Does it hang together

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.

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 $U$ and a lower set $L$ such that at every crossing between the two sets, the $U$ thread is on top. Then $U$ lifts straight off, and the fabric separates.

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

Draw a directed edge from $B$ to $A$ for every crossing where $A$ is above $B$. 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 $e_0$ and $e_1$ and the picks $p_0$ and $p_1$. In plain weave $e_0$ is up on $p_0$ and down on $p_1$; $e_1$ is the reverse.

At the crossing of $p_0$ and $e_0$ the warp is on top, so the edge runs $p_0 \to e_0$. At $p_0$ and $e_1$ the weft is on top, giving $e_1 \to p_0$. At $p_1$ and $e_0$ the weft is on top, giving $e_0 \to p_1$. At $p_1$ and $e_1$ the warp is on top, giving $p_1 \to e_1$.

Four edges, and they form a single cycle: $p_0 \to e_0 \to p_1 \to e_1 \to p_0$. 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.

Now change one square. Make $e_0$ up on both picks and $e_1$ down on both. Every edge now points from a pick to $e_0$ or from $e_1$ to a pick, nothing leaves $e_0$, 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 both selvedges gives a tube; joining at one 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.

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.

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.

Balance forces integrity, and it is a theorem #

The census reports that none of the ninety balanced drafts separates, that a sample at six by six behaved the same way, and calls it a measurement rather than a theorem. It is a theorem, and the proof is four lines of counting.

Suppose a balanced draft on an even repeat n did separate, into an upper set U and a lower set L. Write a for the ends in U and b for the picks in U.

Every U-end lies over every L-pick, because a separation forbids an L thread above a U thread. So each U-end has n − b forced warp-up cells, and balance allows only n/2, giving b ≥ n/2.

Every L-end lies under every U-pick. So each L-end has b forced warp-down cells, leaving at most n − b places for its warp-ups, and balance demands n/2 of them, giving b ≤ n/2.

So b = n/2 exactly, and the same argument down the columns gives a = n/2.

Now finish it. A U-end already has its full quota of n/2 warp-ups among the L-picks, so every cell where a U-end meets a U-pick is warp-down. And every cell where an L-end meets a U-pick is warp-down by the separation itself. So a U-pick is warp-down at all n of its ends — no warp-ups at all — and a balanced pick must have n/2.

Contradiction. No balanced draft separates, at any even repeat.

What the proof says that the census could not #

Three things, and the third is the one worth carrying.

It holds at every size. The census exhausted four by four and sampled six by six; the argument uses only the balance condition and the definition of a separation, so it covers eight, twelve and a jacquard repeat of five hundred alike. The sample was not lucky.

It identifies what the balance is doing. The contradiction arrives because balance is a budget: a thread has exactly n/2 warp-ups to spend, and a separation forces it to spend all of them on one side of the split, leaving nothing for the other. A separation is a demand for a systematic bias, and balance is precisely the statement that no thread has one.

And it says how far the condition can be weakened. Nothing in the proof needs both systems balanced. Following it with only the ends balanced still gives b = n/2 from the two end conditions, and the final step then needs a pick’s quota — so the theorem as it stands wants both. But the two inequalities on b come from the ends alone, so any draft whose ends are balanced can only separate with exactly half the picks in each set, which is already a severe restriction and is checkable by inspection.

That is a genuinely useful weakening for a designer. A draft with balanced ends and an odd number of picks cannot separate at all, because n/2 is not an integer — a one-line test on two numbers, with no digraph and no computation.

And it explains where the failures live #

The proof also accounts for the census’s other observation, that the failures cluster at long floats, without needing a second argument.

A separation forces a whole block of cells to one value — every U-end over every L-pick, every L-end under every U-pick — so a separable draft necessarily contains a solid rectangle of warp-ups and a solid rectangle of warp-downs, each n/2 by n/2. A solid rectangle of that size is a set of long runs, which is a set of long floats.

So long floats are not correlated with separation; they are required by it. The census’s clustering is the theorem’s forced blocks showing up in a float count, and a draft whose longest float is under n/2 cannot separate at all — which at four by four means a longest float under two, and is exactly where the census found nothing.

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.

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.

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.