A leno twists what a weave only crosses

An ordinary woven cloth is held together by friction, because its threads pass over one another and come back rather than through. That is a fact with a number attached — the linking number of any two threads in any weave is nought — and everything a cloth does that a knitted fabric does not follows from it.

A cloth held by friction has a minimum density. Below a certain sett there are not enough crossings, the grip is not enough, and the threads slide about: the cloth is sleazy, its threads shift when it is handled, and an open gauze made as an ordinary weave is not a fabric at all.

There is one woven structure that escapes, it is the oldest special weave there is, and what it does is exactly what the linking number says would be needed.

What a leno does #

A leno or gauze weave carries its warp in pairs. One end of each pair is a standard end and runs straight; the other is a crossing or doup end, and between one pick and the next it is carried from one side of its partner to the other.

So the two ends of a pair swap places at every pick, and over two picks the crossing end has gone once round the standard end. The weft is caught in that winding and cannot slide along the warp, because moving it would have to unwind the pair.

That is a different mechanism from anything else in weaving. It does not depend on the threads being pressed together, and it does not depend on friction at all.

The number #

Take a pair of leno ends over a length of cloth and ask the same question this collection asked of an ordinary weave.

An ordinary warp end and weft pick, with the cloth’s own crimp on both, closed far outside the crossing, give a linking number of four parts in a hundred thousand — nought, with the sampling reporting itself.

A leno pair, closed the same way, gives one half turn per pick: the two ends wind, and the winding accumulates rather than cancelling. Over a hundred picks the pair has wound fifty times.

Nothing about that is approximate, and nothing about it depends on the sett, the yarn, the tension or the finish.

Why it holds where an ordinary weave does not #

The consequence is immediate and is the whole reason leno exists.

An ordinary open cloth is held by friction, and friction is a product of a normal force and a contact. Open the sett and both fall: fewer crossings per unit length, and less crimp because the threads have room, so less normal force at each crossing. The grip falls faster than linearly and there is a sett below which it is not enough.

A leno’s grip does not fall with the sett at all, because the winding is a property of the threading rather than of the spacing. Two ends wound about one another are wound whether they are a millimetre apart or a centimetre.

So a leno holds at densities where an ordinary weave slips, and that is precisely what it is used for: surgical gauze, cheesecloth, agricultural netting, curtain nets, the marquisette a mosquito net is made of, and the leno selvedges woven into the edges of ordinary cloths to stop them fraying.

And why the selvedge use is the giveaway #

The last of those is worth dwelling on because it is a small use with a large implication.

A modern shuttleless loom leaves cut weft ends at both edges of the cloth, and something has to stop them pulling out. The standard answer is a leno selvedge: two or four ends, at the very edge, threaded through a small doup device and wound about one another so that the weft is caught.

Nobody uses a leno selvedge for its appearance or its openness. It is used because it is the only weave that holds a thread that friction is not holding — the cut weft end has almost no crossings left to grip it, which is exactly the condition under which friction fails.

So the trade reaches for the linked structure at precisely the point where the unlinked one runs out, without ever having framed it that way.

What it costs #

A structure that holds without friction is not free, and the costs are worth listing because they explain why leno is a special weave rather than the ordinary one.

It costs a mechanism. A doup is an extra device on the loom: a half-heald, a needle or a rotating leno unit, and it runs slower and breaks ends more often than a plain shed.

It costs warp. A crossing end travels further than a standard end, because it goes round its partner as well as over the weft, so the two ends of a pair take up differently and have to be tensioned separately — usually from a second beam.

It costs cover, in the sense the cover factor measures.

It costs cover. A leno pair occupies more width than two parallel ends, so at a given number of ends per centimetre a leno covers less than a plain weave. That is a feature for a net and a defect for a cloth.

And it costs the pattern. A leno’s structure is visible and regular and does not combine freely with other weaves, which is why leno stripes and leno grounds are a recognisable class rather than a general technique.

Why it is an argument for the rule rather than against it #

An exception can undermine a rule or support it, and which it does depends on whether the exception behaves the way the rule predicts an exception would.

The rule is: a woven cloth is held by friction, so it has a minimum density.

The exception’s properties are exactly what the rule predicts of anything that escapes it. It holds at low density. It is used where friction has run out. It requires an extra mechanism, because ordinary shedding cannot produce it. And it is the only such structure, because there is only one way to make two threads wind about one another with a loom.

If leno held no better than a plain weave at low sett, the rule would be in trouble. It holds much better, and the reason it holds is visible in the structure.

The relatives #

Two structures sit near leno and it is worth saying where they fall, because both are sometimes described as if they were the same thing.

Mock leno is not a leno. It is an ordinary weave whose interlacing pattern groups threads to leave gaps, producing an open, textured cloth that looks like a gauze. Its threads do not wind and its linking number is nought, so it has all the density limits of an ordinary weave and none of leno’s stability. Anybody choosing between them for an open cloth is choosing between two different mechanisms with the same appearance.

Doup or gauze weaves with more than a half turn per pick — full-turn lenos, and the more elaborate crossing structures used in some technical fabrics — are lenos with a larger linking number per unit length, and they hold correspondingly better and cost correspondingly more.

So the family is ordered by the number, which is the useful thing a number does.

What a leno says about open cloth generally #

The rung has a consequence for a question this collection has asked in several places: how open can a cloth be?

The answer given so far has been about geometry and friction. A cloth’s threads jam at one end of the range and slip at the other, and the working sett sits between. How close threads can be set is the jamming end; the slipping end has been treated as a soft limit decided by handling and by finish.

The leno says the slipping end is not a limit at all, but a limit of one mechanism. A structure with a different holding mechanism has a different lower bound, and leno’s is set by something else entirely — by how much the crossing end can be made to travel, and by whether the loom can shed it.

That reframes a family of fabrics this collection has treated as marginal. Surgical gauze, curtain net, mosquito netting, agricultural shade cloth and geotextile mesh are not just very open cloths; they are cloths that changed mechanism in order to be that open, and their design constraints are not the ordinary ones.

The other way to make an open cloth hold #

For completeness there is a third route to an open stable cloth and it is worth naming because it is the commonest of the three.

Bond it. A resin, a heat set on a thermoplastic, or a scrim coating fixes the crossings so that the threads cannot slide relative to one another at all. That is what most industrial mesh is: an ordinary open weave whose crossings have been glued.

Bonding is cheaper than leno and it is not the same thing. A bonded mesh has a linking number of nought and is held by adhesion; cut it and it does not fray, because nothing can slide. A leno is held by its own structure and stays held when the finish is washed out.

Which matters depends on the use. A surgical gauze is washed, autoclaved and used wet, so it is a leno. A shade cloth is not, so it is bonded. Three mechanisms — friction, adhesion and linking — and the choice among them is a choice about what the fabric will be subjected to.

What was counted, and how #

The comparison is computed with the same Gauss double integral used for every linking number in this work, on the same code, checked against the same four arrangements whose answers are known by inspection.

The ordinary crossing is a warp end and a weft pick at the cloth’s own crimp, closed far outside the crossing along paths that contribute nothing, and the number is nought at every crimp from three per cent to twenty and at every number of interlacings from one to eight.

The leno pair is not computed from a solved geometry, because this collection does not have one for a doup: what is computed is the arrangement, and the arrangement’s linking number per pick is a half by construction — it is a half turn, and a half turn is a half.

That is a weaker result than the woven one and is stated as such. The claim here is structural rather than numerical: a leno winds and an ordinary weave does not, and the winding accumulates.

How a doup actually does it, and why no ordinary loom can #

It is worth being concrete about the mechanism, because the reason ordinary weaving cannot produce a linked structure is a fact about looms rather than about cloth.

An ordinary loom raises and lowers warp ends. Each end goes up or down, the shuttle passes through the gap, and the end comes back. Every end stays in its own place across the width of the cloth for the whole length of the piece — that is what a reed does, and it is why a weave can be written as a matrix of ups and downs.

A matrix of ups and downs cannot express a crossing. Nothing in a lift plan says which of two ends is on the left. So no weave that a lift plan can describe has any handedness in it, and a structure with no handedness has no linking.

A doup escapes by adding a degree of freedom the reed forbids: it carries one end sideways past another, through the space the reed would ordinarily keep them in. That is why it needs a special heald, why it runs slower, and why leno is a separate class of weaving rather than a pattern.

So the rule and the exception are both consequences of the same fact about the machine: a loom that can only raise and lower produces only unlinked cloth, and every linked woven structure needs a mechanism that moves an end across.

Which is why the weave matrix is silent about it #

That has a consequence for this collection’s own machinery which is worth recording, because it explains a gap that looked like an oversight.

Every weave on this site is generated from a matrix and then measured. The matrix decides the floats, the interlacings, the layer count and whether the cloth hangs together, and it decides them exactly. What it cannot decide is anything about crossing, because a matrix of ups and downs has no place to put it.

So a leno is not a weave this collection can generate, and that is not a missing feature — it is the same fact as the paragraph above, seen from the software rather than from the loom. Adding lenos would mean a different representation, not a bigger matrix.

That is worth knowing before anybody tries. The natural instinct is to add a third symbol to the point paper; the third symbol would have to say which end went round which, which is a pairwise relation rather than a per-cell value.

Where the model stops #

There is no leno geometry here. This collection can draw and measure any ordinary weave from its own matrix; it cannot yet generate a leno’s thread paths, because a doup’s crossing is not expressible in a lift plan. That is a real gap in the machinery and it is what stops this rung producing the numbers the woven rungs produce.

The tension difference is not modelled. A crossing end and a standard end take up differently, and how differently decides whether a leno can be woven at all. Nothing here computes it.

And the holding is asserted rather than measured. The claim that a leno holds better at low sett is trade knowledge and is not something this collection has computed a force for. Doing so would need the crossing geometry, which is the same gap.

The generalisation #

The rung is a case of a pattern that is worth looking for deliberately.

When a rule has a mechanism, look for the structure that escapes it, and check that the escape uses a different mechanism. A rule with no exceptions is either a very good rule or a rule nobody has tested at its edges. A rule with an exception that works by a different route is a rule that has been tested and has survived.

This collection has one other pair of that shape and has not put them together. The rule that a cloth must hang together as one connected component has an escape too — a double cloth, which is deliberately two components joined at intervals — and the escape uses a different mechanism, namely stitching rather than interlacing. That exception has been treated as a special case rather than as evidence for the rule, and it is both.

What a leno would look like in the collection’s own vocabulary #

Sketching what the machinery would need is worth a paragraph, because it turns a gap into a specification.

A leno’s structure is a braid word rather than a matrix: a sequence of generators, each saying that end i crossed over end i+1 or under it. That is the standard way of writing a braid and it has a well developed algebra behind it, including a way of asking whether two words describe the same structure.

Written that way, an ordinary weave is the identity — no crossings, ever — and a leno is a word in which one generator repeats. The two live in the same formalism and the ordinary weave is the trivial element of it, which is a satisfying place for it to sit.

That formalism would also cover braids, which are the next rung, and it would handle the crossing structures used in technical fabrics that this collection currently cannot describe at all.

It would not replace the matrix. A weave’s floats, interlacings and integrity are matrix questions and the matrix answers them exactly. What a braid word adds is the one thing the matrix cannot hold, which is a handedness — and this work is largely about what that omission costs.

Who found it, and when #

Gauze weaving is ancient — leno structures survive from Egyptian and Peruvian textiles — and the doup mechanism in something like its modern form is medieval.

The observation that leno’s stability is topological rather than frictional is, as far as this collection can find, not stated anywhere in those terms, although it is implicit in every practical account of why gauze is woven that way.

What is this collection’s own is putting it beside the woven result computed on the same instrument, so that the exception and the rule are measured rather than contrasted.

Where the ladder goes next #

The torsion ladder has one structural relative left and it is the third way of holding threads together: a braid, whose strands neither cross and return nor wind in pairs, but travel across the structure and back. A braid is a third way to hold threads, and it is held by something that is neither friction nor linking in the pairwise sense.

Then the ladder closes with what the torsion model can and cannot say, which is where a torsion model stops.