Leno is not a matrix

A weave matrix has an entry for every intersection of warp end j with weft pick i, and the whole apparatus rests on j naming the same thread throughout. Change which thread is at position j between one pick and the next and the notation stops referring to anything.

That is exactly what a leno does, on purpose, as its defining feature.

What a doup is #

The mechanism has been the same for a very long time and it is simpler than its vocabulary.

Warp ends are taken in pairs. One is the standard end and behaves ordinarily. The other is the doup end — sometimes crossing end — and is threaded not directly through a heddle but through a loop, the doup, carried on a half-heddle. The doup can be drawn to either side of the standard end.

Pull it one way and the doup end sits left of its partner; pull it the other and it passes under the standard end and comes up on the right. A pick is then inserted, and the crossing is locked into the cloth because the two ends are wrapped round each other with the weft trapped in the crook.

So a leno is a weaving mechanism with a braiding operation inside it. The warp is not merely lifted and lowered; it is permuted.

Why the matrix fails, precisely #

It is worth being exact about the failure rather than gesturing at it, because the exactness is what makes the rest of the field’s boundary argument work.

A matrix entry W[i][j] asserts something about the intersection of a specific end with a specific pick. For that to be a proposition, the map from column index to thread must be constant in i. In a leno it is not: the end at column j on pick 0 and the end at column j on pick 1 are different physical threads.

Two repairs suggest themselves and both fail in instructive ways.

Index by thread rather than by position. Number the ends once at the beam and keep the numbering. Now W[i][j] means something again — end j, pick i, over or under — and it is a true and complete record of the interlacing. What it has lost is the order, and the order is the whole construction: two leno cloths whose ends interlace identically and cross differently are different fabrics, and this notation cannot tell them apart.

Record the permutation separately. Keep the matrix and add a list of transpositions between picks. This works — it is essentially what the model here does — and the point is that it is no longer a matrix. It is a matrix plus a braid word, which is a different object with different decidable properties, and calling it a weave matrix would be a way of not noticing.

What still works #

Everything in this field has followed the same pattern and this is no exception: the criterion survives, because it never needed the matrix.

The criterion consumes a list of contacts with a verdict about which strand is above at each. In a leno those contacts are of two kinds rather than one. There are the ordinary crossings of picks with ends. And there are the end-to-end crossings, where the doup end passes beneath its standard end — a contact between two warp threads, which no ordinary weave has at all.

Both go into the same graph and the same Tarjan pass decides it. The verdict for a properly crossed leno is one cloth, and the figures on this page assert it.

The control, and the uncomfortable answer #

Here is where this essay stops being a tidy extension of the machinery.

Beside every leno panel above is a control: the identical construction with the crossing removed, which is an open plain weave on the same ends at the same sett. The criterion says it is one cloth.

It is right. There is no separation: every end interlaces with every pick it should, the digraph is strongly connected, and the fabric is topologically sound. And an open plain weave at that sett is a fabric whose wefts slide about under a fingernail and whose structure comes apart in the hand — which is the precise reason leno exists as a construction and the reason curtain gauze, mosquito netting, surgical gauze and geotextile are not simply woven openly.

The criterion returns the same verdict for a gauze and for a fabric that will not hold together in use. That is not a bug and there is no version of the criterion that fixes it, for the same reason there was none for the pile: whether threads slip is a question about friction, and friction is not a topological property.

A leno is a weave and a braid at once #

The end-to-end crossings are worth taking seriously as braid crossings rather than treating them as an oddity, because this site already has the machinery and the connection is real.

Braids were handled at foundation as an interlacement of one strand system with itself, decided by the same connectivity argument on a digraph built from a braid word. A leno’s crossings are exactly that: a word in the transpositions of adjacent ends, applied between picks.

So a leno decomposes into two structures that this site can each describe, layered on one another. The picks and the ends make a weave. The ends among themselves make a braid. And the cloth is what happens when the two are interleaved in time — braid, weave, braid, weave.

That decomposition also says why a leno cannot be reduced to either half. Take away the picks and the remaining braid is a set of independent two-strand twists, not a fabric. Take away the crossings and the remaining weave is the control, which is a fabric that does not hold. Each half is deficient in a different way and the combination is not the sum of them.

Two pairs, not one #

A small constraint in the model turned out to be a real statement about the fabric, and it is the kind of thing worth recording because it was discovered by the code refusing to run.

The first version of the model was asked for a single pair of ends and some picks, and the criterion refused it: many separable components. The reason is immediate once seen. Both ends of a pair lift together, so with only one pair every pick is either over the whole warp or under it — which is a pick floating on the surface, exactly the loose strand the criterion was built to catch.

So a single pair of crossed ends and some wefts is not a cloth; it is a fringe. It takes two pairs before anything interlaces, because interlacing requires two units doing different things. That is not a limitation of the model, it is the model being right about a degenerate case, and the constraint is now stated in the code with the reason attached.

What the encoding loses, item by item #

The pile essay found three of this site’s measurements failing at its boundary, each in a different way. Leno’s boundary is a different one and the losses are correspondingly different, which is worth setting out because two encodings failing differently is more informative than either failing alone.

Float length survives, with care. A run of picks over which one end stays on the face is still well defined — the end is a physical thread and it is on the face or it is not. What has changed is that the float no longer occupies a fixed column, so a float in a leno wanders sideways across the cloth as it runs. Its length is unaffected; its place is not a column index.

Interlacings survive and undercount. The number of times a thread changes face is well defined and countable. But a leno has contacts the count was never designed for — the end-to-end crossings — and an interlacing count that ignores them reports a fabric far looser than it is. The firmness number for a gauze is wrong in a specific direction, and wrong for a reason a reader of the number cannot see.

Cover survives and is beside the point. The fraction of the plane the threads occupy is computable and small, which is correct and is exactly the property the fabric was made to have. A cover factor for a gauze is a true number that answers a question nobody is asking.

Plane group fails outright. A draft’s symmetry group is a symmetry of a periodic pattern of squares, and there are no squares. Whether some analogous classification exists for crossed-warp fabrics is a real question and this site has no machinery for it.

So the pile loses its measurements by leaving the plane, and the leno loses different ones by losing its order. Neither is a failure of the criterion, and in both cases the criterion is the thing that carries.

What was counted, and how #

The construction is built as a named graph, as the pile fabrics were. Strands are the standard ends, the doup ends and the picks.

The crossing schedule comes from one parameter: the number of picks between crossings. A value of one is full-turn gauze, two is an ordinary leno, and zero is the control — the same construction with the doup never moving, which is what makes the comparison fair. The two panels differ in that parameter and in nothing else, which is the whole design of the figure.

At each crossing, an edge runs from the doup end to the standard end, because the doup passes beneath. At each pick, the pair lifts as a unit and adjacent pairs alternate, so the ground is a plain interlacing at the pair level.

Three assertions run. The crossed construction is one cloth. The control is also one cloth — asserted, not merely observed, because the reader is entitled to see this site’s own central check return the same verdict for a curtain gauze and for a fabric that would fall out of the loom. And the two differ in crossings and in nothing the criterion reads, which is the statement the whole essay rests on.

Where the model stops #

The crossing is drawn as a single transposition and a real leno need not be. Two-end crossings are the common case; three-end and four-end crossings exist, and gauze weaves with a standard end crossing several doup ends are a whole family this model does not reach.

Nothing here is about the doup mechanism. Bottom doup, top doup, sley doup, the modern needle-and-half-heddle arrangements: these are how the crossing is made and they matter enormously to whether a leno can be woven at speed. They do not change what the cloth is.

The tension difference is not modelled. A doup end is drawn sideways as well as up and down, so it travels further and needs its own beam or an easer, which is the main practical difficulty of the construction. The graph knows nothing about tension.

And the crossing’s grip is not here at all. That is the next rung, and it is the quantity the control comparison above exists to demand.

The half-cross and the full cross #

The crossing schedule is a parameter and the two values in use have names in the trade, which are worth attaching to the model’s numbers.

A full cross or full-turn gauze crosses at every pick: the doup end changes sides, a pick is inserted, it changes back, another pick is inserted. The ends are wrapped round one another once per two picks and the fabric is maximally locked.

A half cross crosses less often — every second pick is the ordinary case — so the pair spends some picks lying parallel and behaving like an ordinary warp. This is what most decorative leno is, and it is easier to weave because the doup end travels less.

The model takes this as the interval between crossings, and it makes visible something the names obscure: the two are ends of a continuum rather than two constructions. An interval of one is the full cross, two is the half cross, and the limit as the interval grows is the control — the open plain weave with no crossing at all. A leno crossed every twentieth pick is very nearly an open plain weave with a lock every twentieth pick, which is a real thing to want and is roughly what a leno-stabilised scrim is.

The criterion returns one for every value along that continuum, including the limit, which is another way of saying what this essay has been saying: the parameter that decides the fabric is one the criterion does not read.

A leno has two interlacing rates, and the firmness count reads one #

The essay records that an interlacing count applied to a leno undercounts, and it is worth putting a size on that, because the undercount is not a correction — it is comparable with the number being corrected.

A leno has two kinds of contact and therefore two rates. The pairs interlace with the picks, at a rate the ordinary count reads; and the ends cross each other between picks, at a rate set by the crossing interval and read by nothing.

Count both for an open gauze — eight pairs to the centimetre, full cross, so a crossing between every pair of picks.

The weave rate is the pair-level plain interlacing: each pair changes face at every pick, so at sixteen picks per centimetre that is sixteen interlacings per centimetre along each pair.

The crossing rate is one per pair per two picks, which at the same pick density is eight per centimetre along each pair.

So the crossings add about half again to the count, and for a half-cross leno about a quarter. A firmness figure for a gauze computed from the weave alone is therefore low by something between a fifth and a third — which is not a rounding, and it is in the direction that makes the fabric look looser than it is.

Two things follow that the essay’s qualitative note does not give.

The crossing interval is the leno’s own interlacing rate. It plays exactly the role in a leno that the float length plays in an ordinary weave: it is one integer that decides how firmly the cloth is locked, and every property that follows from firmness follows from it. A leno crossed every pick is the firm end and one crossed every twentieth is very nearly the control — and the essay’s own continuum is that integer swept.

And the two rates are independently adjustable, which no ordinary weave allows. A designer can hold the sett and the pick density fixed — fixing the weave rate — and change the crossing interval alone, moving the lock density without touching the cover, the weight or the openness. That is the whole reason a leno exists as a construction: it separates a fabric’s openness from its firmness, and in an ordinary weave those are one decision.

The caution is that adding the two rates treats a crossing and an interlacing as the same kind of contact, and they are not. A crossing is a warp end wrapping another warp end through a half turn, and the grip it supplies is much larger than an ordinary crossing’s — so the arithmetic above understates the crossings’ contribution rather than overstating it, and the true firmness of a gauze is further above its computed figure than a third.

Who found it, and when #

Leno is ancient. Woven gauzes with crossed warps survive from dynastic Egypt and from Han China, and the doup principle is old enough that no origin is recoverable — which is what one would expect of a technique that can be done with a stick and a loop of string on the simplest loom.

The observation in this essay is not about the fabric but about the notation, and it is a modern kind of observation because it needs a notation to be about. Point paper handles leno badly and always has: weaving manuals draw leno with a special convention, arrows or curved lines added to the squared paper, and the special convention is the admission that the squares have run out. What this essay adds is only the reason — the column index has stopped naming a thread — and the consequence, which is that the criterion carries across and the measurements do not.

Where the ladder goes next #

The comparison this essay set up has to be resolved. What holds a pick in supplies the quantity the criterion cannot, and the answer turns out to be exact and independent of the friction coefficient entirely: a plain weave cannot reach a leno’s grip at any sett in any yarn. After that, the criterion’s boundary is stated in general for the whole field.