Where a stitch can hide

Two cloths woven in one repeat become one cloth the moment a single intersection is turned over, and the enumeration says half of them would do it — eight of the sixteen in the smallest repeat, eighteen of thirty-six in the next.

That is the whole of the arithmetic and almost none of the problem. A stitch is a thread appearing on a face it does not belong to. Every one of those eight positions joins the cloth; the question that decides whether the cloth is any good is which of them can be used without the join being visible, and the answer is much smaller than eight and sometimes zero.

A stitch is a thread on the wrong face #

The construction to be defended is the one the previous rung set out. A double cloth has face warp over back weft everywhere and back warp under face weft everywhere, and those two rules are what make it two fabrics rather than one.

A stitch breaks one of them once. The commonest kind brings a back warp end over a face pick: at that one intersection the back system is on top, the missing direction of edge exists in the above-and-below digraph, and the two components fuse.

What is on the surface afterwards is a spot of the back warp’s colour, on the face, at one intersection. In a backed cloth that is the whole design failure — the back exists to add weight and warmth and must not be seen — and in a two-colour coverlet it is a wrong-coloured dot in the middle of a figure.

The mirror case exists and is not its mirror in practice. A face end dropped under a back pick puts a face-coloured spot on the back, and it is covered by the back weave’s weft floats rather than by anybody’s warp floats. A back that is warp-faced has almost no weft float to spare, so the two directions of stitching are not interchangeable: choosing one is choosing which face has floats to lend.

So the question is: at which positions does the face weave lie over the stitch?

The rule, written down so that it can be disagreed with #

The trade’s version is a sentence — stitch under a float, never at an interlacing — and it is right. It is also not something that can be counted, because “under a float” has to be turned into a statement about the matrix before anything can be enumerated, and that translation is a modelling choice rather than a fact.

Two are implemented here, and both are stated in full because the answer moves enormously between them.

The covering rule. A stitch in the gap between face ends a and b, appearing at face pick i, is covered when both a and b are on the face at pick i, and each of them is in a warp float of at least two picks. The first clause puts warp across the stitch on both sides of it. The second stops the cover being a single crossing, which has no length to it and no capacity to lie over anything.

The strict rule. The same, and in addition both neighbours are on the face at the pick before and the pick after — each is in a float of at least three picks with the stitch’s own pick in its interior. This is the rule for a cloth that will be milled or raised, where a float of two is not a roof over anything once the fibres have been made to move.

Neither is the rule. Saying which one produced a number is the whole of the honesty available, and a reader who thinks the second is the only serious one is welcome to read every strict column below and ignore the other.

Plain weave has nowhere at all #

The first consequence is exact and needs no census to see, although the census confirms it.

Plain weave puts opposite states on adjacent ends at every pick — it is the weave in which the parity of end plus pick decides everything. So for any pick and any gap, one of the two neighbouring ends is on the face and the other is not, and there is never a pair of neighbours to hide under. Not one of the four positions in the repeat passes either rule, and enlarging the repeat does not help, because the property is local and the repeat is a tiling of it.

That is the reason a plain-faced backed cloth is a rare thing and a satin-faced one is ordinary, and it is a stronger reason than the usual one. The usual account says a satin hides a stitch better. The arithmetic says plain weave hides one not at all.

What was counted, and how #

Nine face weaves, both rules, every position in each repeat tried in turn. The enumeration runs while the figures are drawn, so a caption and its picture cannot disagree.

For each face weave the count is taken over the n picks and n gaps of the repeat — n² candidate positions, which is exactly the number of places a back end could be raised over a face pick. A position passes when the rule allows it, and nothing else is consulted.

Three things in that table are worth more than the ranking.

The basket is the counterexample to the obvious generalisation. A two-and-two basket has four legal positions, the same as the two-and-two twill — and only two of them can be used at once without two stitches falling in the same gap, against the twill’s four. The basket’s legal positions crowd into two of its four gaps, because the weave has a period of two in the ends and the twill has none. Legal positions and usable positions are different counts, and only one of them is what a designer has.

The satins have a closed form. For a satin of order n, the covering rule allows exactly n(n − 2) positions. Each pick of a satin has exactly one weft-face intersection, so a pair of adjacent ends is spoilt at exactly the two picks where one of them is that intersection — two picks out of n, for each of n gaps. Fifteen on five ends, forty-eight on eight, two hundred and twenty-four on sixteen, and the counted numbers agree with the formula rather than being quoted from it.

And the strict rule is much harsher than it sounds. Under it a two-and-two twill has nothing, a three-and-one twill has nothing, and a five-end satin has nothing either — which is not what a reader who has got as far as “satins are good for this” expects. A five-end satin’s interlacings on neighbouring ends sit two picks apart, and two forbidden windows of three picks each, two apart, leave nothing over in a repeat of five.

Two satins of the same order, and one hides twice as much #

The strict rule has a closed form too, and it contains something the covering rule does not: the move.

Two neighbouring ends of a satin interlace d picks apart, where d is the move’s inverse modulo the order, taken to the nearer end of the repeat. Each interlacing spoils a window of three picks. Two windows three or more picks apart are disjoint and cost six picks between them; two windows exactly two apart overlap in one pick and cost five. So the count is n(n − 6) in the first case and n(n − 5) in the second.

On seven ends that is a factor of two. A seven-end satin of move 3 or 4 has fourteen strictly covered positions; one of move 2 or 5 has seven. The two cloths have the same order, the same float, the same interlacing count, the same shaft requirement and the same everything a weaver measures — which satins are worth weaving is normally settled by how evenly the interlacings scatter — and one of them has twice the room to stitch a backing to.

How few stitches, and where they may be #

The other half of the question is how many stitches are needed at all, and it has an answer that corrects something this site had in print.

For two layers the answer is one, and that has been known here since the integrity ladder was built: the digraph needs exactly one edge in the missing direction, and any of the eight cross intersections supplies it.

For three layers the site said two, on the reasonable ground that three components need more joining than two. It is wrong. The face warp already passes over the back weft — over every layer below it, not merely the next one — so a single stitch between the outermost two layers closes a cycle that runs through every layer in between. One stitch joins a triple cloth, and the exhaustive search over every set of one confirms it: eight of the twenty-four cross intersections do it alone.

What is true is the version with a reach in it. Restrict a stitch to joining layers s apart and the fewest that will do is the ceiling of (k − 1) divided by s — two for three layers stitched only between neighbours, three for four layers, and one at every k whenever a stitch may span the whole thickness. The search finds exactly that number at every case tried, having been told nothing about the formula.

When the two requirements cannot both be met #

Now the two halves together, which is where a design either exists or does not.

Connectivity needs at least one stitch. The covering rule allows a certain number of positions. Every position the covering rule allows is a joining position — a back end raised over a face pick is exactly the reversal that adds the missing edge — so whenever the legal set is non-empty, the minimum stitching can be hidden and the two requirements agree.

When the legal set is empty they cannot. A plain-faced double cloth needs one stitch and has nowhere to put one; under the strict rule a two-and-two twill face, a three-and-one twill face and a five-end satin face are in the same position. The requirement is not “harder” in those cases. It is unsatisfiable, and the answer is a different face weave rather than a better placement — which is the same shape of result the applied field keeps producing, where a specification is two inequalities and the interval between them can be empty.

Legal is not usable, and the difference is a matching #

The basket is treated above as a counterexample and it is more than that: it is the case that shows the count a designer wants is not the count the rule produces.

A stitching plan wants one stitch on every pick and one in every gap — that is what stops the stitching reading as a pattern of its own, for the same reason a satin’s own interlacings do not read as a line. So a plan of n stitches in a repeat of n picks and n gaps is a set of legal positions with no two sharing a pick and no two sharing a gap.

That is a matching in a bipartite graph: picks on one side, gaps on the other, an edge wherever the rule allows a stitch. And a matching of size n exists exactly when Hall’s condition holds — every set of k picks must have legal positions in at least k different gaps between them.

The basket fails it at k = 3. Its four legal positions occupy only two gaps, so any three picks have at most two gaps available and no plan of four exists; the largest matching is two. The twill’s four legal positions occupy four different gaps and one pick each, so its matching is four and every pick gets a stitch.

Two weaves with the same legal count and different usable counts, and nothing in the legal count says so. The distinction has the shape this collection keeps meeting: a bound computed from a local rule, and an achievable figure that needs a global condition on the same set.

Three things follow that the census’s own columns do not carry.

A weave with a period in the ends is the case to check. The basket’s four legal positions crowd into two gaps because the basket repeats every two ends; anything built by doubling a weave will do the same, and the doubling is exactly what a manual teaches as a way to make a heavier cloth. So the constructions most likely to fail Hall’s condition are the ones a designer reaches for on other grounds.

The satins never fail it. A satin has one interlacing per pick and one per end, so its legal positions are spread by construction — every gap is spoilt at exactly two picks and every pick spoils exactly two gaps, which is a regular bipartite graph and therefore has a perfect matching by König’s theorem. Every satin that has any legal positions has a full plan, and that is a stronger statement than the count of positions.

And the matching is what a jacquard designer is actually drawing. Once every end lifts independently the stitching plan is a picture, and the picture has to be a system of distinct representatives whether or not anybody calls it one. A plan that puts two stitches on one pick leaves another pick unstitched, which is a line of unsupported backing across the cloth — the failure the scattering rule exists to prevent, arriving as a consequence of the arithmetic rather than as a separate caution.

Where the model stops #

A covering rule is not a visibility model. Nothing here knows the colour contrast between the two warps, the relative diameters, the sett, or whether the cloth will be raised. A stitch that the rule forbids may be perfectly acceptable in two yarns of one colour, and a stitch the rule allows may show plainly in black on white.

Nothing is said about how many stitches are wanted. The minimum is a topological quantity; a real double cloth is stitched far more often than the minimum, because the layers slide between stitch points and quilted cloths bag and blister. How close is close enough is a mechanical question about friction and drape, and this site does not have it.

The scattered plan is a count, not a pattern. One stitch on every pick and every gap is the arrangement that stops the stitching reading as a check of its own, for the reason a satin’s own interlacings do not read as a line. Whether a particular such plan is well distributed is a question about the arrangement, and the maximum computed here says only that one exists.

And the count is of the face, not of the cloth. The rule is applied to the face weave alone, which is right for a stitch that shows on the face and says nothing about what the back weave is doing at the same intersection.

Who found it, and when #

The rule is old and the count is not. Every weaving manual from the nineteenth century on gives the stitching rule as a sentence and a diagram — stitch where the face floats cover, scatter the points, never on a line — and the drafts in those books are correct.

What is not in them is the quantity. A designer working from the sentence knows a satin face is easier to stitch than a twill; nothing tells them that a plain face admits no stitch whatever, or that two satins of one order differ by a factor of two, because neither statement can be arrived at by looking at drafts one at a time.

The rule was worth writing down partly because a jacquard made the question urgent. Once every end can be lifted independently the stitching pattern is a picture, and a designer drawing a picture has no mechanical feedback about whether it holds — which is the same gap the integrity criterion was built for and the same one that makes an exhaustive enumeration the right tool.

Where the ladder goes next #

The stitch is one intersection and the layer count is one number, and the next question on this anchor is what happens when the two layers exchange rather than being tied — which is how a reversible coverlet gets its pattern, and where every region boundary is a place the connectivity has to be got right.

Below this rung sits the selvedge, which is the other place a double cloth is joined and the one the repeat cannot see. Beside it, on the integrity anchor, sits the ceiling: how many layers a repeat of a given size can hold at all, which turns out to be exactly half its ends.