A knit gives up its fibres more easily

A cotton jersey pills and a cotton poplin does not. They can be spun from the same yarn on the same frame, and everything about the fibre — its staple, its fineness, its tenacity — is identical. The difference is in the structure, and the structure is not obviously doing anything to a fibre end.

The pilling balance says a fabric can only pill if its hairs can reach one another, and the criterion is n_A λ² — the hairs per square millimetre times the square of their own length. A woven cotton sheeting sits at about a half, under the line. A jersey does not.

The cloth #

The criterion has two inputs and a knit changes both.

The first is how much yarn there is per unit area of fabric, because every millimetre of yarn contributes its own hairs. For a woven cloth that is the sum of the two setts, which is a straightforward count of ends and picks.

The second is how much of that yarn’s surface is free, which is where the loop geometry enters. A hair standing between two threads that are pressed together is caught; a hair standing in open air is not.

The claim #

A knitted fabric presents more yarn per unit area than a woven one of the same yarn and holds it under far less inter-thread pressure, so more of its fibre ends stand clear. Both effects push the criterion up, and a jersey clears the entangling threshold that a woven cloth of the same yarn cannot reach. That is why knitwear pills and shirting does not, and it is a statement about geometry rather than about fibre.

There is a second half about which face: a jersey’s two faces have different amounts of yarn showing, so a single fabric can be above the threshold on one side and nearer it on the other.

The yarn per unit area, which is a loop length #

A knit’s dimensions come from its loop: Munden’s constants tie the courses and wales per unit length to the loop length, and the yarn in a unit area is the loop length times the number of loops.

The comparison with a weave is direct. In a woven cloth of a given areal mass, the yarn lies straight and its length per unit area is very nearly the areal mass divided by the count. In a knit of the same areal mass the yarn is also its mass divided by its count — the same length. So the naive comparison gives no difference at all.

What differs is where that length is. A woven cloth’s yarn is buried: at every crossing an end passes under a pick and is covered by it, and the face share says how much of the plan each system owns. A knitted loop is a curve lying mostly on the surface, with its legs exposed on one face and its heads and feet on the other.

So a knit’s yarn is more exposed at the same length, and exposure is what decides whether a hair stands clear or is trapped against a neighbour.

The pressure between threads, which is where the escape happens #

The second effect is larger and is the one this site has the machinery for.

A woven cloth’s threads press on one another. Every crossing is a force: the crimp puts the two systems under tension against each other, and the contact force at a crossing is what grips a thread and what makes a seam hold. That force presses hairs down against the threads they lie on.

A knitted loop is held by bending rather than by tension. A knit is soft because it bends — the whole of its extension is a change of loop shape at almost no yarn strain — and the corollary is that the force between a loop and its neighbours in a relaxed fabric is very small. Nothing presses a knit’s hairs down.

So a knit’s escape fraction is effectively higher than the same yarn’s in a weave, not because the yarn changed but because nothing is holding its ends against it any more. That is the same quantity a size film reduces on a warp and the same quantity a compact spinning frame reduces at the spinning triangle — a mask on the population rather than a change to it.

Which face, and how a jersey differs from itself #

A jersey has two surfaces and they are not the same surface: the technical face shows the legs of the loops, running nearly vertically, and the technical back shows the heads and feet, running nearly horizontally.

The two carry different amounts of exposed yarn per unit area, and the ratio between them is a property of the loop’s shape rather than of the yarn. So one face of a single-jersey fabric sits further above the entangling threshold than the other, and a garment made with the back outward pills differently from the same fabric worn face out.

That is a real and slightly absurd consequence — the same cloth, the same yarn, two pilling grades depending on which way round it is made up — and it is the kind of thing the trade knows as a rule of thumb about which face to use.

Why raising a knit is so effective and so dangerous #

The two facts combine badly in one common product.

A brushed or fleeced jersey is a knit that has been raised, and raising multiplies the hair population by a large factor. A knit is already near or over the threshold, so raising takes it far over — a fleece’s criterion is in the tens, and it pills freely.

That is not a defect of manufacture; it is what the product is. A fleece is a fabric whose entire selling point is a canopy, and a canopy is warm for reasons that have nothing to do with the knit underneath it. The pilling is the same canopy seen through the pilling balance, and the two cannot be separated because they are the same population.

The mitigation used is the one the pilling ladder predicts: weaken the anchors. Fleece is made of polyester and the polyesters used for it are low-tenacity grades, sold on being worse at the thing polyester is bought for.

What a tightness factor should do to it #

The one parameter a knitter has that maps directly onto the argument is the tightness factor, and the direction it moves things is worth setting out even though the size is not available.

A tighter knit — a shorter loop for the same yarn — packs more loops into an area and presses them harder against one another. Both effects are the ones this essay is about, and they point in opposite directions: more yarn per unit area raises the hair count, and more pressure between loops lowers the escape fraction.

So a tightness factor should produce a minimum in pilling rather than a monotone trend, and that is what knitters report: a very slack fabric pills because its surface is loose, and a very tight one pills because it is dense with fibre ends, and there is a range in between where the fabric is best.

A minimum with two opposed mechanisms in it is the same shape as the twist curve this collection found for yarn strength, and for the same structural reason: two effects with different exponents crossing over. The difference is that the twist curve’s two effects are both computed here and the knit’s are not.

Half a woven thread is inside a contact and none of a loop is #

The argument above has two halves and says the second is not computable. One part of it is, and it comes from a number this collection computes for an entirely different purpose — which is worth taking, because it turns the essay’s direction into a bracket.

First the part the essay states and does not check. A jersey at a loop length of 3.5 millimetres has a wale spacing of 875 micrometres and a course spacing of 700, so its yarn per unit area is 3.5 divided by 0.6125, or 5.7 millimetres of yarn per square millimetre of fabric. A sheeting at twenty-eight ends and twenty-six picks to the centimetre, with nine per cent crimp, carries 5.9. The two are the same to within the roundings, which is the essay’s claim arrived at rather than asserted.

Now the exposure. A woven thread has no room to bend measures, for eight cloths, the share of a thread’s own length that lies inside the wrap — the arc where it is pressed against the thread it crosses. It runs from seven per cent on an open scrim to fifty-five on a sheeting. And it reports the corresponding figure for a knitted loop, which is nothing at all: a loop never wraps, so no part of its length is inside a contact of that kind.

That is the exposure difference, in a number, and it was sitting in a neighbouring ladder.

Which puts a jersey on the line rather than over it #

Take the crudest reading of it — that the escape fraction scales with the share of yarn not under a contact — and the knit’s escape fraction is the woven one divided by 1 − 0.55, a factor of 2.2 against a sheeting.

The criterion is linear in the hair count, so a jersey of the sheeting’s yarn lands at

0.5 × 2.2 ≈ 1.1

against a threshold of one.

Just over. Which is the right answer in a way that would have been suspicious if it had come out at ten: a jersey pills, and it pills mildly. It is not a fleece. Its pills are slow to form, they are few, and a moderately tight fine-gauge jersey often does not pill at all — all of which is what a fabric sitting a tenth above a threshold should do, and none of which is what a fabric two orders of magnitude above it does.

It also brackets the answer from below, which is the useful direction. The wrap share accounts only for the yarn that is geometrically inside a contact; the essay’s second mechanism — that a woven contact carries a force some thirty times a knitted one, so it presses down hairs well outside the wrap itself — is entirely additional and pushes the figure up. So 1.1 is a floor, the true value is higher, and the qualitative claim that a knit clears the threshold does not depend on the unmeasured half at all.

Against an open cloth the arithmetic is much weaker, and honestly so. A voile’s wrap share is seventeen per cent, so the same reading gives a factor of only 1.2 — and a voile’s own criterion is lower to begin with. The comparison that works is with a densely set woven cloth, which is exactly the comparison the trade makes when it says a jersey pills and a poplin does not.

What this does not settle #

The proportionality is an assumption and it is the only one, so it is worth naming what would break it. A hair emerging inside a wrap is not merely pressed; it is pinched between two threads at a real force, so it may be held far more firmly than a linear scaling suggests — in which case the woven escape fraction is lower than proportional and the ratio above is an under-estimate again.

And nothing here supplies the tightness dependence, because the wrap share is a woven quantity and has no knitted analogue that varies with the loop length. The minimum the essay predicts is still a prediction.

What has changed is that the knitted criterion is no longer a direction with no number attached. It has a floor, the floor is on the correct side of the threshold, and it was obtained without any contact force at all.

Why the knit’s own arithmetic cannot supply the number #

An honest account has to say what this essay does not compute, and it is the number the argument turns on.

This collection’s knit machinery gives the loop’s dimensions from Munden’s constants and gives the fabric’s extension from the loop’s shape. It does not give a contact force between loops, because a loop has no closure condition — the whole point of that essay is that a knit has no equivalent of Peirce’s h₁ + h₂ = d₁ + d₂, so its geometry is not determined by its own constraints and Munden’s constants are measurements standing where a solution ought to be.

No contact force means no escape fraction, and no escape fraction means the criterion for a knit cannot be computed the way it is computed for a weave. What the essay has is a direction and a mechanism, and the size of the effect is a measurement it does not have.

That is a real gap and it is the sharpest one this ladder leaves. It is recorded as such.

What was counted, and how #

The woven criterion is computed and the knitted one is not, so the assertions here are the woven ones and the argument for the knit rests on them by comparison.

That every woven construction in this site’s table sits under the entangling threshold, so a woven fabric’s freedom from pilling is a structural fact rather than a fibre fact. That the spread across the table is under a factor of two, so no weaver can construct into or out of the regime. And that raising crosses the threshold and singeing takes a fabric two orders of magnitude below it.

The knit’s position relative to the line is argued rather than computed, and the essay says so at the point of arguing it.

Where the model stops #

The knitted criterion is not computed. It needs a contact force between loops, and this site does not have one.

The two faces’ exposed-yarn ratio is a geometric estimate. It comes from the loop’s shape as Munden’s constants describe it, and the loop’s path between the constants is a guess this site has recorded before — the leg is taken as straight over a guessed fraction of a course spacing.

Nothing about the knitting machine is in it. A yarn is abraded by needles, sinkers and guides on its way into a fabric, and that abrasion frees ends exactly as wear does. A fabric coming off the machine is already somewhere along the balance, and where depends on the machine rather than on the structure.

And the tightness factor is absent. A slack knit and a tight one of the same yarn have different inter-loop pressures and should sit at different points, which is a one-parameter family the essay does not draw.

The measurement that would fill the gap #

The knitted criterion is a measurement rather than a computation, and it is a measurement that can be taken with an instrument the trade already owns.

Knit and weave one yarn into two fabrics of equal areal mass. Then run a hair count on each fabric rather than on the yarn — which is what a fabric hairiness tester does, by scanning a specimen edge-on against a light. The ratio between the two counts is the ratio of escape fractions, with the yarn’s own population cancelling out.

That ratio is the number this essay argues about and cannot supply, and it needs no calibration because it is a ratio within one instrument. It would also give the criterion for the knit directly, since the decay length is a fibre property and is the same in both fabrics.

The same experiment run across a series of tightness factors would give the minimum the previous section predicts, and would settle whether the two opposed effects really do cross.

The generalisation #

A structure decides how much of a material’s surface is available, and availability is often the whole of a property.

The transferable shape is that two objects made of the same stuff in the same quantity can differ completely in anything that happens at a surface, because what matters is not how much material there is but how much of it is exposed and how much of what is exposed is pressed against something else. Every surface property — friction, wear, wetting, soiling, catalysis — has that character.

The diagnostic is to compute the exposed area rather than the mass, and then to ask what fraction of the exposed area is in contact with something else. Those two numbers usually differ far more between structures than any bulk property does.

The one place a knit is on the better side of the comparison #

It would be tidy if the knit were simply the worse structure for everything at its surface, and it is not.

Because a knit’s threads are not pressed against one another, a hair that is caught in a knit is caught much less firmly — so a knit sheds more readily as well as fuzzing more readily. That is the removal term in the balance, and it moves in the same direction as the generation term.

The consequence is that a knit reaches its steady state faster than a woven cloth does and sits at a level set by a ratio of two large rates rather than of two small ones. A jersey worn a few times has arrived where it is going; a woven cloth takes much longer and drifts.

Which is why knitwear complaints arrive early and shirting complaints arrive late, and why the two are usually about different things — a jersey is judged on how it looks after three wearings and a shirt on how it looks after fifty washes.

Who found it, and when #

That knits pill more than wovens is universal and is why pilling tests are specified mostly for knitted goods. The explanation given is that knitted structures are “more open” or “less tightly bound”, which is right and is not quantified.

What is added here is the criterion — the specific pure number a fabric has to clear to pill at all — and the observation that the woven side of the comparison is computable and is uniformly below it, so the whole of the difference is in a quantity the knit side cannot yet supply.

Where the ladder goes next #

The knit’s missing contact force is the obvious next piece of work and it is a loop-geometry problem rather than a hair problem, so it belongs to the closure condition a loop does not have.

Meanwhile the same population, on a woven cloth where it can be computed, decides how sharply a print holds its edge and what a filter catches below its own rating.