Textile Structure
how cloth is put together
What cloth is

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.
15 min read 9 figures A weave is a matrixThe float decides

A weaver lifts some warp ends and throws a pick of weft under them. Then lifts a different set and throws another. The record of that operation is a grid: one row per pick, one column per end, filled where the end was lifted.

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. 1 The record, for a two-and-two twill. A filled square means the warp was up for that pick and is therefore on the face of the cloth. Everything printed beside it — the float length, the interlacing count, the number of separable cloths — was counted from that grid rather than read off the picture.

That grid is called a draft, and the thing worth noticing is what it is not. It is not a picture of the cloth, and it is not a diagram of the cloth. It is the instruction, and the cloth is what happens when the instruction is followed. Which means the cloth’s structure is a binary matrix, exactly and without approximation.

Reading one

The conventions are universal and worth ten minutes, because every other source uses them.

Columns are ends — warp threads, running the length of the cloth. Rows are picks — weft threads, running across it. The first pick is at the top and the first end at the left, which is how a weaver reads a draft and the opposite of how a mathematician indexes a matrix.

A filled square means the warp end is on the face at that intersection. An empty one means the weft is. There is no third state and no partial case: at any crossing, one of the two threads is on top, and which one is a fact about the cloth.

The heavy outline is the repeat — the smallest block that tiles the whole fabric. A draft is always understood as tiling: a two-and-two twill is not a four-by-four patch of cloth but an infinite fabric described by a four-by-four rule.

Why the two conventions must not be inverted

A short note on notation, because it is the one place where being clever would be actively harmful.

A filled square means the warp is on the face. That is universal, and it has a physical reason behind it rather than being arbitrary: the draft is a record of which ends were lifted, and a lifted end is one that finishes on top. Filling in the lifted ones makes the paper a picture of the loom’s action, and it makes a warp-faced cloth look dark on the page, which is what it looks like in the hand.

The colour convention on this site follows the same principle. Warp is drawn indigo and weft madder, which are the two oldest dyes in the trade and the pair every historical drafting tradition reaches for — and in denim the warp is literally indigo and the weft literally left pale, so a reader who learns the convention here can read a pair of jeans.

Inventing something friendlier would be the worst decision available. A reader who learned an inverted convention here would misread every other draft they ever met, and the cost of learning the standard one is about two minutes.

Where the floats are in the 3/1 twillEvery 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.3/1 twillwarp floats of 3weft floats of 116 markedruns measured round the repeatfloat 3
Fig. 2 Denim, and why it looks the way it does: three warp on the face for every one weft, so the indigo warp dominates the face and the pale weft dominates the back. The float marks show where the warp runs uninterrupted, which is the same reason the cloth wears white along its creases.

The repeat wraps

That tiling has a consequence which catches people out, and it is the reason every measurement on this site is made cyclically.

A run of warp on the face does not stop at the edge of the repeat. It continues into the next copy, so a draft with two filled squares at the end of a row and one at the start of the next copy has a float of three, not of two and one. Measuring only inside the drawn block gives the wrong answer at every repeat boundary — which is to say, everywhere in the actual cloth.

The right mental model is a torus: the left edge of the repeat is glued to the right edge and the top to the bottom. Every count here — floats, interlacings, symmetries — is made on that torus, and a repeat that is only correct in its interior does not pass.

The 3/1 twillThe 3/1 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.3/1 twilllongest float 316 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4
Fig. 3 The same rule drawn twice over in each direction, with the repeat outlined inside it. Nothing changes at the outline: the diagonal runs straight through it, which is what tiling means and why a float has to be measured round the join rather than up to it.

Two operations, and almost everything follows

Once a weave is a matrix, most of what a weaver cares about is one of two computations on it.

Float length is the longest run of consecutive equal values, down a column for a warp float and along a row for a weft float. Plain weave has floats of one. A two-and-two twill has floats of two. An eight-end satin has warp floats of seven. That single number is behind lustre, drape, snagging, abrasion resistance and how densely the cloth can be set, and it deserves its own essay.

Interlacing count is the number of places a value changes — the number of times a thread crosses from face to back. It is the firmness number, and it runs opposite to float length: the weave with the most interlacings has the shortest floats and the fewest options for being crowded.

Where the floats are in the 5-end satinEvery 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.5-end satinwarp floats of 4weft floats of 120 markedruns measured round the repeatfloat 4
Fig. 4 The floats in a five-end satin, marked where they run. The count is not a description of the drawing; it is the output of walking each column round the repeat and measuring the longest run.

Both are exact. There is no tolerance to choose, no sampling, and no judgement, which is unusual enough in a subject with this much craft vocabulary to be worth saying plainly.

What a shaft costs

There is a second matrix behind the first, and it explains why the historical repertoire looks the way it does.

Ends are not lifted individually. Each end is threaded through a heddle, the heddles are grouped onto shafts, and lifting a shaft lifts every end on it. So a loom with four shafts can only produce drafts in which the ends fall into at most four distinct lifting patterns — any two ends with the same pattern can share a shaft, and any two with different patterns cannot.

The count that matters is therefore not the number of ends in the repeat but the number of distinct columns in the matrix. A twill on four picks needs four shafts. A satin on eight needs eight. A plain weave needs two, whatever its repeat is drawn as, which is why plain weave can be woven on the simplest loom there is.

The plainThe 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.plainlongest float 18 interlacings per repeat1 separable clothrepeat 2 × 2generated, then counted2×2
Fig. 5 Plain weave, drawn over three repeats. Every end is one of two kinds — lifted on odd picks or on even ones — so two shafts suffice however wide the cloth is. That is the whole reason plain weave is the commonest structure ever made.

Everything above four shafts was historically expensive, and everything above about twenty-four required a different machine. That constraint shows in the surviving repertoire more clearly than any aesthetic preference does: the huge families of four-shaft twills and the relative scarcity of anything needing more are a record of what looms could do.

Three families, three rules

Almost every draft in ordinary use is generated by a rule rather than designed square by square, and the rules are short.

Plain weave alternates: filled where the row and column indices have the same parity. Two ends and two picks, and the maximum possible interlacing.

A twill is a sequence of ups and downs, shifted by one end for each successive pick. Written 2/2 or 3/1, the numbers are the runs: 3/1 means three up, one down, which is denim. The shift is what produces the diagonal, and no thread runs diagonally anywhere in the cloth.

A satin places exactly one weft point per pick, stepping by a fixed move. The move has to be coprime with the order or the points revisit ends before covering them all, and it must not be one or one less than the order or the points line up into a diagonal and the weave is a twill. That condition is a real constraint with a famous consequence: there is no regular satin on six ends.

Two drafts, one weave

A subtlety that trips readers comparing sources: the same weave has many drafts.

Shift a draft one pick down and it is the same cloth, started at a different place. Shift it one end across, likewise. Read it from the other face — swapping every filled square for an empty one — and it is the same cloth turned over, which is a genuinely different draft describing genuinely the same fabric. Rotate a square repeat by a quarter turn and, for some weaves, the same cloth appears with warp and weft exchanged.

So a weave is properly an equivalence class of matrices rather than a matrix, and two sources can print different-looking drafts for what everybody agrees is the same cloth. The properties this site computes are all invariant under those operations, which is why they can be quoted without saying which drafting convention was in use.

A draft is a two-coloured patternFor 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.weaveleave the faces aloneexchange themplain1616basket16162/2 twill16163/1 twill16none5-end satin20noneevery candidate operation tried against the repeat
Fig. 6 Which operations leave a draft alone. Some preserve warp-up as warp-up; others exchange the two, which physically means the cloth looks the same from the other side. Only a balanced weave has any of the second kind, and the split is exact.

That last observation is a small result in its own right and it falls straight out of the matrix. A warp-faced weave has more filled squares than empty ones, so no operation can turn it into its own complement — and the essay on drafts as plane patterns takes the idea further.

What the matrix cannot see

Everything the matrix decides, it decides exactly. The list of what it does not know is longer than the list of what it does, and it should be kept in view.

It has no thickness. A draft says nothing about yarn diameter, so it cannot give cover, crimp or the density at which the cloth can be set. Those need a geometric model of the yarn, and there are several which disagree.

It has no friction. Whether a cut edge frays, how the cloth handles, whether a crease recovers — none of that is in the geometry.

It has no twist and no hairiness. Two cloths with the same draft in a worsted and a woollen yarn are different fabrics.

It has no finish. Cloth is scoured, milled, calendered and heat-set after weaving, and the fabric that reaches a shop is not the one that left the loom.

So a claim derived from the matrix is a claim about structure, and every essay here that quotes a number says which model produced it.

The question the matrix answers that nothing else does

Against that list of limitations, there is one question the matrix settles that no amount of looking can.

A draft describes a fabric only if the threads it specifies are tied into one structure. They need not be. The same grid of filled and empty squares can describe two complete fabrics lying on one another and connected nowhere, or a fabric with an end that never goes under anything and can be pulled out with a fingernail.

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. 7 Two drafts that both interlace everywhere and have short floats. One is a single cloth and the other is two, and the bars beside each thread say which. Nothing in the grids themselves distinguishes them.
A thread that is not woven inOne warp end stays on the face across the whole repeat, so nothing holds it down. On point paper it is a solid column and nothing more; in cloth it is a thread that can be pulled straight out.1 strand never interlace2 separable clothsthe marked end lies loosea column with no gap is a thread with nothing over it
Fig. 8 The cruder version of the same failure: a warp end filled down the whole repeat, so it never goes under anything. On point paper it is a solid column and nothing more; in cloth it is a thread that can be pulled straight out.

Deciding which is a connectivity computation on the matrix, and it is what the next essay is about. It is also the reason every weave figure on this site prints its layer count: the check is cheap, the failure is invisible, and a figure that quietly showed the wrong thing would look entirely convincing.

How rare a bad draft is, counted

It is fair to ask whether this is a real hazard or a theoretical one, and the way to find out is to count.

Take every four-by-four draft in which each end and each pick interlaces at least once — the weakest condition under which a draft is worth drawing at all. There are 22,874 of them, and 144 describe more than one cloth. Rather under one per cent, which is small enough to be missed by anybody working from experience and large enough to happen.

How often a draft falls apartEvery four-by-four draft in which each end and each pick interlaces at least once, sorted by its longest float, with the fraction that describe more than one cloth. The counts are produced by running the enumeration rather than by recalling it.22,874 drafts tested · 144 separateof the 90 balanced ones, 0 separatelongest float 10 of 20.0%longest float 20 of 880.0%longest float 3144 of 227840.6%enumerated while the figure was drawn4 × 4
Fig. 9 The census, sorted by longest float. The enumeration runs while the figure is drawn, so the numbers cannot go stale — and the pattern in them is informative: the failures cluster at longer floats and vanish entirely among the balanced drafts.

The second row of that figure is the more interesting one. Of the 90 balanced four-by-four drafts — two up and two down in every end and every pick — not one separates. Balance appears to force integrity at this size, and it survived a sample of balanced six-by-six drafts as well. That is a measurement rather than a theorem, and it is stated that way.

Where point paper came from

Recording a weave as a grid is a mediaeval European device, and it arrived with the drawloom.

Before it, a pattern was carried in the setup of the loom itself and in the memory of the person who had built it. A drawloom encodes the pattern in cords, and a figured cloth might take days to set up; the pattern was a physical object, and copying it meant rebuilding it. Point paper made the pattern portable, which is a much larger change than it sounds — a design could be sent, sold, archived and altered without a loom present.

The next step is the one everybody knows. Joseph-Marie Jacquard’s mechanism of 1804 replaced the drawboy with a chain of punched cards, one card per pick, a hole where an end is lifted. That is the draft made mechanical: the matrix is the cards, and the cloth is the cards executed. Charles Babbage took the idea directly for the Analytical Engine, and the line from a weaving draft to a punched card to a stored program is short and entirely real.

The formal study is much later. Treating a periodic weave as a mathematical object with decidable properties belongs mostly to the 1980s, when Branko Grünbaum and Geoffrey Shephard wrote a series of papers on the geometry of fabrics — including the question of when a weave hangs together, which is the one this site leans on hardest.

Where the ladder goes next

The immediate next step is the check the matrix makes possible: does it hang together, which is the site’s reason for existing.

Then the three foundation weaves and what actually separates them — plain, twill and satin — and the quantity that does most of the separating, which is the float.

What the pictures here cannot show. A draft is a rule for an infinite cloth and every figure shows a few repeats of it. More importantly, a draft shows structure and not fabric: nothing about thickness, friction, twist or finish is in any picture on this page, and those are between them responsible for much of how a real cloth behaves.

Where this goes next

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 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. 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. 5-end satinwarp floats of 4weft floats of 120 markedruns measured round the repeatfloat 4 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.