Textile Structure
how cloth is put together
What cloth is

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.
16 min read 6 figures Does it hang togetherA weave is a matrix

Here are two drafts. Both interlace at every end and every pick. Neither has a float longer than three. Both look like perfectly ordinary small weaves, of the kind that turns up in any pattern book.

One of them is cloth. The other is two cloths lying on top of each other, connected nowhere, which would come apart in the hand.

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. 1 The two drafts, with a bar beside each thread saying which separable cloth it belongs to. On the left every thread is one colour, because there is only one cloth. On the right they divide, and nothing in the grid of filled and empty squares distinguishes the two cases.

This essay is about how to tell, and the answer is that nobody can tell by looking — including people who have woven for forty years, because there is nothing to look at.

What “hangs together” means

Being precise first, because the ordinary-language version is vaguer than the property.

A woven fabric is a set of threads occupying a plane, with a definite over-or-under relation at every crossing — which is exactly what the draft records. Ask whether some of those threads could be lifted away from the rest without cutting anything. If they can, the fabric is not one fabric: it is a stack, and gravity or a fingernail will find that out.

Formally: a fabric hangs together when no non-trivial set of its threads can be separated from the rest by a rigid motion. It is a topological property of the structure, not a strength property of the yarn, and it holds or fails exactly.

The construction that decides it

The test is short enough to give in full, and it is worth having because it is not obvious that the question is decidable at all.

At every intersection, one thread is above another. Suppose the threads could be split into an upper set UU and a lower set LL such that at every crossing between the two sets, the UU thread is on top. Then UU lifts straight off, and the fabric separates.

Now read that as a constraint. Take any crossing where thread AA is above thread BB. If BB is in UU, then AA must be in UU too — because if AA were in LL, there would be an LL thread sitting above a UU thread, which a separation forbids. So membership propagates: B∈UB \in U forces A∈UA \in U.

Draw a directed edge from BB to AA for every crossing where AA is above BB. A separation exists exactly when some non-trivial set of threads is closed under following those edges. And a non-trivial closed set exists exactly when the digraph is not strongly connected.

The cloth hangs together if and only if that digraph is strongly connected. The number of separable layers is the number of strongly connected components, and finding them is one linear-time pass.

Worked through on plain weave

The construction is easier to trust once it has been run on something small, and plain weave is as small as a real weave gets.

Two ends and two picks. Call the ends e0e_0 and e1e_1 and the picks p0p_0 and p1p_1. In plain weave e0e_0 is up on p0p_0 and down on p1p_1; e1e_1 is the reverse.

At the crossing of p0p_0 and e0e_0 the warp is on top, so the edge runs p0→e0p_0 \to e_0. At p0p_0 and e1e_1 the weft is on top, giving e1→p0e_1 \to p_0. At p1p_1 and e0e_0 the weft is on top, giving e0→p1e_0 \to p_1. At p1p_1 and e1e_1 the warp is on top, giving p1→e1p_1 \to e_1.

Four edges, and they form a single cycle: p0→e0→p1→e1→p0p_0 \to e_0 \to p_1 \to e_1 \to p_0. Every thread reaches every other, the digraph is strongly connected, and plain weave hangs together. It could hardly do otherwise — it is the most thoroughly interlaced structure there is — but the point is that the argument produced the answer rather than assuming it.

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. 2 The weave the argument was run on, drawn over three repeats. Two ends and two picks describe the whole of it, and the cycle that ties them together is the shortest such cycle any fabric can have.

Now change one square. Make e0e_0 up on both picks and e1e_1 down on both. Every edge now points from a pick to e0e_0 or from e1e_1 to a pick, nothing leaves e0e_0, and the digraph falls apart. That draft is not a weave: it is a warp end lying over two loose wefts.

Both directions matter

The check is run in both directions on every weave figure here, and the second direction is the one that earns its place.

Detecting more layers than were claimed is the accident. A draft meant as a single cloth turns out to be two, and nothing about the drawing said so.

Detecting fewer is also a failure, and a real one. Double cloth is a deliberate construction — two complete fabrics woven simultaneously, sometimes joined at the edges to make a tube, sometimes stitched together at intervals, sometimes exchanged so the two swap faces and produce a reversible pattern. A draft intended as a double cloth that comes out as a single one has been stitched together somewhere it should not have been.

So a figure does not assert “this holds together”. It asserts a number, and one is the ordinary case rather than the only acceptable one.

Double cloth, verified rather than refused

The clearest evidence that the check is doing something real rather than merely rejecting oddities is that it certifies the construction it might have been expected to reject.

A double cloth is woven with two warps and two wefts at once. The face warp interlaces with the face weft, the back warp with the back weft, and the two systems pass each other without ever exchanging — the face warp always over the back weft, the back warp always under the face weft. What comes off the loom is two complete fabrics occupying the same space.

Run the connectivity test on such a draft and it reports exactly two components, in the right order: everything face reachable from everything face, everything back likewise, and every cross edge running one way. Assert one layer and it throws. Assert two and it passes.

That matters because it is how a check earns trust. A test that only ever says “no” to strange things is a filter; a test that distinguishes an intended two-layer construction from an accidental one is a measurement.

Weavers exploit the same structure in several ways. Joining the two layers at one selvedge gives a tube; joining at both gives a double-width cloth off a narrow loom; stitching them together at intervals gives a quilted fabric with no sewing; and exchanging the layers according to a pattern gives a reversible cloth in two colours, which is how a great deal of coverlet weaving works.

The crude failure, and the subtle one

Two things go wrong, and they are worth separating because only the second is interesting.

The crude case is a thread that never interlaces at all. A warp end filled down the whole repeat is on the face everywhere and under nothing; it lies on the surface and can be drawn out with a fingernail. It is a component of the digraph on its own, because edges point toward it and none point away.

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. 3 A draft with an end that never goes under anything. On point paper it is a solid column and nothing more, which is easy to draw by accident when a repeat is being edited. In cloth it is a thread that is simply not woven in.

That failure at least has a visible signature once a reader knows to look for a solid column. The subtle case has none. The draft in the opening figure interlaces at every end and every pick, so every column and every row has both filled and empty squares, and it still describes two fabrics. The separation is a property of which threads are above which, not of how often anything changes face.

Counting how common it is

A hazard is worth taking seriously in proportion to how often it occurs, and here the question can simply be answered.

Take every four-by-four draft in which each end and each pick interlaces at least once — 22,874 of them. Test each. 144 describe more than one cloth: rather under one per cent.

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. 4 The census, sorted by longest float. Every draft is generated and tested while the figure is drawn, so the numbers are the output of the enumeration rather than a recollection of it.

Two things in that table are worth more than the headline number.

The failures cluster at long floats. Nothing separable appears among the drafts whose longest float is one or two, and the fraction climbs as the floats lengthen. That makes sense: a thread that stays on the face for a long run has few opportunities to be tied down, and integrity is exactly a question about being tied down.

And of the 90 balanced drafts — two up and two down in every end and every pick — not one separates. Balance appears to force integrity at this size. That result also survived a sample of balanced six-by-six drafts, none of which separated either. It is a measurement rather than a theorem: nothing here proves it holds in general, and saying otherwise would be exactly the over-claim this site is supposed to avoid.

Why the eye cannot do this

It is worth being explicit about why experience does not substitute, because the natural reaction is that a weaver would notice.

A weaver reading a draft is checking things a person can check: that the floats are not too long for the intended use, that the interlacing is firm enough, that the pattern repeats as intended, that the shafts required are available. Every one of those is a local property — a statement about a row, a column, or a small neighbourhood.

Integrity is not local. It is a statement about the global connectivity of a relation defined across the whole repeat, and there is no neighbourhood a reader can inspect that settles it. Two drafts differing in one square can differ in layer count, and two drafts that look nothing alike can both be sound.

That is the general shape of the problem this site keeps running into. The properties that are easy to see are the ones a person is already good at judging, and the properties that need computing are precisely the ones with no visual signature at all.

The same question in other structures

Integrity is not a woven-cloth idea, though it takes its sharpest form there.

A knitted fabric is one thread. Connectivity is trivially satisfied — everything is the same strand — and yet a knit fails more easily than a weave, because a single break frees every loop above it. Ravel, fray and run takes that up: the two structures have opposite failure modes and the reason is topological rather than material.

A braid interlaces three or more thread systems obliquely, and the same connectivity question applies with a different geometry. A braid whose strands split into two independent sets is two braids sharing a space.

A nonwoven has no periodic structure at all, so the question changes character completely: coherence there is statistical, a matter of how many fibre-to-fibre contacts there are and how well they grip, and the exact combinatorial test has nothing to work on.

One break, two outcomesThe same single break in a knit and in a weave. In the knit nothing holds the loop above the break, so the failure climbs the wale; in the weave every other thread is still held by the threads crossing it, and one thread comes loose.a knit — the break runs up the wale3 loops freeda weave — the break stays put1 thread freedcounted from the structure, not from experience
Fig. 5 The contrast that makes the point. One break in a knit frees every loop above it, because nothing else holds them; one break in a weave frees one thread, because every other thread is held by the threads crossing it. Both fabrics were perfectly sound before the break.

What a designer does with this

The check is not only a way of catching mistakes; read forwards, it is a design constraint with a shape.

The census says failures cluster at long floats, which is a warning to exactly the kind of weave a designer reaches for when they want lustre or drape. A satin is safe because its single interlacing per pick is placed to scatter, but a fancy weave with long floats arranged carelessly — a large figured repeat, a jacquard design worked out cell by cell rather than from a rule — is the case where the hazard is real.

That is also where drafts are largest and least amenable to being checked by hand. A jacquard repeat may be several hundred ends across; a designer cannot trace the connectivity of a thousand threads by eye, and there is no reason they should have to.

Where the floats are in the 8-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.8-end satinwarp floats of 7weft floats of 18 markedruns measured round the repeatfloat 7
Fig. 6 The floats in an eight-end satin: seven ends of warp on the face for every one of weft. The structure survives because each pick is tied down somewhere and the tie points are scattered rather than aligned — which is the same property that stops the surface reading as a twill.

The practical rule that falls out is not “avoid long floats” — long floats are the whole point of a satin — but make sure every thread is tied somewhere, and check rather than assume it. On a rule-generated weave that is nearly automatic. On a designed one it is not, and the drafts most likely to fail are the most ambitious ones.

Where the criterion comes from

The mathematics of periodic weaves is younger than one might expect. Cloth is prehistoric; the question of whether a given periodic interlacement must hold together seems not to have been posed formally until the twentieth century, and the sustained treatment belongs to the 1980s.

Branko Grünbaum and Geoffrey Shephard wrote a series of papers taking fabrics seriously as geometric objects — classifying periodic weaves by their symmetry, working out which ones are “isonemal” in the sense that the symmetry group acts transitively on the strands, and addressing directly the question of when a fabric hangs together. The connectivity formulation used on this site is the natural computational reading of that question, and while the framing here is the site’s own, the question is theirs.

It is a good illustration of how recent the analysis of a very old craft can be. The same is true of the satin condition, which weavers used as a rule for centuries before anybody wrote down why six ends admit none.

What the check does not do

Three limits, and the third is the important one.

It does not know about yarn. Integrity is a property of the over-and-under relation. A structure that hangs together in the matrix can still be a bad cloth — floats too long to survive wear, a sett too open to hold together in practice, a yarn too smooth to grip. Those are questions about geometry and friction, and so is how much of the surface the threads actually cover. All of it is outside the matrix.

It does not know about edges. A real fabric is finite, and every property here is stated for the infinite periodic cloth. A selvedge, a cut edge and a seam all behave differently, and the check says nothing about any of them.

It does not mean a cloth cannot be pulled apart. A single-cloth structure can still be destroyed: threads can be cut, and a loosely set fabric can be distorted until the crossings slip. Integrity says the structure is topologically one piece, which is a weaker and more precise claim than “strong”.

Where the ladder goes next

The natural next step is the weaves themselves: plain, twill and satin, which are the three rules almost everything else is built from, and every one of which passes the check comfortably.

The natural companion is the quantity the census kept pointing at — the float, which is where the failures clustered and which turns out to sit behind almost every property a fabric has.

And the natural extension is what the eye gets wrong more generally, of which this is the sharpest case: a property with no visual signature at all.

What the pictures here cannot show. Every figure on this page shows a draft, and integrity is not a property of a drawing. The bars beside the threads are the output of a computation, not a feature of the grid, and a reader comparing the two drafts by eye is comparing the annotations rather than anything visible in the weaves themselves.

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. 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 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. 100 ends and 100 picks per inch in bothfine yarncover 78%coarser yarncover 97%cover from diameter and spacing, circular sections 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.