Textile Structure
how cloth is put together
Series

Leno — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A leno and the open plain weave it is not. Two warp ends and the picks they hold. On the left the doup end passes under its partner between picks and comes up the other side; on the right it never crosses, which is an open plain weave at the same sett. The layer count under each panel is computed by the same criterion that decides every other draft, and it returns the same verdict for both.

    Leno is not a matrix

    A doup end crosses under its partner between picks and comes up on the other side, so the end at position three on pick two is a different end on pick three. The entry W[i][j] does not name anything. The encoding is not wrong about leno; it is undefined for it.

    part 1 · fancy
  2. What holds a pick in. The holding force on one weft, as a multiple of the tension applied to its free end. For a plain weave it is the capstan on twice the weave angle, which Peirce's geometry gives at each sett and which falls towards nothing as the cloth opens out. For a leno it is the capstan on a half-turn, which the sett does not enter at all.

    What holds a pick in

    A plain weave grips its weft by the crimp, and the crimp's wrap angle falls towards nothing as the cloth opens out. A leno crossing is half a turn whatever the sett. So the comparison has an exact answer that does not depend on the friction coefficient at all — and no plain weave can ever reach a leno's grip.

    part 2 · fancy
  3. What holds a pick in. The holding force on one weft, as a multiple of the tension applied to its free end. For a plain weave it is the capstan on twice the weave angle, which Peirce's geometry gives at each sett and which falls towards nothing as the cloth opens out. For a leno it is the capstan on a half-turn, which the sett does not enter at all.

    A leno's hole cannot drift

    A filter cloth is rated by its largest hole, and the largest hole grows one micrometre for every micrometre an end moves sideways. What holds an end in place in an ordinary weave is friction at its crossings, and that friction falls smoothly to nothing as a cloth opens — with no threshold to warn anybody. A leno's crossing does not: its ends are wrapped through half a turn by construction, so the grip has no sett in it, and at an open cloth it holds eighteen times what a plain weave manages.

    part 3 · fancy
  4. The fold the trade actually makes. 2 singles of 20 tex cotton at 800 turns a metre, folded at 566 — a ratio of 0.707. That is 1/√2, the ratio at which the fold's surface helix angle equals its singles' — 22.8° — because a fold of 2 singles is √2 times the diameter. The trade's own bracket for 2 folds is 0.6 to 0.75, and it contains this number.

    A leno twists what a weave only crosses

    Every woven cloth's threads have a linking number of zero, and that is why an open cloth slips. A leno is the one woven structure whose warp ends wind about one another, so it is the one whose threads are linked — and it is famously the structure that holds at setts where nothing else does.

    part 4 · fancy
  5. A doup end crossing every pick at 1 mm. One leno pair of 0.25 mm ends over 2 crossing intervals, with picks 1 mm apart and the doup end changing sides every pick, drawn to scale in plan and in section. The standard end runs 1.00 mm per interval; the doup end, passing from 0.250 mm to one side, down 0.250 mm under its partner and up to the other side, runs 1.225 mm — 22.5% more, 11.8% of it from the sideways travel alone. Over a 100 m piece that is 22.5 m of warp, and on a shared beam a pick spacing of slack in 4.4 mm of cloth. What the drawing cannot show is the crimp both ends share over and under the picks, which the comparison cancels.

    The doup end pays for the crossing

    A leno holds its picks because its doup end crosses under its partner and comes up on the other side — half a turn at every crossing, whatever the sett. That half turn is also a length. When 0.25 mm ends cross every millimetre, the doup end travels 22.5 per cent further than its partner: twenty-two metres of extra warp over a hundred-metre piece, and a shared beam a pick short within four and a half millimetres of cloth. The extra falls with the square of the crossing interval, which is why a leno is woven from two beams and its crossings are spaced as far apart as the cloth allows.

    part 5 · fancy
  6. The crossed shed, with the back standard 4 shafts behind the doup. A leno's crossing end in section from the fell to the back rest of an ordinary broad loom, in the crossed shed. The doup 300 mm from the fell lifts it 50.0 mm; its back standard 64 mm further back leaves it 60.7 mm down, beneath its partner. Between the two eyes the end climbs 110.7 mm in 64 mm, and the whole path is 70.2 mm longer than the straight line, 5.85% of the free warp, of which 63.8 mm is that one climb. The same end in the open shed, lifted at its back standard, is 7.2 mm longer, and an ordinary end on the doup's shaft 5.5 mm: the crossed shed is 12.7 times the ordinary shed. Vertical scale exaggerated 3 times. What the drawing cannot show is friction at the eyes, which would keep the extra length in the short span between them.

    An easer gives back the kink the crossed shed puts in

    A leno loom gives its crossing ends slack at every crossed shed, and the obvious reason — the doup carrying its end sideways — is worth a hundredth of an ordinary shed. The length is one heddle further back. In the crossed shed the doup lifts the end on the far side of its partner while the end's own back heddle, a few shafts behind, holds it down, and between those two eyes the end climbs eleven centimetres in six and a half. On a loom that stretches an ordinary end 0.46 per cent, that is 5.85 per cent — nearly thirteen times as much — and no position of the back heddle takes it below four.

    part 6 · fancy
  7. The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres.

    A heddle eye lets the kink through

    A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

    part 7 · fancy
  8. What the crossing end asks for, against what a cam gives it. The length a leno's crossing end needs at each fraction of the crossed shed's opening, against an easer driven in proportion to the shed. The demand is nothing for the first eighth of the opening and then rises steeply, because the kink's extra length is a hypotenuse less its run and grows as the square of the climb. The proportional easer runs ahead of it everywhere, worst at 0.375 of the way open, where it has given 16.5 millimetres more than the end can use.

    A cam easer gives its slack too early

    The essay before it left the crossing end with fifteen millimetres it could not use at half the shed, and three things it might do with them. All three are decidable. It cannot reach the shuttle, because the same eyes that trapped the kink keep the slack behind the harness — and the margin is negative, so that friction is not a nuisance here but the reason a leno weaves at all. It cannot snarl, because the span is under half this account's own threshold. It sags seventy-two millimetres where the easer gives it, and the repair is a cam cut to the demand rather than to the shed.

    part 8 · fancy
  9. The springs a leno easer can be. The stiffest spring, per end, that an easer can carry without any span of the crossing end rising above an ordinary end's working tension at any fraction of the crossed shed, against how far its preload sits above the warp's resting tension of 0.5 N. Behind the harness, where an easing bar acts, it is 1.1 N/m at a preload equal to the resting tension and falls to nothing by 15 per cent above it; in front of the harness it is 4.8 N/m. Everything under a curve works and nothing above it does. The window is a corner: a spring barely stiffer than a dead weight, with a preload matched to the warp's own tension. What the chart cannot show is the bar's mass, which the loom's speed makes matter and which the next views take up.

    A leno easer should be a light weight

    A spring easer gives length when the crossing end pulls, so it cannot give it too early, which was the fault in a cam driven off the shed. The question left was its rate. The answer is that it hardly has one. Behind the harness the easer feels the back span, and two gripping eyes keep that span within fifteen per cent of its resting tension while the kink carries its load. So the spring must hold its span almost constant over a sixty-eight-millimetre stroke: at most 1.1 newtons a metre per end, a dead weight in all but name. At speed the bar's own mass is what limits it, and it falls with the square of the loom's speed: 2.3 grams an end at a hundred picks a minute, 0.6 at two hundred.

    part 9 · fancy
  10. Where the weight on a shared leno easer comes to rest. The tension in the back span of each end of a leno pair — the length between its last heddle and the back rest, where an easing bar acts — at each fraction of the crossed shed, with one bar under both ends weighted at 1.0 N a pair, twice the warp's resting tension, so that it rests on its stop with the shed closed. The two back spans always sum to the weight. Closed, they share it equally. The partner, an ordinary end lying in the bottom of the shed, can take only 11 mm of the bar's travel before it goes slack, which it does at 37.5% of the opening; from there the crossing end's back span carries the whole 1.0 N. On a bar of its own, dashed, the same span stays at 0.50 N. What the chart cannot show is what the crossing end's other spans do with the extra tension, which the next views take up.

    One weight under a leno pair ends on one end

    A leno's crossing end wants about seventy millimetres eased to it at every crossed shed; its partner lies in the bottom of the shed as an ordinary end and wants none. One weighted bar under both is set where the closed shed shares its weight between them, but the partner can take only eleven millimetres before it hangs, and it is slack by three eighths of the shed. From there the crossing end carries the whole weight, twice the rest its own bar would hold it at, and its tightest span ends forty per cent over the working tension while the partner hangs thirteen centimetres behind the harness. No weight serves both, because a bar gives every end it carries one length.

    part 10 · fancy
  11. A leno partner lifted by its doup's loop. The tension in each span of a leno's partner end at the full crossed shed, against how far the doup's loop lifts it above its own straight line 6 mm in front of its heddle on the doup's shaft. The partner is then three spans: fell to loop, loop to heddle, heddle to back rest. With the loop clear it is an ordinary end, 1.07 N in its front span. At a 4 mm lift the front span carries 1.48 N, +38% on the untouched partner, the span between loop and heddle 1.26 and the back span 1.00; with nothing gripping, dotted, all three would carry 1.12. At 8 mm the front span reaches 1.90 N. What the chart cannot show is how far a real loop lifts its partner, which the doup's standard decides and which is swept here rather than computed.

    A leno's slackener eases the partner, not the doup

    In a bottom-doup leno the doup's loop passes beneath the partner, and if it lifts the partner on its way up it puts a second turn six millimetres in front of the partner's heddle. A four-millimetre lift bends the partner through 74 degrees in two places, and its front span goes from 1.07 newtons to 1.48 — a tenth of that from the bend and the rest from grip. The back span, which is all an easing bar can feel, does not move. A slackener that drops the doup's standard four millimetres clears the partner exactly, gives the crossing end six per cent of its easing, and costs the shuttle nothing.

    part 11 · fancy
  12. The partner's peak against how late a slackener starts. The largest tension the partner's front span reaches, against the share of the shed opened when the slackener starts to drop, at a 4 mm rise. It stays at 1.07 N for any start up to 0.748 of the shed, where the partner with no slackener passes that tension. Later than that, the peak is the touched tension at the moment it starts, up to 1.39 N at 0.95, against 1.48 N for no slackener at all.

    A leno's slackener only has to arrive

    A slackener that lets the doup down in step with the loop keeps a leno's partner clear at every moment of the shed, and it looked as though a real one would have to track that closely. It does not. A partner's tension only rises as the shed opens, so a slackener can leave its whole drop until three quarters of the way up and the peak does not move at all. Lateness is paid for elsewhere: starting later than 0.58 of the shed rubs the loop more than having no slackener, and above 234 picks a minute a falling weight cannot start as late as the partner would let it.

    part 12 · fancy

All series