A tucked-in edge asks every weave the odd repeat's question

A weave holds its selvedge with two of its own columns, or not at all turned the question of whether a cloth’s edges hold into a question about the weave’s columns. A shuttle weft turns round the edge end between one pick and the next, and the turn is caught only if that end changes face across the boundary. The turns fall at the right edge on odd boundaries and at the left on even ones, so a weave holds both edges exactly when it has a column of each kind, and two ends re-entered on the right shafts make that true at any width.

That account assumed a weft that turns. On a shuttleless loom it does not. A rapier, an air jet or a projectile carries each pick across from the same side and the pick is cut at both edges. What holds the edge is then something else, and the commonest something else is a tuck-in: a device at each edge that lays the pick’s cut tail back into the shed that follows it, where the next pick binds it.

A tucked tail turns round the edge end exactly as a shuttle weft does. It leaves its own shed, passes the outermost end, and enters the next one, so it is caught only if that end changes face between the two sheds. What is different is how many turns there are.

Every boundary is a turn at both edges #

A shuttle weft reaches one edge per pick. It goes out on pick one and turns at the right; comes back on pick two and turns at the left; and so on, so that each boundary between picks is a turn at one edge only, and the two edges take the boundaries alternately. That alternation is the whole of the earlier essay’s structure: it split the boundaries into two sets, and an end could hold one edge by changing face on one set while doing as it liked on the other.

A tucked edge has no alternation. Pick one is cut at both edges and both of its tails are laid into the shed of pick two. Pick two is cut at both edges and its tails go into the shed of pick three. Every boundary is a turn at both edges, and the starting side, which under a shuttle decided which edge was which, no longer means anything: the picks all come from one side and the tucks are made at both.

So an end that is to hold a tucked edge must change face across every boundary of the repeat. That is the condition the earlier essay found for an odd repeat under a shuttle, where two repeats of an odd number of picks bring every boundary round as a turn at both edges once. The tuck-in asks every weave the question only odd repeats were asked before, and the answer it allows is the same: a column that changes face at every boundary, which is plain weave’s column, up and down on alternate picks.

Of the sixteen columns a four-pick plan can give an end, two hold a tucked edge. The four that held one shuttle edge each — two for the right, two for the left — now hold neither, because they change face at alternate boundaries and every boundary now asks. The ten that held nothing before still hold nothing. And because a repeat of odd length cannot alternate round its cycle, an odd repeat has no column that holds a tucked edge, exactly as it had none that held a shuttle edge; for the 2/1 twill and the five-end satin the tuck-in changes nothing at all.

Fewer drafts can hold their edges on their own shafts #

The census follows at once. A four-by-four draft can hold a tucked edge with two re-entered ends — one at each edge, threaded on one of the draft’s own shafts — exactly when one of its four columns is plain.

10,382 drafts have a plain column and hold. Under a shuttle, 13,238 held with at most two re-entered ends, so 2,856 drafts that could hold a shuttle edge on their own shafts cannot hold a tucked one. They are the drafts that had a right-edge column and a left-edge column but no plain one, and the pairing that rescued them depended on the alternation the tuck-in removes. The 9,636 that needed a shaft of their own under a shuttle need one still.

The named casualty is the 2/2 twill. Its four columns are two of each one-edge kind, which is why the earlier essay could hold it from the left at any width with an extra end at each edge on the right shafts. Under a tuck-in, the hero figure shows what those columns do: every placement of the width catches exactly half of the tucked turns, eight of sixteen, because each of its ends changes face on every other boundary and every boundary now has a turn at both edges.

Every draft that can hold a tucked edge could also hold a shuttle edge, because a plain column is both kinds at once. The tuck-in is therefore strictly harder on a weave than a shuttle is. It takes nothing a shuttle could not hold and adds back nothing a shuttle lost.

One shaft lifted on alternate picks still holds anything #

The repair the earlier essay offered for the drafts that lacked a kind serves every draft here. One extra shaft carrying both edge ends and lifted on every other pick gives each edge a plain column, which changes face at every boundary and so catches every tucked turn. The weave’s own plan is untouched.

What changes is which weaves pay for it. Under a shuttle the selvedge shaft was needed by the 9,636 drafts lacking a kind and by every odd repeat; under a tuck-in it is needed by the 12,492 four-by-four drafts with no plain column and by every odd repeat. The price in plan length is the same as before. The selvedge shaft repeats every two picks, so on an even repeat the plan is no longer — a 2/2 twill keeps its four picks — and on an odd one it doubles to the least common multiple, six picks for a 2/1 twill and ten for a five-end satin. What a dobby stores counts exactly that cost: a dobby’s pattern chain is as long as its plan.

A loom without a spare shaft has one more way out, and it is the one shuttleless looms most often take. A leno pair at each edge holds a cut pick not by turning it but by twisting round it: a leno twists what a weave only crosses found the leno to be the one woven structure whose threads are linked, and what holds a pick in priced its grip as half a turn at every crossing whatever the sett. A leno edge asks nothing of the weave’s columns, because it catches no turn at all — the tail beyond it is simply cut away.

A turn that is not caught slides in #

What happens to a tucked turn the edge end does not catch? Under a shuttle the earlier essay said the loop slides off. With a tucked tail the loop round the edge end is a loop round nothing, so it is drawn inward until it wraps the first end that does change face across that boundary. The edge is then held, on that pick, some ends in from the edge, and the ends outside it are not bound by that tail.

Most drafts that cannot catch a tucked turn at the edge catch it close by. With the best column at each edge, 6,072 drafts catch every tucked turn within one end of the edge and 2,880 within two. A 2/2 twill is in the first group: half its tucked turns wrap the edge end and the other half the end beside it, because neighbouring columns of a twill change face on alternate boundaries.

3,540 drafts never catch some tucked turns at all. They are the drafts with two consecutive picks in the same shed, so that at one boundary no end in the repeat changes face. A tail laid into a shed identical to its own pick’s lies beside that pick through the whole width with nothing between them, and no end wraps it. The hopsack is the named example. Under a shuttle it held its left edge perfectly; under a tuck-in it has a tail at every other boundary that the cloth holds only by the friction of lying in a shed.

That friction is not nothing. A tucked tail lies along the next pick for the tuck’s length, of the order of a centimetre, and is bound by every interlacing that pick makes there — a cloth slips at its least-interlaced thread priced that grip as exponential in the interlacings. The tail is held. What is not held is the edge end, and an edge end bound only every other pick is a long float at the very place a cloth is handled, cut and sewn.

Two picks in every shed at the edge #

The tail has a second cost, which a shuttle edge never had. Over the tuck’s length every shed holds two picks, its own and the tail of the one before, lying side by side. The reed beats them to the same fell as the single picks in the body. Whether they fit is a question for the jamming count behind a cloth’s maximum sett: a repeat of $P$ picks needs $P$ yarn diameters along the warp plus one more for every interlacing an end makes in it, because the end has to pass between the picks where it changes face.

In the tucked zone the same repeat holds $2P$ picks and the same interlacings, so it needs $2P + I$ diameters where the body needs $P + I$. The tails lie flat beside their picks only while the body is woven at no more than $(P + I)/(2P + I)$ of its maximum pick density.

The limit has a ceiling and a floor. An end cannot interlace more often than once a pick, so $I$ is at most $P$ and the limit at most two thirds, reached by plain weave. As floats lengthen $I$ falls towards nothing and the limit towards a half. The twills sit at three fifths, the five-end satin at 58.3 per cent and the eight-end at 55.6.

How close can threads be set puts clothing at about sixty to eighty per cent of maximum, a shirting at sixty to seventy and a windproof cloth at ninety and above. So a plain shirting’s tucked edge is at the edge of fitting, a twill woven like clothing has an edge zone fuller than its room, and a windproof cloth’s tucked edge cannot lie flat at all. The count does not say what an overfull zone does. The reed beats the whole width to one fell, so the doubled zone must either bulge — the tucked edge standing thicker than the cloth — or hold the fell back over the tuck’s length. Thick tucked edges are a known complaint, and the arithmetic says where they begin.

The repair and the fault fall on the same drafts #

Here the two halves of the tuck-in’s cost meet, and they meet badly.

At four picks, a draft lacks a plain column exactly when its tucked zone has the least room. A four-pick column changes face nought, two or four times; nought is not a cloth, and four is plain. So a draft with no plain column is a draft whose every end interlaces exactly twice, and its limit is the floor of the four-pick range, three fifths. All 12,492 four-by-four drafts that cannot hold a tucked edge on their own shafts sit at exactly sixty per cent; all 10,382 that can sit above it, up to two thirds. The weaves that cannot catch their tails are the weaves that have least room for them.

And the repair makes the room smaller. A plain selvedge shaft catches every tucked turn, and if the tucked zone is woven plain to hold it firmly, the zone’s two picks a shed have plain weave’s room at the body’s pick spacing. A plain zone holds fewer picks than a twill one at the same diameter, because it interlaces more, so its doubled picks fit only below a third of $(P + I)/P$ of the body’s maximum: a half for a 2/2 or a 3/1 twill, 55.6 per cent for a 2/1, 46.7 for a five-end satin and 41.7 for an eight-end. The edge weave that catches every turn is the one with least room for the tails it catches.

That is why a tucked edge is usually a compromise rather than a solution. The few ends that must catch the turns are put on a plain shaft; the rest of the tucked zone is left in the body’s weave, or a weave with fewer interlacings, so that the doubled picks have room. Neither half of the conflict is visible in a draft, and both are counted here.

How the counts were made #

The tucked turns are classified exactly. For a repeat of any length, an edge end catches every tucked turn if its column differs across every boundary, the last pick to the first included; the four-by-four census applies that to the 22,874 drafts in which every thread interlaces and records whether each has such a column. The slide is the number of ends passed, counting inward from the edge column, before reaching one whose column differs across the boundary, worst over the boundaries and best over the columns the draft can put at each edge. The fit is the jamming count’s $(P + I)/(2P + I)$ with $I$ the interlacings per end per repeat, and with the zone woven in a second weave, the body’s $P/(P + I)$ at its maximum set against the zone’s doubled room.

What is required of it. The kept, lost and shaft-needing drafts must sum to the census, the shaft-needing count must equal the shuttle census’s 9,636, and the kept and lost together its 13,238. Of the sixteen four-pick columns, exactly two must hold a tucked edge, and every one that does must hold both shuttle edges. A 2/2 twill must catch exactly half its tucked turns at all sixteen placements. Four odd repeats must have no plain column, and one shaft lifted on alternate picks must hold a tucked edge on six weaves, odd and even. Every draft’s limit must lie between a half and two thirds; a satin’s must be (n + 2)/(2n + 2); a plain zone must have less room than the body’s own weave on six weaves; and a draft must lack a plain column exactly when its limit is three fifths.

What the count leaves out #

The tuck’s length is not in it. The census asks which end catches each turn and the fit asks whether two picks lie flat; neither depends on how far the tail runs into the next shed, and the tail’s own grip, which does, is not computed here. A short tuck holds its tail less firmly; a long one carries the doubled zone further into the cloth.

And the jamming count is the simplest one. It treats the yarns as round and the two picks in a shed as two round yarns side by side. Real tails flatten and partly stack, which gives them more room than the count allows; a pick density is a force budget found the beat-up itself pressing a cloth past its geometric jamming point. The limits above are where trouble begins in the simplest geometry, not where it becomes impossible.

Who worked out which part #

Tuck-in selvedges and leno selvedges on shuttleless looms are standard practice, and the thicker tucked edge is a familiar fault with familiar remedies — shorter tucks, finer selvedge ends, fewer ends in the selvedge. The turn criterion is the earlier selvedge essays’, and the jamming count is Ashenhurst’s, taken per repeat so that the picks and the interlacings stay separate.

What is added here is the observation that a tucked edge turns at both edges on every boundary, so that every weave is asked an odd repeat’s question; the census of what that costs the four-by-four drafts, the 2/2 twill among them; how far an uncaught tucked turn slides; the fit of two picks to a shed, with its two-thirds ceiling; and the finding that at four picks the drafts that cannot catch their tails are exactly the ones with least room for them, with the plain repair reducing the room again.

Answered since: how much the doubled zone stands proud #

The fit says where two picks to a shed stop lying flat and does not say what they do instead. A tucked edge stands proud because its warp goes straight gives the answer, and it changes the reading above: the warp is fed as one sheet, so the zone’s ends have only the body’s length to spend, they rise less, and the edge is thicker than the body well below this limit while its pairs stay flat well above it. A held-back fell turns out to relieve nothing once it has formed.