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The thread: Structure before fibre — page 3

Page 3 of 5 of the essays on this thread.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 1400 turns a metre. Its own torque is 2.060 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 0.91 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

A crepe is a yarn that will not lie still

A crepe cloth's pebbled surface is the yarn's own torque acting on a cloth that cannot resist it. The instability that makes a slack yarn snarl says how large the pebbles should be, and the answer is a millimetre — which is what a crepe looks like.

An operation multiplies a spread by its own log-slope. A transformation does not leave a population's spread alone: if y goes as the kth power of x then a small spread in x becomes k times that spread in y, exactly in the limit and nearly so at the CVs a yarn has. So an operation with an exponent below one narrows the population it acts on — a thickness that goes as the square root of a load comes out at half the spread it went in with — and one with an exponent of four widens it fourfold. This is the same derivative that decided every bias in this ladder, read for its magnitude rather than for its curvature, and it is why a finish can be a variance-reducing operation without anyone having chosen it for that. The straight line through the origin is the whole of the rule; the departure from it at the right-hand end is the second-order term arriving, which is where the linearisation stops being one.

After the loom

A finish spends a spread before it spends a mean

Every operation on a cloth multiplies the variation it inherits by its own log-slope, so an operation with an exponent below one makes the cloth more even and one above it makes the cloth less even. Calendering, which is bought for evenness, has an exponent of 1.4.

What a tear asks for is the weakest of a few. A tensile test pulls every thread in the width and averages them. A tear pulls the handful in the triangle at the tip of the cut, and what lets it move is whichever of those is weakest — so the quantity that governs is the minimum of a small sample, and two things follow that no mean can show. Its expectation is below the mean: at CV 15% the weakest of 4 is 0.853 of the mean thread, and the weakest of 40 is 0.718. And its scatter is enormous compared with a tensile test's: 10.9% against the 0.75% a mean of four hundred threads would show. A tear strength that varies by a tenth between specimens of the same cloth is not a badly run test. It is the only answer an extreme of four can give.

Cloth doing a job

A tear asks fewer threads than a pull

A strip test averages a hundred threads and a tear interrogates four. That difference alone accounts for the two things everybody knows about tear testing — that it reads low, and that it scatters — without anything about the cloth being different between the two tests.

The crossings under a 25 mm² presser foot. A 25 mm² foot on a muslin covers 12 ends and 11 picks, which is 132 crossings — and a first guess treats those as 132 chances of finding a thick place. They are not independent chances. Every crossing along one end shares that end's diameter, so the largest crossing is the largest end plus the largest pick, and the number of tries is 23: the threads. Each cell here is shaded by its own two diameters, and the darkest is at the meeting of the darkest row and the darkest column, which is what that identity looks like. The gauge rests on it and reads 0.431 mm, against 0.342 for the cloth's mean crossing — 26% over.

Mechanics and drape

A thickness is a maximum, not a mean

A presser foot rests on whatever is highest beneath it, so the thickness of a fabric is an extreme value — and an extreme grows with how much cloth is asked. The standard specifies the foot's area because the foot's area is in the answer.

Setting and geometry

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.

$A 20 tex cotton yarn and the fibre standing off it. 6 mm of a 20 tex ring-spun cotton yarn with the hair population this site computes from the yarn's own count and staple — 0.89 hairs per millimetre, every one of them drawn. The two axes are at different scales and have to be — the yarn is 167 µm across and its hairs reach past a millimetre, so a picture at one scale is either a bare line or a black rectangle. Along the yarn is 99 pixels to the millimetre and off it is 74, a 1-fold exaggeration of the vertical. Lengths are drawn from the exponential the model predicts, mean 621 µm; the rules mark one, two and three millimetres with the count a hair-counting instrument reports at each, and the hairs crossing each rule in the drawing are the ones those counts are about. At the yarn's own surface the long hairs cover 1.1% of the space beside it, which is why the picture is mostly gap. Nothing here is the short population, which carries most of the protruding length and none of the reach; and a hair reaching past the room the canvas has is drawn to the edge of it, so the very longest few are shortened in the drawing and not in the arithmetic.$ What cloth is

A yarn's surface is a distribution

This collection has computed where a cloth stops, and every one of those numbers is a statement about yarn. What a finger, a plate, a droplet or a ray of light actually meets first is a population of fibre ends standing off the yarn — and it is a population, with a count and a length, rather than a layer with a thickness.

Compound and figured cloths

Hair, nap and pile are one construction

There are three surfaces made of fibre standing off a cloth, and they have been arrived at from three different directions and in three different fields. They are the same object at three settings, and what separates them is not what they are but how much of them anybody decided.

Twist is not torsion: a straight rod, twisted. A rod drawn with a cross painted along it, and the cross turned through 2 full turns from one end to the other. The rod is straight, so it has no Frenet frame at all — a straight line has no osculating plane to turn. The two discs above are the rod's ends seen down its own axis, which is where the twist is an angle rather than a foreshortened wiggle. The twist is the rotation of the painted cross about the rod's own tangent, and it is 2 turns whatever the centre line does. That is the quantity a torsional rigidity resists, and it is why a collection that solved a plane curve had not thereby dealt with torsion.

Pattern and colour

A shadow stripe is two twists

A cloth striped in one colour, where the stripe is visible only because alternate bands are spun the other way round. The pattern is carried entirely by which way the fibres lie on the yarn's surface, and it disappears when the light moves.

Where a muslin's warp jams, over 40 ends. 40 ends drawn at their own diameters, with every neighbouring pair's combined width plotted beneath. A cloth cannot be set closer than its threads will lie, and the pair that decides that is not the average pair — it is the widest one anywhere across the warp, which here is ends 8 and 9 at 205 µm apiece against a mean of 167 µm. Over the 2000 ends of a real warp rather than the 40 drawn here the worst pair is 42% above the mean, and it goes on growing with the width of the cloth: the same yarn in a wider loom jams sooner. The naive estimate that treats every window as an independent try overstates it by 0.48%, which is small enough to say that the overlap between neighbouring windows is not what is going on here.

Setting and geometry

A warp jams where its threads are thickest

The closest a cloth can be set is decided by its worst pair of neighbours, not its average thread — and the worst pair depends on how many pairs there are. The same yarn in a wider loom jams sooner, which makes a jammed sett a property of the machine as well as of the yarn.

$The two diameters of a 20 tex yarn. The pressure inside a twisted yarn is zero at its surface, so the outermost fibres are held by nothing but their own buried ends and some of them stand off as loops and ends. A yarn therefore has two diameters: a mass diameter of 167.1 µm, which is a volume divided by a length and is the one every other calculation on this site uses, and a contact diameter of 217.1 µm, which is what a neighbouring thread, a finger or an air stream meets. The gap is a hair layer of 25.0 µm on each side and it is measured, not computed — nothing here predicts hairiness. What is computed is the consequence, and it is 29.9% of the diameter every cover factor on this site was built from.$ After the loom

Singeing is the cheapest change to a surface

Pass a cloth through a flame fast enough and it loses under one per cent of its mass. What it loses is the part of itself that was doing most of the touching, and lustre, friction, printability, pilling and measured cover all move at once.

$A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument.$ What cloth is

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.

$20 tex, counted. The cross-section of a 20 tex cotton yarn, with every fibre in it drawn. The count is a division and nothing else: a 20 tex yarn spun from 0.17 tex fibre has 117.6 fibres crossing any plane through it, and the yarn is 14.0 fibre diameters across because n fibres packed at 0.6 fill a circle √(n/φ) times as wide. The arrangement is drawn on a lattice and is not claimed: real fibres are not on one, they migrate between the core and the surface as they run, and everything this collection says about a yarn's strength turns on their doing so.$ Setting and geometry

How many fibres make a thread

Every number in this collection began with a diameter, and a diameter is not a measurement — it is a count of fibres, divided. Once the division is written down, three quantities that had nothing to do with each other turn out to be the same number.

What a raising machine can catch in a 2/2 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

After the loom

Raising moves the surface onto the hairs

A raising machine pulls fibre ends up out of the floats, and what a finger or a light then meets is not the cloth's surface at all but a layer of loose fibre standing above it. The criterion for which cloths can be raised is the same criterion, exactly, that decides which cloths have crown line — so raising spends the surface that would have made the fabric shine.

Knits and other structures

A jersey gets taller before it gets shorter

Pull a knit along its courses and the first thing it does is grow along its wales — by two per cent, over the first eighty per cent of extension, before it turns round and contracts. The transverse response changes sign, and there is no material in the explanation at all.

The evenness a cotton yarn cannot be better than. Lay staple fibres down at random and count how many cross a plane: the count is Poisson, its variance is its mean, and the coefficient of variation of the mass per unit length is therefore 1/√n with no material and no machine in it. The lower curve is that floor. At 5 tex there are 29 fibres in the section and the floor is 19.86 per cent; at 5 tex there are 29 and it is 19.86. The upper curve is what a yarn spun at an index of irregularity of 1.35 actually measures, which is the floor times a constant — so the whole shape belongs to the counting and none of it to the spinning. The exponent is exactly −½ and is asserted as such rather than fitted to the curve.

Setting and geometry

A finer yarn is a worse yarn

Fineness is the thing a yarn is priced for, and it is bought with irregularity at a fixed exchange rate. Once the spread is a function of the count, every correction this collection computes becomes a function of the count too — and the cheapest yarn on the shelf is the one the arithmetic describes best.

Where the strain goes, at a 25° twist. Extend a twisted yarn and the fibres in it are not all strained alike. If each stays at its own radius — the affine model — a fibre at helix angle θ takes cos²θ of the yarn's strain, so the fibre on the axis takes all of it and the fibre at the surface takes 82.1% of it. The core reaches its breaking extension first and the yarn cannot realise its own fibres. If instead every fibre migrates between the core and the surface, every fibre has the same mean strain and they all break together. The two are exactly computable — 0.8214 against 0.9509 of what the fibres could give — and the ratio between them, 2/(cos α(1 + cos α)) = 1.158, is what migration is worth. The shading is that arithmetic and the two panels use the same scale.

Mechanics and drape

A straight fibre cannot share the load

Extend a twisted yarn and its fibres are not all strained alike: the one on the axis takes the whole of it and the one at the surface takes cos²α. So they do not break together, and what a wandering fibre is worth comes out as one expression with nothing fitted in it.

Z twist at 1600 turns per metre. A 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 40.0°, and it is the only quantity in this family: the yarn is 13.8% shorter than the fibre in it and carries 59% of the strength the same fibre would give lying straight.

Weaves

The other crepe is in the yarn

A crepe weave puts the texture in the matrix. A crepe yarn puts it nowhere the matrix can see: the cloth is a plain weave, and the surface comes from a thread twisted so hard that it shortens by a seventh and spends the rest of its life trying to untwist.

What it takes to bury a cloth's crowns. The film needed to fill a fabric's surface to a stated level, for plain, 2/2 twill, satin 8 in sheeting, at a film density of 1.2 g/cm³. Burying the crowns entirely takes 187 g/m² on the plain, 186 g/m² on the 2/2 twill, 183 g/m² on the satin 8 — and the ordering is not the ordering of roughness. A weave with plateaux presents a wide flat top that a thin film covers, and a weave with points presents crowns with valleys between them that the film has to fill before it is continuous anywhere. Every gram spent filling a valley is a gram that is not bridging a hole, which is where a coated cloth fails.

After the loom

A coating fills the crowns before it bridges the holes

A film does not sit on a cloth, it fills it — and the volume it has to supply to reach a level is the integral of one minus the bearing area. Burying an ordinary sheeting's crowns takes 185 grams a square metre, which is more than the cloth weighs, and the whole of the weave's influence is spent in the first ten of them.

At 0.1 kilopascals the plate is standing on hair. A flat foot pressed onto sheeting at 0.1 kPa, over 3 mm of cloth. It stops 25 µm above the cloth's own crowns, because that is where the hairs it is bending can carry the load: of the 10 hairs drawn, 9 reach higher than the foot and are laid over under it, and the rest are untouched. The vertical scale is set by the approach and not by the layer — 5188 pixels to the millimetre off the cloth against 192 along it — because the foot's height above the crowns is tens of micrometres and the layer it stands in is more than a millimetre, so a picture at one scale shows the second and not the first. The thickness reported is 0.432 mm against 0.382 mm for the cloth itself, so 12% of the reading is a population and not a fabric. The hair layer carries up to 0.23 kPa before the foot reaches the crowns at all. A laid-over hair is drawn as two straight segments where a real one is an elastica: the corner is a convenience and the height it turns at is the measurement. What the picture cannot show is that a bent hair leans on its neighbours, which the arithmetic behind it does not know either.

What cloth is

A light touch never reaches the crowns

This collection found that a plain weave touches at points and every other cloth touches along lines, and that the difference is an exponent rather than a factor. It is a real result about a real surface, and at the pressures a fabric is actually touched at, nothing ever reaches that surface.

A round thread reflects a beam into a fan. A round thread seen end-on, with a beam arriving from straight above. Every point across the thread has its own normal, tilted by the angle it sits at, and mirrors the beam through twice that angle — so a single direction in becomes a whole fan out, spread across the thread and not at all along it. Seen from the front that fan is a highlight lying along the thread, exactly as long as the length of thread that is straight, which is the float. Nothing about this depends on the fibre. A cylinder of any material returns a fan, and the only way to narrow it is to stop the section being a cylinder — which is what a calender does.

Weaves

A float reflects into a line

A cylinder cannot return a beam to a point. Its normals sweep the whole half-turn across it and nothing at all along it, so a straight thread throws light into a fan — seen as a highlight lying along the thread, exactly as long as the length of thread that is straight. The lobe of a satin is twice as narrow along the thread as across it; the lobe of a plain weave is exactly round.

A seam stands 763 µm proud of a cloth 382 µm thick. A 10 mm seam allowance of 3 plies in a 60 mm panel of one, in sheeting. The seam stands 763 µm above the body of the garment — which is 102 times the depth at which the body cloth first comes into contact with anything at all. So a flat surface rubbed across this garment touches only the seam, over 16.7% of the area drawn, until it has crushed a whole thickness of fabric. Everything this collection computes about where wear lands on a woven surface applies inside that 16.7%, and the other 83.3% is not being touched.

Cloth doing a job

A seam stands proud and wears first

A seam allowance is three plies where the garment is one, so it stands three quarters of a millimetre above a cloth whose own surface has a few micrometres of contact in it. Anything flat rubbed across the garment touches the seam and nothing else — all of the wear on two or three per cent of the area, until a whole thickness of fabric has been crushed.

One canopy, two opposite outcomes, decided by a sign. What a canopy does to a drop, for sheeting raised 32-fold. A rough surface multiplies the cosine of the intrinsic contact angle by its roughness ratio, which here is 2.0 — a hair is a cylinder and contributes πd of surface for every d of shadow. So a fibre that wets at all is driven to complete spreading, and one that does not is driven to a Cassie state sitting on 32.4% solid and air. The dashed diagonal is what the bare fibre would do; the canopy pushes every point away from ninety degrees, in whichever direction it already lay. Raising is therefore not a wetting treatment or a repellency treatment — it is an amplifier, and which one it turns out to be was settled by the chemistry before the raising machine was switched on. What the figure cannot show is which state a real drop reaches, because both are available near the hinge and the one it finds depends on how it arrived.

What cloth is

The hairs decide the sign of the wetting

Raising a cloth is not a wetting treatment and it is not a repellency treatment. It is an amplifier, and which of the two it turns out to be was settled in the dyehouse before the raising machine was switched on — by whether the fibre's own contact angle was above or below ninety degrees.

Why a cuff is ribbed. The force a knit pulls back with, over the range a cuff is used across. It rises the whole way — 0.79 N per metre at 23% to 3.38 at 104% — and it is small throughout, which is the combination a cuff needs and almost nothing else supplies. A rib gets its extension by geometry, folding alternate wales to opposite faces so that its relaxed width is about half its opened one, and it gets its recovery from the loop reconfiguring. Neither is the yarn stretching, which is why a cuff made of a fibre with no elastic recovery at all still works.

Knits and other structures

A rib pulls back on a force the loop supplies

A cuff has to give a great deal at almost no load and come back reliably, and no ordinary material does both. A rib gets its extension from folding, which is geometry, and its return from the loop reconfiguring, which is now a computable force — under four newtons a metre over the whole range a cuff works across.

Folding, and what it takes out of the singles. Two 20 tex singles spun at 800 turns per metre and folded the other way at 540 — a ratio of 0.675, which is the trade's own and is a measurement rather than a derivation. A single held at its ends and wound round its neighbour turns about its own axis once for every turn of the fold, so it is left with 260 turns per metre of its own: its surface fibres lie at 7.8° to its axis rather than the 22.8° they were spun at. Folding untwists. The short strokes are drawn at that residual angle; the two long curves are centre lines and are not the yarn — each strand is itself a bundle of 118 fibres, and the residual angle is what holds them.

Setting and geometry

Folding is untwisting

Wind two singles round each other and each one turns about its own axis once per turn of the fold. So a folded yarn's singles are not the singles that went into it, and at a folding ratio of exactly one half the two helices cancel.

How much every four-by-four draft can shine. All 22,874 four-by-four drafts in which every end and every pick interlaces, at sheeting's construction and a tolerance of 2°, counted by specular area. The range runs from 0.02% to 0.93%, a factor of 60.0, and the distribution is not smooth — it clusters, because the quantity behind it is a count of whole crossings and takes only certain values. The dullest drafts in the catalogue are the plain weaves, which have no plateau at all and shine only from the crowns of their turns; the brightest carry the most float on the face, with the fewest turns interrupting it. Lustre over the catalogue is a length census, and nothing about the yarn enters it.

Weaves

Lustre is a length times a width

The specular area of a cloth factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply. Neither factor knows anything about the other, and over the four-by-four catalogue the first alone spans a factor of sixty.

A hair layer is a balance, so singeing does not stay done. The hair population of a 20 tex cotton yarn under rubbing, started from a singed cloth and from an unusually fuzzy one. Abrasion does two opposite things: it frees ends that spinning left buried, from a supply of 2.23 per millimetre in the surface shell, and it removes hairs that are long enough to be caught. Where the two meet is a fixed point at 1.43 per millimetre, and the cloth goes there from either side with the same time constant — 248 cycles to halve the distance, whichever direction it is travelling. A singeing is therefore undone in a few hundred rubs, because the flame changed the stock and not the balance. What is predicted here is that a fixed point exists, that it does not remember the starting state, and that one rate serves both directions; where it sits relative to the spun level needs two rates the model does not supply, and it is set to reproduce the one thing everyone has noticed, which is that fabrics get fuzzier as they are worn.

After the loom

A hair layer is a balance, not a stock

Singeing takes under one per cent of a cloth's mass and changes its lustre, its friction, its printability and its pilling. It also does not stay done, because rubbing frees fibre ends as fast as it breaks them off, and a flame changes the stock while leaving the balance exactly where it was.

Cloth doing a job

Abrasion takes the hairs first

An abrasion test reports milligrams lost against cycles, and the first milligrams off any fabric are not fabric. On a bare cloth that stage is over in a few thousand cycles. On a napped one it is not over by the end of the test, so a Martindale on a fleece never measures the fleece.

A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

Knits and other structures

A rib climbs a gap

A jersey's yarn crosses one diameter between interlacings because that is what a crossing of two threads is. A rib's crosses the whole distance between the two beds. Nothing else in the model changes, and that one length is the whole mechanical difference between the fabrics.

The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.

What cloth is

What a thickness gauge reads on a knit

The structure says two yarn diameters and the gauge says more, and the gap is not an error in either. A gauge lands on the highest crowns, through a canopy of protruding fibre, under a load that has already begun to compress both — and it does that on a surface that is nothing but crowns.

Setting and geometry

A two-fold yarn is not twice a single

Folding halves nothing. It improves a yarn's evenness by exactly √2 and lowers the floor that evenness is measured against by exactly √2, so the index of irregularity comes out identical — folding does not make a better yarn, it makes a bigger one.

A calender multiplies the highlight by 32, and all of it is width. A 2/2 twill in sheeting pressed at increasing force, with the specular area recomputed at each state from the standard compression model. It rises from 0.91% of the plan to 28.9%, a factor of 32, while the cloth thins from 381.6 µm to 186.1 µm. The gain is not in the length of the crowns: that moves by 5 per cent. It is in their width, which moves by 31.8 times, because pressing puts a flat top on the section and a flat top has one normal rather than a fan of them. The finish does not polish the thread. It changes the dimension of the highlight, from a line to a band, and the arithmetic says so by refusing to put any of the gain in the other factor.

Weaves

A calender buys the width

Press a cloth and its lustre multiplies by twenty-four. None of that comes from the length of its crowns, which moves by six per cent; all of it comes from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refuses to put any of the gain in the other factor.

The strong fibre is the one that pills. Standing pills per unit area by fibre, relative to wool, at one and the same fuzz supply — every row is the same cloth raised the same amount, so the only thing varying is how long a pill survives once it exists. A pill is not made, it is kept: rubbing generates it and rubbing breaks the fibres that tie it down, and a tie survives in proportion to how much force it takes to break. So polyester carries 18 times wool's standing population from the same generation rate, and the ordering here is exactly the ordering of tenacity and nothing else. Wool sheds its pills because wool's ties break. No two real fabrics have the same fuzz supply, which is why a wool knit still pills more than a cotton shirting in practice — the comparison drawn here isolates the tie and says nothing about the generation, and reading it as a ranking of fabrics would be wrong.

Cloth doing a job

A pill is anchored, not made

Every account of pilling starts with how a ball of fibre forms and stops there, which explains why fabrics pill and not why some of them stay pilled. A pill is not a thing that happens; it is a standing population, and the number on a fabric at any moment is a generation rate times a lifetime.

$A print is as sharp as the hairs are long. How far ink carried on a hair reaches past a printed edge into the unprinted cloth, for sheeting in three states. A hair lying near the edge bridges as far as its own length, and the number bridging at least a distance x is (n_A λ/2)e^(−x/λ) — an exponential with the population's own decay length — so the visible feather is a quantile rather than a mean, taken here at one hair per 50 millimetres of edge. As woven the feather is 1911 µm, which is a fifteenth of an inch and coarser than any screen worth engraving: the cloth cannot hold better than 7 lines to the inch whatever the printer does. Singeing caps it at the flame's own reach of 200 µm and takes the cloth to 63 lines — a factor of 10, bought by burning off a fraction of one per cent of the cloth's mass. That is why singeing comes before printing and why nobody prints a fine figure on a raised cloth.$ After the loom

A print is as sharp as the hairs are long

Singeing comes before printing in every finishing route ever written down and the reason given is that the cloth must be smooth. The reason is sharper than that: ink carried on a fibre end reaches as far as the fibre is long, so the feather on a printed edge is a quantile of a hair population and nothing else.

Knits and other structures

Where a two-bed fabric's yarn is

Thirteen named structures, and for each of them the share of its yarn that crosses between the beds — read off its own traverse rather than quoted. It separates the fabrics into three groups, and one of the groups turns out not to be a fabric at all.

The contact force turns as the climb grows. The two components of the contact force against the climb, for a 20 tex cotton jersey at a 3.5 mm loop. The force along the wales is what friction has to hold and the force through the thickness is what holds the fabric open, and the second is bought at the expense of the first. At a jersey's own climb of one diameter they are 37.50 mN and 7.81 mN; at four diameters, which is a rib on an open gap, they are 22.39 mN and 18.61 mN. The friction balance is the ratio: friction has the whole force to work with and only the along-the-wales part to hold, so the coefficient a relaxed knit would need falls from a half to 0.490.

What cloth is

Every fabric's thread lies in a plane

A woven thread's crimp wave lies in a plane at right angles to the cloth. A knitted loop lies in a plane twelve degrees off it. Both halves of this collection turn out to be one picture with one angle in it, and the angle decides how much of a fabric's contact force acts through its thickness.

A designed thin place does not get worse and an accidental one does. The thinnest place a yarn reaches, against how many gauge lengths of it are tested, for a slub yarn and for a randomly uneven yarn of the same 36.3% coefficient of variation. The slub's floor is its base count — 89% of its mean — and it is a horizontal line, because a designed variation has a stated minimum and never goes below it. The random yarn's minimum is an order statistic and falls without limit: 56% over 10 lengths and 22% over 30000. So the advantage is not a number but a function of how much yarn is being asked about, running from 1.6× to 4.0×. What the curve cannot show is the break itself: a yarn's strength at a thin place is not proportional to its linear density there, and the conversion needs a fibre model.

Compound and figured cloths

A designed thin place is kinder than an accidental one

A slub yarn and a badly spun one can carry exactly the same coefficient of variation, and the number tells a mill nothing about which it has. The designed variation has a floor — its base count, and it never goes below it — while the accidental one has a tail that falls further the more yarn is tested. At 36% CV the slub bottoms at 89% of its mean and the random yarn reaches 24%, and the gap widens from 1.6 to 3.7 times as the test grows from ten gauge lengths to ten thousand.

One yarn, two packing factors. The same 20 tex cotton yarn — the same fibres, the same count, the same mass per metre — drawn at a packing factor of 0.45 and of 0.75. Its diameter is 192.9 µm in one and 149.5 µm in the other, a difference of 29.1%, because a diameter goes as the inverse square root of the packing. Every cover factor, every jammed sett and every hole in this collection went through that number, and the site's value of 0.6 was obtained by inverting a rule published for cotton yarns at one particular twist. Nothing here models how packing moves with twist; the figure is here to show the size of the thing that has been held constant.

Setting and geometry

The diameter was quoted at one twist

Every diameter in this collection came from a packing factor of 0.6, and that number was got by inverting a rule published for cotton yarns at one particular twist. Here is what moves if it is wrong by the width of the range real yarns occupy — and which single quantity does not move at all.

Turn the cloth and the highlight changes hands. A 2/2 twill in sheeting turned under a light, with the specular area of each system counted separately at each angle. The warp peaks at 0° and the weft at 83°, a quarter turn apart, and neither returns anything worth seeing where the other peaks. The reason needs no dye and no interference: a warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. A cloth woven with one colour in the warp and another in the weft therefore shows one colour at one angle and the other a quarter turn away, which is the whole of shot silk — a geometric effect that has been sold as a mysterious one for three hundred years.

Weaves

Turn the cloth and the shine changes hands

A warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. Turn the cloth a quarter turn and the weft takes over. That is the whole of shot silk — a geometric effect with no dye that changes and no interference in it.

A fifth of a strong fibre buys most of its pilling. How long a pill survives on a wool fabric as nylon is blended into it, relative to the pure wool. A pill is held by several anchor fibres and survives while any of them holds, so its life is set by the strongest anchor it happens to have — and the chance that a pill with 8 anchors has at least one strong one is 1 − (1 − x)^8, which is already 83% at a fifth. The blend therefore gets 83% of the pure strong fibre's pill life while keeping the whole of the weak fibre's fuzz supply, which is the worst of both. Nine tenths of the way arrives by 30%. No average of the two fibres' properties produces this curve: it is a maximum over a small sample, and a maximum is not an average. It is also why a fifteen-per-cent polyamide in a wool knit is notorious, and the arithmetic says the reputation is deserved.

Cloth doing a job

The strong fibre is the one that pills

A pill survives while any one of its anchors holds, which is a maximum over a small sample rather than an average — and a maximum behaves nothing like an average. A fifth of a strong fibre in a blend buys four fifths of the pure strong fibre's pill life while leaving the whole of the weak fibre's fuzz supply in place.

What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

After the loom

A loop is set and not sprung

The way out of a model that predicts a jersey should spread is to stop treating its yarn as a straight rod bent into a loop. A yarn that has been wetted, heated and dried has taken the loop as its own natural shape — and once the natural shape is the loop, every force downstream becomes computable with the relaxed fabric as the origin.

A bouclé at an overfeed of 0.80. A core of 20 tex running straight, an effect thread of 30 tex delivered 80 per cent faster than it, and a binder of 15 tex over the top. The surplus effect thread has nowhere to go but into loops, and their size is not a free choice: a semicircular loop of radius r consumes r(π − 2) of surplus, so at one loop every 3 mm the radius is 2.10 mm — 0.70 of the spacing. The yarn's resultant count is 89 tex and the sum of its three components is 65, because the effect enters multiplied by its own overfeed. What the drawing cannot show is the load path: the core and the binder are at the yarn's own length and the effect thread is longer than the yarn, so a pull on the yarn is carried by 39 per cent of its mass and the loops stay slack.

Compound and figured cloths

A fancy yarn has its crimp in the wrong thread

A bouclé is made by feeding an effect thread eighty per cent faster than the core it wraps, so the yarn contains more thread than it is long — which is crimp by this collection's own definition. A cloth's crimp is in the thread that carries the load, so removing it is the cloth's first extension. A bouclé's surplus is in a thread that carries nothing, so pulling the yarn stretches the core at once and the loops never straighten. Thirty-nine per cent of the yarn's mass is on the load path, and the other sixty-one is decoration.

Setting and geometry

The sett owns the pitch and the yarn owns the height

Set a cloth twice as close and its surface does not get smoother. The crowns come twice as often, because that is what a sett is, and they stand at very nearly the same height, because the closure condition pins the amplitude to the yarn — so a fine cloth is finer-grained rather than flatter, and the two are confused in every description of handle.

A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

Mechanics and drape

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

Thirty micrometres is a buckling load. The wool fibre diameter at which a protruding end reaches the measured threshold for prickle, against how far it protrudes. A fibre end pressed against skin is a column held at the cloth and free to slide at its tip, so it buckles at 20.19EI/ℓ² and carries no more load than that; below the threshold it bends away and is felt as touch, above it the load stands and is felt as pain. At a two-millimetre protrusion — which this site's own hair model puts in the top fifth of a wool's population — the threshold diameter is 32 µm, bracketed at 29–34 by the range of the measured force. Neither the thirty micrometres nor the two millimetres was put in. The band is the force bracket, and the fourth power in I = πd⁴/64 is what makes the boundary sharp: at a mean of 21 µm and a spread of 24%, 3.2% of the fibres are over it, and it is that few per cent that decides whether a garment can be worn.

Cloth doing a job

Prickle is a buckling load

The wool trade specifies comfort against skin by the percentage of fibres coarser than thirty micrometres, and the thirty is a measured boundary with no derivation attached. It is a column formula: solve for the diameter at which a protruding fibre end stops bending away and starts standing its ground, and thirty micrometres falls out.

A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns.

After the loom

Two knits with one tightness factor are one knit

The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does.

Knits and other structures

Which knitted fabrics lie flat

Curl was explained here by counting face changes between courses, which works for stockinette and garter and reaches nothing else. The same question turns out to be a signed sum over a structure's own grid — and it answers for every fabric a two-bed machine can make, including the ones nobody has a rule for.

Two courses as centre lines, and their closest approach. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, as centre lines, with the closest approach marked. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

What cloth is

The closest approach is not the crossing

Two wavy curves that touch at a point are not necessarily closest at that point. Whether they are depends on one thing: whether they run alongside one another or cross. That distinction decides which of this collection's two fabrics fits together and which does not.

Seeing through a white voile, from the street and from the room. How much of a scene's contrast survives a white voile, whose open area is 49%, looked through from the street into the room and from the room out to the street, against how much brighter the street is than the room. The scene comes through the clear lines of sight and nothing else; everywhere else the viewer sees thread lit from the viewer's own side, which returns light with no image in it. On a bright day, a hundred times brighter outside, 0.5% of the room's contrast reaches the street and 65% of the street's reaches the room. At equal light both are 28% — the curves cross at a ratio of one whatever the cloth — and with the lamps on after dark the room is the side on show, at 63%. What the plot cannot show is the threads' own optics: their reflectance and transmittance are assumed values for a white sheer rather than measurements, and only the crossing point is independent of them.

Pattern and colour

A sheer hides whichever side is darker

A net curtain hides a room by day and shows it at night, and the cloth has nothing to do with which. What a viewer sees through a sheer is an image through its clear lines of sight against a veil of lit thread, and the only thing that decides their balance is how much brighter one side is than the other. At equal light the two views are identical for every cloth there is. And a black sheer of the same openness shows twelve times more of the room by day than a white one.

Setting and geometry

Hairiness goes as the root of the count

A coarse yarn is hairier than a fine one and everybody knows it. What nobody has said is that its hairs are no longer — the count and the length obey different laws, one rises as a square root and the other does not move at all, and the identity behind both was asserted on this site for an entirely unrelated reason.

Warmth is a canopy, and a canopy grows as a logarithm. Thermal resistance of sheeting against how much its hair population has been multiplied by raising, both faces counted. The unraised cloth is given no still air at all, because its hairs cannot reach one another — n_A λ² is under one and there is no canopy to hold air still. Past the threshold the canopy's depth is λ·ln(n_A λ²), so every doubling of the hair population adds the same depth of nap and no more: the steps here are 434 µm apiece, all the way up. At 256× the nap is 3.07 mm deep and worth 39 times the cloth it grows on, which is the whole reason a flannel is warm and a poplin of the same yarn is not — a canopy is two parts in ten thousand fibre, so its conductivity is air's, while the cloth itself is a quarter fibre. What is claimed is a conduction resistance across a depth of nearly still air; whether the air is still is a question about flow and is not asked here.

Mechanics and drape

Warmth is mostly the hairs

This collection established that a fabric's thermal resistance is its thickness and not its fibre, and that twice the thickness is twice the warmth. It never asked what the thickness was made of. On a raised cloth almost none of it is cloth.

A shot effect needs a fibre with no ends. The peak-to-trough contrast of an eight-end satin in sheeting as the cloth is turned in the light, against how much hair stands on it. Bare, the contrast is 37 to one, because a straight thread's normals lie in the plane across it and the warp and the weft therefore reflect a quarter turn apart. A hair layer does two things and only one of them matters: it blocks, which takes the same factor off the peak and the trough and changes no contrast at all, and it returns light of its own, which is added to both. A hair population points every way at once, so its return has no azimuth in it — and adding a constant to both ends of a ratio of 37 destroys the ratio. On an ordinary spun cotton the contrast is already down to 5.1 to one; singeing recovers it to 29; raising kills it outright at 1.00. The one fibre with no staple length is the one fibre with no fibre ends, and every shot fabric ever woven is made of one.

Weaves

A hair layer veils a highlight

An eight-end satin's shine swings by a large factor as the cloth is turned, because a straight thread's normals lie in the plane across it. Put fibre ends on it and the swing disappears — not because the hairs block the light, which changes no contrast at all, but because they return light of their own that has no direction in it.

$A woven filter catches what its rating says it cannot. What fraction of a particle stream is intercepted by the hair layer of a filter cloth, against particle size, at three levels of raising. The cloth's own largest opening is 290 µm, so by geometry it stops nothing smaller than that at all — and the hairs catch a few per cent of particles ten and a hundred times finer, because a particle whose path passes within its own radius of a hair touches it. On a bare cloth the numbers are small; the point is that they are not zero, because a cake grows from the particles that stop, and once a cake exists the cloth is no longer doing the filtering. Raising the same cloth 32-fold takes a ten-micrometre capture from 0.8% to 23%, which is why a napped filter cloth exists. Interception is taken as the bare geometric ratio of the two diameters with no flow model behind it, so every number here is a lower bound.$ Cloth doing a job

A woven filter beats its own rating

A filter cloth's rating comes from the largest channel through it, and by geometry it stops nothing smaller. It stops a few per cent of particles ten times smaller anyway, on the fibre ends standing in its holes — and a few per cent is not filtration. It is exactly enough to start a cake, and after that the cloth is not filtering.

Knits and other structures

A tube and an interlock balance for different reasons

Two structures come out identically flat on a signed count and are as unalike as two knitted fabrics get. One is two jerseys that curl in opposite directions and are joined at the edges; the other is a fabric whose every course crosses. Telling them apart needs a different question asked of the same grid.

What cloth is

Where a yarn is thinnest

A yarn in a fabric is pressed where it crosses and free where it does not, so its section changes along its own length. Every flattening this collection has ever quoted is a single number for a profile that runs from four fifths of a diameter to nearly four.

Compacting a spinning triangle moves one instrument and not the other. What happens to each hairiness reading when a 20 tex cotton yarn is spun compact instead of ring, as a percentage of the ring value. The total falls by 8% and the long hairs by 65%, a ratio of 8.3. The asymmetry is a prediction rather than a fit. Compaction removes ends that were unbound over a long stretch of the spinning triangle, which is the long population and nothing else; the short population is untouched, and it carries about 88% of the length the integrating instrument is adding up. So the instrument that sees everything barely moves and the one that sees only the tail collapses. The model under-states the fall in the total, because compaction certainly does something to the short population too and nothing here models it — the direction of that error is stated and it is the conservative one.

Setting and geometry

The spinning triangle decides the hair

A ring frame converges a flat ribbon of fibres to a round yarn, and for the length of that convergence the fibres at the ribbon's edges are held by nothing. Everything a spinner can do about hairiness is done in that triangle, and the two hairiness instruments respond to it by wildly different amounts.

Knits and other structures

How thick a knit is

Two centre lines pass at a diameter and each has a radius on either side, so a plain jersey is two yarn diameters thick with nothing fitted. It does not depend on the gauge, it lands inside the band of this collection's woven cloths, and it is lower than any gauge will read.

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.

After the loom

A state is a thickness too

This collection's rule is that a fabric dimension quoted without its relaxation state is not a measurement. A knitted fabric has three dimensions and only two of them obey the rule: its thickness is the same in every state, because the interlacing that sets it does not relax.

Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.00, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not.

Mechanics and drape

A thread between two crossings is an elastica

Every crossing force Peirce's thread path can give comes from a tension, and a cloth on a table has none. What presses its threads together there is their own unwillingness to be bent — and recovering that needs a shape nobody had, because a path assembled from an arc and a straight line has a bending moment that jumps.

What the third dimension changes, and by how much. Every number the planar loop model produced, beside the same number with the climb in it, for a 20 tex cotton jersey at a 3.5 mm loop. Four of the five fall and none moves by as much as four per cent, which is the useful part of the answer: the planar model was not wrong about a jersey, it was a projection of the right curve. What it could not have at all is the quantity that is not on this list — the force through the fabric's thickness, 7.81 mN a stitch, which a model with no thickness has nowhere to put.

After the loom

The constants say nothing about thickness

Munden's two constants give a knitted fabric's wale and course spacings from its loop length alone, and the tightness factor collapses every fabric's shape onto one curve. Neither reaches the third dimension: two knits that are one knit in plan are two different thicknesses.

Weaves

A woven cloth asked the same question

A knitted fabric's two adjacent courses occupy the same space by a fifth of a diameter. A woven cloth's two systems overlap by nothing at all in an open cloth and by four and a half per cent in a dense one — and the difference is that they cross rather than run alongside.

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