Textile Structure
how cloth is put together

A weave is a matrix.

Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not a convenient way of writing a weave down — it is the weave, and it means almost every question about a fabric has an exact answer: how long the floats are, how close the threads can be set, and whether the thing holds together at all. The same is true one structure over, where a knitted loop is a curve that can be solved and a yarn is a rod with two stiffnesses and a topology — which is why a knit runs where a weave frays. These are essays about cloth as a structure, from the draft down to the thread, with the arithmetic done rather than asserted.

The weaves everything else is built from. Plain, twill and satin on point paper. Reading across, the floats lengthen and the interlacings fall — and every property a weaver cares about follows from that one trade.
Fig. 1 The three weaves everything else is built from, drawn on the point paper a weaver would use — a filled square means the warp is on the face. None of these was drawn by hand. Each was generated from its own rule and then examined: the floats counted, the interlacings counted, and the cloth itself checked for the failure that a draft cannot show, which is whether the threads are tied into one fabric or several.

Added 3 November 2026

7 essays · everything that arrived when, newest first

The partner's peak against how late a slackener starts. The largest tension the partner's front span reaches, against the share of the shed opened when the slackener starts to drop, at a 4 mm rise. It stays at 1.07 N for any start up to 0.748 of the shed, where the partner with no slackener passes that tension. Later than that, the peak is the touched tension at the moment it starts, up to 1.39 N at 0.95, against 1.48 N for no slackener at all. Compound and figured cloths

A leno's slackener only has to arrive

A slackener that lets the doup down in step with the loop keeps a leno's partner clear at every moment of the shed, and it looked as though a real one would have to track that closely. It does not. A partner's tension only rises as the shed opens, so a slackener can leave its whole drop until three quarters of the way up and the peak does not move at all. Lateness is paid for elsewhere: starting later than 0.58 of the shed rubs the loop more than having no slackener, and above 234 picks a minute a falling weight cannot start as late as the partner would let it.

7 figures A bay window's drops cut from one roll, three ways. Four drops of 250 cm and two of 140 cm cut from one roll of a 64 cm repeat, drawn along the roll to scale with the cloth lost between drops shaded. A straight match loses 76 cm at best. A half-drop cut in hanging order loses 154 cm. The same half-drop in its best order loses 44 cm, by keeping the long drops together, where each loses 6 cm, and changing class only after a short drop, which prefers to. What cloth is

A bay window wastes less than a wall

When a job's drops are all one length, a half-drop's best cutting order has one forced move, because the cutter has to change class once whatever the drops would prefer. Mix the lengths, as a bay window or a window over a radiator does, and each length sets its own price for what follows it. Six drops then lose a quarter of a repeat a gap instead of three tenths. Whether every drop can have its way is the bridges of Königsberg on two islands, and for a straight match mixing the lengths does almost nothing: its order can only choose which drop is cut last.

7 figures A fibre end prickles only inside a window. The lengths at which a wool fibre end can press on skin with the measured threshold force, against its diameter. Above the rising curve the end is too long a column and lies over before it reaches the force. Below the falling curve it is too short to sink far enough into a skin of 30 kPa to reach it. The two meet at 23.2 µm and 1.08 mm, and no fibre finer than that prickles at any length. A 30 µm fibre prickles between 0.83 and 1.79 mm. Cloth doing a job

A fibre end prickles only inside a window

A fibre end pressed against skin has two ways to fail to hurt. Too long, and the column lies over before it reaches the force a nerve answers to. Too short, and the tip cannot sink far enough into something as soft as skin to push that hard. The two limits run opposite ways with diameter and they cross, so a fibre below one diameter cannot prickle at any length at all — and that diameter has no protruding length in it. It also settles a sum left unweighed: a crop at 1.1 mm makes a wool cloth two and a half times worse, and one at 0.6 mm removes the prickle.

8 figures The ring across eight dips, for each fibre. The ring's depth after each of eight twenty-second dips, where an empty fibre left in the bath for good would fill by 20% a dip. instant fibre: 14.0% after one and 23.5% after eight; τ = 20 s: 13.0% after one and 19.6% after eight; τ = 100 s: 23.1% after one and 26.1% after eight; τ = 400 s: 35.5% after one and 36.6% after eight. An instant fibre earns its ring by saturating the skin dip by dip. A slow one starts near its final depth, since it never fills its skin. After the loom

A slow fibre starts its ring deep

Every account of an indigo dip here lets a cotton fibre take its dye at once. Give the fibre a clock of its own and the ring does not simply run deeper, which is what the question seemed to ask. A clock of a few seconds makes the ring shallower, by nearly a fifth at most, and only past about twenty-six seconds does it run deeper than an instant fibre's. Two sections tell the accounts apart that one cannot: an instant fibre earns its ring dip by dip, from 14 to 23.5 per cent of the radius over eight dips, and a fibre with a hundred-second clock starts at 23 and ends at 26.

7 figures One surplus, three depths, one slope. A 300 mm width of three cloths under 5 newtons per metre, each shortened by 1.0%, drawn at one scale in both directions. a chiffon has wrinkles 43 mm apart and 1.4 mm deep; a shirting has wrinkles 127 mm apart and 4.1 mm deep; a coating has wrinkles 249 mm apart and 8.0 mm deep. All three leave the plane at the same steepest slope, 11.5°, because that slope is set by the surplus alone. Mechanics and drape

A stiff cloth's wrinkle is deeper and no steeper

Stiffness sets how far apart a wrinkle's crests lie, and the surplus length it has to hide sets the rest. Scale a wrinkle up and its slopes do not change, so one per cent of surplus leaves a chiffon, a shirting and a coating all at 11.5 degrees, 1.4 to 8 millimetres deep. The steepness has no cloth in it — and steepness is what a light reads.

7 figures A 2/2 twill line's ends where the reed leaves them. A light warp line 3 ends wide in a dark 2/2 twill, with the ends drawn at their true places at a cover of 60% and both boundaries traced. Set evenly the edge swings 2.00 thread spacings. With 3 ends to a dent, closed by 40% of a spacing so that they touch, and the line on the dent's edge, it swings 1.20. One end along in the same reed it swings 2.40, more than if it had been set evenly. The weave and the reed are the same in the last two. Only where the line starts has changed. Pattern and colour

A stripe's edge is cleanest inside one dent

No weave can draw a line with a straight edge, since the thread at the boundary must go under somewhere. A cloth has one freedom its weave lacks, which is the spacing of its ends, and it looked as though crowding a stripe's outer end could halve the swing. It cannot, in most cloth: two ends stop at a diameter apart, so the swing falls only as far as the cover. The freedom that costs nothing is another one. A line's narrowest unbroken width is exactly its swing plus one, so it fits one dent of a reed, and the same reed gives 1.2 thread spacings or 2.4 according to which end of the dent the line begins on.

6 figures

The other 1 in the same group

Four ways in

besides the list

Series

A field says what an essay is about. A series says what else there is to say about one idea — the distinct arguments against it, from the one that introduces it to the one that assumes the rest.

83 ideas

The weave index

Every draft in the essays, measured the same way — float, interlacing, layers, plane group, warp face, densest sett. Not one number in it was typed in, and two of the rows are not one cloth.

24 drafts

Refutations

Every commonplace claim about cloth that has been put to a number, with the verdict and the quantity that settles it — computed rather than quoted.

89 claims tested

Concepts

The named objects themselves — float, crimp, jamming, the wale — and every essay that touches each one. The third axis, after fields and series.

562 objects

The nine fields

what the site covers, in the order the argument builds it

What cloth is Weaves Setting and geometry Knits and other structures Mechanics and drape Pattern and colour Compound and figured cloths After the loom Cloth doing a job All nine

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531 essays · all of them, paginated

The 5-end satin. The 5-end satin 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. Weaves

Plain, twill and satin

Three rules, and everything else in weaving is a variation on them. What separates the three is not appearance but one trade — how often a thread changes face against how far it runs when it does not.

6 figures 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.

7 figures 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.

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

The loop

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

6 figures 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.

6 figures The plain. The plain 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. What cloth is

A fabric is a structure, not a material

Cotton is a material. Cloth is an arrangement, and almost everything a fabric does follows from how the threads are put together rather than from what they are made of.

6 figures 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.

7 figures A draft is a two-coloured pattern. For each weave, the symmetries that leave warp-up as warp-up and those that exchange the two. Only a balanced weave has any of the second kind, because exchanging warp and weft is only a symmetry when there is as much of one on the face as the other. Pattern and colour

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

7 figures A fabric to fill and a fabric to load. Fill time against the sett of a woven reinforcement, with the two limits a part imposes. Stiffness wants fibre, which wants a close sett; the resin has to arrive before it gels, and the channels between the tows — which is what carries the flow — close as the sett rises. The interval between the two is where a fabric can exist, and it narrows with the size of the part. 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.

6 figures Colour and weave — 2/2 twill. One interlacement, threaded in two colour orders. What a reader sees is the colour of whichever thread is on the face, so the visible pattern can be something the draft gives no hint of. 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.

7 figures 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.

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

Relaxation is the crimp coming back

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

6 figures

All 531 essays · search them

Threads running through

themes, not chapters

A weave is a matrix

Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not an encoding of a weave — it is the weave, and it makes almost every question about it decidable.

111 essays

Does it hang together

A draft can look entirely reasonable on point paper and describe cloth that falls into two layers, or has threads lying loose on the surface. The failure is invisible and the test is exact.

65 essays

The float decides

How far a thread travels on the face before it goes under is the quantity behind lustre, drape, snagging, abrasion and how close the cloth can be set. Almost nothing else has that reach.

44 essays

Exactly this many

There is no regular satin on six ends, and the reason is a one-line argument about common factors rather than a rule of thumb. Counts in this subject are more often theorems than tallies.

248 essays

A mechanism, not a material

Cloth on the bias stretches half as long again while no thread in it stretches at all. The deformation is kinematic — a trellis closing — and almost every explanation given for it is wrong.

351 essays

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.

330 essays

Structure before fibre

Most of what a fabric does follows from how it is put together rather than from what it is made of. The same yarn gives a shirting or a satin, and they behave nothing alike.

272 essays