Textile Structure
how cloth is put together
Pattern and colour

A cloth is more opaque than it is closed

Opacity is not cover — this collection established that already and left the discrepancy attributed to the thickness of the threads. Part of it is not in the threads at all. A hair standing in a hole blocks light exactly as well as a thread does and costs nothing in air.
16 min read 6 figures Measured, not claimedA mechanism, not a materialStructure before fibre

Worth reading first: Opacity is not cover · A yarn has a diameter for every instrument · Half the air goes through a tenth of the holes.

Opacity is not cover established that a fabric blocks more light than its geometric cover factor says it should, and attributed the gap to the threads having thickness: a ray arriving at an angle meets more thread than a ray arriving square, and a thread is a cylinder rather than a flat ribbon so it obscures its own width plus a little. Both are true and both are computed there.

There is a third contributor and it is not in the cloth. A hair standing in a hole blocks a ray of light exactly as well as a thread of the same width does.

A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument.
Fig. 1 The fraction of the space beside a twenty tex cotton yarn that is occupied by hair, against height. At the yarn’s own surface it is under a tenth and it falls exponentially. Every one of those obstructions is in front of whatever is behind the cloth, and none of them is in the cover factor.

The cloth

A cover factor is a sett times a diameter and it is the arithmetic behind a great deal of this site: jamming, the hole between four threads, air permeability, wicking. In every one of those it is the right quantity, because every one of those is a question about where the material is.

Opacity is not such a question. Opacity asks what fraction of a ray’s path is obstructed, and an obstruction does not have to be load-bearing. A fibre a hundredth of a millimetre thick, standing in the middle of a hole and held by nothing, obstructs the same fraction of the ray as the same fibre would if it were part of a thread.

So the cover a cloth has and the cover a cloth appears to have are two contours of the same profile, and the second one is looser.

The claim

The hair layer adds several points of optical cover to an ordinary woven cloth and adds no measurable air permeability at all. The two quantities a single cover factor was standing in for therefore move in opposite directions under a finishing operation: singeing makes a cloth measurably more transparent and not measurably more permeable, and raising does the reverse.

The arithmetic behind the asymmetry is worth the essay on its own. A projected area is a sum and a flow resistance is a series with a bottleneck, and those two kinds of arithmetic respond to a dilute obstruction in opposite ways.

Why a sum and a series behave differently

Put a very dilute obstruction in front of two different questions.

For light, everything counts. The fraction of the plan not obscured is the product of the transmissions of the layers, and each layer’s transmission is one minus its own projected area. A layer of a few per cent projected area removes a few per cent of the transmitted light, and it does so whether it is at the top of the cloth or the bottom, whether it is fibre or thread, whether it is attached to anything or not. Every shadow is worth its own area.

For air, only the tightest section counts. Half the air goes through a tenth of the holes is this site’s own statement of it: flow through a fabric is dominated by its largest channels, and a fourth-power dependence on the channel’s width means the resistance is set almost entirely by the narrowest place in the widest hole. A layer that is nine parts in ten gap does not narrow anything.

So the same fibre standing in the same hole is a full contribution to one question and near enough nothing to the other. That is not a subtlety about hairs; it is the difference between an integral and a maximum, and it is why one cover factor cannot serve both.

The depth belongs to the fibre and the density to the yarn. Over a 5-fold range of cotton yarn counts, the hair layer's decay length moves by 6.2% and its population moves by 2.37-fold against a square root of 2.24. Both follow from one cancellation. The shell's share of the section is 4d_f/D, the migration period is a fixed number of yarn diameters, and λ = ½(kD)(4d_f/D) — the yarn's diameter divides out and leaves λ = 2k·d_f, a length belonging to the fibre alone. The density has no such cancellation and goes as √(nφ)/L. The two small departures visible here are not two facts: they are the shell's second-order term, and they are the same number to the last bit of a double. The consequence for a spinner is that a coarse yarn is hairier and its hairs are no longer, so everything that depends on reach — pilling, prickle, a printed edge — is decided by the fibre and not by the count.
Fig. 2 How far the population reaches, which is the quantity a sum cannot have. The depth of the hair layer belongs to the fibre — it is set by how long a fibre is and how much of it can protrude — while the density belongs to the yarn. Singeing and raising move the second and barely touch the first, which is why they price so differently in the two currencies this essay uses.

How many points it is worth

Per unit area the hairs present a projected area equal to their total length times a fibre’s diameter. For the long population on a sheeting that is about one per cent; grossed up by the measured split between the long and short populations it is eight or nine.

Eight or nine points of optical cover, on a cloth whose geometric cover is in the seventies. That is not a rounding: it is roughly the difference between a shirting a reader would call sheer and one they would not, and it is bought and sold — a finisher who singes a cloth for printing is also, unavoidably, making it look thinner.

The same arithmetic run on a raised cloth gives an optical depth above two, which means the fabric transmits under a seventh of what its holes would let through. A raised cloth is nearly opaque at a cover factor that would be see-through unraised, which is a real property of flannel and is usually described as the nap “filling in”.

What it does to a measurement of cover

There is a practical consequence for anybody measuring cover rather than computing it.

The three contours a yarn’s diameter has reappear here as a difference between two measurements a mill already takes. A cover computed from a count and a sett is the mass contour. A cover measured by transmitted light is a much looser one. The gap between them is proportional to the sett times the hair coverage, so dividing it by the sett recovers the coverage directly.

That is a hairiness measurement hiding inside a discrepancy, using two instruments any mill has. It is not proposed as a method; it is the observation that the discrepancy carries information which is currently discarded, in the same way the gap between a fabric’s two thicknesses does.

Whether a cloth's hairs can reach one another. n_A λ² for each construction in this site's table — the hairs per square millimetre times the square of their own length, which is the pure number that asks whether a hair can touch its neighbour. It is a count times an area, so it has to be a pure number. Every one of them is under one, which means no ordinary woven cotton cloth has a hair layer at all: it has isolated whiskers on a bare surface. The dashed line is the threshold. The spread across the whole table is only 1.9-fold, because the density goes as the sett times the root of the count and those move in opposite directions as a cloth is made finer — so construction is almost powerless here, and everything that crosses this threshold does so by finishing rather than by weaving.
Fig. 3 The canopy criterion across this site’s constructions. The relevant point for opacity is the flatness: a coarse open cloth and a fine close one carry nearly the same hair coverage, because the density’s square root in the count is cancelled by the sett’s inverse root. So the optical correction is nearly the same for every cloth while the geometric cover it corrects varies over the whole table.

That flatness has a consequence worth drawing out. Because the correction is nearly constant and the thing being corrected is not, the correction matters most where the cloth is most open — a voile picks up the same eight or nine points as a duck, and on a voile that is a much larger fraction of what is there. An open cloth is proportionally much further from its own arithmetic than a close one, which is the same direction the hair-layer cover correction took, and for a related reason.

The two finishing operations, priced in both currencies

Set the four combinations out and the asymmetry becomes a table anybody can act on.

Singeing removes the obstruction and leaves the bottleneck alone. A singed cloth transmits more light and passes the same air. For a printer that is a pure gain, because a printed edge feathers along the hairs and the operation buys a factor of ten in resolution. For anybody selling a sheer fabric on its opacity it is a loss, and it is a loss that arrives with no compensating change in handle, weight or permeability to explain it.

Raising multiplies the obstruction and leaves the bottleneck alone. A raised cloth is nearly opaque and just as windy, which is the pairing the wind essay is about from the thermal side: a flannel is warm, opaque and not windproof, and all three follow from a dilute layer being a full contribution to a sum and none to a bottleneck.

Calendering does something different again and is the control case. It flattens the threads, which raises the geometric cover and narrows the channels, so it moves both quantities and moves them the same way. That is what a change to the cloth looks like, and it is how a reader can tell that the other two are changes to something else.

A finer fibre gives more hairs, each of them shorter. At a fixed 20 tex yarn count, what the fibre's own fineness does to the hair layer. A finer fibre means more fibres in the section and a thinner surface shell, so the count of hairs goes up by 2.12-fold across the range and their length falls by 1.88-fold — and the two very nearly cancel, so the total protruding length moves by 13%. The geometry therefore says a finer cotton spins a hairier yarn, and the trade says the opposite. The disagreement is not smoothed over here. It lands entirely in the escape fraction, which the geometry does not supply: a finer fibre is more flexible and has more neighbours to catch it. That is the clearest statement available of where this model's one measured constant is doing real work, and the honest reading is that the constant is not a constant.
Fig. 4 Where the material comes from, and why so little of it does so much. A finer fibre gives more hairs, each of them shorter: the count rises and the length falls, so the mass hardly moves while the obstruction — a total length times a diameter — moves a great deal. The long population is a tenth of a per cent of a cloth’s areal mass and several points of its opacity.

Why this is not a claim about appearance

The word opacity is doing careful work here and the boundary has to be stated, because it is the boundary this whole site keeps against a neighbouring subject.

What is computed is a projected area. A fraction of the plan is obstructed by material, and the material is a set of cylinders of stated diameter. That is a geometric count of shadows and it is the same construction the shine ladder uses for a specular fraction.

What is not computed is anything about light. No scattering, no absorption, no wavelength, no refractive index, and nothing about how a fabric looks. A dyed cloth and an undyed one have identical geometry and completely different appearance, and nothing in this arithmetic can tell them apart or should try.

The distinction matters because the veiling this essay computes has a second effect that is genuinely optical — the hairs return light of their own, diffusely — and that effect belongs to the shine ladder rather than here. It is what destroys a shot effect, and it is a different quantity from the obstruction.

What was counted, and how

The obstruction is the population’s total protruding length per unit area times the fibre’s own diameter, which is a product of two quantities this ladder computes independently: the areal hair density from the yarn’s arithmetic and the sett, and the fibre diameter from the count.

Nothing is fitted to an opacity measurement, and the essay’s number is therefore a prediction rather than a description. What is asserted is that raising raises the obstruction and singeing lowers it, and by how much — a raised cloth’s optical depth is required to be more than twenty times a singed one’s, which it is by a wide margin.

The permeability half is asserted nowhere, because this site’s own permeability machinery is not run through a hair layer. The claim that the hairs cost nothing in air is an argument from the fourth-power dependence rather than a computation, and it is stated as such.

What the same argument says about a coated cloth

There is one more place where a sum and a bottleneck part company, and this ladder has already met it from the other side.

A coating fills the crowns before it bridges the holes found that the volume a coating must bury before it can span a hole is far larger than a block model predicts, because threads are round and the plan is more open than the cover factor says over most of the descent. That is a volume integral, and a volume integral is a sum.

The hair layer sits above that surface and is drunk by the coating before any of it reaches the crowns. Per unit area it is a small volume — the same tenth of a per cent of the cloth’s mass — but it is a large surface, because a cylinder has πd of surface for every d of shadow, and a coating wets surface rather than filling volume.

So a hairy cloth takes more coating to reach the same film thickness and more of it ends up in the wrong place. The essay above cannot compute how much, because nothing here models a liquid. What it can say is which direction the error runs and that it is the direction that would be attributed to the coating’s viscosity rather than to the cloth.

The hair population of a 20 tex cotton yarn. How many hairs on a 20 tex ring-spun cotton yarn stand at least a given height off it, per hundred metres, on a logarithmic count axis. The line is straight, which is the whole claim: the distribution is exponential, and it is exponential because a fibre end lands at a random phase of an irregular migration, so the length between the end and the last time the fibre was pulled inside is a memoryless residual. A perfectly regular migration would give a uniform distribution and a curve that stopped. The decay length is 621 µm, and it is 56 fibre diameters — a property of the fibre and not of the yarn. The counts marked at one, two and three millimetres are what a hair-counting instrument reports, and they are the counts it does report on yarns of this description. What the figure cannot show is the short population, which lies to the left of everything drawn and carries most of the protruding length.
Fig. 5 The population behind every number in this essay. The obstruction is its total length times a fibre diameter, so it is the area under this curve rather than any point on it — which is precisely why singeing, which truncates the tail, changes the obstruction so much less than it changes anything that needs reach.

Where the model stops

The short population’s projected area is grossed up rather than computed. The whole obstruction figure rests on one measured ratio, which means the eight or nine points should be read as a number with a factor of one and a half of uncertainty either side.

The hairs are taken as cylinders at random orientation. A hair lying flat along the cloth’s surface obscures nothing that the thread underneath it was not already obscuring, and a hair standing perpendicular obscures its full length only from directly above. The model takes the projected length as the full length, which over-states the obstruction for a viewing direction along the hairs and under-states it across them.

And no ray is traced. The transmission is one minus a projected area rather than an integral along a path, so multiple obstruction is handled by an exponential and the layer is treated as thin. At the optical depths a raised cloth reaches, that approximation is being pushed.

The permeability claim is a bound and not a calculation. The hairs certainly add some resistance to flow, and nothing here says how little.

The measurement that would settle the size of it

The essay’s number rests on one measured ratio and there is a cheap way to remove it.

Take a cloth, measure its transmitted-light cover, singe it, and measure again. The difference is the obstruction that was removed, and it is the whole of the long population plus whatever fraction of the short one the flame reached. No model is needed to interpret it and no calibration is required, because it is a difference between two readings on one instrument on one specimen.

The same experiment run on the permeameter must give nothing, which is the essay’s other half and is the more surprising of the two to a reader who has just watched the cloth become visibly thinner.

Doing both on the same specimen is a two-instrument, two-state experiment that takes an afternoon and settles a claim this collection can only compute. It would also give the short population’s share directly, which is the one number in the whole hair model that is a measurement by difference and would then be a measurement in its own right.

One hairiness reading does not fix the other. Every yarn on this curve has exactly the same total protruding fibre length — the quantity an integrating hairiness meter reports — and they differ in how that length is distributed. The count of hairs at least three millimetres long runs from 170 to 4354 per hundred metres, a factor of 26, across decay lengths real yarns actually have. The two instruments read two functionals of one population: the first moment N₀λ and the tail N₀e^(−3/λ). A correlation between them can exist only if λ is fixed across the yarns being compared, and λ is a fibre property, so it is not. That is the whole of why the trade's two hairiness numbers have never agreed, and it is arithmetic rather than instrumentation. What the figure cannot show is which of the two predicts anything: the tail does, because pilling, prickle and a printed edge all need reach.
Fig. 6 And why the measurement this essay proposes has to say which instrument made it. One hairiness reading does not fix the other: a count of hairs crossing a beam and a total projected length are different functionals of the same population, and two yarns can agree on one and differ by half on the other. An opacity correction computed from the wrong one is computed from a number that does not determine it.

A hair over a thread obscures nothing

The obstruction is computed as the hairs’ total projected area, and it should not be. A hair lying over a thread is shadowing something already shadowed, so it contributes nothing whatever to the transmission — and hairs emerge from threads, so a great many of them are doing exactly that.

A hair leaves a thread at a random point and points in a random direction, so the fraction of its projected length that falls over open plan is very nearly the open fraction itself, 1 − cover. The useful obstruction is therefore

(1 − cover) × the hairs’ projected area,

which on a sheeting at a cover of 0.7 is thirty per cent of the figure the essay quotes. Eight or nine points becomes two and a half or three.

That is a factor of three, and it is in the direction the essay’s own caution about orientation points without naming this as the reason. The correction is not about how a hair is angled. It is about what is underneath it.

Which makes the effect an open-cloth effect twice over

The essay already notes that the correction matters most where the cloth is most open, because the base it is added to is smaller there. The shadowing factor says the correction is also larger there, and the two compound.

cloth cover useful obstruction as a fraction of the base
voile 0.30 6.0 points +20%
shirting 0.50 4.3 +8.6%
sheeting 0.70 2.6 +3.7%

A voile’s optical cover exceeds its geometric cover by a fifth; a sheeting’s by under four per cent — a fivefold spread in the relative correction, from a hair population that is nearly identical across the whole table.

So the essay’s flatness result stands and its consequence sharpens. The hair population barely varies with the construction, and what it is worth optically varies by five times, because a close cloth has already covered the places the hairs would have covered.

Which is why singeing a sheer cloth is so visible and singeing a sheeting is not. A finisher who singes a voile takes a fifth off its apparent cover; one who singes a duck takes a few per cent. That is a real difference in how much the operation is noticed, and it is usually attributed to the sheer cloth being easier to see through in the first place.

And it corrects the measurement the essay proposes

The two-cover recipe — measure the optical cover, subtract the geometric one, divide by the sett to recover the hair coverage — needs the shadowing factor dividing out as well, or it reports a coverage that is low by 1 − cover.

hair coverage = (optical cover − geometric cover) ÷ [sett × (1 − geometric cover)].

That is still two instruments a mill already has and two numbers it already takes, and it now returns the same answer on an open cloth and a close one — which is the test of whether the recipe is measuring the yarn or the construction.

If the corrected figures agree across a range of setts, the method is measuring the hair population. If they do not, the shadowing model is wrong and the disagreement says by how much, which is a more useful outcome than a single number from a single cloth.

The generalisation

A quantity that appears in two different kinds of arithmetic is two quantities.

The transferable shape is the one this essay opened with. When a property is computed by summing over everything present and another is computed by finding the worst bottleneck, a dilute contribution changes the first and not the second — and any single parameter standing in for both will be right for one of them and wrong for the other, with no warning.

The diagnostic is to ask whether the arithmetic downstream is an integral or a maximum. Cover factor feeds both. So does a pore size, which is a maximum for filtration and a mean for wicking; so does a thickness, which is a maximum for a gauge and a mean for a resistance. Each of those has cost this collection an essay.

Who found it, and when

That a fabric looks more closed than it is measures is standard practical knowledge, and that singeing makes a cloth look thinner is stated in every finishing manual as a caution rather than as an effect to be quantified.

This site’s own opacity essay computed the thread-geometry contributions and did not have a hair layer to add. What is new here is the third contribution and the argument that it is asymmetric — that a hair is a full contribution to the sum and no contribution to the bottleneck.

Where the ladder goes next

To the second optical effect, which is the one that is not an obstruction. A hair layer veils a highlight finds that blocking a specular ray changes no contrast at all, because it takes the same factor off the peak and the trough — and that what destroys a shot effect is the hairs’ own return, which has no azimuth in it because a hair population points every way at once.

And sideways, to the same obstruction doing something useful: a woven filter beats its own rating, where the material standing in the holes catches particles that the holes themselves cannot.

Where this goes next

How open a batiste is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this batiste it is 40.1 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 39.0° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 6.42 per cent open — 6.3 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none. Opacity is not cover The covering rule counts a thread as a bar that stops everything, and a single fine cotton thread held to a window plainly does not. What a cloth transmits is the open area plus whatever comes through the threads, so the covered fraction is a lever rather than a barrier — and the lever is longest exactly where the rule says the cloth is most closed. Nothing here computes a thread's transmittance, and saying why is the useful half. A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument. A yarn has a diameter for every instrument Conservation of volume gives a yarn one diameter and every other route gives a different one. The disagreement is not experimental scatter: a yarn's outside is a coverage that falls away exponentially, and each instrument stops at whatever contour of it will trigger the instrument. Where a muslin's air goes, as the cloth is set closer. A permeability computed from the average hole says nothing about which holes the air uses, and once the holes have a spread the answer is: not many of them. Two curves, both over the same population of holes — the share of the flow carried by the widest tenth, and how few of the holes carry half of it. At 16 ends per centimetre the cloth is nearly democratic: the widest tenth takes 12% and half the air needs 45% of the holes. At 44 the widest tenth takes 53% and half the air goes through 8.8%. Both inputs move together as the cloth closes — the spread in the holes rises because the spacing is fixed and the diameter is not, and the exponent rises because the pressure drop stops being inertial — so the concentration rises faster than either. Half the air goes through a tenth of the holes A permeability computed from the average hole says nothing about which holes the air uses. Once the threads have a spread, the answer is: not many of them — and in a close cloth, half the flow leaves through less than a tenth of the openings.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Air permeabilityCanopy criterionClear openingCover factorHair coverageHair layerOpacityOpen areaSingeingVeil