Textile Structure
how cloth is put together
Theme

The thread: A mechanism, not a material

Cloth on the bias stretches half as long again while no thread in it stretches at all. The deformation is kinematic — a trellis closing — and almost every explanation given for it is wrong.
A trellis sheared 30°. The net at an angle, with every thread segment exactly the length it started at. The extension along the diagonal is the bias stretch, and it is a change of shape rather than a change of length. Mechanics and drape

The bias is a mechanism

Cloth cut at forty-five degrees stretches by a third and springs back, while the threads in it stretch by nothing at all. Almost every explanation given for this is wrong, and the right one is not about elasticity.

 The knitted loop. One thread, bent into a course of loops, each of them drawn through the loop below. Nothing here is straight, which is why a knit extends in every direction while a woven cloth extends only on the bias. Knits and other structures

The loop

A knit is one thread bent into loops, each drawn through the loop below. Nothing in it is straight, which is why it stretches in every direction while a woven cloth stretches only at an angle.

 The same thread, on the loom and off it. One warp end in section over eight picks, drawn twice at the same scale. The thread is the same length in both panels; in the lower one more of that length is in the bends, so the cloth it spans is shorter. Nothing has been lost — the length has moved into the crimp. The thread's thickness and the height of its bends are drawn larger than scale so the interlacing is legible; the two panels' spacings are not. After the loom

What comes off the loom is not the cloth

Every number on this site so far describes a fabric in one particular condition — held under tension, stretched to the reed width, never wetted — and none of them says so. That condition lasts until the cloth is taken off the machine, which is the first thing that happens to it.

 Pull it lengthways and it narrows. The same cloth before and after a small extension along the warp. No thread has stretched: the extension came out of the warp crimp, that crimp went into the weft, and the fabric is narrower for it. Setting and geometry

Crimp, and why cloth narrows when it is pulled

A thread in cloth is longer than the cloth it crosses. Pull the fabric one way and that extra length is taken out of one direction and put into the other, so the cloth gets narrower without a single fibre stretching.

 The same thread, on the loom and off it. One warp end in section over eight picks, drawn twice at the same scale. The thread is the same length in both panels; in the lower one more of that length is in the bends, so the cloth it spans is shorter. Nothing has been lost — the length has moved into the crimp. The thread's thickness and the height of its bends are drawn larger than scale so the interlacing is legible; the two panels' spacings are not. After the loom

Relaxation is the crimp coming back

A cloth that shrinks in the wash has not lost any thread. The thread is exactly as long as it was; more of its length is now spent going up and down rather than along, and the difference is the shrinkage, computable to the last figure from the woven geometry alone.

 How far the bias goes, and where it stops. Extension along the bias against shear angle, with the angle at which the threads jam marked for three settings. The curve is geometry and so is the wall — a more closely set cloth reaches it sooner. Mechanics and drape

The locking angle

The bias runs out at an angle that yarn diameter and thread spacing decide between them. It is the number behind whether a cloth will go round a curve, and it has nothing to do with how strong the fabric is.

 Why one curls and the other does not. A knitted loop is not symmetric front to back. Worked every course the same way, the asymmetries add along the edges and the fabric rolls; worked alternately, consecutive courses point opposite ways and cancel. Knits and other structures

Why stockinette curls

A knitted loop is not symmetric front to back. Work every course the same way and the asymmetries add up along the edges; alternate them and they cancel. That is the whole difference between a fabric that rolls and one that lies flat.

 A cloth laid over a sphere. Every thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes. Mechanics and drape

Why clothes need darts

A flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. The amount of shear is decided by the curvature, and when it exceeds what the threads allow, something has to be cut out.

 How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it. After the loom

Why the warp shrinks more

A woven cloth almost always loses more length than width, and the reason is not in the cloth. It is in the machine — the warp is held under tension for the whole of weaving and the weft is held for a fraction of a second — so the two systems arrive at the finishing works with different amounts of crimp missing.

 The cantilever test. A strip of cloth pushed out over an edge until its tip has drooped to the stated angle. The overhang at that moment gives the bending length, and cubing it with the mass per unit area gives the flexural rigidity. Mechanics and drape

Bending stiffness and the drape coefficient

Two standard measurements try to say how a fabric hangs. One measures a length and cubes it; the other measures an area and, on the geometry, turns out to be answering a different question from the one it is asked.

 The tricot lapping. A warp-knit lapping drawn from the guide bar's movement. Each thread is coloured by the group of wales it belongs to, so a lapping that leaves the wales independent shows as several colours. This one joins them into 1 group. Knits and other structures

Warp knitting, which is a different thing entirely

Every wale has its own thread, and if the thread never leaves its wale the fabric is a set of independent cords. Whether a lapping makes cloth is decided by a coprimality condition — the satin theorem, in a knit.

 What a pre-shrunk label promises. Three lengths of the same cloth: as woven, as it leaves the compressive-shrinkage machine, and where it will finally settle. The residual shrinkage quoted on a label is measured against the second of these and the total against the first, so the two numbers are not the same quantity and cannot be subtracted. After the loom

Pre-shrinking is a subtraction done in advance

A compressive-shrinkage machine takes four per cent out of a cloth before anybody buys it, and the label then quotes what is left. The two numbers are fractions of different lengths, so they cannot be subtracted — and the difference between doing that correctly and incorrectly is most of the number.

 1×1 rib — alternate wales to the back. A knitted fabric seen in section across the wales. Alternate wales pulled to the back fold the fabric like a concertina, so its relaxed width is a projection; pulling it wide unfolds the section and no yarn changes length while it happens. Knits and other structures

Rib and interlock

A rib fabric is a plain knit folded like a concertina, and its enormous widthwise stretch is the fold opening out. Nothing in it is elastic, and the extension available is a cosine.

 A cloth laid over a sphere. Every thread segment is exactly one pitch and none has stretched. What has changed is the angle at each crossing, and the amount is decided by the surface — a developable one costs nothing and a curved one costs more the further the cloth goes. Mechanics and drape

Shear locking in a composite preform

Laying a woven reinforcement over a mould is the bias mechanism doing useful work, and it stops dead at the angle where the threads jam. Where the cloth wrinkles is a geometric prediction with a radius attached.

 What the drape coefficient answers to. The drape coefficient against the number of folds, and against how far the hem has come in. Over the range a real specimen shows, the fold count barely moves it; the hem radius moves it across almost its whole range. Mechanics and drape

A drape coefficient is one number for a directional thing

A fabric bends more easily one way than the other — a factor of two is ordinary. The drape test reports a single percentage, and the quantity that carries the directionality is the fold count, which the coefficient is almost blind to.

 The same thread, on the loom and off it. One warp end in section over eight picks, drawn twice at the same scale. The thread is the same length in both panels; in the lower one more of that length is in the bends, so the cloth it spans is shorter. Nothing has been lost — the length has moved into the crimp. The thread's thickness and the height of its bends are drawn larger than scale so the interlacing is legible; the two panels' spacings are not. After the loom

The cloth gains weight by losing size

Mass per unit area is the number a fabric is bought by, and it rises by eleven per cent when a cloth relaxes — with nothing added, nothing removed and no thread changed. The quantity is a ratio, and finishing moves its denominator.

 Four ways a fabric gets longer without stretching. Extension available from each mechanism, computed from the geometry that provides it. None of these numbers involves a yarn changing length; every one of them is a shape changing, and they differ by an order of magnitude. Knits and other structures

Why a knit recovers and a woven does not

Four fabrics get longer without a single yarn stretching, and the four mechanisms are worth wildly different amounts. Three of them are recoverable and one is very nearly not, and which is which follows from where the extension came from.

 Wrinkles 127 mm apart. A 300 mm width of a 120 g/m² cloth whose bending length is 20 mm, held under 5 newtons per metre across it and compressed. It cannot carry the compression in the plane, so it leaves the plane, at a wavelength the bending rigidity and the tension settle between them: 127 mm, which is 2.4 wrinkles across the width. The amplitude is drawn and is not computed — this arithmetic sets the spacing and says nothing about the depth. Mechanics and drape

A cloth cannot carry a push

The net model that runs this site's mechanics has no bending stiffness at all, so it buckles under any compression whatever, into wrinkles of any wavelength whatever. What picks the wavelength is the competition the net leaves out — and the answer is a quarter power, which is why a wrinkle is so hard to change.

 What a tuft's wrap is worth at μ = 0.3. The two classical ways of binding a cut pile into its ground, with the holding force each provides as a multiple of the tension applied to the free end. The model is the capstan equation and the friction coefficient is measured rather than derived, so the ratio between the two matters and neither absolute number should be quoted alone. Compound and figured cloths

How a tuft is held

A V-fastened tuft and a W-fastened tuft are both attached, and the integrity criterion returns the same verdict for both. One of them is specified for hotel corridors and the other sheds. What separates them is not a topological property at all — it is an angle, and it needs a coefficient somebody has to measure.

 The ratchet a wool fibre is. A fibre with its scales, and the two strokes of one cycle of agitation. The push is the same in both directions; the distance is not, because the scales resist tip-first motion more than root-first. Every cycle therefore nets a displacement in one direction, and no amount of further agitation undoes it. After the loom

The ratchet that makes wool felt

A wool fibre is covered in scales pointing one way, so it slides more easily root-first than tip-first. Agitate it and the motion is symmetric while the result is not — every cycle nets a displacement in one direction, and no amount of further agitation undoes it.

 A beat at 19.1 mm from grids at 0.5 mm. Two grids at 0.5 and 0.5 mm pitch, the second turned by 1.5°, over a 30 mm window. The dashed rules are one predicted beat period apart. The pattern between them is 38 times the pitch of either grid and neither grid has anything at that scale. Pattern and colour

Watered silk is a beat

Fold a ribbed cloth on itself and press it, and a figure appears at a scale neither ply has — fifty times the rib pitch, wandering across the piece. It is the difference of two wavevectors, it is enormous because the angle is tiny, and no two pieces match because no two are folded at the same angle.

 What a balanced cloth wastes under pressure. The fraction of a balanced fabric's fibre that is along for the ride, in a stress field of each ratio. A closed cylinder is exactly two to one — the ratio of the two areas the pressure acts on — so a balanced cloth reaches its limit around the circumference with the axial system at half its capacity, and a quarter of the fibre is doing nothing. Cloth doing a job

An inflated cylinder wants an unbalanced cloth

Balance is a virtue in almost every other cloth. Under pressure it is a defect with a size — a closed cylinder carries exactly twice the stress around its circumference as along its axis, so a balanced fabric reaches its limit in one direction with a quarter of its fibre doing nothing at all.

 Corduroy: floats of 6 ends, cut. 3 wales of an extra weft floating over 6 warp ends each, cut at their midpoints so that each half stands away from the ground. The pile height is half a float's length and nothing else decides it, so a finer sett at the same float count gives a shorter pile; the wale spacing is the float plus its binding ends. Compound and figured cloths

Corduroy is a cut float

An extra weft floats over some ends, a knife runs along the cloth, and each half of the float stands up. So the pile height is exactly half the float's length — which makes corduroy the one pile fabric whose surface is decided entirely by a quantity this site has been counting since its first essay.

 Milling, on the nonwoven's own scale. Migrated fibre accumulating with agitation, measured against the stick-percolation threshold this site uses for nonwovens. Above the line the fibre network holds on its own, which is why a milled cloth can be cut without fraying — the weave is no longer the only thing keeping it together. After the loom

Milling holds the cloth a second time

This site's integrity criterion returns exactly the same answer for a melton as for the loose twill it was woven as, and it is right both times. What has changed is that a second network now holds the fabric together, made of migrated fibre rather than of thread crossings — and it is the one that decides whether a cut edge frays.

 The angle a pressurised hose wants. A fixed length of yarn wound on a cylinder at a stated angle, with the radius and length the geometry gives, beside the volume it encloses as the angle varies. The maximum is at arctan √2 — 54.74° — with no material constant anywhere in it, and the same angle comes out of balancing the hoop and axial stresses, which is a different calculation with the same answer. Cloth doing a job

The angle a hose wants

A braided hose has one angle at which pressure neither lengthens it nor shortens it, and the angle is arctan √2 — 54.74° — with no friction coefficient, no modulus and no fitted constant in it. Two arguments that share no algebra arrive at the same number, and which side of it a hose was braided on decides which way it moves.

 The number the drape test does not record. A 150 mm specimen over a 90 mm pedestal, for fabrics from limp to stiff. The curve is the fold count the buckling argument predicts — three quarters of a power of the specimen's radius over its own bending length — and it runs from 2 folds to 11. The number beside each mark is the drape coefficient the same specimen would report, which moves by a few points across the whole range. Mechanics and drape

The nodes a drape test throws away

A drape test lays a circular specimen over a pedestal, photographs the shadow and reports one number. The specimen also falls into a definite number of folds, which is a buckling mode set by the fabric's own bending length — and the standard method observes it, does not record it, and reports the number it is least sensitive to.

 Which dentings leave a mark. Every combination of ends per dent and weave repeat, with how many ends the grouping takes to come back into step. A small number means the reed treats every repeat the same way and the grouping shows as a stripe at the dent pitch; a large one means the grouping walks across the weave and there is nothing periodic for the eye to find. The rule is one word: dent so the two share no factor. Pattern and colour

The reed leaves its own mark

A reed does not space a warp evenly. It groups it, several ends to a dent, and the grouping beats against the weave repeat — so a denting that shares a factor with the repeat treats every repeat identically and shows as a stripe, and one that does not is invisible.

 Everywhere a sheeting can go. Every state a sheeting of 28 × 26 threads per centimetre in 25 and 25 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 4.03 per cent of extension is available along the warp, and reaching it costs 6.98 per cent of the width. Mechanics and drape

A cloth extends by moving its crimp

Everybody says the extension available along the warp is the warp's own crimp. On six of eight ordinary cloths it is not — a close sheeting has 14.61 per cent of warp crimp and reaches 4.03 per cent, because the limit lives in the weft.

 A membrane is cut smaller than it is. Warp and fill compensation against how far the crossings flatten under prestress. With the thread lengths and the cloth's thickness both fixed there is no strain available at all — the closure condition determines the state — so the whole of the compensation is the crossings squeezing and the crimp coming out. The direction that arrives with less crimp gives back less, which is why a warp compensates less than a fill. Cloth doing a job

A membrane is cut smaller than it is

A tensioned fabric roof is cut to a pattern smaller than the shape it will take, because stressing it makes it grow. At a fixed cloth thickness a prestress can only interchange crimp — one direction grows and the other shrinks — so everything that makes both directions grow is the crossings flattening, which is the one quantity here this site cannot compute.

 Pull it lengthways and it narrows. The same cloth before and after a small extension along the warp. No thread has stretched: the extension came out of the warp crimp, that crimp went into the weft, and the fabric is narrower for it. Setting and geometry

What crimp interchange actually conserves

Pull a cloth lengthways and it narrows, because the crimp moves from one system to the other. Inextensibility says that much and no more — it is one equation short of an answer, and the second equation everybody uses is an assumption with a name.

 What a shrink-resist treatment has to do. Net displacement per cycle of agitation as a treatment closes the gap between the two friction coefficients. The chemistry is sold as gluing the scales down; what it has to achieve is arithmetic — make the fibre slide equally well both ways and the ratchet has nothing to rectify. After the loom

Shrink-resist is one number

Machine-washable wool is sold as a coating that glues the scales down. What the treatment has to achieve is narrower and more exact — make the fibre slide equally well in both directions, and the ratchet has nothing left to rectify, whatever the friction happens to be.

 Terry at a let-off of 5 to one. Towelling has two warp beams. The ground warp is held at ordinary tension; the pile warp is let off 5 times as fast, and the excess has nowhere to go but up. The loop height follows from the ratio and the pick spacing, and no float length appears in it. Compound and figured cloths

Terry needs two beams

A towel's pile is not woven longer. It is fed longer — from a second warp beam let off several times faster than the first — so the loop height follows from a ratio and a pick spacing, and the float length that decides a corduroy does not appear in it anywhere.

 The same yarn, flattened. One yarn's cross-section at five degrees of flattening, all drawn at one scale and all of the same area. Nothing is added: the yarn is wider because it is thinner. The cover rises for that reason alone, which is why a calendered cloth is more opaque than the cloth that went into the machine. After the loom

Calendering is the cloth arriving at the other model

This site has carried two thread sections side by side since its foundation — Peirce's circle and Kemp's racetrack — and has been careful to say which produced any number. They are not two opinions about one yarn. They are one yarn on either side of a finishing machine.

 The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica not available here. What cloth is

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

 Everywhere a muslin can go. Every state a muslin of 24 × 22 threads per centimetre in 20 and 20 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 6.59 per cent of extension is available along the warp, and reaching it costs 21.82 per cent of the width. Mechanics and drape

Pulled both ways, only one can give

A cloth at constant thread length has one degree of freedom, so its reachable states are a curve rather than a region. Equal extension in both directions meets that curve at exactly one point — the state the cloth is already in — so the amount available is nought.

 Where a bias cut's waste actually is. A bolt of cloth with panels placed at a stated angle, the ones that fit drawn and the rest of the cloth left shaded. Identical panels at a single angle tile the plane exactly, so the interior of the bolt loses nothing at all and the whole of the waste is at the two selvedges. The inset is the single-panel bounding box, which is the picture the usual account of bias cutting draws. Cloth doing a job

The bias cut and the selvedge

A bias-cut square costs exactly twice its own area, and every other shape costs more. That is an exact result about one panel — and it is not where a cutting room's waste comes from, because identical panels at one angle tile the plane. The loss is at the two selvedges, so it falls as the cloth gets wider, which no account in terms of the diagonal can explain.

 single float as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here. Knits and other structures

The float in a knit

Five rungs of this anchor have taken the float to be a length of thread on the surface with nothing holding it down. A knit has one too, and it behaves the same way in the light and the opposite way in the hand — because a woven float lengthens its thread and a knitted one shortens its fabric.

 Z twist at 800 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 22.8°, and it is the only quantity in this family: the yarn is 4.1% shorter than the fibre in it and carries 85% of the strength the same fibre would give lying straight. Setting and geometry

Twist is one angle

Every model of cloth treats a yarn as a cylinder with a diameter. It is a bundle of loose fibres, and what makes it behave like a cylinder is twist — which is a helix, so the whole subject is one angle, and the trade's twist factor turns out to be the only combination of count and turns that decides anything.

 One cloth, before the knife. Double plush: two ground cloths woven face to face with a pile warp shuttling between them. Uncut, the pile is the only thing either ground touches the other through, and the integrity criterion says one cloth. Cut down the middle, every pile end is severed and the same criterion says two. The whole manufacture is the deliberate destruction of an integrity the criterion otherwise exists to confirm. Compound and figured cloths

Velvet is cut apart

Two ground cloths are woven face to face with a pile warp shuttling between them, and while it does the whole thing is one cloth — the criterion says so, and the pile is the only reason it is true. Then a knife runs down the middle and the same criterion says two. The manufacture is the deliberate destruction of cloth integrity.

 A ratio that will not hold still. The exchange rate between the two directions for 4 cloths, against how far each has been extended along the warp. Every curve is above one half everywhere it is drawn, and every curve rises — so a single number for a cloth's ratio has to name the state it was read at, which a material's ratio does not. Mechanics and drape

A cloth's Poisson ratio is not a material's

The ratio of a cloth's contraction to its extension has a name everywhere else in mechanics, and it breaks every rule the name comes with — above one half on all eight cloths measured, doubling across a four per cent span, and not reciprocal between the two directions.

 How much of the curvature a cloth can take without being cut. The dart angle a spherical cap still demands after the fabric's own shear has absorbed what it can, against how closely the cloth is set. The total the cap demands is fixed by Gauss–Bonnet and is the same for all of them; what changes is how much of it the trellis can supply before its threads jam. An open cloth drapes a hemisphere with no dart at all; a closely set one has to be cut from the start. Cloth doing a job

A hemisphere costs one full turn

The total angle a pattern must remove to cover a hemisphere is exactly 360 degrees, and it is the same for a hat and for a stadium dome. What changes with the fabric is how much of that the cloth can supply by shearing instead of by being cut — and that is a property of the sett, computed from the angle at which its threads jam.

 The shed in section. A warp line 1200 mm from the fell of the cloth at the left to the back rest at the right, with 8 shafts at 300 mm and every 16 mm behind it. Each shaft lifts its ends in proportion to its own distance from the fell, which is what gives the same 30 mm clear opening at the reed from every shaft. The resulting extension is 0.460 per cent at the front shaft and 0.722 per cent at the back, a ratio of 1.57. Vertical scale exaggerated 4 times. Mechanics and drape

The shed is an extension

Every model of a finished cloth treats a thread as inextensible, and every one of them is about a fabric that has left the loom. On the loom the warp is stretched thousands of times a minute by the shed itself, and the amount is exact trigonometry with nothing fitted in it: the square of the shed's tangent, times the shaft's own distance from the fell, over twice what is left behind it.

 What holds a pick in. The holding force on one weft, as a multiple of the tension applied to its free end. For a plain weave it is the capstan on twice the weave angle, which Peirce's geometry gives at each sett and which falls towards nothing as the cloth opens out. For a leno it is the capstan on a half-turn, which the sett does not enter at all. Compound and figured cloths

What holds a pick in

A plain weave grips its weft by the crimp, and the crimp's wrap angle falls towards nothing as the cloth opens out. A leno crossing is half a turn whatever the sett. So the comparison has an exact answer that does not depend on the friction coefficient at all — and no plain weave can ever reach a leno's grip.

 A thread withdrawn from a plain weave and from a satin. One pick of a sheeting being pulled from 1.60 mm of cloth, in a plain weave and in a five-end satin, at a friction coefficient of 0.30. The drawn width of each thread is the tension in it at that point, to a common scale; the faint rules are where the thread actually turns, which is at its interlacings and nowhere else. A plain weave turns 2.80 times per millimetre and a satin 1.12, so the plain weave's tension compounds 2.78-fold over this length against the satin's 1.46. What the drawing cannot show is that the wrap angle is taken from a plain-weave geometry in both panels: a satin's crimp is genuinely smaller, so its real turns are gentler than these and its grip weaker still. What cloth is

A thread is gripped where it turns

The usual arithmetic for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.

 A fashioned edge at 1 wales in 2 courses. A knitted panel narrowing by 1 wale every 2 courses, drawn at the fabric's own aspect: a wale is 1.2791 times as wide as a course is tall, which is Munden's ratio of the two published constants. The edge therefore stands at 32.60 degrees from the wale, and that angle is the same in every yarn, at every gauge and at every loop length. Marks show where the 8 transfers fall. Knits and other structures

A fashioned edge has a quantised angle

A knitted panel is shaped by transferring loops, so its edge steps by whole wales at whole courses and its angle is the arctangent of a fraction. The available angles turn out to be the same for every plain knit there has ever been — in any yarn, at any gauge, at any loop length — because the constant they scale by cancels the loop out. There are eighteen of them, and 16.67° between the last two.

 What a raising machine can catch in a 3/1 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 spends the cloth's strength

Fibre standing up on the surface is fibre no longer in the load path. A nap is warmth bought with tensile strength, and the exchange rate runs along the same axis as everything else the float decides — which means a fabric cannot be optimised for both ends of it.

 The strain across a harness. The warp strain each shaft of a 24-shaft harness puts into its own ends, from 0.460 per cent at the front to 1.728 per cent at the back. Shafts within a budget of 1.0 per cent are drawn in one colour and those outside it in another; the budget is reached at shaft 13. Mechanics and drape

The back shaft works hardest

How hard the loom is on a warp end is the product of two numbers from different worlds — how often its column changes sides, which is a property of a binary matrix with no millimetre in it, and the strain of the shaft it happens to be on, which is a property of a machine with no weave in it. Neither knows about the other, and the threading that joins them is a decision nobody makes on structural grounds.

 A leno and the open plain weave it is not. Two warp ends and the picks they hold. On the left the doup end passes under its partner between picks and comes up the other side; on the right it never crosses, which is an open plain weave at the same sett. The layer count under each panel is computed by the same criterion that decides every other draft, and it returns the same verdict for both. Compound and figured cloths

The criterion cannot see friction

This site's central check is exact, decidable in linear time, and structurally incapable of distinguishing a carpet from a fabric that sheds. That is not a defect to be repaired — a separation exists or it does not, and there is no margin in it — and the fancy weaves are precisely the constructions that live in the gap.

 The sett moves the flux and not the height. A 20 tex cotton yarn woven at every sett from 8 to 34 threads per centimetre. Above: the hole between the threads lifts from 27 to 234 mm as the cloth closes, while the space between the fibres lifts 6.37 m at every one of them — so the cloth's maximum is the flat line, and the sett does not touch it. Below: the permeability of those holes falls by a factor of 292 over the same range. Both curves are monotone, so there is no optimum — only an interval, ending at the jam at 34.6 threads per centimetre. Setting and geometry

The sett decides how much, not how high

Every essay on wicking so far has found the sett deciding something. This one finds it deciding nothing at all: a cloth's maximum rise is 6.37 m at eight threads per centimetre and 6.37 m at thirty-four, because the sett cannot reach inside a yarn.

 Stick, slip, and the ratio between the two coefficients. The force in a thread as a cloth is agitated, with a static coefficient of 0.300 and a kinetic one of 0.225 — a ratio of 0.75, which is what fibre on fibre measures. The force climbs until it reaches the static limit, the contact breaks away, and while it is sliding it resists only at the kinetic limit. So a cloth that is being shaken can be left anywhere in the narrower band, and a cloth at rest anywhere in the wider one. What the trace cannot show is how much this buys: the band does not narrow in the ratio of the coefficients, because the restoring force stiffens away from the minimum, and the real narrowing is nearer 0.87 than 0.75. What cloth is

Two coefficients, not one

Every friction on this site is a single number, and the account of why a cloth relaxes better when it is agitated depends entirely on there being two. Separating static from kinetic changes what a resting state is: a cloth at rest is held by one coefficient and a cloth being shaken by the other, and the band it can be left in narrows — by less than the ratio, because a cloth's restoring force is not linear in its extension.

 A cloth does not mind a hole. Strength left after a hole, against the width of the hole, for a woven cloth and for a film of the same width. The cloth's threads carry their own load, so it loses exactly the threads the hole removes and the loss is linear. The film is a continuum, so a hole of any size at all costs it the stress concentration factor — quoted from Kirsch's elasticity solution, not derived here — and the size of the hole does not enter. Cloth doing a job

A cloth does not mind a hole

A hole in a film costs it two thirds of its strength whatever the hole's size, because a continuum concentrates stress at an edge. A cloth's threads carry their own load and hand almost nothing to their neighbours, so a hole costs exactly the threads it removes — a loss that is linear in the hole, independent of the sett, and zero for a slit along the load.

 Flexes per end in the same check, ground a pick along. One bar per warp end above the draft of the same check, ground a pick along, each the number of times that end changes sides in one repeat — which is the number of times the shed drags it through its heddle eye. The counts run from 6 to 10, and they are the same numbers that give the cloth its interlacing rate of 0.500 per intersection. Mechanics and drape

A figure is harder on its warp

Every basic weave flexes all its ends exactly as often as each other — plain, twill, satin and sateen alike, and it is a one-line theorem. Figure one of them on another and that evenness goes, or does not, depending on where the ground weave was started relative to the figure. The same invisible offset that decides whether a fine figure holds together decides, at a coarse one, how unevenly the loom works the warp.

 The reed is not the sett. Twelve ends held at the reed's pitch above and at the cloth's pitch below, for a sheeting whose weft crimp is 14.61 per cent. The count is the same in both rows and only the spacing changes: the cloth is 12.75 per cent narrower, so a reed at 24.43 ends per centimetre produces a cloth at 28. The crimp comes from the Peirce solution at this cloth's quoted construction. Setting and geometry

The reed is not the sett

A reed holds the warp at a pitch, and the cloth that leaves it is narrower — by exactly the weft's crimp, with nothing fitted and nothing approximated. On a close balanced sheeting that is 12.75 per cent, on an open scrim 1.87, and a weaver who allowed one figure for both would be wrong by a factor of nearly seven.

 A cotton fibre dry and wet. One cotton fibre in its dry state and saturated with water, both drawn at the same scale in both directions. It is 20% wider and 1.2% longer, so its cross-sectional area rises by 44% if the section stays similar to itself. The length difference is drawn and is nearly invisible, which is the point: a swelling that were the same in both directions would make a cloth bigger and change nothing about its structure, and this one changes every ratio of a diameter to a spacing in the cloth. What the drawing cannot show is the section: a cotton fibre is not a cylinder, and the directly measured area swelling of 40% to 42% does not agree with the square of the width change, which is a fact about the fibre. What cloth is

What water does to a thread

A cotton fibre in water is a fifth wider and a hundredth longer. If it grew equally in both directions a wet cloth would simply be a bigger cloth and nothing structural would follow; because it does not, every ratio of a diameter to a spacing in a cloth moves, and they all move the same way.

 Heddles per shaft: stripe. The threading of a narrow satin stripe on a broad plain ground, over a warp of 1,200 ends. Each bar is one shaft and its length is the heddles on it. The draft needs 10 shafts however they are loaded; the heaviest carries 500 and the lightest 25, a factor of 20.0. Spending 20 shafts instead brings the heaviest down to 100. Compound and figured cloths

Where the heddles go

A draft says how many shafts it needs and says nothing about how the ends divide between them. On a satin stripe over a plain ground the two ground shafts carry twenty times what the stripe shafts do — and the only cure is to spend shafts, which turns the threading into an allocation problem with an exact answer.

 The loop that costs nothing to extend. A plain knitted loop at rest and extended by 35 per cent, with the arcs marked. The arcs' radius is the diameter of the yarn the loop wraps, 0.167 mm, and it is set by contact rather than by the fabric's dimensions — so extending the fabric lengthens the legs and bends nothing further. The bending energy is 0.0176 N·mm at both, and the model therefore asks no force at all for an extension a woven cloth would refuse. What the drawing cannot show is what a real knit's first few per cent do cost, which is friction and yarn flattening and is not a bending property. Knits and other structures

A knit is soft because it bends

Ask the same energy question of a woven cloth and a knitted one and the answers are not different by a factor — they are different in kind. A woven cloth's bending energy changes the moment it is extended. A knitted loop's does not change at all, exactly, over the whole of its extension, because its arcs are held to a radius by contact rather than by the fabric's dimensions.

 The two calculations a yarn's stiffness admits. A bundle of 9 fibres bent with the fibres free to slide and with them locked together. Free, the rigidity is the sum of the fibres': 0.00141 N·mm² for a 30 tex cotton yarn. Locked, it is the fourth power of the yarn's own diameter: 0.689. The ratio is the fibre count over the square of the packing factor, 490, and nine fibres are drawn where the yarn has 176. What the drawing cannot show is where a real yarn sits between them, which is a question about friction rather than about fibre. Mechanics and drape

A yarn's stiffness is a bracket, not a number

Two calculations are available for how stiff a thread is in bending, and both are exact. One treats the fibres as free to slide and gives the sum of their stiffnesses; the other treats them as locked and gives a solid rod. They differ by the fibre count, which for an ordinary cotton yarn is a factor of five hundred — and no measurement of the fibre narrows it by anything at all.

 What the two setts can be set to. Two rules on one scale from 8 to 40 threads per centimetre. The upper carries the 49 warp setts a metric reed catalogue reaches at one to four ends per dent; the lower carries the 4825 pick densities a change-wheel take-up reaches. The mean spacing is 0.656 threads per centimetre in the warp and 0.0066 in the weft, a ratio of 99. The widest gap in the reed's range is 2.10 threads per centimetre. Setting and geometry

The setts a loom can reach

Transposing a draft gives a perfectly good draft, and every count this site takes off a matrix either is symmetric under exchanging warp and weft or has a mirror twin. The loom is not symmetric at all: over the range ordinary cloth is woven in, it can choose a pick density 99 times more finely than a warp sett — and the warp sett cannot be changed once the warp is drawn in, at any granularity whatever.

 The two routes a poplin has to a strain. A poplin drawn in section at three places: as woven, at the end of what its crimp can supply, and past that. Between the first two the warp's crimp falls from 8.97% to 4.85% and the weft takes on what it gave up, and the thread length is 0.4953 mm in both — nothing has stretched, and the cloth is 3.93% longer. Between the second and the third the geometry cannot move because the weft's straight run has vanished, so the cloth's extra 2.0% is the thread's extra 2.0%. What the drawing cannot show is which of the two a piece of cloth has had: the first two states look different and the last two look the same, and it is the last two that differ in whether the cloth comes back. What cloth is

A cloth gives back less than it took

Everything this collection computes about a deforming fabric is reversible, and no fabric is. The repair is not a new material property: a woven cloth has two routes to a strain, one of them costs its threads nothing and comes back in full, and where the first route runs out is a number about the sett with no fibre in it at all.

 Mock leno, 3 threads to a bundle. Threads that interlace identically have no weft passing between them, so nothing holds them apart and they lie touching. The reed still sets the average spacing, so the space they leave collects at the bundle's edge — a hole 0.60 by 0.60 mm, made without one thread crossing another. Drawn to scale on a fixed 9 mm square of cloth at 20 threads per centimetre and a 0.3 mm yarn. Weaves

A hole with nothing crossing

A real leno holds its holes open by crossing one thread over another, which is a topological arrangement and cannot come undone. A mock leno makes the same holes by grouping threads that nothing separates, and everything about it is friction.

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