Textile Structure
how cloth is put together
Series

Leno — the series

7 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

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