Hair, nap and pile are one construction

Three parts of this collection have each produced a surface made of fibre standing off a cloth, and none of them noticed the other two.

The pile came first, among the fancy weaves: a third thread system, woven in, cut or looped, held by a W or a V through three picks. The nap came with the finishing: fibre pulled out of a float by a raising machine, standing loose above the cloth. The hair layer arrives here, from the yarn’s own arithmetic: fibre ends that were never held in the first place.

The cloth #

They look like three subjects and they are described in three vocabularies. A pile is a weaving problem, with beams and wires and a cutting blade. A nap is a finishing problem, with wire teeth and a drum speed. A hair layer is a spinning problem, with a triangle and a staple length.

Set them beside one another as populations — a density, a length, an anchor — and the vocabularies fall away.

The claim #

Hair, nap and pile are one object at three settings. All three are a density of protruding fibre with a length distribution and an anchor force, all three are governed by the same canopy criterion, and the quantities span large factors rather than being different in kind. What distinguishes them is which of the three numbers was decided on purpose.

That is a claim about how this collection is organised as much as about cloth, and it is the kind of claim a site built on one criterion ought to be able to make.

The three, measured against each other #

Hairs on a bare cloth: about one and a third per square millimetre, mean length six tenths of a millimetre, anchor whatever the twist happened to leave. The criterion n_A λ² is about a half — under the line, so they are whiskers rather than a layer.

A raised nap: the same length, because raising does not change what a fibre end is, and a density sixty-four times larger. The criterion is thirty-four, and the canopy is two millimetres deep.

A woven pile: about twelve tufts per square millimetre and three millimetres long, held by a construction. The criterion is over a hundred.

A large factor in density, a factor of five in length, and a strictly increasing criterion. All three assertions are in the model, because a ladder that claimed to be one object with three settings ought to be monotone in the quantity that decides whether the object exists.

The anchor is the number that tells them apart #

Density and length are quantitative differences. The anchor is a difference in kind, and it is the one a designer chooses.

A hair’s anchor is an accident. It is held by whatever length of it happens to remain buried, and what grips it is friction over that length under a radial pressure that falls to zero at the yarn’s surface — which is why it is there at all. Nobody chose it and nobody can specify it.

A nap’s anchor is the float it came out of. Raising spends the cloth’s strength: the fibre is pulled out of a yarn that was carrying load, so the anchor is the remaining grip on a fibre that has already been partly extracted. It is weaker than a hair’s, and it is why a nap wears away.

A pile’s anchor is designed and is measured in newtons. What holds a tuft computed it: a W-tuft through three picks, a V through two, and the difference between them is a specification a carpet is sold against.

So the ladder runs from an anchor nobody chose, through one that is a by-product of an operation chosen for something else, to one that is the point of the construction. Three settings of one object, and the interesting axis is not the length.

What each rung costs the cloth it grows on #

There is an economic axis running alongside the anchor axis and it points the other way, which is what makes the ladder a set of real choices rather than a ranking.

A hair layer is free and unwanted. It costs nothing to produce because nobody produces it, and most of what this ladder computes about it is a nuisance — a veiled highlight, a feathered print, a pill, a prickle. The only things it gives away are the ones nobody was buying.

A nap is paid for in strength. The fibre standing above the cloth came out of the yarns below it, and raising spends the cloth’s strength irreversibly: the cloth is weaker by roughly what the nap weighs, and the nap wears off while the weakness stays. That is the worst exchange rate on the ladder and it is why raising is done to fabrics whose strength is not the point.

A pile is paid for in yarn. A third thread system is extra material, extra warp beams, and a slower loom; nothing is taken out of the ground cloth at all. It is the most expensive rung and the only one whose cost buys something permanent.

So the ladder runs from free and useless, through cheap and temporary, to expensive and durable, which is a sensible thing for a ladder to do and is not how the three subjects read when they are treated separately.

What the continuum says about the constructions between #

Reading the three as a continuum immediately raises the question of what is in between, and this site already has the answers scattered across three fields.

A brushed knit sits between a nap and a hair layer, with a density set by a raising machine and an anchor set by a loop that was never under much tension in the first place. It has the worst anchor of anything on the ladder, which is why fleece sheds.

An uncut terry loop sits between a nap and a pile. Terry needs two beams because its loops come from a slack warp; they are a third system with a designed anchor and no blade, so they are a pile in every respect except that their tops are curved. And that curvature is exactly what puts them back on the crown side of the surface argument — an uncut loop bears like any other curved thread, and a cut pile bears at a step.

A needled felt sits past the pile end: fibres driven through the cloth by barbed needles, anchored by entanglement rather than by weave, at a density and depth beyond anything here.

And a chenille yarn is the ladder turned inside out — a yarn that is already a fabric, carrying its own pile, woven in as an ordinary thread.

Five constructions this collection has treated as five subjects, arranged on one axis.

Where the surface argument breaks, and it breaks once #

The surface ladder found one surface in this collection that is not a set of crowns, and it is the top of this ladder.

A pile is the only surface with no crowns: a cut pile is made by a blade, so every tuft ends in one plane and the bearing curve is a step — a finite area at zero depth, where every other fabric’s contact vanishes with the load. That is a third exponent, zero against a half against one, and it is a property of the tool rather than of the weave.

The other two rungs do not do that. A hair layer and a nap are populations with exponential length distributions, so their contact opens gradually and their bearing curve has no step in it at all. What they have instead is the crossover — a pressure below which nothing reaches the cloth — and a cut pile has that too, at a much higher pressure.

So the ladder is continuous in density, length and anchor, and it is discontinuous in one thing: whether the tops were cut. That discontinuity is a blade, and a blade is not a matter of degree.

What the three share, which is the criterion #

Everything in this ladder that needed a canopy needed the same pure number, and the ladder is the clearest demonstration that it is the right one.

n_A λ² is a count times an area. It asks whether a member of the population can reach its neighbour, and every property that needs a layer rather than a set of whiskers turns on it: the still air that makes a fabric warm, the entangling that makes a pill, the roughness ratio that amplifies a contact angle, the bed that intercepts a particle.

All three rungs are described by it and only two of them clear it. That is a useful thing for a criterion to do — a criterion that every case satisfied would be telling nobody anything.

And it explains why the three subjects were treated separately for so long. The bottom rung is invisible: an ordinary woven cloth’s hair layer does none of the things a layer does, so nothing about it invited comparison with a nap. Only when the criterion was written down did the bottom rung turn out to be the same object as the other two, sitting just under a threshold.

The one property that runs the wrong way up the ladder #

Almost everything improves as the ladder is climbed — the anchor gets stronger, the layer gets deeper, the warmth rises — and one thing does not.

Uniformity. A hair population is exponential in length, so it is extremely non-uniform; a nap is the same population multiplied, so it is exactly as non-uniform; and a cut pile is one length to the accuracy of a blade.

That looks like the top rung winning again, and for contact it is: a cut pile’s bearing curve is a step, and a step is the only surface in this collection whose pressure concentration is bounded at every load. But for anything that needs reach it is the opposite. A population with an exponential tail has members far longer than its mean, and those members are what bridges, entangles, intercepts and prickles.

So a cut pile cannot pill and cannot feather a print, not because its fibres are held better but because they are all the same length and none of them reaches past the others. A carpet’s surface is a set of tufts that never touch across their tops.

That is the ladder’s least obvious result and it follows from nothing but the shape of the length distribution. A blade does not merely shorten a population; it collapses its variance, and a collapsed variance removes every property that lived in the tail.

What was counted, and how #

Two assertions, and they are about the ladder’s structure rather than about any rung.

That each step up closes its canopy harder than the last — a strict ordering of the criterion across the three, which would be broken by any rung being mis-parameterised.

And that the three differ by large factors in both density and length, rather than being three names for nearly the same thing. That is the check on the claim’s other half: an argument that three constructions are one object has to show that they are also genuinely different, or the observation is empty.

The pile’s numbers are the fancy-weave table’s and not this ladder’s. They are tufts per square millimetre and a pile height, both of which are construction data rather than computed populations, and the essay says so.

Reading the ladder back through itself #

It is worth using the ladder as an index of what has actually been established, because it sorts the results by which rung they belong to.

Everything about the bottom rung is about a population being small. The veil that flattens a shot effect, the feather on a print, the seeding of a filter cake, the prickle threshold — all of them are properties of a few members reaching far, on a layer that does nothing as a layer.

Everything about the middle rung is about a canopy existing. The still air, the wetting amplification, the entangling that makes a pill, the crossover that puts a hand’s pressure above the cloth. All of those need n_A λ² above one and none of them happens on a bare woven cotton.

And the top rung belongs to the fancy weaves, where a blade produced the only bearing curve in this collection with a step in it.

So the twenty essays of this ladder divide almost exactly along the criterion, and the division was not designed — the criterion was written to answer a question about warmth and turned out to sort everything else.

Where the model stops #

The pile’s length has no distribution. A cut pile is cut to one length by a blade, so its population is a spike rather than an exponential — which is the ladder’s discontinuity and is also why the criterion, which uses a mean length, is not strictly the same quantity at the top rung as at the other two.

The anchor forces are not on one scale. A hair’s grip is a friction over a buried length, a nap’s is the same after partial extraction, and a tuft’s is a measured newtons. They are the same physical quantity and this collection has never computed the first two.

The intermediate constructions are named and not measured. A brushed knit, a terry loop, a needled felt and a chenille are placed on the axis by argument rather than by populations, because three of the four have no hair model at all.

And the bare rung is a woven cotton’s. A knit’s is larger, by an amount this ladder could not compute, which would move the bottom rung up and might move it over the line.

The generalisation #

Three subjects with three vocabularies are often one object with three settings, and the way to find out is to write down what each of them is a quantity of.

The transferable move is to strip the process language — weaving, raising, spinning — and ask what the resulting thing is: here, a density, a length and an anchor. Objects described by the same three numbers are the same object, however differently they were made, and the differences that remain are the interesting ones.

The corollary is the one worth carrying into this collection’s own organisation. A criterion is more useful when some cases fail it, because the failures are what identify the boundary; and a family that was invisible for years turned out to be invisible precisely because it sat just under one.

Who found it, and when #

Piles, naps and hairiness are three literatures and they do not cite one another. The pile literature is weaving and carpet manufacture; the nap literature is finishing; the hairiness literature is spinning and quality control.

The observation that they are one object is not, as far as this collection can tell, anywhere. It became available here only because all three were computed against the same criterion, which was written for a different reason — to decide whether a hair layer holds air still.

Where the ladder goes next #

The pile anchor has been measured and the other two have not, and the gap is the obvious next piece of work: a nap’s anchor is a partially-extracted fibre and this site has the grip arithmetic to compute it. That would put all three rungs on one scale in the quantity that actually distinguishes them.

Below that, the bottom rung’s own position needs the knitted case, which is a contact force between loops this collection does not have. Both are the same shortfall in a different field: a surface’s anchor is a friction problem, and friction problems are where this site’s models stop.