A crease is a fold the crimp cannot supply

Fold a cloth and the outer face has further to go than the inner. Fold it sharply enough and something has to give, and the whole question of why some fabrics crease and others do not is a question about what that something is.

A cloth has two ways of finding the length. It can move crimp — the warp on the outside of the fold straightens while the warp inside takes on more, which is the same crimp interchange that lets a cloth extend for nothing. Or it can strain its fibres. The first comes back and the second largely does not, so which of them supplies the fold is the whole of whether the fold is permanent.

This rung establishes that the first gives out very early, and then uses the fact to settle something else.

The cloth-level route, and how quickly it ends #

The arithmetic is one line. A sheet of thickness t bent to radius R needs its outer face longer than its middle by t/2R. Set that equal to the interchange budget and the radius at which the budget is exhausted is t over twice the budget.

For the eight cloths in this collection’s table it comes out between 1.83 millimetres for the batiste and 11.0 for the cheesecloth, on the computed thickness; between 1.29 and 7.85 on the measured one. Every one of them is millimetres.

A pressed crease is not millimetres. It is a fraction of one — a trouser crease, a folded collar, a bag in a knee — so on any fold worth calling a crease, the cloth-level route was spent an order of magnitude ago and everything past it went into the fibres.

The two mechanisms are not alternatives, and that is worth being exact about. Every fold uses the coarse route first and the fine route for whatever is left; a gentle fold uses only the coarse one and comes back; a sharp fold uses the coarse one for a millimetre’s worth of radius and then hands over.

The fine route, and the question it runs into #

Once the fibres are carrying it, the surface strain at a fold is the classical one: a filament of diameter d bent so that its axis lies on a circle of radius R has its outer surface stretched by d/2R.

Which d? That is exactly the question this collection has been unable to settle since it first gave a yarn a bending stiffness. A yarn is a bundle, and whether its fibres may slide past one another at a bend is the difference between two extreme models: at the free bound each fibre bends about its own middle and the yarn’s rigidity is the sum of its fibres’; at the coherent bound they are locked and the yarn bends as a solid rod. The ratio between the two rigidities is the fibre count, several hundred for an ordinary yarn, and no amount of care about the fibre narrows it.

Read as a strain the same two assumptions say something much narrower, and this appears to be the first time this collection has asked them to.

At the free bound the strain is the fibre’s diameter over twice the radius. At the coherent bound it is the yarn’s. The ratio is the ratio of the two diameters, which is √(N/packing) where N is the fibre count — so a rigidity bracket of 327 corresponds to a strain bracket of 14.0. Both identities are asserted rather than stated, because they are the claim that the two brackets are one fact told twice.

Fourteen is a narrow enough bracket to decide something.

The fold with no measurement in it #

There is one more thing to remove before the argument closes, and it is the assumed crease radius. How sharp is the fold an iron makes? Nobody in this collection knows, and the answer would be a measurement of an appliance rather than of a fabric.

It turns out not to be needed. A fold cannot be tighter than the yarn it is made of. The two crowns of yarn on the inside of the fold cannot pass through one another, so the fold’s radius is bounded below by the yarn’s own radius, and at that bound the arithmetic collapses:

$$\text{coherent strain} = \frac{d_{\text{yarn}}}{2 \cdot d_{\text{yarn}}/2} = 1$$

exactly, for any yarn of any fibre at any count. And the free bound becomes the ratio of the two diameters, which is √(packing × fibre tex ÷ yarn tex) — 7.14 per cent for a 20 tex cotton yarn of 1.7 decitex fibres at a packing of 0.6.

The refusal, and what it settles #

A coherent yarn at its own tightest fold demands a fibre strain of one: a fibre at twice its original length. No fibre in this collection’s table survives that, and none comes within a factor of two of surviving it — the closest is nylon, whose measured breaking extension reaches forty per cent, and one hundred is two and a half times forty.

A creased cloth does not fall to pieces.

So the fibres at a fold slide past one another, and the yarn there is at or near its free bound. That is not a fit, an estimate or an inference from a curve. It is a refusal: the alternative predicts an outcome that does not occur, at every radius from the geometric floor upwards, for every fibre in the table.

This collection had two independent measurements of where in the bending bracket a real yarn sits, and both landed a few times above the free bound — a couple of per cent of the way up a bracket five hundred wide, read on the logarithmic scale a bracket of that size has to be read on. One was a cantilever test on a hanging strip. The other was the width of the band a washed cloth comes to rest in. Neither has anything obviously to do with the other.

The crease is a third, and it agrees. It is also the cheapest: it needs no instrument, no specimen and no curve-fitting, only the observation that ironing a shirt does not destroy it.

How far from a crease the argument reaches #

The refusal is strongest at the tightest fold and weakens as the fold opens out, and the honest way to say so is to compute where it stops.

A coherent yarn stops breaking a given fibre at a radius of the yarn’s diameter over twice the breaking extension. For cotton that is 0.835 millimetres — five yarn diameters — and for nylon 0.241, or 1.3 diameters. So the argument covers every fold sharper than about five yarn diameters for a cotton cloth and about one and a half for a nylon one.

That is a comfortable margin for anything anybody would call a crease and no margin at all for a soft drape. A cloth hanging in folds is bent at radii of centimetres, its coherent strain there is a fraction of a per cent, and nothing about how its fibres behave can be read off a hanging fold. The two regimes are genuinely different and the argument belongs to one of them.

Why the fold’s own strain lands where it does #

There is a coincidence in the numbers that is worth pointing at, because nothing arranged it and it is the reason the whole ladder is usable.

The free-bound strain at the tightest fold is √(packing × fibre tex ÷ yarn tex). For the ordinary combinations — a fibre of one or two decitex spun into a yarn of ten to fifty tex at a packing near 0.6 — that expression lands between six and twelve per cent. It is a square root of a ratio of two counts, and both counts are set by entirely unrelated considerations: fibre fineness by what the plant or the spinneret produces, yarn count by what the cloth is for.

Six to twelve per cent is exactly the range in which fibres differ from one another most sharply. Below two per cent almost everything recovers; above twenty almost nothing does; and the breaking extensions of the common textile fibres are scattered right through the middle of the interval a fold happens to impose. Cotton breaks between six and ten and is strained 7.14. Flax breaks between 1.5 and three and is strained 9.49. That both numbers come out of a count and a packing factor rather than out of a measurement is what makes the coincidence worth pointing at.

So a crease is a discriminating test, and it is discriminating by accident. Had spinning produced yarns a hundred times coarser relative to their fibres, every fold would be trivial and no fabric would crease; a hundred times finer, and every fold would break fibres and every fabric would crease equally. The fact that some cloths crease and others do not is a fact about a coincidence of two counts.

What the handover radius is made of, and the construction it implies #

The radii at which the cloth-level route gives out run from 1.83 millimetres to 11.0 across the table, which is a factor of six between cloths of the same fibre. That spread is worth taking apart, because both of its ingredients are things a designer chooses.

The radius is the thickness over twice the interchange budget. Thickness goes roughly as the yarn diameter, since a plain-weave cloth is two threads deep at a crossing, and the diameter goes as the square root of the count. The budget goes with the crimp, since interchange is crimp moved from one system to the other and there is only as much of it as the cloth has.

So the handover radius is, to within the accuracy of that sketch, proportional to the yarn’s diameter and inversely proportional to the crimp. A fine yarn hands over later; a heavily crimped cloth hands over later; and a thick, flat, low-crimp cloth hands over almost at once.

That is a construction rule for crease resistance with no chemistry in it at all:

Fine yarn. Halving the count divides the diameter by 1.41 and the handover radius with it, so the cloth stays on the free route down to a sharper fold.

High crimp. A cloth set to interchange freely has a larger budget, and the budget is in the denominator. Plain weave has more of it than a satin at the same sett, which is one of the reasons a satin creases along a float and a poplin does not crease at all in the same place.

Low thickness. The one that is not free, because thickness is also what a cloth is bought for in most of the applications where creasing is complained about.

Read against the table the rule holds: the batiste is the finest and the thinnest and folds free to 1.83 millimetres, and the cheesecloth is coarse and lofty and gives out at 11.0.

And the rule stops exactly where creasing starts. Every one of those radii is millimetres and a crease is tenths, so improving the handover radius by a factor of two moves the boundary from four millimetres to two and changes nothing about what an iron does. The cloth-level route decides whether a gentle fold leaves a mark — a bag at an elbow, a fold in a bolt, the shape a garment takes overnight — and contributes nothing whatever to a pressed crease.

That is the cleanest statement of what the two routes are for. The coarse route decides whether a fold happens at all; the fine route decides how bad it is once it has. A construction optimised for the first is a cloth that resists casual folding and creases exactly as badly as any other when pressed, which is why the two complaints — it wrinkles and the crease will not come out — are different complaints with different remedies, and why a finish that fixes one is under no obligation to touch the other.

What was counted, and how #

The strain bracket is computed from the two diameters and checked against the fibre count in two ways: the ratio must equal √(N/packing), and the rigidity bracket must equal N divided by the square of the packing. Both are asserted to a relative tolerance, because they are identities and an identity that has stopped holding is an arithmetic error rather than a discovery.

The tightest-fold coherent strain is asserted to be exactly one, which is the kind of assertion worth making precisely because it looks trivial: it is the whole of the parameter-free step, and a factor of two hiding in a radius would land there.

The refusal is asserted per fibre, with a margin of two, and the margin is reported. Flax fails the coherent bound by a factor of 33 and nylon by 2.5, and stating the worst case is what makes the assertion a claim rather than a formality.

Where the model stops #

The tightest fold is a geometric floor and not what an iron makes. A real crease is somewhere above it — how far above depends on the pressing force and the cloth’s own bending stiffness, which this collection could compute and has not, because doing so would put an appliance into the argument and the argument is better without one.

The cloth-level route is treated as exhausted rather than as continuing. A real cloth at a fold can still find a little room by flattening its sections, which this collection prices separately, and that is a third route between the two — cheaper than stretching a fibre and dearer than moving crimp. Including it would push the handover to a slightly sharper radius and would not change the order of magnitude.

Every number is for a plain weave. A float lets a warp end cross a fold with no turn in it at all, which is a different question and has its own rung.

And nothing here says what fraction comes back. The strain at the fold is computed; what the fibre returns of it is a measurement, and at 7.14 per cent it is a measurement four of the seven common fibres do not have. That is the subject of the rung beside this one.

The generalisation #

A bracket that is useless in one quantity can be decisive in another, if the second is a lower power of the first. The bending bracket is a factor of several hundred in a rigidity and a factor of fourteen in a strain, because rigidity goes as the fourth power of a diameter and strain as the first. Nothing was learnt about the yarn between the two statements; the same ignorance was expressed in a variable where it does less damage.

That move is available more often than it is used. Before accepting that a quantity cannot be bracketed usefully, it is worth asking which other quantity the same uncertainty appears in and at what power.

The second lesson is about settling a question by refusal. A model that predicts something impossible has been tested, and the test is free. The coherent bound predicts that ironing a shirt breaks it. No experiment was needed, no specimen was cut, and the conclusion is stronger than a measurement would have been, because it does not depend on anybody’s calibration.

Who found it, and when #

Peirce’s geometry and Kemp’s racetrack are the ancestors of the cloth-level route, and the observation that crimp supplies a fold’s length difference is implicit in every account of fabric bending. The bending bracket between free and coherent yarn is standard in textile mechanics and is usually credited to the work on yarn flexural rigidity of the 1950s and 1960s.

Reading the same two assumptions as a strain rather than as a stiffness, and closing the question with the geometric floor on a fold’s radius, appears to be this collection’s own. The reason it has not been done is probably that the two literatures are separate: fabric bending is measured on a cantilever and crease recovery on a folded specimen, and the quantity that connects them is a fibre strain neither test reports.

Where the ladder goes next #

The strain is computed and what comes back is not, and the next rung finds that the answer splits in two: some fibres break at their own tightest fold and others survive it and fail to return, and no single ordering covers both.

Sideways, the weave decides how much crimp is available at the fold’s own position, and a satin has places with none at all. And the same coarse-then-fine handover governs a pile crushed under a foot, where the coarse route never runs out and the fibres are never strained.