branch survives, and it survives marginally: 5.97 against a floor of 6.

At 1 mm — a fold in a sleeve — the coherent branch is 8.4 per cent, which is inside cotton’s own breaking range. Both branches are survivable and the fold gives no information about which is happening.

At 2 mm — a wrinkle — the coherent branch is 4.2 per cent and the free branch is 0.30. Both are comfortably survivable, and they differ by fourteen times in a quantity that decides everything about whether the fold comes out.

So the crease’s settlement is a boundary case, not a general result. It works because the crease radius is the one place the bracket’s upper branch runs into a hard physical limit, and that limit is a fibre breaking rather than anything about bending.

A wrinkle cannot settle it, and it needs the answer just as badly.

Crease recovery predicted from the fibre, against the test. A specimen creased through 180° has a curvature imposed on its fibres; a fibre returns a measured fraction of a strain; curvature is proportional to strain at a fixed radius. So the angle recovered should be 180° times that fraction, with no sett, no weave and no friction anywhere in it. Against the reported crease recovery angles the prediction is within a factor of two everywhere and is high for some fibres and low for others. That is the useful shape of the disagreement rather than a disappointment: the two mechanisms left out push in opposite directions — friction between yarns holds a fold in after the fibres have finished pulling on it, and delayed recovery lets it out over the hours after the test ends. What the bars cannot show is the four fibres missing from them, whose recovery has never been measured at the strain their own tightest fold imposes.
Fig. 2 What the trade measures instead: the crease-recovery angle, which is the outcome of the whole chain and cannot be inverted into any one link of it.
The tightest fold a 20 tex cotton yarn can be given. A cloth folded as sharply as it can be folded. The two yarn crowns on the inside of the fold cannot pass through one another, so the fold's radius is the yarn's own — 0.084 mm for a 20 tex cotton yarn — and there is no measurement of an iron anywhere in the argument. At that radius a fibre free to slide is strained 7.14%, which is √(packing × fibre tex ÷ yarn tex), and the whole yarn bending as a rod would be strained exactly one hundred per cent. Cotton's measured breaking extension is 6.0% to 10.0%, so the free bound does not survive and the locked one cannot. What the drawing cannot show is the fibres inside the yarn, which is exactly what the argument is about — the picture is the same either way and the strain is fourteen times different.
Fig. 3 The tightest fold a twenty-tex cotton yarn can make, from the first rung: the radius at which even the free branch reaches the fibres’ breaking extension. Every crease anybody presses is near it and every wrinkle is far away.

What the difference costs at a wrinkle radius

The factor of fourteen is not academic, because recovery falls with strain and the two branches land in completely different parts of the recovery curve.

Cotton’s recovery is measured from two per cent strain to five: 74 per cent at 2 per cent strain and 45 per cent at 5. So:

On the coherent branch a two-millimetre wrinkle strains the outer fibres 4.2 per cent, and cotton recovers about half of it. A permanent set of two per cent of the fibre’s length, in a cloth that has merely been sat on.

On the free branch the same wrinkle strains them 0.30 per cent, which is below the range anybody has measured recovery over at all — and at strains that small the elastic recovery of every fibre in this collection’s table is very near complete.

So the two branches predict “the wrinkle stays” and “the wrinkle comes out”, and there is nothing in between. That is the sharpest possible form of an unresolved bracket: the two answers are not different sizes of the same effect, they are different phenomena.

And it has an obvious consequence for what wrinkling is. If the free branch were right, cotton would not wrinkle at all — and cotton wrinkles, famously and unmistakably. So a wrinkle is on the coherent branch, or on something between the two, and the crease’s settlement does not transfer.

Which is an argument the crease’s own rung could not make

That inference is worth being careful about, because it runs backwards from a fact about the world to a fact about the model, and this collection is usually the other way round.

The chain is: cotton wrinkles; the free branch predicts complete recovery at a wrinkle’s strain; therefore a wrinkle is not on the free branch.

And the crease’s rung showed a crease is on the free branch, because the coherent one would break the fibres.

So the same yarn is on different branches at different radii, which is not a contradiction and is a real physical statement: at a tight fold the fibres are forced to slide because nothing else is possible, and at a gentle one they are gripped by twist and friction and bend together.

That is a transition, and it has a radius. Where it lies is exactly the number neither rung can supply — it is a friction question, and the criterion this collection is built on is blind to friction, and so is every model downstream of it.

What can be bracketed is where it is not. It is below one millimetre, because a wrinkle at one millimetre wrinkles; and it is above 0.084 mm, because at the tightest fold a cotton yarn can make even the free branch is at breaking. A range of about ten to one, and no way to narrow it from inside this collection.

The pile is the third case and it settles nothing either

There is a fold in this collection at a radius smaller than either of these, and looking at it completes the picture.

A crushed carpet pile is a tuft bent under a foot, and its fold radius is set by the tuft’s own diameter rather than by any cloth’s thickness. The finding there was that the fibres are strained by an order of magnitude less than anything anybody has measured — so the pile’s recovery is not a fibre-recovery question at all, and the tuft comes back for a different reason.

So the ladder now has three folds and three regimes:

  • a pile, at a radius where even the coherent branch is negligible;
  • a wrinkle, where both branches are survivable and they differ by fourteen times;
  • a crease, where the coherent branch is impossible.

Only the last one is decidable, and it is decidable by exclusion rather than by anything positive. The other two are the cases where the model has two answers and the world has one, and neither can be made to yield.

That is the shape of the whole bracket problem and it is worth the sentence: a bracket collapses only where one of its ends is forbidden, and forbidding is a fibre property rather than a bending one. So the radius at which a bracket becomes decidable is set by whichever fibre is in the yarn, which is why linen and wool sit at opposite ends of the crease ranking and why the crease’s argument is a boundary case rather than a theorem.

A 5.0 mm cotton pile standing and crushed. A pile tuft standing 5.0 mm proud and the same tuft pressed flat, which turns it through a right angle over its own length and so bends it to a radius of 3.18 mm. Its fibres are strained 0.187% — an order of magnitude below the smallest strain anybody has measured a recovery at, so this file declines to say what fraction comes back. The consequence is that a crushed carpet is not held down by its fibres: what keeps a pile flat is the tufts leaning on one another and the friction where they touch. Below 0.469 mm the fibre does enter its measured range, which is the difference between a carpet and a velvet and has nothing to do with what either is made of. What the drawing cannot show is the neighbouring tufts, which are the mechanism.
Fig. 4 The third fold: a pile tuft under a foot, at the radius its own diameter sets. Every strain in it is far below the range recovery has been measured over, which is what makes it the third undecidable case rather than a confirmation of either branch.

Why the trade’s two requirements are contradictory

The practical form is the thing anybody who has worn trousers knows and it now has an arithmetic behind it.

A crease pressed across a seam is a different fold again, and it is the next rung. Within one cloth the problem is sharper than that.

A trouser wants a permanent crease down the leg and no permanent wrinkles behind the knee. Those are the same fold at two radii, in the same cloth, of the same fibres, and every treatment that acts on the fibre acts on both.

Resin crosslinking, which is the standard crease-recovery finish, raises the elastic recovery of the fibre at every strain. That is exactly what makes a wrinkle come out — and it is exactly what stops a crease staying in. The trade’s own name for the resulting fault is a crease-resistant trouser that will not hold a press, and it is not a failure of the finish; it is the finish working.

So the two requirements are not merely in tension, they are the same quantity with opposite signs, and no fibre treatment can separate them.

What can separate them is locality. A crease is applied at one line, with heat, moisture and pressure, and a wrinkle happens everywhere without any of the three. Every mechanism that actually delivers both — a resin applied only at the crease, a heat-set thermoplastic fibre pressed in one place — works by being local rather than by being selective.

That is a genuine conclusion and it follows from an arithmetic identity rather than from chemistry: a treatment that changes recovery cannot distinguish two folds of the same cloth, so a treatment that must distinguish them has to be applied to one and not the other.

A wool yarn at a fold, at both ends of its bending bracket. The same 60 tex wool yarn bent to a radius of 0.156 mm under the two assumptions this site's bending bracket is drawn between. If the fibres slide freely past one another each bends about its own middle and the surface strain is 7.07%; if they are locked the bundle bends as a rod and the outermost fibre is strained 100%. The ratio is 14.1, which is the ratio of the two diameters and therefore the square root of the fibre count over the packing — so the bracket of 333 this site carries in a stiffness is a bracket of 14.1 in a strain. What the drawing cannot show is which of the two a real yarn does, and the answer is settled by a refusal: at the tightest fold a cloth can make, the locked bound asks for a strain of one, and a creased cloth does not fall apart.
Fig. 5 The same bracket for a sixty-tex wool. Every number has moved and the ratio has not: the bracket is the square root of the fibres in the yarn, so a coarser yarn of coarser fibres widens it by less than either change alone would suggest.

Flax is the fibre that shows what a failed settlement looks like, and it is worth the third drawing. Its breaking extension is 1.5 to 3 per cent, which is a quarter of cotton’s, so at the tightest fold a flax yarn can make even the free branch is over it — the fibres break whichever way the yarn bends, and there is no branch left to be right.

That is not a settlement by exclusion; it is both branches excluded, and it is the arithmetic behind linen’s place at the bottom of the crease ranking. A linen crease is not a fold the fibres survive and fail to return from — it is a fold they do not survive.

A flax yarn at a fold, at both ends of its bending bracket. The same 20 tex flax yarn bent to a radius of 0.084 mm under the two assumptions this site's bending bracket is drawn between. If the fibres slide freely past one another each bends about its own middle and the surface strain is 9.49%; if they are locked the bundle bends as a rod and the outermost fibre is strained 100%. The ratio is 10.5, which is the ratio of the two diameters and therefore the square root of the fibre count over the packing — so the bracket of 185 this site carries in a stiffness is a bracket of 10.5 in a strain. What the drawing cannot show is which of the two a real yarn does, and the answer is settled by a refusal: at the tightest fold a cloth can make, the locked bound asks for a strain of one, and a creased cloth does not fall apart.
Fig. 6 The bracket for flax, the fibre the ranking puts worst. Even the free branch is over its breaking extension at the tightest fold — which is the one case here where a crease settles nothing because neither branch survives.

So the argument’s shape depends on exactly two numbers per fibre, and they are the two the second rung ranks on: how far the free branch strains a fibre at the fold it is being asked for, and where that fibre breaks. Cotton’s pair make the settlement work marginally, wool’s make it work comfortably, and flax’s break it altogether.

The strain at a fold against the strain each fibre breaks at. For each fibre, the surface strain at the sharpest fold a 20 tex yarn of it can make — which is √(packing × fibre tex ÷ yarn tex), with no measurement of a crease in it — beside its own measured breaking extension. cotton and flax are strained past the low end of their breaking range, so some of their fibres break at the fold, and that is what a linen crease is. The rest survive, and among them viscose and wool return less than three fifths of what they were given, which is the other way a cloth creases. Both columns are needed: viscose survives with a factor of two to spare and is among the three worst by measurement. What the bars cannot show is wool, which the census puts in the wrong column because the recovery figures are the immediate ones and wool's is the most delayed of any fibre here.
Fig. 7 The crease strains against the fibres’ own breaking extensions, from the second rung. Every fibre here is being asked the same question at the same radius, and the answer sorts them — which is what a bracket collapsed to one branch makes possible.

The other fibres, where the boundary case moves

The crease’s settlement depends on a fibre’s breaking extension being small enough to rule the coherent branch out, and that quantity varies enormously across the fibres this collection carries.

fibre tightest fold free strain there breaking extension
flax 0.084 mm 9.5% 1.5–3%
cotton 0.084 mm 7.1% 6–10%
polyester 0.088 mm 6.7% 12–30%
wool 0.090 mm 12.2% 25–35%

Flax is over its breaking extension even on the free branch, at its own tightest fold — which is why linen creases worst and is the second rung’s finding.

Wool and polyester are comfortably under it, so for them the free branch is safe at any radius and the coherent branch is what has to be checked. At a crease radius the coherent branch is 84 per cent and both of them break long before that, so the settlement holds for them too.

So the argument’s form is the same for every fibre and its margin is not. Cotton settles it by six per cent against six; wool settles it by twelve against twenty-five, which is not a settlement at all — the free branch is comfortable for wool at the crease radius, so nothing rules the coherent branch out except that eighty-four per cent exceeds thirty-five.

That is still a settlement and it is a much weaker one. It says the yarn is not perfectly coherent; it does not say it is perfectly free, which is what cotton’s marginal case does say.

And it is a caution about generalising a boundary case. The first rung’s argument reads as general and is a statement about one fibre at one count, and the reason it works is a coincidence of two numbers rather than a fact about bending.

What was counted, and how

The bracket is memory.js’s creaseBracket, unchanged, at the site’s own default twenty-tex cotton at a packing factor of 0.6. Its two branches are d/2R with the two diameters, and the ratio between them is the ratio of the diameters — which is why it is constant in the radius and why saying so is worth a line rather than a table.

The recovery figures are the collection’s own measured table, and the range each fibre has been measured over is carried with it. That is what makes the finding above statable: the free branch at a wrinkle radius lands outside the measured range, at 0.30 per cent against a floor of 2, so the claim “every fibre recovers completely there” is an extrapolation and is marked as one.

The inference from cotton wrinkling is stated as an inference. It is not a measurement this collection made; it is an observation everybody has, used to reject a branch. That is a legitimate move and it is the only one available, and it is flagged because the collection’s usual direction is the other way.

And the transition’s bracket is asserted as a bracket. Below one millimetre and above 0.084 — a range of about ten to one, which is what can be said and no more.

What would settle it, and why nobody has

The transition radius is the number the whole ladder is missing, so it is worth saying what would produce it.

A direct measurement is available in principle. Bend a yarn round a known radius and measure the length of its surface against the length of its axis. If the fibres slide, the surface is longer by the fibre’s own diameter over twice the radius; if they do not, by the yarn’s. The two predictions differ by fourteen times and would be trivially distinguishable.

The reason nobody does it is that the measurement is on a yarn a fifth of a millimetre across at a radius of a tenth, and the quantity being measured is a length difference of a few micrometres over a millimetre of arc. That is a hard optical measurement and it is not one anybody in the trade has a use for.

An indirect one is easier and is what the trade actually does. The crease-recovery angle test folds a specimen under a stated load for a stated time and measures how far it opens. That number is the outcome of the whole chain — bracket, strain, recovery, time — and it is what a specification quotes.

So the trade measures the answer and this collection wants the mechanism, and the two are not related by anything computable. A crease-recovery angle cannot be inverted into a bracket, because it depends on the recovery curve and the hold time as well.

That is the honest position and it is worth stating plainly rather than as a limitation. The quantity that would close this ladder is measurable, is not measured, and has no commercial reason to be — which is a different kind of gap from the ones this collection usually records, and a more permanent one.

Where the model stops

The two branches are limits and the truth is between them. A real yarn’s fibres neither slide freely nor are perfectly bonded; they are gripped by twist and by friction, and the effective bending body is somewhere between a fibre and a yarn. Every number here is one end of a bracket rather than an estimate.

Recovery is measured at a strain and a fold is not a uniform strain. The outer fibres of a bend are at the quoted strain, the neutral axis is at nothing, and what a fold recovers is an integral over that distribution. The comparison here uses the extreme, which is the right thing for asking whether anything breaks and the wrong thing for asking how much comes out.

Nothing here is about time. A crease is pressed for seconds and a wrinkle is held for hours, and both fibre recovery and permanent set are strongly time-dependent — so two folds at the same strain held for different times do not behave alike. The site has no rate model and this is the largest missing thing in the whole ladder.

And nothing here is about moisture. Cotton’s recovery is very different wet and dry, which is why a crease is pressed with steam and why a shirt wrinkles worst in humid weather. The recovery table is at one condition and does not say which.

Who found it, and when

The bending bracket is a standard difficulty in textile mechanics and its two limits are in every account of fabric bending: the fibres slide, or they do not, and real cloth is between.

That a pressed crease resolves it is the first rung’s, and it is a good argument. What is added here is its scope, and the scope is narrow: the argument works because one branch is impossible at the crease radius, and there is no other radius at which any branch is impossible.

The contradiction between crease retention and wrinkle recovery is trade knowledge — it is why durable-press trousers were a technical problem for decades — and the explanation usually given is chemical. The arithmetic says it is not chemical at all: it is one quantity being asked to take two values, and no chemistry can do that.

Where the ladder goes next

Every fold in this ladder has been a fold of one cloth. A garment’s creases fall across its seams, and a seam is a stack of layers that have been stitched together — so the outer layer of the stack has further to go round the fold than the inner, and the length has to come from somewhere.

Where this goes next

The tightest fold a 20 tex cotton yarn can be given. A cloth folded as sharply as it can be folded. The two yarn crowns on the inside of the fold cannot pass through one another, so the fold's radius is the yarn's own — 0.084 mm for a 20 tex cotton yarn — and there is no measurement of an iron anywhere in the argument. At that radius a fibre free to slide is strained 7.14%, which is √(packing × fibre tex ÷ yarn tex), and the whole yarn bending as a rod would be strained exactly one hundred per cent. Cotton's measured breaking extension is 6.0% to 10.0%, so the free bound does not survive and the locked one cannot. What the drawing cannot show is the fibres inside the yarn, which is exactly what the argument is about — the picture is the same either way and the strain is fourteen times different. A crease is a fold the crimp cannot supply A fold needs its outer face longer than its inner, and a woven cloth's way of supplying a length is to move crimp. That runs out at a radius of millimetres, and a pressed crease is tenths of one — so the fold is handed to the fibres. How hard it strains them turns on a question this collection has been unable to settle for two fields, and a crease settles it by refusing. The strain at a fold against the strain each fibre breaks at. For each fibre, the surface strain at the sharpest fold a 20 tex yarn of it can make — which is √(packing × fibre tex ÷ yarn tex), with no measurement of a crease in it — beside its own measured breaking extension. cotton and flax are strained past the low end of their breaking range, so some of their fibres break at the fold, and that is what a linen crease is. The rest survive, and among them viscose and wool return less than three fifths of what they were given, which is the other way a cloth creases. Both columns are needed: viscose survives with a factor of two to spare and is among the three worst by measurement. What the bars cannot show is wool, which the census puts in the wrong column because the recovery figures are the immediate ones and wool's is the most delayed of any fibre here. Which fibres crease, and why there are two answers Run the sharpest fold a yarn can make against every fibre's breaking extension and its recovery, and the ranking that falls out is the trade's own — linen worst, cotton next, wool and the melt-spun filaments best. It falls out of two columns rather than one, because two fibres in the bottom three get there by different mechanisms, and a treatment that fixes one does nothing for the other. Elastic recovery against strain, for seven fibres. The elastic recovery of seven fibres at the strains it is reported at: extend to a stated strain, unload, read the strain returned immediately. Each fibre's points are joined and the line stops where the measurements stop, which is the point of the figure — a fibre strained past the last point on its own line is a fibre this collection declines to answer for. Nothing here is measured below one per cent of strain, and a woven cloth just past its own interchange budget is at a thread strain of a few hundredths, so the region that matters most for a fabric is the region nobody has reported. Recovery falls monotonically for every fibre, which is what lets the unmeasured region be bracketed between the lowest measured value and one rather than extrapolated. What the plot cannot show is the delayed recovery, which is excluded by the convention and is largest for the fibre with the best reputation for recovering. Recovery is measured and nothing predicts it A fibre's stiffness is a bracket this collection can compute the ends of. What fraction of a strain it gives back is not: it has to be looked up, the tables are thin, they stop exactly where a fabric needs them, and the most attractive explanation for the ordering they show turns out to have no signal in it at all.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Bending bracketBreaking extensionCreaseElastic recoveryFibre finenessFold radiusWrinkle