Textile Structure
how cloth is put together
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.
14 min read 6 figures Structure before fibreA weave is a matrix

Take a length of cotton yarn and pull it. It breaks at a few per cent extension, in a way that is entirely a property of cotton. Now take a knitted cotton fabric and pull it. It extends by half as much again and springs back, and no cotton anywhere in it did anything of the kind.

The weaves everything else is built fromPlain, 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.plainfloat 1 · 8 interlacingsfirmness 1.002/2 twillfloat 2 · 16 interlacingsfirmness 0.505-end satinfloat 4 · 20 interlacingsfirmness 0.40a filled square means the warp is on the facecounted from the matrices, not from the pictures
Fig. 1 Three ways of putting the same yarn together, drawn on the point paper a weaver would use. A filled square means the warp is on the face at that intersection. Reading across, the floats lengthen and the interlacings fall — and almost every difference between the resulting cloths comes out of that one trade.

The second measurement is not about cotton. It is about arrangement, and the whole of this site follows from taking that seriously.

What a fabric is made of, and what it is

The distinction is worth drawing sharply because the trade routinely blurs it.

Fibre is the material: cotton, wool, polyester, glass. It has a stiffness, a strength, a moisture regain and a density, and those are properties of the substance.

Yarn is fibre assembled — twisted, or extruded, or spun — into something long and coherent. Twist is already a structure rather than a material: a singles yarn holds together because twist presses the fibres against one another and friction does the rest, and a yarn with no twist is a bundle that pulls apart in the hand.

Fabric is yarn assembled again, into a plane. Woven, knitted, braided or entangled, and each of those is a different structure with different consequences.

Three levels, and at each of them the properties of the level above come mostly from the arrangement rather than from the substance. That is not a slogan: it is why the same cotton gives a shirting and a satin, and why the two behave nothing alike.

The problem a fabric solves

Making a plane out of a line is not obvious. A sheet of paper is a plane made from fibres held by adhesion and hydrogen bonding; a sheet of plastic is a plane made by extruding a plane. Neither is a fabric, and the difference is why cloth exists at all.

Adhesion gives a sheet that is strong, cheap and useless for clothing. It does not drape, it does not stretch, it creases permanently, and it tears in a straight line from any nick. Every property that makes cloth wearable comes from the threads being separate — free to move against one another within limits.

Interlacement supplies exactly that. The threads are held in place by friction and by geometry rather than by being stuck to one another, so the cloth is coherent at rest and free to change shape when pushed. That combination is what a fabric is for.

A warp end in section — plainOne warp thread drawn through the cloth, with the weft threads it crosses shown end-on. The thread is longer than the cloth it spans, and the excess is the crimp — measured here from the drawn path rather than quoted beside it.the cloth this thread spansplainwarp crimp 22.3%8 interlacings per repeatso the crimp shown exceeds a real cloth'sthread thickness exaggerated for legibility3 face changes
Fig. 2 One warp thread in section, with the weft threads it crosses shown end-on. Nothing here is glued. The thread is held by going over and under, and by the friction of the crossings, and it can slide within that constraint — which is what lets the cloth move without coming apart.

The loom, in one paragraph

The machinery is worth a moment, because the vocabulary comes from it and the vocabulary is otherwise arbitrary.

A loom holds one thread system under tension: the warp, running the length of the cloth, hundreds or thousands of ends side by side. Each end passes through a heddle, and the heddles are grouped onto shafts. Lifting a shaft raises the ends threaded on it, opening a gap called the shed, and a single pick of weft is thrown across the shed and beaten up against the cloth already made. Lower, lift a different combination, throw again.

The 2/2 twillThe 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.2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4
Fig. 3 The record of that operation: which ends were lifted for each pick, one row per pick, one column per end. A filled square means the warp was up and is therefore on the face. The repeat is outlined, and the whole cloth is that repeat tiled.

So a weave is, quite literally, a sequence of instructions about which ends to lift — and the natural way to record a sequence of yes-or-no decisions is a grid of filled and empty squares. Point paper is not a diagram of the cloth. It is the instruction, and the cloth is what happens when the instruction is followed.

Two consequences follow immediately. The number of distinct weaves available on a loom with a given number of shafts is finite and countable. And a weave that requires more shafts than the loom has cannot be made on it, which is why so much of the historical repertoire is four-shaft work.

Friction, which is doing more work than it looks

An idealised woven cloth would have frictionless crossings. It would also fall apart the moment it was cut, because nothing would stop a thread from sliding out of the structure at a raw edge.

Friction at the crossings is what makes a cut edge stay put long enough to be sewn. It is also what makes a cloth resist shear before it starts to move, what makes a woven fabric hold a crease imperfectly and recover imperfectly, and what makes the difference between a fabric that feels crisp and one that feels limp when the two have identical drafts.

None of that is in the geometry, which is why this site is careful about what its models claim. The weave matrix decides floats, interlacings and whether the cloth holds together, and it knows nothing about friction at all.

The two directions are not alike

One asymmetry deserves stating early because it surprises people and it runs through everything.

Warp and weft are not interchangeable. The warp is under tension for the whole of weaving — hours or days of it — and is abraded by the heddles and the reed every pick. So warp yarn is stronger, more tightly twisted, and often sized with a temporary coating that is washed out afterwards. The weft is under almost no tension and can be softer, bulkier and weaker.

The cloth inherits that. A woven fabric is usually stronger along the warp, extends differently in the two directions, and shrinks differently when it is washed. None of this is in the draft — the matrix treats the two directions symmetrically — and all of it matters when the cloth is cut.

Pull it lengthways and it narrowsThe 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.as wovenwarp crimp 8.0% · weft 4.0%pulled 3% along the warpwarp crimp 4.9% · weft 7.1%warp thread 108.0 and weft 104.0 — unchanged in bothboth thread lengths checked, not assumed3% pull
Fig. 4 One consequence, and a counterintuitive one. Pull a cloth along the warp and it narrows, because the extension has to come out of the warp crimp and the crimp it gives up is taken by the weft. No thread has stretched: both thread lengths are the same before and after, which the figure checks rather than assumes.

That is the first appearance of a habit this site keeps: where a fabric appears to do something a material could not, the explanation is usually that a length has been rearranged rather than changed.

Four ways to make a plane from a line

Woven cloth is the most familiar and it is not the only answer.

Weaving crosses two thread systems at right angles: a warp held under tension on the loom and a weft carried across it. The result is nearly inextensible in both thread directions and remarkably extensible at forty-five degrees to them, which is a mechanism rather than a material property.

Knitting bends a single thread into loops, each drawn through the loop below. The result extends in every direction without needing a bias, because a loop can change shape long before a thread has to change length. It also fails differently, which is the price.

Braiding interlaces three or more thread systems obliquely, giving a structure that is naturally tubular and that changes diameter as it changes length — the property a Chinese finger trap is built on.

Entangling makes a nonwoven: fibres laid down at random and locked by needling, hydro-entangling, heat or adhesive. It is the cheapest structure and the one with the least anisotropy, which is sometimes exactly what is wanted.

The four are genuinely different rather than variations. A property that holds for one very often fails for another, and a sentence beginning “fabric does” usually means “woven fabric does”.

Why the same yarn gives different cloths

Here is the claim in its strongest form: hand a weaver one yarn and the cloths that come back can differ more from each other than either differs from a cloth in a different fibre.

What an interlacing costsEach 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.weaveinterlacingsdensest settingplain1.0020 per inch2/2 basket0.5027 per inch2/1 twill0.6724 per inch2/2 twill0.5027 per inch3/1 twill0.5027 per inch5-end satin0.4029 per inch8-end satin0.2532 per incha geometric jamming model, circular sections, yarn diameter 1/40 inch
Fig. 5 Several weaves in the same yarn, with their interlacing counts and the closest each can be set. Every thread that changes face has to bend, and a bend takes room, so the weave that interlaces most is the one that has to be set most openly.

Plain weave interlaces at every intersection. A satin interlaces once in five or once in eight. That single difference decides how firmly the cloth handles, how much it can be crowded, how it reflects light, how it wears, and how it drapes. It is not a small effect at the margins of a fibre’s behaviour; it is the dominant term.

The corollary is uncomfortable for a great deal of marketing copy. A claim about a fabric that mentions only the fibre has left out the variable that matters most, and thread count is the standing example.

The one property nobody can see

Everything so far has been about properties a careful person could work out by handling a cloth. There is one that nobody can, and it is the reason this site has a gate.

A draft describes fabric only if the threads it specifies are tied into a single structure. They need not be. A grid of filled and empty squares can perfectly well describe two fabrics lying on top of one another, each complete in itself, connected to the other nowhere. Weavers make such cloths deliberately — double cloth is a real and useful construction — and the point is that a draft giving one by accident looks exactly the same on paper.

Whether the cloth is one clothTwo drafts. Both interlace everywhere, both have short floats, and both look like perfectly ordinary weaves. One is a single fabric and the other is two fabrics lying on each other, and the bars beside each strand say which layer it belongs to.the bars mark which separable cloth each thread belongs toan ordinary 2/2 twillone clothevery end and pick interlacestwo separable clothsconnectivity of the above-and-below relation, computed
Fig. 6 Two drafts. Both interlace at every end and every pick, both have short floats, and both look like perfectly ordinary weaves. The left one is a single fabric. The right one is two fabrics lying on each other, and the bars beside each thread say which of the two it belongs to.

The test is exact and it is not visual. At every intersection one thread passes above another; if the threads split into an upper set and a lower set such that at every crossing between them the upper thread is on top, the upper set lifts off. The essay on integrity works it through, and every weave figure on this site carries the verdict.

What makes it decidable

The reason a site can make sharp claims about this subject is that the structure of a woven cloth is discrete.

At every intersection, one thread is on the face and the other is not. There is no third possibility and no partial case. So a weave is a matrix of ones and zeros over a repeat that tiles the fabric — and that is not an encoding of the weave, it is the weave.

Almost every structural question then becomes arithmetic. How long is the longest run on the face? How often does a thread change face? Does this draft describe one cloth or two? All three are computed here rather than judged, and the third one matters most because it is invisible in the drawing.

Where structure stops explaining

Being clear about the limit is more useful than being enthusiastic about the principle.

Fibre decides absorbency, thermal behaviour and chemistry. No arrangement makes polyester take up moisture the way wool does, and no weave makes cotton melt. A great many comfort properties belong to the fibre and to the finish rather than to the structure.

Yarn decides surface and bulk. A worsted yarn and a woollen yarn of the same count, in the same weave, give quite different cloths — one lean and smooth, the other hairy and full of air. That is a yarn-structure effect and it sits below the weave.

Finishing changes everything after the loom. Cloth is washed, scoured, milled, calendered, resin-treated and heat-set, and the fabric that reaches a shop is not the one that came off the machine. Some of what is casually attributed to weave is actually finishing.

So “structure before fibre” is a claim about which variable dominates the mechanical behaviour of a fabric, not a claim that fibre is unimportant.

What the models here do and do not know

Two models carry the arguments on this site, and both are narrow on purpose.

The weave matrix is exact about what it covers. Floats, interlacings, integrity and the symmetry of a draft come out of it with no tolerance and no approximation. It has no notion of thickness, twist, friction or stiffness, and two cloths with the same matrix can behave very differently.

The pin-jointed trellis treats a woven cloth as inextensible threads free to rotate at their crossings. It says what is geometrically possible — how far the bias goes, where the threads jam, what a curved surface costs — and it says nothing about force. It has no bending stiffness and no friction, so it cannot predict hand, or how a fabric hangs under its own weight, or the shape a wrinkle takes.

Where a number appears in these essays, the model that produced it is named. That is a habit rather than a formality: the same cloth has different crimp under Peirce’s circular-section geometry and under a racetrack section, and quoting either without saying which is how disagreements between sources become permanent.

How old this is

Weaving is older than writing, older than the wheel, and very nearly as old as agriculture. Impressions of woven cloth appear on fired clay from around 27,000 years ago at Dolní Věstonice in Moravia — the fabric itself is long gone, but the pattern it pressed into the clay survives, and it is unmistakably twined and plaited work.

That is worth holding on to when the subject looks technical. The structures came first by many thousands of years; the analysis is recent. Point paper as a way of recording a weave is a mediaeval European device. The idea of treating a draft as an object with computable properties is twentieth-century, and the specific check this site is built around — whether a periodic weave hangs together — belongs to the 1980s, when Branko Grünbaum and Geoffrey Shephard wrote a series of papers taking fabrics seriously as mathematical objects.

Nobody needed any of it to make cloth. What it buys is the ability to say why, and occasionally to catch a draft that a very experienced eye would pass.

Where the ladder goes next

The immediate step is to take the matrix claim literally: the draft is a matrix, point paper is how it is written down, and reading one is a skill worth twenty minutes.

The immediate payoff is the check that motivates the whole site: does it hang together, where a draft that looks entirely reasonable turns out to describe two fabrics rather than one.

And the immediate surprise is the bias, where a cloth of inextensible threads stretches by a third and no thread in it stretches at all.

What the pictures here cannot show. Every figure on this page is geometry. Friction, twist, hairiness and finish are all absent from it, and they are between them responsible for a great deal of how a real fabric feels. A drawing of a structure is not a drawing of a cloth.

Where this goes next

2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4 The draft is a matrix Point paper is not a diagram of cloth. It is a grid of yes-or-no decisions about which threads to lift, which makes a weave a binary matrix and almost every question about it arithmetic. the bars mark which separable cloth each thread belongs toan ordinary 2/2 twillone clothevery end and pick interlacestwo separable clothsconnectivity of the above-and-below relation, computed Does it hang together A draft can interlace everywhere, have short floats, and describe two fabrics lying on each other rather than one. Nothing about the drawing says so, and the test that does is exact. shear 30°bias +22.5%across -29.3%area 87%locks at 60°every segment checked against its own lengthno thread stretches 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.