The strong fibre is the one that pills

Wool knitwear pills and sheds. Polyester knitwear pills and keeps. Everybody in the trade knows both halves, and the explanation offered — that polyester fibres are stronger — is correct and is usually left there as a fact about a material rather than followed into its arithmetic.

The arithmetic is worth following, because it produces a result nobody would guess and that the trade has been acting on for sixty years: adding a little of a strong fibre to a weak one is nearly as bad as making the whole thing out of the strong fibre.

The cloth #

A blend is normally reasoned about by averages, in the way a cloth’s areal weight is. A fabric that is eighty per cent wool and twenty per cent nylon is expected to behave like wool with a fifth of a nudge toward nylon: a fifth of the strength gain, a fifth of the abrasion resistance, a fifth of whatever else.

That reasoning is right for the properties that are sums. A tensile strength is a sum over fibres. An areal mass is a sum. A thermal resistance is a sum.

It is wrong for the properties that are maxima or minima, and this collection has met one of those already from the other direction: a bundle is weaker than its threads, because a bundle breaks at its weakest place and a minimum over a sample is nothing like a mean. A pill is the same shape with the inequality reversed.

The claim #

A pill survives while at least one anchor holds, so its life is set by the strongest anchor it happens to have. With several anchors, the chance of having at least one strong fibre rises very steeply with the strong fibre’s share — and the pill life follows it. At a fifth of a strong fibre, four pills in five already have one, and the blend has bought four fifths of the pure strong fibre’s pill life while keeping the whole of the weak fibre’s fuzz supply.

That is the worst of both, and it is where the notorious fifteen-to-twenty per cent polyamide in a wool knit sits.

The order statistic, written out #

Let a fraction x of the fibres be the strong one and let a pill be held by z anchors drawn at random from the population.

The probability that a pill has no strong anchor is (1 − x)^z. So the probability that it has at least one is 1 − (1 − x)^z, and with eight anchors that is 0.34 at five per cent, 0.57 at ten, 0.73 at fifteen and 0.83 at twenty.

A pill’s life is set by its strongest anchor, so its expected life is the weak fibre’s life plus the difference, weighted by that probability. The whole curve is therefore the shape of 1 − (1 − x)^z, which rises steeply and flattens early — nine tenths of the way to the pure strong fibre’s life arrives by thirty per cent.

Nothing in that is an average. A mixing rule of any kind gives a straight line from one pure fibre to the other; this curve is convex from the start and has most of its rise in the first fifth.

Why the generation term does not come to the rescue #

The obvious objection is that a blend also changes how much fuzz there is, and it does — but in the direction that makes things worse.

Nylon and polyester are filaments in their pure form and are cut into staple to be blended, so a blended fibre has a staple length chosen to match the wool it is going with. It contributes ends at the same rate. Meanwhile the wool is still there, still supplying its own ends, still crimped and still entangling readily — and wool’s scales make it the readiest entangler of any fibre in the table.

So the generation term is very nearly the wool’s, because the wool is four fifths of the fabric and is the more readily entangled of the two. The blend has the wool’s fuzz supply and nearly the nylon’s pill lifetime.

That is the arithmetic behind a piece of received wisdom in knitwear — that a small nylon addition for strength is paid for in pilling — and it says the payment is much larger than the addition.

Why the trade sells a deliberately weaker fibre #

The remedy follows immediately and it is one of the odder products in textiles.

Low-pilling polyester is polyester engineered to be weaker. Its tenacity is lowered, usually by reducing the draw ratio, so that its anchors break sooner. It is sold at a premium for knitwear and its selling point is that it is worse at the thing polyester is bought for.

The model says exactly why that works and why nothing else would. Generation cannot be lowered enough — the fuzz comes from the yarn’s construction and the fabric’s, and a knit is above the entangling threshold by a wide margin. Lifetime can be lowered directly, and lifetime is proportional to the anchor’s breaking force.

A material sold on being weak is the strongest evidence available that the lifetime term is the operative one, and it predates any model of it by decades.

How many anchors, and why it matters so much #

The whole curve’s shape depends on z, the anchor count, and z is the parameter this ladder does not compute.

At z = 1 the curve is a straight line and the blend behaves like an average. At z = 4 the eighty-twenty blend has 0.59 of the strong fibre’s life; at z = 8, 0.83; at z = 16, 0.97. The steepness is entirely z’s, and the practical consequence — that a small addition is nearly as bad as a large one — needs z to be at least a handful.

There is a reason to think it is. A pill of a few tenths of a millimetre across contains tens of fibres and sits on a cloth whose surface presents many yarn segments; the fibres still rooted are the ones that were rooted before the ball formed, held by the friction over a gripped length that holds any fibre in any yarn, and there is no mechanism selecting a small number of them. A handful is a conservative estimate.

And z is measurable, by the route the previous rung proposed: a fabric’s plateau pill count against its fibre’s tenacity across a few fibres determines the product of z and the tenacity, and the tenacity is known.

The same shape, twice, with the sign flipped #

It is worth putting the two order statistics side by side because this collection now has both.

A bundle is weaker than its threads: a bundle of parallel fibres of varying strength does not break at the mean strength, because when the weakest fails its load is shared among the rest and the next weakest follows. The minimum governs, and adding a few weak fibres to a strong population costs more than their share.

A pill: the maximum governs, and adding a few strong fibres to a weak population buys more than their share.

Both are properties of small samples from a mixed population, both are invisible to any mixing rule, and both are decided by the extreme rather than the middle. A cloth is a population and not a thread is the essay that set this site up to notice such things, and its own worked cases were a thickness gauge reading a maximum and a warp jamming at its worst adjacent pair.

A fourth case, and the first where the extreme is favourable to nobody.

What a specification for a blend would have to say #

The curve has one immediate consequence for how a blend is written down and it is uncomfortable.

A blend proportion is not a dial. Everything a designer wants from the strong fibre — the strength, the abrasion resistance, the dimensional stability — arrives roughly in proportion to the share, because all three are sums. Everything unwanted arrives on the order-statistic curve, which is nearly complete at a fifth.

So the blend proportions that look like a sensible compromise are the ones with the worst ratio of benefit to cost. Five per cent nylon buys five per cent of the strength and a third of the pilling; twenty buys a fifth of the strength and four fifths of the pilling. There is no proportion at which the trade is good, and the curve’s convexity means it gets worse as the addition gets smaller relative to what it delivers.

The only escape is the one the trade takes: change the strong fibre’s tenacity rather than its share. That moves the height of the curve rather than its shape, and it is available because a fibre’s draw ratio is a manufacturing parameter.

What was counted, and how #

Four assertions and the first two are about the curve’s shape rather than its values.

That a fifth of a strong fibre buys more than three quarters of the pure strong fibre’s pill life. That nine tenths of it arrives before a third. Both are statements about 1 − (1 − x)^z at z = 8 and would fail at z = 2.

That the strong fibre carries more standing pills than the weak one from the same generation rate — the pure-fibre ordering, which is the essay’s premise.

And the gate from the rung below, re-asserted here in the fibre that most needs it: a polyester sheeting whose hairs cannot reach one another returns exactly zero standing pills. That is the assertion that stops the essay being read as “polyester pills”, which is false — polyester in a woven shirting does not pill, and the construction is why.

Where the knee is, exactly, and what it costs to sit before it #

The curve’s shape is 1 − (1 − x)^z and its two useful landmarks can be written down rather than read off a figure.

The half point is at x = 1 − 2^(−1/z), which for any z above about four is very nearly ln 2 ÷ z — 8.3 per cent at eight anchors, 15.9 at four, 4.2 at sixteen. So the share of strong fibre at which half the pills already have a strong anchor is about seven tenths of one over the anchor count, and nothing else enters it.

And the efficiency of an addition is the ratio of what it buys to what it costs, which is x divided by 1 − (1 − x)^z, since the strength arrives in proportion to the share and the pilling arrives on the curve. That ratio is 1/z as the share goes to zero and one when the fibre is pure, and it climbs monotonically between them.

Read the last column and the received practice inverts. The worst possible addition is the smallest one. A one per cent nylon addition buys one per cent of the strength and eight per cent of the pilling penalty, which is the worst bargain available anywhere on the curve; the commercial fifteen-to-twenty per cent sits at about a quarter efficiency; and a half-and-half blend buys twice as much strength per unit of pilling as an eighty-twenty does.

That is a strange instruction and it follows from the arithmetic without any additional assumption: if a strong fibre is going in at all, put a lot of it in. A touch of nylon is the one thing that cannot be justified, because the whole of its cost is incurred in the first few per cent and none of its benefit is.

The usual reasoning goes the other way — a little of an expensive or unwanted fibre, to get some of the benefit without spoiling the hand — and it is reasoning by averages applied to a property that is not an average.

How weak a low-pilling fibre has to be #

The other lever moves the curve’s height rather than its shape, and the arithmetic says how far it has to move.

A blend’s pill life relative to the pure weak fibre is 1 + P × (r − 1), where P is the chance of at least one strong anchor and r is the ratio of the two tenacities. Standard polyester at 0.55 newtons per tex against wool at 0.12 gives r = 4.6, and an eighty-twenty blend at eight anchors therefore has about four times wool’s pill life — which is what makes such a blend notorious.

Now set a target. To keep a twenty per cent blend within half again of pure wool’s pill life, 0.83 × (r − 1) must be under a half, so r must be under 1.6 and the additive’s tenacity must be under about

0.19 newtons per tex.

That is a third of standard polyester’s and only half again wool’s own — a fibre that has given up nearly all of the tensile advantage it was added for. Which is exactly what low-pilling polyester is, and it explains why the product is such an extreme intervention rather than a mild one. The order statistic is unforgiving: the additive has to be nearly as weak as what it is going into, because eighty-three per cent of the pills are going to find it whatever it is.

It also says what would happen to a milder version. A fibre at half standard tenacity, 0.28, gives r = 2.3 and a blend at 2.1 times wool’s pill life — still twice as bad as the wool it was meant to protect, for a fibre that has already lost half its strength. There is no useful middle setting on this lever either, which is the same convexity showing up in the remedy as in the problem.

Where the model stops #

The anchor count is a parameter and the whole curve’s steepness is its. The essay says so at every use and the sensitivity is stated rather than hidden.

Anchors are drawn at random from the blend and they are not. A blend has a spread of its own, and a spread is not a mixture. A blend is not perfectly intimate; fibres of one component cluster, and a pill formed at a wool-rich place has wool-rich anchors. Clustering would flatten the curve toward the average, and how much depends on a blending quality nothing here measures.

Lifetime is taken as proportional to tenacity. An anchor fails by fatigue under many small cycles rather than by a single overload, so the right quantity is a fatigue life and not a breaking force, and the two are not proportional across fibres. Wool’s fatigue behaviour is particularly unlike its tensile behaviour.

Nothing pulls out. An anchor can fail by being pulled out of the yarn as well as by breaking, and pull-out is governed by friction over a gripped length rather than by tenacity — which for a smooth strong filament may well be the operative failure mode and would break the essay’s ordering.

And the generation term is held fixed across fibres, which no two real fabrics do. Every comparison here is at equal fuzz supply and the figures say so.

The measurement that would falsify it #

The curve is a specific shape and it is not the shape any competing account gives, so it is falsifiable with one series.

Blend one wool with one nylon at five, ten, fifteen, twenty, thirty and fifty per cent, knit six identical fabrics, and grade them for pilling at a fixed cycle count. A mixing rule predicts a straight line. This model predicts a convex curve with its knee inside the first fifth, and the two are not close: at twenty per cent they differ by a factor of four in the distance travelled from pure wool.

The experiment needs no calibration, no absolute pilling scale and no comparison between laboratories, because it is a shape rather than a level. It is also, as far as this collection can tell, an experiment nobody has published — the blend proportions used commercially are the ones that work, and series across the whole range are rare because the intermediate proportions have no market.

The absence of such a series is itself weak evidence for the model, in the sense that a trade which had found the response linear would be using smaller additions.

The generalisation #

When survival requires only one member of a small sample to hold, a minority controls the outcome, and it controls it long before it is a majority.

The transferable form is that the operative statistic is the extreme rather than the mean, and that the extreme of a small sample from a mixture reaches the stronger component’s value at a mixing fraction of roughly 1/z rather than at a half. Redundancy is the everyday name for it: a system that survives while any one of several elements holds is a system whose behaviour is set by its best element, and a small admixture of a better element is nearly as good as a wholesale replacement.

The corollary, which is the one this essay is about, is that it is nearly as bad as one, too, whenever the property being controlled is undesirable. Robustness and stubbornness are the same arithmetic.

Who found it, and when #

That pilling propensity tracks fibre strength is standard and dates from the arrival of the synthetics in the 1950s; low-pilling polyester followed within a decade. That small nylon additions to wool worsen pilling is knitwear practice and is stated in every technical manual as a caution.

What is added here is the order statistic: that the mechanism is a maximum over a handful of anchors, that this predicts the curve’s convexity and the position of its knee, and that it is the same shape as the bundle argument this site has already made, with the inequality the other way round.

Where the ladder goes next #

Out of pilling and into the other thing a fibre end does to a person. Prickle is a buckling load takes a single protruding fibre, treats it as a column, and finds that the thirty-micrometre diameter the wool trade specifies to falls out of a column formula and a measured nociceptor threshold with nothing fitted — and that it is again the tail of a distribution, not its mean, that decides.