The hairs are what touch

Every diameter in this collection has been a mass divided by a length, turned into a width by a density and a packing factor. It is a good number, it is the right number for a great many things, and it has one property that ought to have been suspicious from the start: it is a property of the inside of the yarn.

What touches a neighbouring thread is not the inside. It is the surface, and the surface of a spun yarn is not a cylinder.

The claim #

A spun yarn has two diameters and this collection has only ever used one of them.

• The mass diameter is a volume over a length. Every cover factor, jam, hole and thickness here is computed from it.
• The contact diameter is what another body meets, and it exceeds the mass diameter by twice the hair layer — about thirty per cent for an ordinary 20 tex cotton.
• The gap is an absolute addition, so its effect on a cover factor grows in exact proportion to the sett. A close cloth is further from its own arithmetic than an open one, which is the wrong way round from every other correction in this collection.

Why there are hairs at all #

The reason is not that spinning is imperfect. It is in the pressure field.

The radial pressure inside a twisted yarn is ½σ(cos²θ® − cos²α), which is greatest on the axis and exactly zero at the surface. It has to be: pressure at a radius is the accumulated inward pull of everything outside it, and at the surface there is nothing outside.

So the outermost fibres are held by nothing at all where they are. What holds them is that their other parts are buried further in — and a fibre whose end happens to lie at the surface, with a short length of it above the plane where the pressure went to zero, is held by nothing over that length. It stands off.

Two consequences follow from the shape rather than from the zero. Hairiness is a surface property, so it scales with the yarn’s circumference rather than with its mass; and it is a property of fibre ends, so it scales with how many ends per unit length there are — which is the fibre count divided by the staple length. A finer or shorter fibre gives more ends and more hairs, at the same yarn count.

What it does to a cover factor #

Cover factor is a sett times a diameter, so an addition of 2h to the diameter adds 2h × sett to the cover.

That is not a proportional correction. It is an absolute one, multiplied by the sett — so it is small in an open cloth and large in a close one.

This matters most where a cover factor is used as a threshold rather than as a description, and this collection has three of those:

The jam. How closely threads may be set is a statement about diameters touching, and what touches is the contact diameter. So a warp jams sooner than the mass arithmetic says — which is one of the reasons a practical maximum sett is always below the computed one, and is a separate mechanism from the thickest-pair extreme.

The air. A cloth’s resistance to air comes from the size of its holes, and a hole is a spacing less a diameter. Subtracting the contact diameter rather than the mass one makes every hole smaller, and because the exponent on a hole is between two and four, a thirty per cent error in the diameter is a much larger error in the flow. This is the largest single unquantified term in this collection’s permeability arithmetic.

The opacity. Opacity is not cover, but both are about what fraction of the plane is obstructed, and a hair layer obstructs partially. A hairy cloth covers better than its construction says and does so with a soft edge rather than a hard one, which is why a raised or a hairy cloth looks denser than it measures.

Worse for a fine yarn, again #

The hair layer is an absolute thickness — a fibre standing off a surface stands off by some fraction of its own length, and that has nothing to do with the yarn’s count. So its relative effect goes as the reciprocal of the diameter.

This is the third quite separate respect in which the collection’s arithmetic is least reliable for the finest yarns. The evenness floor rises as the count falls, so the population corrections get worse; the strength becomes an extreme over a longer tail; and now the diameter itself is furthest from what anything actually touches.

The three have different mechanisms and the same direction. Taken together they say something worth stating plainly: a fine cloth is not a scaled-down coarse one. The arithmetic that describes a duck to two per cent describes a voile to twenty.

Which diameter did Peirce measure? #

There is an awkward question buried in the above and it is worth asking directly, because this collection’s own constant depends on the answer.

The diameter used throughout is obtained from a packing factor of 0.6, and that packing factor was obtained by inverting Peirce’s rule — one over twenty-eight times the square root of the cotton count. Peirce did not compute that constant from a density; he fitted it, to yarns, by measurement.

So which diameter was measured? If the measurement was optical — a yarn viewed against a lit background and its width read off — then it included some of the hair layer, and the packing factor derived from it is too low, because a wider diameter at the same mass implies looser packing. If the measurement was made by pressing yarns together and counting how many fitted in a width, it is nearer the contact diameter still.

This is not resolvable from inside the collection and it is stated rather than solved. What can be said is that the two readings differ by about the size of the hair layer, that a thirty per cent difference in diameter is a large one, and that the constant’s provenance matters more than the constant’s precision — which is a general point about inherited numbers and a specific warning about this one. It is the same doubt, arriving from a different direction, as the question of which twist the packing factor was measured at.

Friction is a hair property #

There is one more consequence and it changes how a number this collection uses a great deal should be read.

Yarn-on-yarn friction appears in the tuft-holding arithmetic, in the seam-slip arithmetic, in the tear model and in the capstan calculation that runs through all of them. It is carried as a range — two coefficients rather than one — and treated as a property of the fibre.

It is not a property of the fibre. Two yarns crossing meet through their hair layers before they meet at all, and what is being measured in a yarn-on-yarn friction test is the shearing of two brushes of fibre ends against each other. That is why the reported ranges are so wide, why the same fibre gives different values in different yarns, and why the number changes when a yarn is singed or waxed without the fibre having changed at all.

The practical rule that follows: a friction coefficient quoted for a fibre is a friction coefficient measured on some particular yarn of it, and substituting it into an arithmetic about a differently made yarn is an extrapolation nobody flags.

The hairs in the loom, and the size that lays them down #

A weaver’s experience of hairiness is not about cover at all. It is that adjacent warp ends stick to one another.

Two hairy ends lying side by side in the reed interlock through their hair layers. When the shed opens, an end that should go up is held by its neighbour going down, and the shed does not clear: the pick is laid across a thread in the wrong place, which is exactly the fault a mispick is, or the end breaks. On a hairy warp this happens often enough to stop the loom.

Warp sizing is the answer, and its mechanism is precisely this essay’s. Size is a film applied to the warp before weaving that glues the protruding fibres down against the yarn’s body — it does not strengthen the yarn nearly as much as it is credited with; it converts a brush into a cylinder. Sizing is a way of making a yarn’s contact diameter equal to its mass diameter, temporarily, and it is washed out after weaving.

Two facts about weaving practice follow from that and are usually given separately. Wefts are not sized, because a weft is thrown rather than dragged past its neighbours and never has to clear a shed. And filament warps need little or no size, because a filament yarn has no fibre ends and therefore no hair layer at all — which is the sharpest available demonstration that the whole problem is about ends rather than about fibres.

Why a fine warp is sized more heavily, which is not the hairs #

The sizing argument above says that size converts a brush into a cylinder, and it explains why filament warps need almost none. It does not explain the other thing every sizing table says: that the percentage of size on the yarn rises as the count falls — six to eight per cent on a coarse warp, twelve to fifteen on a fine one.

The obvious reading is that a fine yarn is hairier and needs more. It is not hairier; the hair layer is an absolute thickness and a fine yarn has fewer protruding ends, because the ends per unit length go as the fibre count and the fibre count is the yarn’s count over the fibre’s.

The real reason is a ratio this collection meets everywhere. Size is applied to a surface and quoted against a mass, so the add-on that a given film thickness represents is the surface area per unit mass — which for a cylinder is the circumference over the cross-section, or four over the diameter. The diameter goes as the square root of the count, so

size percentage ∝ 1 ÷ √tex.

Against a 60 tex warp, a 20 tex one needs √3 = 1.73 times the percentage for the same film. Read against the trade’s own figures — fourteen per cent against eight — the ratio is 1.75. The whole of the difference is geometry, and none of it is a statement about how hairy the two yarns are.

That is worth separating carefully, because the two mechanisms point the same way and are usually conflated.

The hair layer decides whether sizing is needed at all. A filament yarn has no protruding ends and needs no size to clear a shed; a spun yarn does, at any count.

The surface-to-mass ratio decides how much. Given that a film is wanted, its cost as a percentage of the yarn’s weight is fixed by the count and by nothing else.

And the two have different consequences when a third variable moves. A yarn spun from a finer fibre at the same count has more protruding ends and therefore needs a better film — but its surface per unit mass has not changed, so the percentage does not move. A yarn spun to a finer count has fewer ends and a worse ratio, so it needs a thinner film at a higher percentage. Both are counter-intuitive and both fall straight out of separating the two questions.

The practical form is the one a sizing house would recognise. A size percentage is not a measure of how much protection a warp is getting, because the same percentage is a much thinner film on a coarse yarn than on a fine one. What is comparable between counts is the film thickness, which is the percentage times the diameter — and it is the quantity nobody quotes.

What was counted, and how #

The proportionality is asserted, not observed. The gap between the two cover factors must grow at every step of the sett sweep, and the ratio of the gap at the ends of the range must equal the ratio of the setts to twelve decimal places — because the claim is that the correction is absolute rather than proportional, and a claim about the form of a correction is checked on its exponent.

The hair layer is an input everywhere it appears. Nothing here predicts it; every figure and every number names the layer it used. The value of 25 µm is at the lower end of what is reported for a ring-spun cotton, which makes every consequence quoted here a conservative one.

And the excess is computed at three counts so that the reciprocal relation is visible rather than asserted from one case.

Where the model stops #

Hairiness is not predicted and that is the whole of the gap. It is measured — the standard instruments count fibre ends crossing a beam, or report a total protruding length per unit of yarn — and the measurement does not decompose into a layer thickness without an assumption. Representing a hair layer as a uniform annulus is the crudest possible model of a very irregular thing.

A hair layer is not solid. For contact it behaves like a compressible brush rather than like an added diameter: two hairy yarns pressed together interpenetrate, so the effective addition under load is less than under no load, and it is load-dependent in a way nothing here captures. The figures should be read as the no-load case.

For air it is not an annulus either. Fibres crossing a channel obstruct it partially and add drag out of proportion to the area they block, so treating the hair layer as a solid enlargement of the thread is likely to understate its effect on flow — which is the opposite direction from the contact case.

Nothing here says which fibres protrude. The reasonable expectation is that they are disproportionately the short ones, since a short fibre is more likely to have both ends near the surface and less likely to be gripped over a useful length — which would make the hair layer a property of the short-fibre fraction rather than of the staple.

And the two diameters are not a bracket. They are two different quantities answering two different questions, and there is no sense in which the truth is between them: the mass diameter is exactly right for a mass and exactly wrong for a contact.

What the two diameters are each right for #

Set out as a rule, because the mistake this essay is about is using one where the other belongs.

Use the mass diameter for anything that is a mass, a volume or a count of material: an areal weight, a cloth’s mass per square metre, the fibre count in a section, a crimp computed from thread lengths, and every one of the population corrections. In all of those the diameter is standing in for an amount of substance, and the hair layer contains almost none.

Use the contact diameter for anything that is a meeting: a jam, a friction, a thickness under a light foot, a cover as it appears, a hole as air meets it, and the sett a warp will actually run at. In all of those the diameter is standing in for where one body stops and the next begins.

And be careful with anything that mixes the two. A cloth’s thickness is the clearest case: under a heavy presser foot the hairs are flattened and the reading approaches the mass diameters plus the crimp, while under a light one it includes the hair layers of both systems. That is one of the reasons a thickness is a maximum rather than a mean and why every thickness standard specifies a pressure as well as an area — the pressure decides which of the two diameters is being measured.

Where the ladder goes next #

Into the operation that removes them. Singeing takes a tenth of a per cent of a cloth’s mass away and changes its lustre, its friction, its measured cover and its printability — the largest ratio of consequence to material removed in the whole finishing trade, and the reason is entirely in the pressure field above.

Sideways, into the yarn constructions that suppress hairiness by holding the surface fibres down rather than burning them off. Folding traps each single’s surface against its neighbour, which is the one place where folding tightens this collection’s arithmetic instead of loosening it.