The pattern seam · a theorem you can feel in your hands

Shake the Cloth

A weave draft is nothing but a grid of over-and-unders, and some grids are not cloth at all. Shake them and they come apart. Below is a draft that looks like satin and behaves like a fringe: press the button and three whole warp threads drop out of it, because gcd(2,6) = 2 and those three threads were never bound to anything. Then the browser enumerates every draft up to four by four and tells you exactly how rare cloth is.

That is the whole subject in one gesture. A draft (weavers call the picture a drawdown) is a binary matrix: warps run down, wefts run across, and each cell records which of the two passes over the other. The mathematical question, first asked in print by Branko Grünbaum and G. C. Shephard in 1980, is which of those matrices are fabric at all. Robert Thomas, writing in the same tradition, puts the definition plainly:

"a fabric is a prefabric that hangs together, that is, that does not fall apart in the sense that some warps or some wefts or some of each can be lifted off the remainder because they are not bound into a coherent network by the interleaving defined by the preferential ranking."

Make that arithmetic. Write M[i][j] = 1 when warp j passes over weft i. A set of warps W and wefts F can be lifted clean off the rest exactly when two conditions hold, and both are forced by geometry rather than chosen:

The fabric falls apart if any such pair exists other than the two useless ones (lift nothing, lift everything). Watch the animation again with that in mind: nothing ever has to pass through anything. The threads that leave were already on top of every thread they cross that stays. The animation is not an illustration of the proof; it is the proof.

Why checking one thread at a time is enough

There are 2²ⁿ candidate pairs, which is not a search you would want to run. You never have to. The two conditions read as forced implications:

So each single strand generates a smallest liftable set containing it: seed that strand and close under the two rules. Every liftable set containing that strand contains its closure. Therefore the fabric falls apart if and only if some strand's closure is not the whole cloth, and there are only 2n strands in a repeat to try. Two further things fall out for free. First, both rules quantify over all rows (respectively all columns), so a closure is automatically a union of whole periodic thread classes: no aperiodic separation can hide from this test. Second, the liftable sets are closed under union and intersection, which is a two-line check: if j lies in W₁ ∪ W₂ then it lies in one of them, say W₁, and any i outside F₁ ∪ F₂ is outside F₁, so M[i][j] = 1 already. That is the algorithm the button above runs, and it is what makes the decision cheap: T. C. Enns published an efficient algorithm for exactly this in 1984, four years after Clapham settled the decision problem.

One corollary you can test immediately, because it disposes of a whole family at once: every single-riser draft hangs together. If each pick has exactly one warp raised and each warp is raised on exactly one pick (a permutation matrix), seed any warp j; it is under all but one weft, so all but one weft joins; each of those wefts drags in the one warp it rides under; between them they name every warp. The closure is everything. Every regular satin is a permutation matrix, so every regular satin hangs together, and the reason a six-end "satin" fails is not that it is a bad permutation. It is that it is not a permutation at all.

The same question, asked as a graph

There is a second way in, and it is worth having because it agrees with the first for entirely different reasons. Draw one node per strand, and at every crossing draw an arrow from the thread on top to the thread underneath. Because exactly one thread is on top at each crossing, this is a complete bipartite tournament on the 2n strands: Clapham's object, from a 1985 note in Discrete Mathematics. A set A of strands can be lifted off exactly when no arrow points into A from outside it, because an arrow into A is a strand outside it lying on top of one inside it, which would have to be dragged along. And a digraph has a nonempty proper subset with no incoming arrows precisely when it is not strongly connected. So: a fabric hangs together if and only if its tournament is strongly connected.

That is a completely different algorithm, two graph searches forward and backward from any single node, resting on a completely different idea. The check panel at the foot of this page runs both procedures over all 65,536 four-by-four drafts on load and reports the number of drafts on which they disagree. It is zero, and that zero is recomputed in front of you.

What "on six shafts" actually means

A loom does not let you address crossings one at a time. Warp threads are gathered onto shafts; a shaft lifts as a unit; a treadle lifts a chosen set of shafts; and a pick presses one treadle. Three matrices, and the cloth is their product over the boolean semiring (+ is OR, × is AND):

Two ends threaded on the same shaft always rise together, so they have identical drawdown columns. That gives a sharp constraint, not a rule of thumb: the minimum number of shafts a drawdown needs is exactly its number of distinct warp columns, and the minimum number of treadles is its number of distinct picks. Paint a drawdown below and the browser will factor it: group the identical columns into shafts, group the identical picks into treadles, read the tie-up off the intersections, then multiply the three matrices back together and check every cell.

The wheel, and the wall it hits at six

The three great weave families are one family. Take a repeat of n and a step s, and put the riser of pick i on warp s·i mod n. Step 1 is a straight twill, its risers marching in a diagonal you can see across a pair of jeans. Step n-1 is the same twill leaning the other way. Anything strictly between them scatters the risers so no diagonal forms, and that is satin: the reason it is smooth and shiny is that its binding points are hidden under long unbroken floats of thread that reflect light as a sheet.

Draw the step on a wheel and the arithmetic becomes visible. Mark the n warps around a circle and step round by s each time. You trace a star polygon, and you return to the start after n / gcd(s,n) moves. The path visits every warp exactly when gcd(s,n) = 1. Otherwise it closes early, leaving whole warps that the risers never touch at all.

So there it is, and it is not deep mathematics: it is the definition of coprime. What makes it worth a page is that an elementary fact about the number six has a consequence you can feel through a shirt. Handwoven states it flatly, without the arithmetic: "True satin isn't possible on six shafts because the counter cannot be spaced evenly." Four is in the same position. Every other repeat from five to twenty-four admits at least one regular satin. Six does not, and no amount of loom skill changes that.

What weavers do about six

They break the rule that the step be constant, and keep only the thing the step was for: no two risers in consecutive picks may land in the same warp or a neighbouring one, since adjacent binding points would start a visible diagonal and spoil the surface. Formalise that and enumerate all n! single-riser orders. The browser does it below for n = 4 to 8.

Four gives zero: on four shafts there is no single-riser order at all that keeps its risers apart, regular or otherwise, which is why a "four-end satin" always has adjacent binding points somewhere. Six gives a nonempty answer, and every one of those orders hangs together. The order usually printed in weaving texts for irregular six-end satin, 1, 3, 6, 4, 2, 5, is one of them; you can check it in the list above and shake it in the first instrument.

How rare is cloth?

Here is where the sophisticated objection arrives: surely any reasonably alternating pattern holds, so this is a curiosity about degenerate cases. The way to settle that is not to argue. It is to count every single draft.

Run the four by four and look at the last two columns. The overwhelming majority of failures are trivial: a thread that never interlaces. Strip those out and the residue at four by four is very small, and it has a shape. Press the button below and the browser will pull every subtle failure out of the sixty-five thousand, sort them by symmetry, and draw one of each.

Every subtle four by four failure splits two warps and two wefts away from two warps and two wefts. That is not a curiosity: it is the signature of double cloth, the technique in which a weaver deliberately arranges the fall-apart so the loom produces two separate fabrics at once, joined only where they are made to swap layers. Pockets, tubes, double-faced blankets and the pouches of a rug loom are all built on the failure this criterion detects. At four by four there is literally nothing else: a draft is cloth, or it has a thread hanging loose, or it is two cloths.

The frontier, part one: what happens as the repeat grows

The census fraction rises. At two by two it is one eighth; at four by four it is about a third. The obvious question is where it goes, and this is the point at which honesty costs something: we could not find this enumeration published anywhere as of 2026-07-24. The sequence 2, 102, 22730 returns nothing from OEIS, alone or in combination (three query variants, re-run at build time). That is an absence of evidence, not a discovery claim; the literature on fabric combinatorics is small but old, and a 1980s paper we have not read may well contain it.

What we can do is make the asymptotics operable. There is an exact handle on the trivial failures. A given row is constant with probability 2^1-n²·2^n²-n, that is 2^1-n; there are n rows and n columns, so a union bound gives about 4n·2^-n. The exact count follows by inclusion and exclusion, with one wrinkle worth the arithmetic: if some rows and some columns are all constant, they meet, so they must share a single value, which contributes a factor of 2 rather than 2^r+c. The slider below computes that exact count and puts it against a live sample of the true failure rate.

So the curation instinct that a hanging-together draft is a rare and special object is exactly backwards past a certain size. It is rare at two, uncommon at three, a minority at four, and a majority at five, which is the largest repeat this page enumerates exhaustively. Past five the claim rests on two weaker things: the exact loose-strand count, which bounds the failure rate from below, and the sampler above, which estimates it. Both say the majority holds and grows, and by twelve the sampler finds essentially no way left to fail but leaving a thread out. That is evidence, not enumeration. The interesting drafts are not the ones that hold. They are the small handful, like the double cloths, that fail without a loose thread.

Below is the exhaustive five by five result. It is the one number on this page that is not computed live by default, because it is thirty-three and a half million drafts. The button will run it here if you want to watch; on a desktop browser it takes ten to twenty seconds, longer on a phone, and it should land on exactly the same figures.

The frontier, part two: none of this is why your shirt holds together

Everything above treats threads as rigid, infinitely long, perfectly slippery lines, and "falls apart" as a rigid lift-off in a fabric with no edges. Real cloth has selvages, hems and finishing; real threads are hairy and grip each other; a six-end step-two draft on a real loom with bound edges does not disintegrate, it just sheds three sleazy floating threads that a fingernail will pull out. The combinatorics tells you when a fabric has no resistance to separation. It says nothing at all about how much resistance a hanging-together fabric has.

That is not a gap you can close by being cleverer with matrices; it is a different subject. Warren, Ball and Goldstein showed in 2018 that even the individual yarn is a frictional object: tension in a staple yarn (one spun from short fibres, which is most of them) is transmitted by collective friction, with a percolation transition above which the transmitted tension is in principle unbounded. Below the threshold the yarn is not strengthless; it fails ductilely, by fibres slipping past each other at around a millinewton per fibre. Above it, failing means breaking fibres, at twenty to a hundred and thirty millinewtons each, and the authors put the jump across the transition at perhaps a hundred-fold increase in tensile strength. A thread is already a fabric that hangs together by friction, one level down.

The instrument below is honest about the seam. The left column is exact combinatorics read off your draft from the first instrument. The right column is a deliberately crude model, and every number in it carries the model badge. It is not calibrated against any measurement and its units are arbitrary.