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.

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