Theme

The thread: Measured, not claimed

Thread count, breathability, the strength of satin: the trade is full of numbers that do not mean what they are taken to mean. Where a claim is refuted here, it is refuted by a computation.

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.

$How much fibre a woven reinforcement holds. A unit cell of a woven reinforcement in section: flat tows, the thickness they add up to, and the wave the warp makes to cross them. The bar below is the fibre volume fraction against the same tows laid flat in two plies and against the packing factor of a tow, which is the most any cloth of them could be. The vertical scale is exaggerated so the interlacing is legible; the horizontal scale is the cloth's own.$ Cloth doing a job

The crimp is the price of being cloth

A woven reinforcement is bought for stiffness along its fibres, and the usual explanation of what it gives up — fibre content, lost to the crimp — has the sign wrong. At a given thickness a woven fabric holds slightly more fibre than two flat plies of the same tows. What the crimp costs is stiffness, and the float length is the knob.

$The same thread count, twice. Two cloths with identical thread counts and different yarn. The count is the same number in both; the fraction of the surface the threads actually occupy is not, and that fraction is what thread count is usually taken to mean.$ Setting and geometry

Thread count is not quality

It counts threads. It says nothing about how much of the cloth they cover, it can be inflated by counting plies, and it ranks fabrics in nearly the wrong order. The quantity it is mistaken for is cover factor, and that one is computable.

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.

Cloth doing a job

A fabric to fill and a fabric to load

A reinforcement has to hold as much fibre as possible and still let a resin through it, and the two demands are the same decision pulling opposite ways. Both bounds are computable, the interval between them narrows as the square of the part, and past a certain size it is empty — which is the most useful thing the arithmetic says.

Pattern and colour

Colour and weave

Thread the same weave with a different order of coloured ends and a pattern appears that is nowhere in the draft. Houndstooth is an ordinary two-and-two twill, and nothing about its interlacement is unusual at all.

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.

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.

Where the floats are in the 5-end satin. Every longest run of warp on the face, marked. A float is smooth because nothing interrupts it, which is the same reason it snags: there is a length of thread lying on the surface with nothing holding it down.

Weaves

The float decides

How far a thread runs on the face before it goes under is one number, and it sits behind lustre, drape, snagging, abrasion, tear strength and how densely the cloth can be set. Almost nothing else in the subject has that reach.

A 2/2 threading, run out to three widths. The same four shafts threaded for repeats of increasing width. The threading line runs up and back down; the harness never grows, and the number of ends it carries is bounded by the loom's width rather than by its shafts.

Compound and figured cloths

The harness does not grow

Nearly every account of a loom says the shaft count limits the repeat. It limits the repeat of a straight draw, and of nothing else — a reversed twill on ninety-six ends weaves on the same four shafts as the twill it was made from, and the ratio grows without bound.

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.

Two sections, one yarn. The same yarn given a circular cross-section and a racetrack one of equal area, both jammed. The spacing the two models allow is nearly the same; the cover and the cloth thickness they predict are not.

Cloth doing a job

A tow is not a yarn

Every geometric model on this site assumes a thread with a diameter. A reinforcement tow has a width and a thickness instead, and the two are eight to one — which moves the jammed sett, the crimp, the cover and the cloth's thickness at once, in four different directions, from one change.

What an interlacing costs. Each weave's interlacing count beside the closest it can be set in the same yarn. The two run opposite ways, because a thread that changes face often has to bend often and a bend takes room.

Setting and geometry

How close can threads be set

There is a maximum. Push more threads into a cloth than the geometry allows and they simply will not go, and the limit depends on the weave as much as on the yarn.

The 2/2 twill. The 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.

Pattern and colour

The diagonal is not a thread

A twill's diagonal is the most looked-at feature in weaving and the most misdescribed. No thread in the cloth runs diagonally. What runs diagonally is a pattern of which thread is on top, which is a different sort of object.

The draft for herringbone, as a loom holds it. The herringbone written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

Compound and figured cloths

What a dobby stores

A pattern chain does not store picks. It stores lifts — the distinct sets of shafts a draft ever raises — and for most drafts worth weaving that number is very much smaller than the number of picks, which is the second half of the reason a wide repeat is affordable.

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.

After the loom

A cloth cannot shrink past its own crimp

However hard a loom held the warp, there is a limit to what relaxation can take back, and it is not a fitted constant or a measured one. It is the crimp itself, read as c over one plus c, and it is the only result in this field with nothing empirical in it at all.

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.

Weaves

Interlacings and firmness

Every time a thread changes face it has to bend, and a bend takes room. That one sentence decides how densely a cloth can be set, how firm it feels, and why the two run in opposite directions.

The twelve groups a draft can have. Every four-by-four draft in which each thread interlaces, classified by its plane symmetry group. Twelve of the seventeen groups occur; the five that do not are the ones needing a three-fold rotation, which no grid of warp and weft admits at any size.

Pattern and colour

The seventeen groups a draft can have

Twelve of them, in fact. The five missing ones all need a three-fold rotation, and no grid of warp and weft admits one at any size — which makes it a theorem rather than an artefact of the census.

Six ways of saying how fine a yarn is. The same four yarns written in six count systems, all placed on one axis of tex. A direct system states mass per unit length and rises as the yarn coarsens; an indirect one states length per unit mass and falls. Both kinds are in daily use and neither says which it is.

Setting and geometry

The yarn count systems, and why there are several

Half the ways of saying how fine a yarn is get bigger as it gets finer and half get smaller. That is not carelessness — and one of the constants buried in the oldest rule of thumb turns out to be a measurement nobody wrote down.

Three different weaves, one cloth. Drafts drawn from one collision class, with the blind intersections marked, and the single surface all of them produce. Where the two crossing threads are the same colour the weave leaves no trace, so the cloth cannot report what it is.

Pattern and colour

Colour and weave as a two-colour problem

Where the two threads crossing are the same colour, the intersection looks identical whichever is on top. Half of them are, so 22,874 drafts collapse onto 256 surfaces — eighty-nine weaves apiece, and no way to tell them apart.

A web at 6 fibre lengths squared per unit area. Straight fibres dropped at random positions and random angles. The largest connected group is drawn solid; everything not joined to it is drawn faintly. Whether that group reaches both edges is what decides whether this is a sheet or a heap.

What cloth is

Nonwovens, and what holds them together instead

A web of fibres laid down at random has no repeat, so the exact test for whether a fabric holds together has nothing to work on. What replaces it is a threshold, and the threshold is sharp.

Setting and geometry

Peirce against the racetrack, measured

Two models of a yarn's cross-section, given the same yarn and the same closure condition, agree on how densely the cloth can be set and disagree by nearly half on how thick it is. Which quantity is being asked about decides whether the choice of model matters at all.

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.

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.

Cloth doing a job

The hole between four threads

A woven cloth's holes are all the same size. That is not an approximation — it is what a repeat means — and it is the whole reason a woven filter is specified by one number while a nonwoven needs a curve. The number is the spacing less the diameter, which this site has been computing since its first questions about setting.

The 2/2 twill both ways round. The same twill stepped one end to the right and one end to the left. Every quantity the matrix can produce is identical for the two, so the direction is real in the cloth and invisible in the arithmetic.

Weaves

Twill direction, and how it is named

The two twills are mirror images, and every quantity the matrix can produce is identical for both. What separates them is the yarn, which has a handedness of its own — and the angle, which is forty-five degrees only in a square-set cloth.

Figure and ground on 8 ends. A damask is a satin and its own complement. The figure is warp-face and the ground is weft-face, they carry the same longest float and need the same shafts on the same threading, and the whole of the pattern is carried by which system is on top.

Compound and figured cloths

A damask is its own complement

The pattern is carried by direction alone. Figure and ground are the same satin, one warp-face and one weft-face, with the same longest float, the same shaft count and the same threading — so the cloth's most famous effect costs it no structural difference whatever between the two areas.

Cloth doing a job

A filter cloth has two jobs

Hold the soil back and pass the water. The first wants a close sett and the second wants an open one, so a filter cloth is not a fabric but an interval — one that is sometimes a single sett wide, and for a fine enough soil is empty. The empty case is the useful one, because it says the answer is not a woven cloth at all.

Setting and geometry

Balance, and what an unbalanced cloth does

How much of the surface each thread system takes is the mean of the weave matrix, and it is one of the very few quantities in this subject that can be read straight off. Almost nothing that follows from it can.

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.

How many four-by-four weaves there are. The same census counted four ways. A draft is a notation; shifting the repeat's origin, turning the cloth over and turning it end for end all change the matrix and not the fabric. Each bar is the number of distinct objects left once those identifications are made, counted by canonical form and checked against Burnside's lemma.

Pattern and colour

How many cloths are there

Twenty-two thousand eight hundred and seventy-four four-by-four drafts. Four hundred and twenty-six four-by-four cloths. And the site's own separation rate — the fraction of drafts that look like fabric and are not — more than doubles when fabrics are counted instead of notations, which is the opposite of what was expected.

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.

Setting and geometry

The weave decides the sett, and two models disagree about it

Ashenhurst's rule and Peirce's geometry both answer how densely a cloth can be set, and for a plain weave they are fifteen per cent apart. Only one of them reaches the other weaves at all, and it is the cruder one.

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.

2/2 twill: written at 4×4, repeating on 4. On the left the draft as written, with the smallest rectangle that tiles it outlined. On the right the same draft with every translation that leaves it unchanged marked from the top left corner: there are 4 of them, so a fundamental domain holds 4 intersections rather than the 16 the rectangle claims and the 16 the point paper carries. Nothing in the drawing on the left says so.

What cloth is

What a repeat repeats

Point paper is ruled in squares, and the rectangle a draft is written on is a decision by whoever drew it. The unit a cloth actually has is smaller — two intersections for a plain weave, four for a 2/2 twill — and it is not a rectangle at all.

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

3 cloths on 6 ends. A repeat of 6 ends and 6 picks holding 3 complete cloths, each with 2 ends and 2 picks of its own. The bars beside the strands say which cloth each belongs to; the ceiling at this size is 3, and the longest float is 5 because the face warp passes over every pick below it.

What cloth is

How many layers a draft can have

Of the 22,874 four-by-four drafts this site sweeps, 22,730 are one cloth and 144 are two. None is three, and none can be — a repeat of n ends holds at most n halved cloths, because every cloth needs two ends and two picks of its own before it interlaces at all.

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.

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.

$Where the twenty-eight comes from. The setting a cotton cover factor of 28 prescribes, and the setting at which the threads would cover the surface geometrically, against yarn count. They are one curve: the trade's scale is Peirce's diameter with the units taken out. Below them is the sett at which each weave actually jams, which is a fixed fraction of it.$ Setting and geometry

Where the cover factor comes from

A cotton cover factor is threads per inch over the root of the count, and the scale says twenty-eight means a covered surface. The twenty-eight is not a convention: it is the reciprocal of Peirce's yarn diameter, and the trade's practical ceilings of fourteen and twenty-two fall straight out of it.

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.

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.

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 one cut costs. The length of thread set loose by a single abrasion cut through a warp float, with each weave measured at the densest setting that weave allows in the same yarn. The bars are inches of freed thread; the note beside each is the float in picks and the setting that float permits.

Weaves

Floats and abrasion

A satin is said to wear badly. It does not wear quickly — its flat face spreads the rubbing over more thread than a plain weave's crowns do. What it does is fail badly, and those are different quantities moving in opposite directions.

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.

Yarn per needle, by structure. Wale spacings of yarn per needle position, at one loop length. A knitted loop is 4.30 of them — Munden's constant, measured rather than derived — a float across one needle is exactly one, and a tuck is taken as a stated multiple of a loop. Everything else follows by counting, so the percentages beside the bars are not estimates.

Knits and other structures

What a tuck costs

A knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop. That much is a count. What it does to the fabric's size is not, and this essay is careful about which is which.

A tartan sett in 2/2 twill. A pivoted sett of 70 threads used in both systems, which is what makes a tartan a tartan rather than a check. On the left, the cloth at block scale: the squares on the diagonal have one colour in both systems and the rest are mixtures. On the right, one mixture rectangle at thread scale, where the weave decides. In 2/2 twill the reflection survives on 50.0 per cent of a mixture's intersections, against a ceiling of 75 per cent that no weave can reach. The three tints stand for the three colours of the sett and carry no other meaning here.

Pattern and colour

A check is two stripes and a tartan is one

A tartan is quoted as one sett of thread counts because the warp and the weft carry the same order. That is said to make it symmetric about its diagonal, and the arithmetic says it cannot be — the ceiling is three quarters, and a 2/2 twill reaches one half.

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.

What the geometry says about a tear. A slit in a woven cloth, with the ends ahead of the tip gathered into the group that will break together. How many can gather is the opening divided by the slack in each gap, and the slack is the thread spacing less the thread diameter — an expression with no weave in it at all.

Weaves

Does a loose weave tear better

The trade says a twill tears stronger than a plain weave of the same yarn, because fewer interlacings let the threads group. The geometry of that grouping has no weave in it at all — and where the geometry does speak, it predicts the opposite.

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.

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 on this site 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.

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.

rib-float in section. A two-bed structure seen in section across the wales, over 3 repeats of course 1. The front bed's loops sit on the upper line and the back bed's on the lower one, offset by 0.5 of a needle pitch because the gating is rib. Of the 2 floats in the repeat, 2 lie in the gap between the beds and 0 on a surface. The upper panel is the fabric at the machine and the lower one is the same course with the beds closed up, which is what happens when the fabric is cast off — and neither panel is a relaxed fabric, because the loops are drawn as arches of one size and a real one settles wherever the yarn's bending leaves it.

Knits and other structures

A second bed changes what a float is

Every float so far has been on a surface, because every knit so far has had one needle bed. Put a bed behind it and the yarn runs in the gap between them — and over the whole enumeration of two-bed structures, 1,248 floats of 1,272 lie inside the cloth, on no surface at all.

Swelling, at the same count. The same yarn before and after mercerisation, drawn to one scale. Its linear density has not changed — the same grams per kilometre go into the cloth — but the fibre occupies more volume, which is a lower packing factor and a larger diameter. Every consequence in this field follows from that single number.

After the loom

Mercerising is a packing factor

Cotton held in caustic soda swells, and everything the treatment is famous for follows from one number in this site's diameter calculation. The lustre it is actually sold for does not, and saying which consequences are computed and which are not is the whole of the discipline here.

Z twill, Z twist. A square of cloth with two directions on it. The broad lines are the twill, whose angle 45.0° from the warp comes from the setts alone — 24 ends and 24 picks per centimetre. The fine lines are the surface fibres of the warp ends, at 25.3° from the warp because the yarn is twisted 900 turns per metre. Between them is 19.7°, and that is the whole of the rule.

Setting and geometry

Twist and the twill line

The oldest rule of thumb in weaving says to weave a Z twill from S-twist warp for a bold line and from Z-twist warp for a subdued one. Both halves of it are geometry: the twill's angle comes from the two setts, the fibres' angle comes from the twist factor, and the rule is their difference.

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 arithmetic this site has used 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.

Every cloth at 150 grams. The counts and setts that all weigh 150 g/m² at a crimp of 7 per cent and a balance of 1. Hollow marks are past the jam — arithmetic rather than cloth. Among the 10 that can be woven the cover factor runs from 0.28 to 0.77, a factor of 2.77, and every one of them is the fabric the specification asked for.

Setting and geometry

What a fabric weighs

Every fabric is sold by its weight in grams per square metre, and the number is a sum of four products in which no term appears alone. One equation, four unknowns: a hundred and fifty grams describes an open coarse cloth and a close fine one, and the cover factors differ by a factor of nearly three.

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.

Cloth doing a job

What holds a thread in a seam

A seam fails in two quite different ways, and the one the trade worries about is not the stitches breaking. It is the cloth's own threads sliding out of the weave beside the seam — and what resists that is friction at the crossings, accumulating multiplicatively, so a satin gives a seam a thousandth of the grip a plain weave gives it.

The draft for 2/2 twill, as a loom holds it. The 2/2 twill written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

Weaves

How many shafts a draft needs

Every quantity this site has counted so far is a property of the cloth. This one is a property of the loom — the number of distinct columns in the matrix — and it is very probably the strongest single predictor of which of the twenty-two thousand four-by-four drafts anybody ever wove.

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.

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 rung of this ladder 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.

The stitch density that makes the strongest seam. Seam strength against stitch density, as the two limits that decide it: the sewing thread crossing the seam, which rises with the stitches, and the fabric the needle perforates, which falls. The seam is the lower of the two, so the optimum is where they cross — and whether the fabric line falls at all is decided by the clear gap between threads against the width of the needle.

Cloth doing a job

The stitch that weakens the seam

More stitches per centimetre put more sewing thread across a seam and more holes through the cloth beside it, so seam strength rises, crosses and falls. Whether the fabric line falls at all is decided by the clear gap between two threads against the width of the needle — the same arithmetic a filter cloth is specified by, doing a different job.

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.

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.

Float lengths in a honeycomb. The same draft twice. On the left, filled where the warp is on the face — which is all point paper says. On the right, every intersection shaded by the length of the float it belongs to, from one at the palest to 6 at the strongest. The gradient on the right is the whole mechanism of a relief weave and it is invisible on the left.

Weaves

A honeycomb gets its cells in the wash

The obvious mechanism is take-up on the loom, and the arithmetic says it is wrong: every end of a diamond passes through the long floats and the tight ones alike. What is left is finishing, and a cell is a region that wanted to shrink less than the cloth around it.

After the loom

A knit's dimensions come from its loop

A relaxed plain knit's courses, wales and stitch density depend on the loop length and on nothing else — not the yarn count, not the fibre, not the machine gauge. The constants are measured rather than derived, and the interesting thing about the published set is that it does not quite satisfy its own arithmetic.

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.

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.

How much a draft agrees with itself. On the left the draft; on the right its correlation at every offset, one cell per offset, with the offset of nothing at the top left. Warp-up counts as plus one and weft-up as minus one, so the number in each cell is agreements minus disagreements out of 64. The correlations away from the origin sum to exactly -64, whatever the draft — structure can be moved about and not removed.

Weaves

A crepe cannot be structureless

A crepe weave is designed to have no line in it anywhere. The correlations of a draft with itself sum to a number fixed by the repeat alone, so structure can be spread and never removed — and on eight ends the floor turns out to be half the repeat, set by a fact about binary words with nothing textile in it.

After the loom

A dimension without a state

Every essay in this field turned on the same omission, and collecting them produces a result none of them had on its own — the quantities a cloth's geometry determines divide cleanly into those a finishing works can move and those it cannot, and the division is not the one the vocabulary suggests.

A tear breaks threads or pulls them out. The grip a cloth has on a thread at the tip of a tear, against the sett, with the thread's own strength drawn across. Below the line the thread slides and the yarns group, which is the trade's explanation of why a loose weave tears well; above it the thread breaks where it is and grouping never happens. The essay that asked it could not compute this: it needed a friction, and the fancy weaves supplied one.

Cloth doing a job

A tear stops where the grip is

This collection asked whether a loose weave tears better and had to answer that the geometry could not say — it supplies a grip count and a slack, and neither is a force. The fancy weaves brought a friction. With it the question has an answer — a cloth's threads slide up to a computable sett and break above it — and a ripstop grid has a bound with no free parameter in it.

The step between a figure and its ground. Three figures on a plain ground, all in one sheeting's threads at its own setts. A thread presses on the thread it crosses only where it turns, and it turns at its interlacings — so the pressing a region receives per unit area is the contact force at one turn times the turns per unit area, and the second factor is a property of the matrix exactly. A five-end satin turns two fifths as often as a plain weave, is pressed two fifths as hard, flattens less, and stands 55 µm proud of it. What the rows cannot show is that both regions are given a plain weave's weave angle: a satin's crimp is genuinely smaller and its turns genuinely gentler, so the real step is larger than this, by an amount not computed here.

Pattern and colour

A figured cloth has a step in its surface

A damask is one cloth in one set of threads at one sett, and it is not flat. A thread presses on the thread it crosses only where it turns, so a region that turns less often is pressed less often, flattens less and stands thicker — and the step is a ratio of interlacing rates, read off the matrix with no yarn property in it.

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.

Two pore systems in one cloth — 24 threads per centimetre. A plain weave of 20 tex cotton in section, at 24 threads per centimetre, so the yarn is 167 µm across and the clear hole between two picks is 250 µm. That hole's hydraulic radius is 124.8 µm. Inside the yarn, fibres 14 µm across packed at 0.6 leave spaces of hydraulic radius 2.33 µm — 53 times finer, and by Jurin's law 53 times higher: 6.37 m against 119 mm. The yarn's interior is magnified 6 times and the two discs at the foot are the only part drawn at one scale.

Cloth doing a job

How high a cloth wicks

A woven cloth has two capillary systems and they are a factor of twenty to a hundred apart. The one every diagram draws — the hole between four threads — lifts 119 mm. The one nobody draws, inside the yarn, lifts 6.37 m.

The weight against the constant nobody has measured. Areal weight against the stitch-density constant, for four two-bed structures at 20 tex on a 0.35 cm loop with a tuck taken as 1.15 of one. Every line is exactly straight through the origin, because the weight is tex times yarn per repeat times k_s divided by the area of the repeat and there is no fitted constant anywhere in that division. The three vertical rules are the only measured values this site has — Munden's published k_s for plain single jersey in three relaxation states — and none of them is the right value for any structure drawn here. Where each line should be read is the whole of what is missing.

After the loom

The constants do not compose

The yarn in a knitted structure is a sum of what each element takes, exactly, over structures nobody has measured. The size the structure relaxes to is not a sum of anything — and the whole gap between the two is one number per structure, which would cost forty-five fabrics each to obtain.

A knit's restoring force, and the column that does not move. A plain knit of 20 tex cotton at a loop length of 3.5 mm, over the extension range its own geometry admits. The bending energy stored in one loop is the same number at every extension — the loop's arcs are held to a radius by the thread they wrap rather than by the fabric's dimensions, so extending the fabric does not bend anything more. The frictional resistance at the interlocks is not zero: it is μ times the force pressing there, times 2.34 interlocks per millimetre of width. So a knit's resistance to extension is dissipative rather than elastic, which is why it does not spring back and why its dimensions depend on how much it has been agitated. What the rows cannot show is the interlock force itself, which this site does not have for a knit and which is recorded as missing.

Knits and other structures

What stops a knit extending

A knitted loop's bending energy does not change as the fabric extends — exactly, over the whole range its geometry admits. Something resists, and it is not stiffness. It is friction at the interlocks, which is dissipative rather than elastic, and that single fact accounts for why a knit does not spring back, why a softener changes its dimensions and why the constants its size is quoted with contain no yarn property at all.

Compound and figured cloths

Where a stitch can hide

One reversed intersection turns two cloths into one, and half the intersections in the repeat would do it. Almost none of them may be used — a plain-faced double cloth has nowhere at all to put a stitch, a five-end satin has fifteen places or none depending on which rule is asked, and two satins of the same order differ by a factor of two.

Plain weave, doubled. Plain weave, then the same weave with its picks grouped in 2, its ends grouped in 2, and both — which are a warp rib, a weft rib and a hopsack — and a 2/2 twill beside them for comparison. The rules under the drafts bracket the threads that lift together on every pick and so lie touching. Each doubling multiplies the cloth's own unit by 2: 2, then 4, then 8 intersections. The hopsack and the twill interlace equally often and are drawn from different rules. The setts under each draft are the densest that weave may be set at with a 0.25 mm yarn, and a rib's two are 0.750 apart where every other weave here is square.

Weaves

A cord is a stripe with no colour in it

Warp rib, weft rib and hopsack are plain weave with its threads doubled, in the warp, the weft, or both. Six of the nine measures this site takes off a matrix cannot tell a 2/2 hopsack from a 2/2 twill, and the three that can are not the ones a weaver quotes.

The harness a strain budget buys. How many shafts stay inside a 1.0 per cent warp-strain budget, against the clear shed opening the loom needs at the reed: 12 mm gives 43, 16 mm gives 36, 20 mm gives 28, 24 mm gives 21, 30 mm gives 13, 36 mm gives 7, 44 mm gives 1. The shed's tangent enters the strain squared, so the opening is much the strongest thing a loom builder controls.

Compound and figured cloths

The harness has a depth

Why does a dobby carry sixteen or twenty-four shafts rather than two hundred? The usual answers are about the mechanism — how many jacks a box can drive, how many hooks a dobby has — and they are real limits that are not the binding one. A stated tolerance on warp strain is a stated distance from the fell, and a stated distance is a whole number of shafts. One per cent buys thirteen.

A front on a thread with 8 per cent crimp. Three rows at one scale. The top row is the warp end laid out straight, with the wetted front marked at four equal quarters of its own length — which is where Washburn's law puts it at four times whose square roots are evenly spaced. The middle row is the same thread crimped at 8 per cent, so it covers 92.6 per cent of the paper the straight one did. The bottom row is the cloth, and the four fronts on it are the four above pulled back by 1.08. A coefficient is a length squared over a time, so it comes down by 1.1664 — exactly (1 + c)², with no property of the liquid or the fibre in it. The thread's thickness is not drawn and neither is the liquid: a meniscus in a 2.33 µm pore is finer than any line on this canvas.

Cloth doing a job

Wicking is slower along a crimped thread

A front travelling up a warp end travels the thread's path, which is longer than the cloth by exactly the crimp. So the wicking coefficient measured on the fabric is the yarn's own divided by (1 + c)² — 14.3 per cent lost at eight per cent crimp, whatever the liquid.

The resting band, not the resting point. The bending energy of a sheeting along its own constant-thread-length locus, with the band in which friction can hold it shaded. The minimum is a single state; the band is 10.9 per cent of length wide, because the cloth stops sliding as soon as the energy it can release falls below the 0.0756 N friction takes to move a crossing. What the drawing cannot show is which end of the band a given piece of cloth stops at, which depends on the direction it arrived from and is what makes relaxation hysteretic.

After the loom

A cloth relaxes until its threads stop pushing

The finishing field treats the relaxed state as a place a cloth arrives at. With an energy along its own locus and a friction at its crossings it is not a place but a band — and which point of the band a piece of cloth stops at depends on which side it came from, which is why washing it twice gives two answers.

A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.40 N per crossing the sections flatten to aspect ratios of 1.75 and 1.84, the cloth thins to 0.263 mm, and the warp runs flat for 0.109 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

Mechanics and drape

A flattened thread is a record of a force

A yarn in cloth is not round, and this site has modelled the flattening for as long as this collection has run with the amount of it left as a number somebody chose. Give the section a stiffness and ask what the cloth prefers, and the answer is a circle — at every stiffness, for every balanced cloth in the table. Flattening does not happen by itself; it happens because something pressed.

Weaves

The three basic weaves do not generate the rest

Every weaving manual opens with the same sentence: there are three basic weaves, and everything else is derived from them. The complete catalogue of the smallest interesting repeat is in hand, so the claim can be checked instead of repeated. Starting from plain weave, every twill and every satin, and applying every derivation the manuals name, reaches nine of the 426 cloths that exist there.

Setting and geometry

Peirce and Kemp are one cloth at two moments

This site has run two thread sections side by side since the setting field was built — a circle and a flattened racetrack — and said honestly that they disagree and that the disagreement is the point. They are not rival descriptions of the same fabric. They are descriptions of the same fabric before and after something pressed it, and the difference between them is a pressure that can now be named.

How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

Compound and figured cloths

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

The bracket that closes and the bracket that does not. A 25 tex cotton yarn at a packing factor of 0.6. Its bending rigidity lies between 0.00117 N·mm² — the sum of its fibres', with them free to slide — and 0.478, a solid rod of its own diameter: a factor of 408, and both ends are derivations. Its resistance to being squashed out of round has an upper bound of the same kind, 369 N/mm² for a solid section, and no lower bound at all, because fibres free to slide resist a change of shape with nothing. That is why the aspect ratio of a flattened yarn has been a free parameter here since the setting field was built: a quantity bounded below by zero cannot be estimated from its bounds, and has to be measured.

Mechanics and drape

The stiffness with no lower bound

A yarn's bending rigidity lies between two derivable ends and the ratio is the fibre count — wide, but closed. Its resistance to being squashed out of round has an upper bound of the same kind and a lower bound of exactly nothing, because fibres free to slide resist a change of shape with nothing at all. That is why nobody could ever compute the aspect ratio of a flattened thread.

The resting band with two coefficients in it. The range of extensions a sheeting can be left in, at rest and while being agitated, at three ratios of kinetic to static friction. The upper bar of each pair is the stuck band, held by the static coefficient; the lower is the band a cloth being shaken can be left in, held by the kinetic one. Agitation narrows the band but by less than the ratio of the coefficients: at 0.75 the narrowing is 0.868. The restoring force is not linear in the extension, so cutting the friction by a quarter does not move the band's edges by a quarter of the way in. What the bars cannot show is where a given piece of cloth actually stops inside its band, which depends on which side it came from.

After the loom

Why agitation helps a cloth relax

Every standard relaxation procedure agitates: tumble it, wash it, steam it, work it. The explanation given is that agitation lets the fabric find its own dimensions, which is true and is not a mechanism. The mechanism is that a sliding contact resists less than a stuck one — and putting a number on it shows the effect is real, is smaller than the obvious arithmetic suggests, and does not account for what a relaxation procedure achieves.

From a calender's line load to a force at one crossing. A sheeting through a nip loaded at 30 N per millimetre of bowl width, with the cloth in contact over 5.0 mm. The pressure is the first divided by the second, 6.00 N/mm², and the force at one crossing is that pressure times the area a crossing owns — the product of the two thread spacings, 0.1374 mm². So the crossing carries 0.824 N, the sections flatten to 2.42 and 2.63, and the cloth thins from 0.388 mm to 0.218 mm. What the drawing cannot show is that two cloths through the same nip are not given the same treatment: the area a crossing owns varies fivefold across this site's table, and it is a factor in the force.

After the loom

A calender spends the compression for good

Calendering was described on this site as moving a cloth from one thread-section model to another, which was right and had no number in it because the amount of the move was a free parameter. It is a pressure now — and the same nip setting turns out to give two cloths quite different treatments, because the force at a crossing is the pressure times the area a crossing owns.

The size of the capstan correction. The ratio of the capstan crossover to the crossover a sum of independent contacts gives, for every cloth in the table at a friction coefficient of 0.30. It is exactly ln(1 + z)/z, where z is the thread's breaking load times the wrap angle, over the contact force — a quantity with no friction coefficient in it at all. That is why the earlier result that μ·L* is exactly constant survives this correction to twelve figures: μ was only ever in the factor outside the logarithm. The correction is largest for the duck, whose coarse strong yarn makes z large, and smallest for the batiste. What the rows cannot show is that a real cut edge frays at a friction nobody measured on that particular cloth.

Cloth doing a job

How far a cut edge frays

A seam allowance, a fray width and a tuft's bound length are the same number wearing three hats, and the earlier estimate of it was four times too long. Correcting it moves the whole table across the boundary an ordinary allowance sits on — from four cloths holding and four slipping, to all eight holding — and turns a specification argument into a different one.

The crimp ratios a dense shirting can have. Poplins in 15 tex warp and 20 tex weft at 22 picks, from 32 ends per centimetre to 52. Each bar is the interval of warp-to-weft crimp ratios the construction admits at all: the warp must supply at least the thickness the weft cannot reach, and at most what it can reach itself. The rule at one is this site's standing default. It sits inside the interval up to 44.18 ends per centimetre and outside it beyond — so for a dense shirting an equal division of the crimp is not merely the wrong assumption but a geometric impossibility. The 44-end poplin this site's own cloth table called impossible for a long time sits a fifth of an end below that limit, which is why the solver failed on it: its feasible interval was real and narrow, and a bisection on the whole range walked away from it.

Setting and geometry

The cloth that was called impossible

This site's own table of fabrics carries a note saying a real 44-end poplin has no solution in its geometry at all, and that the poplin row was therefore set at 32 ends. The cloth solves. What had no solution was the search — a bisection that treated a state it could not reach as evidence of having gone too far, and walked away from the answer every time.

Mechanics and drape

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

Weaves

A float presses on nothing

The rung below expected the float correction to change how a satin's hold compares with a plain weave's, by something like the ratio of their interlacing rates. It does not change the comparison at all — the interlacing rate leaves the answer outside the logarithm and divides straight out of any ratio. What it changes is the absolute answer, by a factor of four, for every weave alike.

The two halves of a beat-up force. The force the reed must apply per metre, for a sheeting, against the number of picks that are still sliding against the warp. The elastic half — the warp tension times the crimp's elasticity with respect to the pick spacing — is 573 N/m and does not depend on the zone at all; that cancellation is exact and is the result the rung below established. The frictional half is 1133 N/m per sliding pick and is nothing but zone. Against a reported 400–1500 N/m, that leaves room for at most 0.82 picks sliding — so the fell region a weaver can see, ten to fifty picks deep, is not the same quantity as the picks that are still moving. What the rows cannot show is that this is a static friction throughout, and a beat-up is a blow.

Setting and geometry

The half of the beat-up that is all zone

The elastic half of the beat-up force is exact and the length of the beat-up zone cancels out of it, which is this site's own result and disagrees with every practical account of weaving. The frictional half is nothing but zone — and requiring the total to match the force a loom is actually built to apply puts the number of picks still sliding at about one.

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

The hairs are what touch

The pressure inside a twisted yarn falls to exactly zero at its surface, so the outermost fibres are held by nothing and some of them stand off. A yarn therefore has two diameters — the one its mass gives and the one a neighbour meets — and only the first is in the arithmetic.

The crown line of every four-by-four draft there is. All 22,874 four-by-four drafts in which every end and every pick interlaces at least once, at sheeting's construction, counted by how much horizontal crown line each carries per square millimetre. The bar at zero holds 2 of them — the two plain weaves, and nothing else in the catalogue. Every other draft has a float somewhere, and a float is a plateau, and a plateau is a line of constant height. So the whole catalogue divides into two drafts that touch at points and 22,872 that touch along lines, with no intermediate case, because a float is either present or it is not.

What cloth is

Two drafts of twenty-two thousand

Every four-by-four draft there is, measured by how much horizontal crown line its surface carries. Two of them carry none — the plain weave and its translation, and nothing else in the catalogue — and the quantity turns out to be smallest for the most balanced cloths and largest for the most one-sided, which is the opposite of what a float count suggests.

A damask's figure and ground trade places when the cloth is turned. A satin 8 figure on a sateen 8 ground in sheeting — one cloth, one set of threads, one sett, and the ground is the figure's own complement. Their total specular areas are within a few per cent of one another, so neither is intrinsically the brighter. What differs is the direction: the figure's crowns run with the warp and the ground's with the weft. So the contrast between them is 2.0-to-one with the light coming from 8° and 0.47-to-one from 90° — it reverses, exactly, a quarter turn apart. That is what makes a damask visible in one colour, and it is not the step in its surface: the step is fifty micrometres and returns no light at all under a diffuse illumination, while this contrast is a factor of 2.0 and is present whenever there is a direction in the light.

Pattern and colour

A figure shows by its shine, not its step

A damask is one cloth in one colour and its pattern is plainly visible. This collection attributed that to the step in its surface — fifty micrometres of relief, computed from the interlacing rates. The step is real and returns almost no light. What makes the figure visible is that its crowns run at right angles to the ground's, so the two trade places when the cloth is turned.

single jersey, as loops. Three courses of the same structure drawn as yarn. A knitted needle takes a new loop through the one below; a tuck takes the yarn into the loop below and leaves it on the needle, so that loop is held for another course; a miss floats straight past. Six needles is as many as a loop diagram can carry, which is why the array beside it exists.

Knits and other structures

A jersey has two surfaces

The face of a plain knit shows the legs of its loops, which run along the wale; the back shows the heads and feet, which run across it. So the two faces carry their crowns at right angles — the same situation as a damask's figure and its ground, in a fabric with no warp, no weft and no float.

$What it costs to touch a cloth. The pressure needed to bring a stated fraction of the plan into contact, for plain, 2/2 twill, satin 8 in sheeting. Reaching two per cent of the plan takes 2.83 kPa on a plain and 0.05 on a satin 8, a factor of 55. The stiffness in this figure is fitted and is labelled as such. A yarn's resistance to being squashed out of round has no lower bound at all — a bundle of fibres free to slide is a fluid in cross-section — so no bracket exists to compute this from, and what is used is the value this collection fitted to measured fabric thickness. Every curve moves together across its published range, which is why the ratio between weaves survives and the absolute values are quoted with the fit named.$ What cloth is

How much of a cloth is touching

Press a fabric against a flat plate with the weight of a hand and ask what fraction of it is actually in contact. The bearing curve answers, and the answer is about four per cent — of which the great majority is not the cloth's surface at all, but the hairs standing off it, which nothing in this arithmetic can see.

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

A cloth is more opaque than it is closed

Opacity is not cover — this collection established that already and left the discrepancy attributed to the thickness of the threads. Part of it is not in the threads at all. A hair standing in a hole blocks light exactly as well as a thread does and costs nothing in air.

Whether a cloth's hairs can reach one another. n_A λ² for each construction in this site's table — the hairs per square millimetre times the square of their own length, which is the pure number that asks whether a hair can touch its neighbour. It is a count times an area, so it has to be a pure number. Every one of them is under one, which means no ordinary woven cotton cloth has a hair layer at all: it has isolated whiskers on a bare surface. The dashed line is the threshold. The spread across the whole table is only 1.9-fold, because the density goes as the sett times the root of the count and those move in opposite directions as a cloth is made finer — so construction is almost powerless here, and everything that crosses this threshold does so by finishing rather than by weaving.

Knits and other structures

A knit gives up its fibres more easily

Knitwear pills and shirting does not, and the fibres are often the same fibres. The difference is a count of yarn per unit area and a pressure between threads, and both of them push a knit over a threshold that a woven cloth of the same yarn cannot reach.

$The presser foot sinks 1.8 µm into a 2/2 twill. A thickness gauge presses a flat foot onto the cloth at 1 kPa and reads the gap. It does not read the geometric thickness. The foot sinks until the area it is touching can carry the load, and that is 2.57% of the plan at a depth of 1.8 µm — so a 2/2 twill in sheeting whose outside stands 381.6 µm apart measures 379.8 µm. How far the foot sinks is a property of the draft, because the bearing area near the top is, and a weave with plateaux stops the foot in a fraction of the distance a plain weave lets it travel. The transverse stiffness used here is fitted to measured fabric thickness rather than predicted: across its published range the reading moves between 377.0 µm and 380.6 µm.$ What cloth is

A thickness gauge reads the draft

A presser foot does not stop at the top of a cloth. It sinks until the area it is touching can carry the load, and how far that is depends on the shape of the bearing curve near the top — which is a property of the weave. So there is a weave term inside a measurement nobody thinks of as a weave measurement, and it is worth about one per cent.

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.

Pattern and colour

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.

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

Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left.

Knits and other structures

The relaxed knit is not at a minimum

Differentiate a loop's bending energy along the fabric instead of across it and the answer should be zero, because a relaxed fabric is one nothing is pulling. It is not zero. It is tens of newtons a metre, downhill in both directions at once — and the three relaxation states everybody measures run the wrong way up the slope.

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.

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.

The surface of an 8-end shading. The height of the cloth's own surface at each tone of an 8-end shading, for the two chains, on a sheeting at 0.50 N in the end. A thread presses on the thread it crosses only where it turns, so a region that turns more often is pressed more often and finishes thinner — and firmness rises towards the midtone of a shading. The spread chain therefore sinks 84 µm between its ends and its midtone and the consecutive chain 43 µm, with 2 of its steps at exactly one thickness against the spread chain's 0. The tone scale is level by construction and the surface under it is not. What the plot cannot show is the light: a step in the surface reads as a line under a raking beam whatever the tone is doing, which is why a relief nobody specified is visible at all.

Pattern and colour

A tone ramp is a valley, and the satin digs it

A shading's tone is exact and its lustre is measured; its thickness is neither, and nobody specifies it. A firmer weave is pressed harder at every crossing and finishes thinner, so an eight-end shading sinks eighty-four micrometres between its ends and its midtone — about a third of the cloth's whole thickness, on every cloth tried. Build the same chain on a twill instead of a satin and the sag is exactly nothing.

Knits and other structures

What a knit gives when it is pulled

How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight.

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 460 — a ratio of 0.575, 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 340 turns per metre of its own: its surface fibres lie at 10.1° 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.

Cloth doing a job

A sewing thread is a different animal

It is folded, balanced, lubricated and finished, and every one of those is an answer to a requirement no weaving yarn has. The lubricant is the interesting one: it makes the thread sewable by lowering the friction that was holding its own fibres together.

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

What cloth is

Two hairiness meters read two moments

The trade has two instruments for yarn hairiness and thirty years of failing to predict either from the other. They are not measuring the same thing badly. One reports the first moment of a distribution and the other reports a tail probability, and two functionals of one curve are related only through a parameter neither of them reports.

$A cloth loses its strength long before it loses its mass. Rubbing a 2/2 twill in sheeting down, plotted against how much of its own solid volume has gone. The lower curve is the fraction of the plan the rubbing is touching; the upper is the fraction of the warp's section that has been cut away. They are wildly different because the wear is spread and the damage is concentrated: material comes off the whole surface, but it comes off every thread at the same place, and a thread breaks at its thinnest place. At one per cent of the mass gone the section is already 5% smaller. That is why a fabric that looks barely worn fails a strength test, and why abrasion resistance measured as mass loss and abrasion resistance measured as residual strength are two different quantities that are quoted as one.$ Cloth doing a job

A cloth loses its strength before its mass

Rub a fabric and it sheds material from all over its surface, but it sheds it from every thread at the same place — and a thread breaks at its thinnest place. So the strength gone is always several times the mass gone, the ratio is computable from the bearing curve, and it is worst for the weave whose crowns are points.

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.

Knits and other structures

How far a knit could go if its yarn were the limit

The yarn in a stitch allows three hundred and twenty per cent course-wise extension before the straight line between two interlacings reaches the thread spanning it. A jersey jams at about a hundred. The factor of three is the finding: what stops a knit stretching is not the loop running out of yarn.

The pressure a 23° twist puts on its own fibres. A twisted yarn squeezes itself. Every fibre under tension at a radius pulls inward with sin²θ of its tension per unit length, and integrating outward to the surface — where the pressure is zero by definition, because there is nothing outside to push against — gives p(r) = ½σ(cos²θ(r) − cos²α) in closed form. At a surface angle of 23° the pressure on the axis is 7.6 per cent of the core fibre's own axial stress and the mean over the section is 3.61 per cent of it. The shading is that closed form and the curve is the same function plotted; the shape is what matters, and its one uncompromising feature is the zero at the edge. The outermost fibres are held by nothing, which is why a yarn is hairy, why a surface fibre is the one that comes away on a finger, and why singeing changes a yarn's behaviour out of all proportion to the mass it removes.

Mechanics and drape

What grips the end of a fibre

A twisted yarn squeezes itself, and the squeeze holds the fibre ends in. Write the slip and the break out side by side and the fibre's own strength cancels — and so does the load on the yarn — leaving a gripped length that depends on fineness, friction and the twist and on nothing else.

Both halves of the twist curve, at 28 mm staple. The falling curve is obliquity and is exact — the affine end of the bracket, computed from the helix and nothing else. The rising curve is cohesion and is the half this collection had declined: a fibre end is gripped by friction under the twist's own radial pressure, the fibre's strength and the yarn's load cancel out of the comparison, and what is left is a critical length that depends on the twist through a pure function of the angle. Their product has a maximum at a twist factor of 3101 — 693 turns per metre at 20 tex, a surface angle of 20° — which is inside the range spinners use. The scale of the rising curve is fitted, through a contact efficiency of 0.05, and moving it moves the optimum; what it cannot move is the ordering between two staples or two fibres, which is what the two claims made from this figure are about.

Setting and geometry

The other half of the twist curve

This collection computed the falling half of the strength–twist curve exactly and declined the rising half as being out of reach. It is not out of reach. With the grip derived rather than assumed, the optimum comes out at a twist factor — and the same twist factor at every count, which is why the trade quotes twist factors at all.

$Two cloths touch on a fraction of what one cloth does. A 2/2 twill in sheeting pressed against a flat plate, and the same cloth pressed against another piece of itself. At an approach of 24.9 µm the single surface is touching 15.87% of the plan and the pair 3.130% — a factor of 5. The reason is that a gap between two rough surfaces is the sum of two depths, so both surfaces have to be near their own maxima at the same place, and the chance of that is the product of two small numbers. The pair's curve is the convolution of the two height distributions, computed exactly on histograms rather than fitted to a Gaussian — because a woven surface is bimodal and nothing about it is Gaussian.$ After the loom

Friction is two surfaces, not one

A gap between two rough bodies is the sum of two depths, so two cloths face to face touch on the convolution of their height distributions rather than on either of them. At the approach a light touch produces that is twenty times less contact than the same cloth against a plate — which is why a fabric's friction against a plate and against another fabric are two different measurements.

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.

A colour order against a 2/2 twill. The visible face of a 2/2 twill under 2 colour orders, drawn at the repeat the divisor arithmetic allows and outlined at the repeat the surface has. A filled cell is a dark thread on the face, which is the warp's colour where the warp is up and the weft's where it is not — so none of these patterns is in the draft, and the draft is the same in all of them. The colour period and the weave repeat beat exactly as a reed's grouping beats against a weave: the surface repeats on the least common multiple of the two, which here is 8×8 and 4×4. What the panels cannot show is colour: the two threads are drawn as filled and empty, and two colours of similar value make a pattern far weaker than this.

Pattern and colour

A colour order beats the weave it is threaded on

The reed's grouping beats against the weave repeat and the arithmetic is a least common multiple. A colour order is a second grouping of the same warp and the arithmetic is identical — but where the reed's beat is a fault to be dented out of a cloth, the colour order's beat is the pattern the cloth is sold for. Across 472 colour orders on four weaves the divisor bound is the surface's exact repeat in 470 or more, and the handful that beat it have no pattern left at all.

plain over 5-end satin: what each layer's warp does. A warp end of each layer of a double cloth, in section over the same span of cloth, with the length of warp each eats drawn beneath. The crimps come from Peirce's geometry asked about a thread that bends at its own average rate rather than at every crossing, and they are 14.35% for the plain and 2.03% for the 5-end satin. They are not the same, so the two layers consume warp at different rates: a hundred metres of cloth takes 114.4 m of warp from one and 102.0 m from the other. A beam delivers one rate, so the difference has nowhere to go and accumulates with the length woven — one pick spacing after 4 mm. What the sections cannot show is the yarn's thickness, which is exaggerated so the path is legible; the crimps beside them are computed at the real diameter and are not read off these drawings.

Compound and figured cloths

Two layers need two beams

A layer weaving a metre of cloth eats one plus its crimp metres of warp, and a beam delivers one rate. A plain face over a five-end satin back differs by twelve percentage points of crimp, which is twelve metres of warp over a hundred-metre piece and one pick spacing of slack after four millimetres of weaving. The difference has nowhere to go and does not settle — so the only double cloth that can share a beam is two layers of the same weave at the same sett.

Mechanics and drape

The singles inside a ply are not the singles

Folding leaves each single with a fraction of its own twist, which should ruin its grip on its own fibres. It does not, because the ply's helix presses on the singles exactly as a single's helix presses on its fibres — and across the whole practical range the two very nearly cancel.

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.

The magnification a cloth will carry. The largest magnification a moiré can be read at, against the irregularity of the cloth making it. The points are measured: a grating whose spacings are drawn from a seeded lognormal is laid against a perfect one, the fringes are found from the phase difference, and the gain is recorded at which their count first departs from what the ideal beat predicts. The product of the irregularity and that gain comes out at 0.82 to 0.84 across every irregularity tried, so the ceiling is 0.84 divided by the coefficient of variation — the solid curve. The dashed curve is the accumulated-wander model, in which position errors random-walk and the ceiling goes as the inverse square; it is wrong by a factor of 42 at a two per cent irregularity. What the plot cannot show is what a cloth's spacing irregularity actually is: it has not been measured, and the curve is therefore a prediction with an unmeasured input.

Pattern and colour

A moiré is a vernier, and it magnifies the error too

Two gratings a per cent apart in pitch beat at a hundred pitches, so a moiré reads a pitch difference at a hundred times — which is what a vernier is. The magnification is free and its ceiling is not: the fringes split when one thread's own spacing error reaches 0.84 of the pitch difference the beat is built on, so the usable gain is 0.84 divided by the cloth's coefficient of variation, inversely and not inverse-squarely. At an ordinary yarn's spacing irregularity, a moiré carries a magnification of five.

Which end of the bracket a thread in cloth is at. A yarn bends as a solid rod while its fibres cannot slide and as a loose bundle once they can, and the crossover is a curvature: the coherent state demands an axial force in the outer fibre that has to be built up by friction under the twist's own radial pressure. The curves are the crossover radius against twist at four contact efficiencies, the top one being 1.0 — the claim that fibres touch along their whole length, which nobody makes. The rule at the bottom is the radius a thread is bent to by its own crimp in a cloth, about 0.25 mm. Every curve is above the rule by at least 19-fold, so a thread in cloth is at the free end of its bracket at every twist and every efficiency, and the collection's habit of using the lower bound is a result rather than a convention.

Mechanics and drape

Twist decides where in the bracket

A yarn's bending rigidity can only be bracketed, and the bracket is three hundred wide. What decides where a yarn sits in it is whether its fibres can slide — and for a thread bent by its own crimp in a cloth, the answer is not close.

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 site's own 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 anchor fibres that hold it, and an anchor 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 anchors 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 anchor 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.

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.

$Two exponents in a compression curve. How far a flat plate sinks into plain, 2/2 twill, satin 8 in sheeting, against the pressure it is applying, on logarithmic axes where a power law is a straight line. The measured slopes over the light end of the range are 0.50 for the plain, 0.67 for the 2/2 twill, 0.67 for the satin 8 — against two thirds predicted for any weave carrying a float and one half for a weave carrying none. The prediction is one line of algebra: pressure is a stiffness times a strain times a bearing fraction, the bearing fraction is a square root of depth for a plateau and linear in it for a point, so the pressure is the three-halves power in the first case and the square in the second. Nothing is fitted to produce it; the slopes are measured afterwards and compared. At the heavy end every curve bends, because the crowns have merged and the cloth has stopped being a surface and started being a solid — which is a different regime with a different law, and it belongs to the compaction of a fibre mass rather than to the geometry of an interlacement.$ Mechanics and drape

A cloth compresses along its own bearing curve

A fabric's pressure–thickness curve is always fitted with an empirical power law and the exponent is reported without explanation. It is not empirical. At light loads it is two thirds for any weave carrying a float and one half for a weave carrying none, and the two numbers come out of one line of algebra with nothing fitted 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.

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 rib is quietest at a gap of two diameters. The through-thickness force of a one-by-one rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 14.8 mN at 5 diameters against 7.0 mN at two.

Knits and other structures

A rib is quietest at two diameters

Open the beds of a rib and everything about it should get stronger. It does not. The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters and rises on both sides of it, because a crossing's climb and an interlacing's own climb cancel there.

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.

Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Pattern and colour

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

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.

Weaves

A crepe is flat in its draft and not in its surface

A crepe weave is chosen by pushing the draft's correlations as flat as they will go. That criterion turns out to tie: on the base this collection uses, 4,416 of the 5,040 rearrangements reach the floor. Sixteen of them additionally spread their crown line perfectly evenly — and the crepe actually drawn is not one of the sixteen.

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.

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.

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

What holds a nap in a knit

A fibre buried in a woven cloth breaks rather than slides once about thirty-four millimetres of it is held, and a cotton staple buries about fourteen. In a knit the same figure is over a metre — seventy-five times the burial available — so nothing is ever close, and a raised knit sheds for the whole of its life.

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.

What holds a thread in, as two factors. The two quantities whose product is the grip on a buried thread, for a woven poplin and a jersey of the same yarn. Each contact in the knit is lighter by 8.0 times, and the contacts are further apart by 5.6 — one per half loop length against one per thread spacing. They multiply rather than competing, so the grip per millimetre of buried thread is 31 times weaker in the knit, and the crossover length at which a thread breaks rather than slides moves with it: 11.6 mm in the cloth and 461 in the knit. The two factors are drawn separately because their product is two orders of magnitude and a bar chart of it would put the knit's bar below the width of a line.

Cloth doing a job

The knitted pilling criterion gets its number

Knitwear pills and shirting does not, and the standing explanation here has been that a knit presents more exposed yarn under less pressure between its threads. The second half had no number. It has one now, and it is bigger than the argument needed: the grip on a buried fibre is thirty times weaker in a knit than in a woven cloth of the same yarn.

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.

What cloth is

A fabric is a population of contacts

Only a fifth of a knitted fabric's yarn is inside a diameter of its neighbour. So a fabric's friction lives in a fifth of its length, and every calculation this collection makes about withdrawal, slippage and fraying has assumed it lives everywhere.

A jacquard harness 130 cm wide under a 150 cm fall. A jacquard harness in front elevation, to scale: hooks spread over 40 cm, cords fanning down 150 cm to a comber board 130 cm wide, and hanging straight from the board to their mails. The centre cord is vertical and lifts its mail by the whole 10 cm hook lift; the edge cord leans 16.7° and lifts its mail 9.60 cm, 96.0% as far, and its bend at the board adds 9% to the hook's load at a friction coefficient of 0.3. What the drawing cannot show is the tie that decides which hook feeds which hole, which in a real mount is not the simple spread assumed here.

Compound and figured cloths

A jacquard harness needs three half-spans of height

A jacquard has no front shaft and no back one, so every end takes the same shed — as long as every cord hangs straight. Across the width they cannot: the hooks sit in a machine a few tens of centimetres wide and the comber board is as wide as the cloth, so an edge cord leans and its mail rises by the difference of two hypotenuses rather than by the hook's lift. Holding the edge shed within five per cent of the centre's takes a fall of about three times the edge cord's sideways reach, and that is a height a room has to have.

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.

Mechanics and drape

A force is what an energy does when a crossing moves

The force a thread presses its neighbour with is the rate its bending energy changes as the crossing is displaced. Solved as a constrained minimisation, that number arrives with the shape rather than after it — and the same force, recovered a second time from the curve's own equilibrium, agrees to a tenth of a per cent.

Setting and geometry

A woven thread has no room to bend

Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.

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.

Knits and other structures

A jersey's course has no writhe

The mechanism everybody quotes for why a hard-twisted jersey leans is that the fabric relieves the yarn's twist by writhing. This collection's own solved course has a writhe of minus six parts in a million, and it is not small — it is exactly zero, by a symmetry, and the symmetry is a statement about what the model left out.

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.

A repeat across a 1800-end warp. Four repeat widths laid across the same 1800-end warp, drawn at the warp's own scale. The pale bands at the two ends are the selvedge threading, 24 ends each, which weaves its own firmer weave and is not part of the design. Between them the body is ruled into whole repeats, alternating so they can be counted, and the marked bands at the two sides are the remainder — the part of a repeat that did not fit, split between the two selvedges because the trade centres the pattern. None of these four repeats divides the body exactly, and the leftovers run from 2 to 24 ends. What the drawing cannot show is what the break looks like: a quarter of a repeat at the selvedge reads as a border and half of one reads as a mistake, and where the line between those falls is a judgement.

What cloth is

A repeat has to fit the width

A repeat tiles the plane and a warp has two edges, so somewhere between them a repeat is cut through. The set of repeat widths that divide a warp exactly is the set of divisors of its body, and a body of a few thousand ends has a few dozen — two to eight per cent of the candidates. So a designer choosing a repeat for any reason except the width chooses one that does not fit, and the leftover averages half a repeat, split between the two selvedges.

$20 tex yarn in one layer and in 2. Sections across the width, to scale, of a cloth of 20 tex cotton at a cover of 0.8, and of the same yarn per area divided into 2 layers two ways: by count, 10.0 tex at the same sett, and by sett, 20 tex at 1/2 of the ends. Divided by count the cloth is 1.41 times as thick with a cover of 0.57 in each layer; divided by sett it is 2.00 times as thick with a cover of 0.40. At the free end of the yarn's stiffness bracket both are exactly as stiff as the single cloth; at the coherent end the first is 0.50 times as stiff and the second 1.00. What the sections cannot show is crimp, which thickens every layer by an amount the weave decides.$ Compound and figured cloths

A double cloth is only softer if its yarn is set

A double cloth is sold as weight without stiffness: two light cloths in place of one heavy one. Divide the same yarn into two layers and the cloth is √2 or twice as thick, but at the free end of a yarn's stiffness bracket — where an unset yarn sits — its bending rigidity does not move at all, because it is the number of fibres across the width times the stiffness of one. Only a set yarn makes the double cloth the softer, and stitching the layers together pushes it the other way.

Pattern and colour

A figured sheer is a negative from one side

A net curtain with a pattern in it carries two patterns, one for each side, and by day they are opposites. A more open figure in a white voile is a dark figure a quarter below its ground from the street and a light one five per cent above it from the room, and after dark the two views trade places exactly. The figure vanishes from the street at one light ratio and from the room at another. And a figure can be made that the room cannot see at all while the street sees it nearly black — along one line of openness and thread tone, and never from both sides at once.

How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip.

Mechanics and drape

What a loop presses with

A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

Cloth doing a job

A seam must give what the knit gives

A knitted seam fails because it is too short, not because it is too weak: the thread in it is nearly two thousand times stronger than the load it carries. What decides whether it survives is one line of geometry — the extension a seam can reach is twice the fabric's thickness times the stitches per unit length.

$A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.$ Setting and geometry

How dense a knitted fabric is

A fabric's areal weight is what the trade specifies and it says nothing about bulk. Divide it by a thickness and the answer is a density — 0.40 grams a cubic centimetre for a jersey, a quarter of the fibre it is made of — and that quarter, the share of the volume that is not air, is the number every other property follows.

What leaving the plane costs, which is less than nothing. Bending energy a stitch against how far the yarn climbs between one interlacing and the next, for a 20 tex cotton jersey at a 3.5 mm loop. The curve falls. A climb is not an extra bend added to a curve that was otherwise unchanged; it is part of the straight line the thread has to span, so a longer climb leaves less slack to be spent on curvature. At a jersey's own climb of one diameter the energy is 24395 nJ against 25060 for the planar model — 2.7% lower, where an earlier estimate on this site put the ride at three per cent of the loop's bending and had the sign the other way round. At a rib's climb of three or four diameters the fall is 33%.

After the loom

A rib's relaxation is not its bending either

A jersey does not settle where its bending energy is least, and this collection has said so for several rungs. A rib does not either, and the model now says how far from least it would have to go: sixteen yarn diameters of bed gap before the yarn runs out, against the two or three a machine is set to.

What links what, in the two ways of making cloth. The linking number between two adjacent courses, for a knitted tube of 12 wales, for the same tube as this collection's model draws it, and for a woven cloth's two thread systems. The fabric's is 12 — one for every needle loop drawn through the loop below. The model's is -0.0000, because it places the interlacing at a point where two centre lines pass a diameter apart and two curves passing beside one another are not linked. The woven cloth's is -0.0000 and always will be, at any crimp and for every weave. That last row is not a defect of any model: a woven cloth really is unlinked, and it is the reason it frays where a knitted fabric runs.

Knits and other structures

A point cannot link

A knitted fabric of n wales has a linking number of n between every pair of adjacent courses. This collection's model of the same fabric has zero, and it has zero because the interlacing was declared to be a point where two centre lines pass a diameter apart — which is a near miss, and a near miss is not a knot.

What a group of 2 threads behaves as. A group of 2 threads of 250 µm with nothing separating them, beside the single thread of the same width and the single thread of the same yarn, all drawn to one scale. The bars are the group's four readings as ratios. Cover is a width and adds, so the group covers exactly what a 500 µm thread would — with 50 per cent of its yarn. Bending rigidity is a second moment and does not add: 2 threads free to slide give 12.5 per cent of the thick thread's and the same 2 fused into one body give 62.5, so a real group is somewhere between and where depends on friction. Against the thread of the same yarn the two readings point opposite ways: the group covers 1.41 times as much and bends 0.50 times as stiffly if free. What the drawing cannot show is the friction that decides where between the two limits a finished cloth sits.

Weaves

A group is one thread for cover and two for bending

The rung below leaves a limitation standing: two ends with nothing between them lie touching, and whether they behave as one thread of twice the diameter was said to depend on twist, hairiness and finish. Three of the four measures have exact answers with no friction in them and no two agree. A pair covers exactly what a double-diameter thread covers, with half its yarn, and bends at between an eighth and five eighths of its rigidity — and at equal yarn it covers forty per cent more and bends half as stiffly.

What matching a pattern costs a cutting room. The cloth a matched panel needs beyond its own length, against the pattern repeat, for 6 panels of 70 cm. Every panel of a patterned cloth must start at the same phase of the repeat or the pattern breaks at the seams, so a panel's cut length is rounded up to a whole number of repeats. The bar is the allowance a cutting room budgets — a whole repeat a panel, because a panel's length is not a multiple of anything — and the mark is the waste actually expected, which is half a repeat. The two differ by (L + r)/(2L + r), which is between a half and two thirds and is nearer two thirds the larger the repeat. The rows marked in the second colour are the repeats that happen to divide the panel exactly and waste nothing at all, which is what makes the real cost jagged rather than smooth. What the bars cannot show is nesting: a cutting room lays many panels on one length and a short panel can sometimes be taken from another's waste.

What cloth is

A repeat has to fit the panel, and the panel is cut

The warp's width is fixed at warping and a piece's length is not, so the fitting problem in the two directions is not the same problem. Along the length a repeat has to fit a *panel*, because every panel of a patterned cloth must start at the same phase or the pattern breaks at the seams — and the allowance is a whole repeat per panel. A ten-centimetre repeat costs a seventy-centimetre panel twelve and a half per cent and a sixty-four-centimetre repeat costs it forty-eight.

Compound and figured cloths

The doup end pays for the crossing

A leno holds its picks because its doup end crosses under its partner and comes up on the other side — half a turn at every crossing, whatever the sett. That half turn is also a length. When 0.25 mm ends cross every millimetre, the doup end travels 22.5 per cent further than its partner: twenty-two metres of extra warp over a hundred-metre piece, and a shared beam a pick short within four and a half millimetres of cloth. The extra falls with the square of the crossing interval, which is why a leno is woven from two beams and its crossings are spaced as far apart as the cloth allows.

A 1.55 mm net 50 mm behind a 0.3 mm voile, from 0.6 m, 1.5 m, 4 m. A 1.55 mm net 50 mm behind a 0.3 mm voile, drawn across 40 mm of the near layer at true pitch as seen from 0.6 m, 1.5 m, 4 m. Two grids this different beat through a harmonic: the net's k-th against the voile's first, for the k nearest the ratio of their pitches as the eye sees them. From 0.6 m that is the fifth, in register every 6.2 mm; From 1.5 m that is the fifth, exactly in register, with no fringe; From 4 m that is the fifth, in register every 14.9 mm. What the strips cannot show is how strong each family is, which falls with the harmonic, nor the net's second family of threads at right angles.

Pattern and colour

A net over a voile beats through a harmonic

Two identical sheers hung apart make a moiré by perspective alone. A net in front of a voile is not two identical sheers — its mesh is five times the voile's pitch — and it beats anyway, through the net's fifth harmonic, which is a grid 3.3 per cent coarser than the voile. With the net behind, that is a pair of sheers with its coarser layer at the back, and the fringes vanish at exactly 1.5 metres. Closer in, the harmonic changes, the fringes dissolve into a texture twice the net's pitch and re-form, and they vanish again at 17 centimetres.

Mechanics and drape

What a loop model still cannot say

A planar rod with a natural curvature and point contacts gets a knit's forces, its modulus and its extension. It does not get torsion, it does not get the third dimension the interlacing actually needs, and it does not stop adjacent courses passing through one another — which is why its extension ceiling sits three times beyond any jersey.

A knit is warm because of where its yarn is not

Warmth is a thickness of still air, and until a knitted fabric had a thickness there was nothing to compute. It has one now, and the answer is that a rib's warmth is a machine setting: opening the beds from two diameters to five nearly trebles the fabric's resistance without changing a gram of yarn.

Two numbers that stay at zero however large the fabric gets. The linking number of two adjacent courses, and the writhe of one course per wale, for tubes of 6 to 20 wales. Both sit at zero and stay there: the largest departure anywhere on the plot is 1.5e+1, which is the sampling. A quantity that should grow with the fabric and does not is the cleanest kind of null result: the model has the geometry of knitting and none of its topology, and making the fabric bigger does not make the topology appear.

Knits and other structures

Five symptoms of one omission

Five things this collection recorded as unexplained, found in four different ladders over three years of work. They are the same defect seen from five directions, and the defect is one sentence written for good reasons with no visible cost at the time.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 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 1.59 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.

Setting and geometry

Why a slack yarn snarls

Let go of a twisted thread and it wraps on itself. That is not the yarn being badly behaved: it is a buckling, it has a criterion, and the criterion turns a nuisance into an instrument for measuring the one constant this collection cannot pin down.

How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

After the loom

A yarn that has been set has no torque

Every torque on the torsion ladder assumes a yarn is elastic in twist for ever. It is not: a steamed yarn's residual torque is gone, its twist is unchanged, and the process that removes one without the other is the trade's whole answer to liveliness.

A warp rib's cord, at doublings of 2 and 4. One warp end drawn in section along the cloth, over the pick groups of a warp rib, at doublings of 2 and 4 and at one scale. The cord's crests are the pick groups and its two dimensions come from different places. The height is the weft's own crimp amplitude and Peirce's closure condition caps it at the two yarn diameters together — 500 µm here — so it runs 304 µm at a doubling of 2 and 361 at 4, which is 61 and 72 per cent of the ceiling. The pitch is the doubling times the pick spacing and has no ceiling at all. So a larger doubling gives a taller cord and a wider one, and wider faster: the aspect falls from 0.40 to 0.29. What the section cannot show is what a finish does, which flattens the cord without changing either the pitch or the ceiling.

Weaves

A cord's height has a ceiling and its width has none

A warp rib's cord is a wave, and its two dimensions come from two different places. The height is the weft's own crimp amplitude, and Peirce's closure condition caps it at the two yarn diameters together — 500 µm for a quarter-millimetre yarn, of which a 2/2 rib reaches 304 and a 6/6 rib 391. The pitch is the doubling times the pick spacing and has no cap at all. So a bolder rib is taller and wider, and wider faster: the aspect falls from 0.41 to 0.22.

What cloth is

A profile draft is a notation whose alphabet is weaves

The rung below measured four notations for a single weave and left open the notations for something larger. A profile draft is the first of them: a grid of blocks, each carrying a figure weave or a ground weave. Its image is enumerable and it is tiny — every pair of two-by-two weaves against every assignment of two-by-two blocks reaches 306 of the 22,874 interlacing four-by-four drafts, which is 1.34 per cent. And the 306 are the ones anybody weaves.

A bouclé cloth jammed at its loops and at its count, overfeed 0.8. Sections across 36 mm of plain-woven cloth in a bouclé of 89 tex — a 20 tex core, a 30 tex effect overfed 80% with loops every 3 mm, and a 15 tex binder — drawn to scale. The loops stand 2.10 mm off the core, so the yarn's outline is 4.43 mm across while the diameter its count implies is 0.352 mm, 12.6 times smaller. Jammed at the outline the cloth takes 1.13 ends a centimetre and weighs 20 g/m²; jammed at the count, 14.2 and 252 g/m². What the drawing cannot show is where between the two a real cloth jams, which is how far its loops interleave with their neighbours'.

Compound and figured cloths

A bouclé is set by its loops and weighed by its count

A bouclé yarn has two diameters: the one its count implies, a third of a millimetre, and the one its loops occupy, over four millimetres. A cloth jams where its yarns touch, and a bouclé's yarns can touch at either — so its jamming sett is a bracket twelve and a half times wide, from 1.1 to 14 ends a centimetre. The weight follows the count wherever in that bracket the cloth is set, which puts the open end at a twelfth of the close end's weight with four per cent of its area covered by yarn.

Setting and geometry

How much yarn has to hang

The tension a thread needs to stay straight, converted into the only unit anybody has an intuition for: the length of the yarn's own weight. One bound says two metres and the other says six hundred, and everybody who has handled thread already knows which.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. 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.

Knits and other structures

The fabric that does not fit

Every solve in this collection minimises an energy over a centre line, and a centre line has no thickness. Nobody had checked whether the fabric that comes out of it can be built. It cannot: two adjacent courses of the relaxed jersey approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere.

After the loom

The twist a fabric gives back

A T-shirt that hung straight in the shop has its side seam round the front after three washes. The torque was there all along, the setting had hidden it, and the water gave it back — which makes spirality a finishing failure rather than a knitting one.

What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

Cloth doing a job

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

The crossed shed, with the back standard 4 shafts behind the doup. A leno's crossing end in section from the fell to the back rest of an ordinary broad loom, in the crossed shed. The doup 300 mm from the fell lifts it 50.0 mm; its back standard 64 mm further back leaves it 60.7 mm down, beneath its partner. Between the two eyes the end climbs 110.7 mm in 64 mm, and the whole path is 70.2 mm longer than the straight line, 5.85% of the free warp, of which 63.8 mm is that one climb. The same end in the open shed, lifted at its back standard, is 7.2 mm longer, and an ordinary end on the doup's shaft 5.5 mm: the crossed shed is 12.7 times the ordinary shed. Vertical scale exaggerated 3 times. What the drawing cannot show is friction at the eyes, which would keep the extra length in the short span between them.

Compound and figured cloths

An easer gives back the kink the crossed shed puts in

A leno loom gives its crossing ends slack at every crossed shed, and the obvious reason — the doup carrying its end sideways — is worth a hundredth of an ordinary shed. The length is one heddle further back. In the crossed shed the doup lifts the end on the far side of its partner while the end's own back heddle, a few shafts behind, holds it down, and between those two eyes the end climbs eleven centimetres in six and a half. On a loom that stretches an ordinary end 0.46 per cent, that is 5.85 per cent — nearly thirteen times as much — and no position of the back heddle takes it below four.

A white thread 70% open figure behind a net 60% open, from both sides. A lozenge figure in the inner curtain of a pair, drawn as the street sees it and as the room sees it through a plain net 60% open, with the street 30 : 1 as bright as the room. From the street the figure's contrast against its ground is −7.0% where a single curtain would have given −24.6%; from the room it is +6.9% where a single curtain would have given +5.4%. Each panel is shaded relative to its own brighter region, both at one gain, so the two steps are to scale against each other, and the contrast is stretched 9 times to be visible at all — the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the light between the two curtains, which is 19.80 times the room's illuminance from the street and 0.57 from the room, nor the thread optics, which are assumed.

Pattern and colour

A net in front gives the figure to the room

The commonest double curtain is a plain net outside a patterned one, and the plain net does not merely dim the pattern. It weakens the figure from the street by a factor of three and a half and strengthens it from the room by a quarter, so a pattern four and a half times stronger outside than in becomes one the room sees slightly better. A figure designed to be invisible from indoors reappears at nearly three per cent, and the day-and-night exchange a single curtain obeyed exactly stops holding at all.

How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

Mechanics and drape

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

Setting and geometry

A snarl comes in one size

The radius a twisted thread coils to is twice its bending rigidity over its torque. Write the torque out and the bending rigidity cancels completely, leaving a number that depends on the twist and on the ratio of two stiffnesses — and on nothing else about the yarn at all.

The section the fabric asks for, beside the one the model drew. A 20 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.167 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 78% of it, which is the closest the fabric's own adjacent courses come to one another — 0.130 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett.

Knits and other structures

The flattening nobody fitted

A yarn in cloth is not round, everybody knows it, and nothing has ever predicted how flat. This collection's own knitted geometry turns out to require a flattening of four fifths — from a solve that knew nothing about flattening, made no allowance for it, and would have been written the same way if the idea had never occurred to anybody.

The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

After the loom

A wet knit's yarn is flatter

A knitted fabric's own geometry demands a flattened yarn, and how flat depends on how much room the fabric leaves. A wet cotton yarn is a tenth thicker than a dry one at the same length, so a wet fabric leaves less room — and asks its yarn to be flatter by an amount the geometry gives.

How hard each grouping holds its own threads. The grip a cloth has on one of its own threads inside a 10 mm seam allowance, for six members of the doubled family at 24 threads per centimetre and a friction coefficient of 0.3. Grip accumulates multiplicatively at every crossing — the capstan equation on Peirce's own weave angle — so it is exponential in the crossings, and the crossings are the interlacing rate times the intersections in the allowance. The family's interlacing rate has a closed form, (a + b)/2ab, which is half the sum of the two reciprocals — so the two groupings enter symmetrically and each one saturates. The dashed line is the thread's own strength: a cloth whose grip falls short of it lets the thread slide out rather than break, which is seam slippage. A 2/2 hopsack is below it at this allowance and a 1/4 warp rib is above, on cloths whose firmness differs by an eighth. What the bars cannot show is the friction coefficient, which is measured and is not a constant of cloth; the ordering holds at every value anybody reports and the sizes do not.

Weaves

What nothing separates comes out together

The doubled family's interlacing rate has a closed form — half the sum of the two groupings' reciprocals — so a seam's grip on its own threads is the exponential of a harmonic mean, and it saturates in each grouping separately. At a ten-millimetre allowance a 2/2 hopsack holds a thread at seventy-four times the applied tension and a 1/4 warp rib at two hundred and eighteen, on cloths whose firmness differs by an eighth.

Two lots of one cloth matched at the top over a 250 cm drop. Two lengths of the same patterned cloth, woven with the warp at tensions of 0.6 and 0.7, which relax by 6.80% and 7.94% along the length, hung side by side over a 250 cm drop and matched at the top, with a mark at every 32 cm repeat. The whole drop is drawn to scale on the left and the last mark at six times the scale on the right: lot B's repeat is 31.61 cm, and 7 repeats down its mark is 27.3 mm from lot A's. A 3 mm match holds for the first 25 cm. What the drawing cannot show is how large an offset an eye accepts across a seam, which the tolerance stands in for.

What cloth is

Two lots of one cloth drift apart down a drop

A pattern repeat is a count of picks, and its length in the finished cloth is whatever that count's loom length became after the cloth gave back its crimp. Two lots woven with the warp held a tenth harder give back 1.13 points more of their length — so matched at the top, two lengths from the two lots are thirty millimetres apart at a 250 cm hem, and visibly apart a quarter of a metre down. The size of the repeat does not enter, and no cutting allowance can take it out.

Bouclé loops on 3 mm binder spacing at overfeeds of 30%, 80%, 150%, 300%, 500%. Loops of an overfed effect thread drawn to one scale, each between two binder points on a straight core, as the curve of least bending energy with its ends along the core. Overfeed 30%: 1.30 times its base in thread, 1.10 mm tall, a bell, against a semicircle of 0.79 mm; Overfeed 80%: 1.80 times its base in thread, 1.95 mm tall, a bell, against a semicircle of 2.10 mm; Overfeed 150%: 2.50 times its base in thread, 2.92 mm tall, overhanging, against a semicircle of 3.94 mm; Overfeed 300%: 4.00 times its base in thread, 4.83 mm tall, overhanging, against a semicircle of 7.88 mm; Overfeed 500%: 6.00 times its base in thread, 7.25 mm tall, overhanging, against a semicircle of 13.14 mm. The dashed arcs are the semicircles, drawn where they fit. What the drawing cannot show is the binder's own path and thickness, which the loop's feet are taken to sit exactly on.

Compound and figured cloths

A bouclé loop is an elastica, not a semicircle

A bouclé's loop was taken to be a semicircle, and the semicircle cannot be right: it meets the core at a right angle, and at eighty per cent overfeed it is wider than the gap it stands in. Solved as what it is — a length of thread leaving the core along the core at two binder points — the loop is taller than the semicircle below an overfeed of two thirds and shorter above it, overhangs at 119 per cent and closes its neck at 559. Nothing of the fibre is in its shape, and the bouclé's jamming bracket at eighty per cent is 11.7 wide rather than 12.6.

Pattern and colour

A curtain is gathered so that it is seen edge-on

A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

Mechanics and drape

What friction has to hold in a relaxed knit

A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

Where a fold's two moments cancel, and where the trade folds. The two moments about a fold's own axis, for 2 singles of 20 tex cotton at 800 turns a metre. The falling curve is what the singles' own residual twist supplies, which the folding takes out of them; the rising one is what bending each single onto its helix costs. They cross at 161 turns a metre, a ratio of 0.201, and the closed form for that crossing is C/(B+C) — the ratio of the two stiffnesses and nothing else. The shaded band is where the trade actually folds, 0.6 to 0.75 of the singles twist. The balance point is nowhere near it, by a factor of three.

Setting and geometry

The folding rule is not a torque balance

Fold a two-fold yarn at about two thirds of its singles twist. This collection has carried that as a bracket copied from the trade and derived nowhere. It is now derivable, the derivation gives a fifth rather than two thirds, and reaching two thirds would need a fibre stiffer in torsion than in bending.

Knits and other structures

A flattening that follows the tightness factor

Eighteen solved fabrics — five loop lengths, three relaxation states, three counts — and the flattening each one's geometry demands falls on a single curve against one dimensionless group. Nothing about the fibre or the count survives except through that group, which is what turns an arithmetical result into a structural requirement.

$A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot.$ Cloth doing a job

A knot is nothing but contact

A knot has no fastening in it. Nothing is glued, hooked, sewn or threaded through a hole: a thread is bent round itself until the friction where it presses on itself is more than the load. That makes a knot the purest contact problem in the subject, and the place to look first for what contact does.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 24.2 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.184 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.140 mm — 0.761 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 76% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

After the loom

What wetting does to the bending limit

A rod cannot be bent to a radius below its own. A relaxed knitted loop sits at twice that limit, and a wet yarn is a tenth thicker in the same loop — so wetting moves a fabric a tenth of the way towards a bend it cannot physically take.

3 widths of cloth at 2% skew and 0% bow, hung level. 3 widths of a 140 cm patterned cloth hung side by side and joined at their selvedges, with three of its pattern rows drawn across each. The cloth's weft is skewed by 2% of the width and bowed by 0%. Hung level, every width starts its rows at the same height, so each seam shows the skew's 28 mm step. The rows' displacement is drawn 8 times its true size against the width. What the drawing cannot show is the ceiling, pole and hem a real hanging is seen against, which is what a tilt is read by.

What cloth is

A skew steps at every seam and a bow at none

A finished cloth's weft rows are seldom square to its selvedges, and on a patterned cloth the weft rows are the pattern rows. A skew — rows straight but slanting — puts its whole rise across every seam where two widths meet: two per cent on a 140 cm width is a 28 mm step, and matching the steps out tips the whole hanging instead, 112 mm across four widths. A bow — rows curving between level selvedges — matches perfectly at every seam and scallops each width in between. A three-millimetre match needs a skew under a quarter of a per cent, and down the seam a lot difference overtakes it.

What a passer-by is looking at, for rooms of four reflectances. The street's view of a window behind a sheer white thread 49% open at 30 : 1, split into the part that is an image of the room, arriving through the holes, and the part that is the cloth's own threads lit by the street. a room reflecting 8% contributes 0.4% of the view; a room reflecting 20% contributes 1.0% of the view; a room reflecting 35% contributes 1.8% of the view; a room reflecting 60% contributes 3.1% of the view. The veil is identical in every row because the cloth and the street are: only the furniture changes. What the chart cannot show is where in each room that reflectance sits, and a dark room with a lit lamp in it is not its own average.

Pattern and colour

What a sheer hides is decided by the furniture

Every result these essays have produced assumed the room and the street reflect the same three tenths of the light on them. That was never a fact about cloth. A room reflecting a twentieth is hidden two hundred and thirty-eight times better than the street it faces and one reflecting four fifths only fifteen times, with the same curtain at the same window — and the day-and-night exchange they rested on, exact for equal scenes, is out by more than half the glow for an ordinary pair of unequal ones.

A bouclé loop at 80 per cent overfeed, pressed to four heights. One loop of a bouclé at 80 per cent overfeed on 3 millimetre binder spacing, drawn to scale at 100, 80, 50, 20 per cent of its free height, with the force it pushes back with beneath each. The pressed loop is the same elastica the free one is, with one condition added: at its apex the tangent is along the core again, so a pressed loop is two half-loops each with its rise prescribed. At 100 per cent it stands 1.95 millimetres and pushes with -0.00 millinewtons; At 80 per cent it stands 1.56 millimetres and pushes with 12.19 millinewtons; At 50 per cent it stands 0.97 millimetres and pushes with 10.02 millinewtons; At 20 per cent it stands 0.39 millimetres and pushes with 4.90 millinewtons. What the drawing cannot show is the stiffness bracket: the force is the lower bound, with the fibres free to slide.

Compound and figured cloths

A loop has a maximum force in it

A bouclé loop pressed by its neighbour is the same elastica the free one is, with one condition added — at its apex the tangent lies along the core again, so a pressed loop is two half-loops with their rise prescribed and needs no contact solve at all. Solved, the eighty-per-cent loop pushes back with nothing at its free height, twelve millinewtons at four fifths of it, and two and a half at a tenth. A curve with a maximum in it is a softening spring, and the promise this account made — a cloth's position read from its beat-up — cannot be kept, because the whole bracket sits at one pressure.

Mechanics and drape

A loop is a plane curve in another plane

A knitted loop was solved as a flat curve because two curves in one plane cannot pass through one another and a loop must. Letting it out of the plane turns out to change nothing about its shape: the loop is still planar, and its plane is the fabric's turned through twelve degrees.

Three rules for how hard to fold a yarn, and which one the trade uses. For 2, 3, 4 folds of 20 tex cotton at 800 turns a metre: the ratio that makes the fold's surface helix angle equal its singles', the ratio that sets the fold's net moment to zero, and the ratio the trade actually folds at. The surface rule is 1/√n exactly, because a fold of n singles is √n times the diameter. The torque balance is C/(B+C), which for this fibre is 0.201 and is the same at every count, every twist and every number of folds. The trade's brackets contain the surface rule in all three rows and the torque balance in none of them.

Setting and geometry

The folding rule is a surface angle

Fold at two thirds for two singles, six tenths for three, a half for four. Those are one over the square root of the fold count, they are what makes a fold's surface twist angle equal its singles', and all three of the trade's brackets contain the number exactly.

Where a knot breaks

It breaks at the entry, before the knot has done any gripping at all. Two quantities run along a knot's path and only one of them rises; the other falls from the first millimetre; and their sum is largest where the thread arrives.

Knits and other structures

A loop bends at twice its own radius

A rod of radius r cannot be bent to a centre-line radius below r without occupying its own space. A knitted loop's tightest bend is at 2.04 yarn radii — twice the hard limit, and falling as the fabric tightens. That is a ceiling on how tight a knit can be, from contact alone.

The section the fabric asks for, beside the one the model drew. A 24.2 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.184 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 76% of it, which is the closest the fabric's own adjacent courses come to one another — 0.140 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett.

After the loom

The diameter that does need a state

A fabric dimension quoted without its relaxation state is not a measurement. The rule was applied to the two plan dimensions and then to the thickness, and it was never applied to the one dimension that is not the fabric's at all — the yarn's.

Weaves

A cloth slips at its least-interlaced thread

A weave's firmness is quoted as one number, the interlacings per crossing averaged over the whole repeat. A cloth does not fail on average. A thread pulled through a seam or out of a cut edge is held by its own crossings, the grip is exponential in them, and the thread with fewest goes first. In every four-by-four draft but plain weave some thread interlaces twice a repeat — the fewest possible — whatever the average says, and a satin stripe on a plain ground averages 0.87 while its satin ends grip at a seventh of the average thread.

The edge of a warp line in a 2/2 twill, at 3 and 4 threads. A light line in a dark 2/2 twill, drawn at 3 and 4 threads wide over 3 repeats with both of its boundaries traced crossing by crossing. The line has no gap at either width, and neither boundary is straight: at the crossings where the outermost thread of the band is under the ground, the edge retreats to the next thread in. At 3 it swings 2 threads with a period of 4; At 4 it swings 2 threads with a period of 4. What the drawing cannot show is distance, at which a swing of one thread width is below what an eye separates and a swing of three may not be.

Pattern and colour

An unbroken line is not a clean one

A line of colour has two boundaries and neither is straight, in any weave there is. The thread at the edge must go under somewhere, and where it does the edge retreats to its neighbour — so the boundary steps, and by exactly one thread less than the narrowest unbroken line the weave draws. Over 22,874 drafts there are three widths and three swings and no draft anywhere else, and widening the line past its narrowest unbroken width leaves the edge precisely where it was.

A 100-gram bouclé in section, with the depth to its own load path. A bouclé cloth of 100 grams a square metre in section: the core and binder along the bottom, carrying everything, with the effect thread's loops standing 0.31 millimetres above them. The cloth's surface is the loops and nothing else — its loop envelope is 12.6 times its count diameter — so a rubbing surface meets effect thread first and reaches the core only after that depth. 60.7 per cent of the yarn's mass is effect thread on no load path and 39.3 per cent is core and binder. What the section cannot show is the third dimension: the loops of neighbouring yarns lie between these and are pressed by them.

Compound and figured cloths

A bouclé wears from the loops down

A cloth loses its strength before its mass, because a woven cloth's crowns are the very threads that carry the tension. A bouclé inverts it exactly. Its surface is its loops, its loops hang off a core that carries everything, and 61 per cent of its yarn is on no load path at all — so a rub takes three tenths of a millimetre of thread that does nothing before it reaches anything that does, and the cloth is ruined to look at while it is still as strong as it was woven.

The four-by-four catalogue's crown line, counted three ways. Every one of the 22,874 interlacing four-by-four drafts, binned by how much horizontal crown line it carries, under three counts: both systems summed, which is what the published census reports; the warp alone; and the weft alone. A bearing curve sees one system, because a plate meets whichever crown stands higher and meets nothing else until it has sunk past the step between them. Under the summed count 2 drafts carry none; under the warp alone 494 do, and under the weft alone 494. What the histogram cannot show is which drafts moved, which is most of them.

What cloth is

The census counted two systems and a surface has one

Two drafts of twenty-two thousand touch at points, and the two are the plain weave. That is a count of the crown line both systems carry, and a bearing curve sees one: a plate meets whichever crown stands higher and meets nothing else until it has sunk past the step. Counted the way a surface is read, 494 drafts touch at points rather than two — and which 494 depends on a crimp division already called a convention rather than a measurement.

Mechanics and drape

What leaving the plane costs

Every number the flat loop model produced, beside the same number with the climb in it. Four of the five fall, none moves by four per cent, and the estimate that priced the third dimension beforehand had the sign the wrong way round for a reason worth naming.

The fold the trade actually makes. 2 singles of 20 tex cotton at 800 turns a metre, folded at 566 — a ratio of 0.707. That is 1/√2, the ratio at which the fold's surface helix angle equals its singles' — 22.8° — because a fold of 2 singles is √2 times the diameter. The trade's own bracket for 2 folds is 0.6 to 0.75, and it contains this number.

Setting and geometry

What a balanced yarn is balanced about

A specification that says a yarn is balanced does not say which of two conditions it means, and the two have different numbers, different dependences and different consequences. One of them is what folding achieves and the other is what folding is said to achieve.

Cloth doing a job

A knot halves a yarn and says why

The rule is quoted for every rope and every knot and derived nowhere. It comes out at forty-six per cent for a cotton — but only if the yarn's fibres bend individually. A yarn bending as a solid section has already spent five times its breaking strain before any load arrives, so it could not be knotted at all.

The extension ceiling, with the yarn given a thickness. As a jersey is pulled along its courses the wale spacing grows and the course spacing has to fall, because the yarn between two interlacings is a fixed length. The upper curve is how small the course spacing may be before the yarn runs out; the lower is how small it may be before two adjacent courses occupy the same space. The geometric ceiling is 322% and the contact one 299% — 7% lower. That is the result and it is a negative one: a measured jersey extends by about a hundred per cent, so contact between courses is not what puts the computed ceiling three times beyond a real one. The candidate this ladder was written to test is ruled out.

Knits and other structures

Contact is not why a jersey stops

The model says a jersey can be pulled to three hundred and twenty per cent along its courses. Real ones stop at about a hundred. The recorded diagnosis was that nothing stops adjacent courses passing through one another — and giving the yarn a thickness closes seven per cent of a gap of two thirds.

A cotton yarn at a fold, at both ends of its bending bracket. The same 20 tex cotton yarn bent to a radius of 0.084 mm under the two assumptions this site's bending bracket is drawn between. If the fibres slide freely past one another each bends about its own middle and the surface strain is 7.14%; if they are locked the bundle bends as a rod and the outermost fibre is strained 100%. The ratio is 14.0, which is the ratio of the two diameters and therefore the square root of the fibre count over the packing — so the bracket of 327 this site carries in a stiffness is a bracket of 14.0 in a strain. What the drawing cannot show is which of the two a real yarn does, and the answer is settled by a refusal: at the tightest fold a cloth can make, the locked bound asks for a strain of one, and a creased cloth does not fall apart.

After the loom

A wrinkle cannot settle what a crease settles

The first rung of this ladder found that a pressed crease decides a question this collection had been unable to settle for two fields — whether a bent yarn's fibres slide or bend as one body — and it decides it by refusing: the coherent branch would strain the fibres by eighty-four per cent and cotton breaks at six. A wrinkle is the same fold at twenty times the radius, and there both branches are survivable. So the bracket that a crease collapses stays fourteen times wide at every radius anybody actually creases a cloth at accidentally.

The selvedge turns of a 2/2 twill, 4 ends wide, from the left. A strip of 2/2 twill 4 ends wide over 8 picks, the first thrown from the left, with the weft's turn between every pair of picks drawn at the edge it reaches. 0 of the 8 turns are caught, where the edge end is on the other face on the second pick, and 8 slip. Across all its edge placements the weave catches every turn at 8 of 16. What the drawing cannot show is how far a slipped loop travels, which the beat-up and the weft tension decide.

Weaves

A selvedge holds only where its edge end changes face

A shuttle weft goes out on one pick and back on the next, and between them it turns round the end at the edge. The turn is caught only if that end is on the other face on the second pick; otherwise the loop has nothing to wrap and slides off. Plain weave catches every turn at every width. A 2/2 twill catches them at half its widths, and only if the first pick is thrown from the right side. A 3/1 twill, a hopsack and every satin catch them nowhere, and of the 22,874 four-by-four drafts, 9,636 cannot hold a selvedge at any width at all.

A comber board for 60 ends a centimetre, in side elevation. A jacquard's comber board seen from the side, with the fell of the cloth at the left and the back rest at the right. The board carries one hole per end; at 60 ends a centimetre the ends are 0.167 millimetres apart and a cord with a mail on it needs 0.9, so the holes are ruled in 6 rows staggered fore and aft, 6 millimetres apart — a harness 30 millimetres deep. Each row's ends are strained by its own distance from the fell: 0.460 per cent at the front and 0.524 at the back, a spread of 0.0636. What the elevation cannot show is the sideways fan of the cords above the board, which is a separate and much larger geometry.

Compound and figured cloths

A jacquard's harness has a depth after all

A jacquard was said to escape the shaft loom's depth entirely, because every mail hangs at the same distance from the fell. Every mail does, if the comber board has one row of holes — and it cannot. At sixty ends a centimetre the ends are a sixth of a millimetre apart and a cord with a mail on it wants most of one, so the holes are ruled in six rows thirty millimetres deep. The escape is real and it is a factor of eight rather than a release, and it closes as the cloth is set finer.

The bearing crowns of 2/2 twill and 2/2 hopsack, over 3 repeats. The cells at which the warp is on the face, drawn over 3 repeats of each draft — which is the surface a plate meets, since the other system is a step below it. 2/2 twill has 1 component in its repeat and a path that runs the whole way across the cloth, in both directions; 2/2 hopsack has 2 components in its repeat and no path across the cloth at all. Both carry the same length of crown line by the bearing count, and one is a ridge while the other is a field of islands. What the drawing cannot show is the depth of the gaps between them, which is the step to the second system and is a few micrometres.

What cloth is

Four drafts in five have no path along their own crowns

A 2/2 twill and a 2/2 hopsack carry exactly the same length of bearing crown line, which the surface census noted and could not explain. One of them is a ridge running diagonally across the cloth without a break; the other is a field of square islands with no path between them. Counted over the whole catalogue, 4,016 of 22,874 drafts have a crown path that reaches the far side, 1,616 have one in both directions, and 130 have crowns with no neighbour at all.

Mechanics and drape

The force that holds a knit open

Resolving a knitted loop's contact force out of the fabric's plane leaves a fifth of it pointing through the thickness. That fifth is 7.8 millinewtons a stitch, fifteen kilopascals over the area a stitch occupies, and it is the whole reason a jersey has a thickness rather than a plan.

Setting and geometry

A cabled yarn is a fold of folds

The rule that sets a fold's twist is one over the square root of the number of components. Apply it twice and a cabled yarn's three twist levels are fixed by two integers — which is a prediction with no free constants, about a class of yarn the trade quotes no rule for at all.

Cloth doing a job

Which yarns knot well

A knot's efficiency depends on three things and only one of them is the knot. The other two are the fibre's fineness and its breaking strain, and across this collection's own table those give efficiencies from six per cent to eighty-three at exactly the same bend.

The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

Knits and other structures

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

A 4-layer stack round a 180° fold. 4 layers of 260 µm cloth taken round a 180 degree fold, drawn as concentric arcs at their own separation. The outer layer runs round a larger radius than the inner one, so it must be longer — by the angle times the separation, which is 2.45 mm here and 0.82 mm at every interface. In a loose stack that length is found by the layers sliding over one another at the fold. In a stitched seam they cannot: the stitches pin them every 3 mm, so the slip a fold needs is 82 per cent of the distance between two stitches and has nowhere to come from. What the arcs cannot show is what happens instead, which is that the fold opens out to a radius the stack can manage.

After the loom

A crease cannot cross a seam

The outer layer of a folded stack has further to go than the inner, by the fold's angle times the stack's own thickness. A four-layer seam of quarter-millimetre cloth taken through a half turn needs its outer layer to be 2.45 millimetres longer than its inner — and a stitch line every three millimetres has pinned them. So the fold opens out where it crosses the seam, which is what a trouser crease visibly does, and the arithmetic gives the radius it opens to.

Compound and figured cloths

A damask is the only figure that costs its beam nothing

Figure and ground consume warp at different rates, and the difference accumulates down the length of the figure. An eight-end satin figure on a plain ground puts its two regions fourteen per cent apart, which on a loom absorbing a millimetre of slack bounds the figure at eight and a half millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were — and their crimps are identical rather than close, to the last bit of a double.

What cloth is

A cotton's own water is a twentieth of what a cloth holds

A wet cloth keeps water in three places and only one of them is the fibre. A sheeting saturated holds 113 per cent of its own dry weight: 7.5 per cent of that inside the cotton as regain, 39 per cent in the channels between the fibres of its yarns, and 54 per cent in the holes four threads bound. The same construction in polyester, whose regain is a fortieth of cotton's, holds 105 per cent — an eight-point difference from a fortyfold one, because absorbency is a geometry with a fibre in it rather than a fibre with a geometry round it.

How many cloths any one cloth derives into. The 426 four-by-four cloths sorted into the orbits the manuals' derivations cut them into. 12 orbits hold 1 cloth; 83 orbits hold 2 cloths; 62 orbits hold 4 cloths. The largest orbit in the whole catalogue holds 4, so no cloth derives into more than 3 others by any sequence of the named operations, however long. The derivations generate a group of 256 elements and it cuts the catalogue into 157 pieces. What the bars cannot show is which cloths are in which orbit, which is the next figure.

Weaves

No cloth derives into more than three others

Every weaving manual opens by saying the three basic weaves generate the rest. This collection counted the reach and found nine of 426, and left the nine as a count. It is not a count: every derivation the manuals name is a relabelling of the grid or a complementation of it, both invertible, so they generate a group — and that group cuts the 426 cloths into 157 closed pieces of which the largest holds four. The claim is not merely wrong about how much derivation reaches; derivation cannot reach more than four cloths from anywhere, by any sequence of operations, however long.

How little eccentricity a curl needs. The radius a jersey would curl to, against how far the fabric's neutral surface fails to bisect its loops. The model's own loop is bisected exactly — its crest sits half a diameter behind the mid-surface and its trough half a diameter in front — so it carries no curling moment at all, and the curve here is what any departure from that would buy. The moment is proportional to the eccentricity, so one solve settles the whole curve. A radius of three millimetres, which is about what a fine jersey rolls to, needs 5.6% of a yarn diameter — 9.4 micrometres on a yarn 167 micrometres across. That is why curl is easy to see and hard to model: the whole of it lives inside a rounding error on the geometry. Only the course-wise edges are drawn, because bending about the other axis turns the thread's end tangents out of the fabric and a free thread given turned ends buckles clean out of it — 3.2 times the fabric's own thickness, for 37% off its energy. That configuration is not available to a thread with neighbours, so the second moment is refused rather than computed.

Mechanics and drape

How little asymmetry a curl needs

A model with a thickness can finally be asked why stockinette rolls. It answers that it does not — its curling moment is exactly zero, by a symmetry — and the useful part is what that costs to break: five per cent of a yarn diameter buys the whole of the curl anybody has ever seen.

Setting and geometry

What a high-twist yarn costs a cloth

Twist buys strength up to a point and then loses it, and everything else it does is a cost. A crepe twist is chosen knowing that, and the trade's twist limits are a balance among five quantities that this collection can now put beside one another.

A tube of an elastic band knitted 157 mm round, up one leg. A tube knitted 157 mm round from an elastic band, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 40%, 20.0 mmHg; lower calf 29 cm, stretched 85%, 32.1 mmHg; calf 36 cm, stretched 129%, 39.4 mmHg; below the knee 34 cm, stretched 116%, 37.6 mmHg. The calf is pressed 1.97 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by.

Cloth doing a job

A tube of one size presses the calf harder than the ankle

A band presses a limb with its tension over the limb's radius, so it is easy to conclude that a band grips hardest where the limb is thinnest. That is true of a band held at one tension, and no knitted tube is. A tube knitted to one size is stretched further wherever the leg is thicker, and its tension rises faster than the radius does: an elastic tube pressing an ankle at twenty millimetres of mercury presses the calf at thirty-nine, and a cotton jersey tube presses its calf three and a half times as hard as its ankle. A stocking graduated the other way has to pull hardest where it presses less.

Which satin orders are row-free, from 5 ends to 40. Every satin order from 5 to 40, marked where the best move's lattice has two shortest steps of equal length rather than one — which is the condition under which the interlacings do not line up into a row. The row-free orders are 5, 10, 13, 15, 17, 24, 25, 26, 29, 34, 35, 37: twelve of the 35 orders that admit a regular satin at all. An n-end satin floats over n − 1, so a float limit is a ceiling on the order, and the ceilings for limits of 8, 12, 16 are drawn. What the strip cannot show is the spread, by which the orders are ranked and which decides which move is best within each.

Weaves

A float limit leaves one row-free satin

A satin is chosen so that no diagonal forms, and an earlier essay found that most of them fail: the interlacings lie on a lattice, every lattice has a shortest step, and the marks line up along it unless two steps tie — which happens at twelve of the thirty-five orders from five to forty. The other constraint was named and not applied. An n-end satin floats over n − 1, so a yarn that will not carry a float longer than eight admits four orders in all, and exactly one of them is row-free: the five-end satin, which is the one everybody already weaves.

The suction water supplies, against the pressure twist supplies. The pressure pressing a yarn's fibres together, against its twist, with the suction inside the menisci of a damp yarn's own pores drawn as a level. The suction is 0.0624 newtons a square millimetre — 2γ over the pore radius of 2.3 micrometres — and it does not depend on the twist at all. The twist's pressure crosses it at about 186 turns a metre, so water is worth as much as the first 186 turns and nothing more: at 600 turns it is 11 per cent of what the twist already supplies. What the chart cannot show is the saturation, at which the suction vanishes because a full yarn has no meniscus in it.

What cloth is

A wet fibre is stiffer and a wet yarn is not locked

The intuitive reason a damp cloth stiffens is that water pulls the fibres together — a meniscus is curved, the pressure inside it is below atmospheric, and the suction presses the assembly exactly as twist does. Computed, that suction is worth the first 186 turns a metre of twist and nothing after them: six per cent of what an ordinary yarn's twist already supplies, and nowhere near enough to stop the fibres sliding. What wetting actually does is fatten the fibres, and a fibre's bending rigidity goes as the fourth power of its diameter — so a wet cotton fibre is 2.07 times as stiff with no contact in the argument at all.

The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres.

Compound and figured cloths

A heddle eye lets the kink through

A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

Pattern and colour

A net of three directions beats a voile one way at a time

A tulle's threads run three ways at sixty degrees and a voile's run two ways at ninety, so a net hung over a voile could show one family of fringes, three, or a lattice of them. It shows two, at right angles, and they are nothing alike. Along the voile threads that lie parallel to one of the net's, the net beats exactly as a one-directional net does: six-millimetre fringes from three metres. Across them no set of the net lies anywhere near, and the only slow beat comes from a line of points two sets make together at √3 over the net's pitch — fringes four times wider and a fifteenth as strong, with a null at 8.8 metres where the strong family has none.

Mechanics and drape

What a knit gives up when it is pressed

The woven half of this collection has had a compression curve for several rungs — a thickness that falls under load, a bearing area that grows, a pressure at every point. The knitted half had a plan and no depth. It has a relaxed thickness and an initial slope now, and the two fabrics turn out to resist for different reasons.

Setting and geometry

What a sett is when the yarn is not round

A jamming condition says how close threads can be set, and it says it in terms of a diameter. A thread in a cloth does not have one diameter: it has a wide one and a narrow one, and which of them a jam is about depends on which way the threads are jamming.

The course helix of a 30-inch machine with 96 feeders, unrolled. A knitted tube from a 30-inch machine with 96 feeders unrolled flat, one round wide, with its wales drawn straight along it and its courses climbing across it. The machine lays 96 courses in each turn, so each course climbs 96 course spacings — 48 mm of fabric — in one round of 2,262 wales, 162 cm: an angle of 1.70° from the tube's cross-direction. The course drawn heavy is followed round one turn. The vertical scale is exaggerated 6 times. What the drawing cannot show is the hand of the helix, which is set by the direction the cylinder turns.

Knits and other structures

A circular machine leans its courses whatever the yarn

Spirality is blamed on the yarn, and the yarn is most of it. The rest is the machine's. A circular knitting machine's needles make wales that run straight along the tube, and its feeders lay one course each per turn, so every course climbs its feeder count in every round: on a 96-feeder machine, 48 millimetres round 162 centimetres of tube, an angle of 1.7 degrees. The helix has no machine size in it, its hand is set by the way the cylinder turns, and it survives every remedy aimed at the yarn — a steamed yarn, a plied one, S and Z on alternate feeders, and a rib.

An inflated tube in section at 0.5, 1, 1.6 times its wrinkling moment. The cross-section of an inflated tube of 100 mm radius at 50 kPa, bent by 0.5 times, 1 times, 1.6 times its wrinkling moment of 78.5 N·m, with the outside of the bend at the top. Each stroke is the wall's axial tension at that point: the pressure's even 2.5 kN/m with the bending's cosine added. At half the wrinkling moment the inside is still in tension; at the wrinkling moment it falls to nothing; past it the inside has gone slack over a dashed arc and the rest carries both the pressure's end force and the moment. What the drawing cannot show is the wrinkles themselves, whose wavelength a cloth's bending stiffness decides and this section leaves out.

Cloth doing a job

An inflated beam wrinkles at a moment with no cloth in it

An air-filled tube can be used as a beam because the pressure pulls its cloth taut along its length, and a cloth that cannot carry a push can carry a bending moment for exactly as long as that pull outweighs it. The inside of the bend goes slack at πpr³/2 and the tube folds at πpr³ — seventy-nine and a hundred and fifty-seven newton metres for a tube a fifth of a metre across at half a bar — and there is no property of the cloth in either. The cloth decides how far the beam bends on the way, and how much pressure it can be pumped to.

Weaves

An irregular satin scatters where a regular one lines up

A regular satin's marks lie on a lattice, so its closest pairs all run along one vector and make a row. An irregular satin has no lattice at all, so its closest pairs may point several ways at once — and at seven and nine ends, where every regular satin at the best spread has a row, an irregular one reaches the same spread with its closest pairs scattered over four directions and no more than a third in any one. At eight and eleven ends there is no such satin: the best spread is reached by regular satins alone, and irregularity has nothing to offer.

6 tapered panels laid across a 150 cm cloth three ways. 6 panels 14 cm across the top, 32 cm across the bottom and 75 cm long, laid across a cloth 150 cm wide, drawn to scale. Turned end for end alternately, 6 fit side by side and the 6 take 75 cm of cloth. Laid all one way in lanes, 4 fit and they take 150 cm. Laid all one way with alternate columns shifted half a length, 5 columns fit and they take 150 cm. What the drawing cannot show is a real marker's other pieces, which fill the gaps these leave.

After the loom

A nap is paid for by the taper of the pattern

A raised cloth's fibres lean, so a panel turned end for end shows a different amount of fibre and every piece of a garment has to lie the same way along the bolt. What that costs is not a property of the cloth. A rectangle costs nothing laid one way; a tapered panel costs (1 − r)/(1 + r) of extra cloth in lanes, where r is its narrow width over its wide one; and the best any one-way lay can do is exactly half of that, because a trapezoid's difference body is a hexagon and hexagons tile. On a real width it arrives in whole panel lengths: six skirt gores take 75 centimetres two ways and 150 one way.

A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it.

Mechanics and drape

A thread has a second stiffness

Every mechanical number here came from one material constant: how hard a thread is to bend. A thread also resists being twisted, nothing here has ever used that, and the ratio between the two turns out to be the only stiffness number about a yarn that can be known at all.

Setting and geometry

What a flattened yarn does to its cover

A cover factor is a sett times a diameter, and it decides how much of a cloth is thread and how much is hole. A flattened yarn is a third wider than a round one of the same area, so a cloth of flattened yarn covers more at the same sett — and every opacity, permeability and shade computed from a round diameter is wrong in one direction.

What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 900-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 12 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.

Cloth doing a job

A grade charges by the length and a cutter pays by the panel

The four-point system scores a fault by how far it runs — one point to three inches, four beyond nine — and caps a linear metre at four points however many faults it holds. A cutting room pays by how many panels the fault lands in. Below a panel's length every fault costs exactly one panel while its score runs from one to four; above it the panels grow without bound and the score does not move at all. Two fifty-metre pieces built to the same 267 points a hundred square metres lose 33 per cent of their panels and 92.

Knits and other structures

A tuck decides whether a third of two-bed fabrics lean

A jersey leans because every loop is on one bed and a rib does not because its loops are mirrored across two, and the count that says so treated a tuck as nothing. A tuck is a loop of the same lively yarn wrapped round a needle of one bed, and nobody here has measured how much of a knit loop's lean it carries. It matters. Of the 1,135 two-bed fabrics a two-needle, two-course frame can make, 135 are balanced whatever a tuck carries and 612 lean whatever it carries — and 388, a third, are balanced under one answer and lean under the other. A half-cardigan leans a third of a jersey if a tuck carries nothing and not at all if it carries a full loop's torque, which makes it the instrument that would settle the question.

Mechanics and drape

Twist is not torsion

A curve has a torsion and a material has a twist, they share a word, and only one of them is what a torsional rigidity resists. Getting them the wrong way round would have made this collection conclude that a knitted loop carries no twist, on the strength of a theorem that says nothing of the kind.

Setting and geometry

The count that decides how flat

A knitted fabric's demanded flattening is a function of its tightness factor, and a tightness factor is the square root of a count over a loop length. So a coarser yarn at the same loop is flatter — which is a prediction about a spinner's choice that nobody has framed as one.

The first 12 metres of a bolt, cut three ways. A bolt of 50 metres carrying 40 faults, with its first 12 metres drawn: the fault positions above, and below them the 900-millimetre panels a cutter gets cut blind, cut with the same rigid tiling slid to its best offset, and cut around the faults with the map in hand. Over the whole bolt the three yield 28, 30 and 40 sound panels of 55. Sliding the tiling buys 2; breaking it buys 12, which is 43 per cent more cloth from the same roll. What the strip cannot show is the width, across which the same argument runs again with a different panel dimension.

Cloth doing a job

A fault map is worth most where the grade is worst

A cutter who knows where the faults are can slide the marker or break it, and only one of those is worth anything: sliding a rigid tiling to its best offset recovers two panels of fifty-five, and letting the tiling break recovers twelve — two panels in five more cloth from the same roll. The gain has a maximum in the middle of the range, because there is nothing to recover on a clean bolt and nothing to be done on a ruined one. And the prediction the grading essay made, that a map is worth most on a bolt whose faults are bunched, is false: bunching leaves clear runs for the blind cutter too.

Mechanics and drape

The one fibre whose answer is known

A table of measured constants is worth what its worst row is worth, and nobody can tell which row that is. This one has a member whose answer was known before anybody measured it, and the ordering of the other nine turns out to say something about how fibres are made.

What the count moves at 150 grams. Plain cotton cloths that all weigh 150 g/m², from 20 tex to 200 tex, with sett and crimp solved together. Thickness, which is two yarn diameters, rises 3.16 times across the line; the cover factor of each thread system falls 2.75 times; their product with one plus the crimp is the same number at every count, because the weight has fixed the volume of fibre and the count only decides whether it is laid out flat or stacked up.

Setting and geometry

A weight fixes the fibre and not the drape

Every plain cotton cloth of 150 grams a square metre contains the same fibre, and the count decides only how it is arranged. Across the counts that can make that weight, thickness rises threefold and cover falls in step, so their product holds still. The bending length does something stranger: at the bound a woven yarn actually sits near, it depends on neither the count nor the weight, only on the fibre.

Linked: a knitted interlacing: one loop drawn through the next. Two closed curves and the Gauss linking integral taken over them, which returns -1.0002 at 200 segments a curve. A knitted interlacing: one loop drawn through the next. A linking number is an integer, so a value coming back at a few thousandths of one is the discretisation reporting itself rather than a fabric that is slightly linked. The two arrangements are the two ways of making cloth: a knitted fabric's courses link and a woven cloth's threads do not, at any crimp and for ever.

Mechanics and drape

What a closed thread cannot choose

A thread whose ends are held has a quantity it cannot change without breaking: the total number of times its material winds about its own axis, plus the number of times that axis winds about itself. The two can trade, and everything a twisted yarn does when it is let go is that trade happening.

Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one.

Mechanics and drape

Where a torsion model stops

A second stiffness was added because an earlier ladder named its absence as the first thing to disbelieve. It settled four things, refuted one trade explanation, and left the question it was built for exactly where it found it.

What flattening costs, against what the loop's bending is worth. The energy of squashing a 20 tex cotton to the 78% the fabric's geometry demands, as a multiple of the whole bending energy of one stitch, at three lateral rigidities. The free bound is exactly zero: a bundle of fibres free to slide resists a change of shape at constant area not at all, so flattening is free and the bracket on it has no floor. The coherent bound is 36 times the loop's bending, so a yarn that could not rearrange could not be knitted into this fabric at all. The fabric flattens, so it is reading the free end — which is the fourth ordinary observation on this site to land at that end of the bracket.

Mechanics and drape

Flattening is free and impossible

The fabric demands a flattening and the yarn has to supply it. At one end of this collection's oldest bracket the deformation costs exactly nothing; at the other it costs thirty-six times the whole bending energy of a stitch. The fabric flattens — which is the fourth everyday observation in one phase to land at the same end.

What flattening a section does to the ratio of the two stiffnesses. C/B is 2G/E for a circular section, because a circle's polar second moment is exactly twice its flexural one. A flattened section has two different flexural moments — easy about the long axis, hard about the short one — and the polar moment is still their sum, which is the perpendicular axis theorem and holds for any section whatever. So a flattened thread has three constants rather than two, and the multiplier on C/B depends on which way it is being bent. At the 0.78 a knitted fabric's own geometry demands, the easy direction multiplies the ratio by 1.322 and the hard one by 0.804. A knitted loop bends in the easy direction, so the collection's headline ratio is a lower bound for a yarn in cloth.

Mechanics and drape

The section that changes both stiffnesses

A thread's two rigidities are in the ratio 2G/E, and that is a fact about a circular section: a circle's polar second moment is exactly twice its flexural one. A yarn in cloth is not circular, so a yarn in cloth has three constants rather than two — and the ratio a whole ladder rests on is a lower bound.

Mechanics and drape

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

Mechanics and drape

A fabric reads its own bracket four ways

A yarn's stiffness is unknown to a factor of three hundred, and no laboratory measurement has closed it. Four unrelated everyday observations — a snarl, a knot, a flattened yarn and a cloth's own thickness — all say the same thing about which end of it a yarn sits at.

Mechanics and drape

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

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