A leno's hole cannot drift

A filter cloth is rated by its largest hole. Push one end sideways by δ and the gap on one side becomes g − δ while the gap on the other becomes g + δ; the sum is unchanged, so the cloth’s open area is unchanged to twelve decimal places and a percent-open-area measurement returns exactly what it did before. The largest hole has grown by δ, with no factor between them — one micrometre of rating for one micrometre of drift.

So the question is what holds an end where the reed put it, and in every ordinary weave the answer is friction.

This collection has already computed how much of it there is, and found the answer disquieting. A pick is held in by the wrap of the ends over it, the wrap angle is the weave angle Peirce’s geometry gives, and an open cloth’s weave angle goes to zero: at eight threads per centimetre it is 7.7 degrees. The capstan turns that into a grip of 1.084, meaning the crossing holds eight per cent of the thread’s own tension and no more.

That number is worth setting against what holds a pick in, which is the same capstan asked about the other thread system, and against the force at a relaxed cloth’s crossings, which is where the normal force comes from.

It goes to nothing smoothly, with no threshold to warn anybody. There is no sett at which a cloth suddenly stops holding together; there is a continuous decline, and a cloth is called sleazy when somebody notices.

The claim #

A leno’s spacing is held by its geometry and a plain weave’s is held by its friction, so the openest cloths — which are the ones anybody wants to filter or screen with — are exactly the cloths whose ratings a plain weave cannot keep.

The grip a leno supplies is e^(μπ), which is 2.566 at μ = 0.3 and has no sett in it. The grip a plain weave supplies is e^(2μθ), and θ is the weave angle.

What is actually held is the grip less one, since a grip of one is a crossing that holds nothing. Compared that way:

The ratio grows as the cloth opens, which is the whole practical point: the advantage is smallest where nobody needs it and largest where everybody does.

The argument #

The mechanism is worth stating carefully, because the leno is more stable is easy to say and hard to make exact.

A plain weave’s end is held by two things and both are the same thing. It presses on the picks above and below it, and the friction at those contacts resists a sideways push. The normal force comes from the thread tensions bending round each other, and the angle through which the thread bends is the weave angle — so the friction available and the crimp are the same quantity looked at twice. Open the cloth and the thread has room to lie straight, the angle falls, the normal force falls, and the friction goes with it.

A leno’s doup end does not lie beside its partner. It passes under it and comes up the other side. That is a half turn — π, exactly, by construction — and Peirce’s geometry cannot produce a weave angle of ninety degrees at any spacing, because a thread turning through a right angle has doubled back. So 2θ < π always, and no plain weave, in any yarn, at any sett it can be woven at, and at any friction coefficient, grips as hard as one leno crossing. The μ cancels; the closest approach is 57 per cent, at the tightest sett Peirce’s closure will solve.

That comparison is this collection’s, from the field that built the leno, and it was made about holding a pick. The same arithmetic holds an end against a sideways push, and it is the sideways push that decides a rating.

What the criterion says, and what it cannot #

This collection’s own integrity criterion is worth asking, because it gives an answer that is right and unhelpful, and the unhelpfulness is instructive.

A leno gauze is one cloth. So is the open plain weave it is compared against, built on the same ends and the same picks with the doup left uncrossed. The criterion returns one for both — deliberately, and it is asserted, because a reader is entitled to see the site’s central check give the same verdict to a curtain gauze and to a fabric that would fall out of the loom.

That refusal to distinguish is the same one the criterion makes between a tuft bound under one pick and a tuft bound under three.

The criterion is topological and the difference here is metric. It consumes contacts and a direction at each, and it asks whether the above-and-below relation is strongly connected. Both cloths pass. What separates them is how hard it is to move a thread without breaking any contact, and that is a question about friction and geometry that no connectivity argument can reach — the same gap the criterion has whenever it is asked about friction.

What was counted, and how #

Peirce’s circular geometry for the weave angle, this site’s own solver, so the number is the one the rest of the collection would give. The capstan for the grip, quoted as a published result. The friction coefficient is an argument with a stated default of 0.3 and every number is quoted at the value it was computed at.

The one claim that needs no coefficient is the comparison, and it is the strongest thing here: since 2θ < π at every construction, e^(2μθ) < e^(μπ) for every positive μ, so the ordering holds at any friction whatever. A statement that survives its own free parameter is worth more than a statement about a particular yarn, and it is why the closest-approach figure of 57 per cent is quotable while the ratios in the table are quoted at a stated μ.

The drift arithmetic itself asserts two things and both are the point: that the open area does not move when an end does, to twelve decimal places, and that the largest hole grows by exactly the displacement, with a slope of 1.000000000 micrometres per micrometre.

Why every mesh is a leno, and what that costs #

Bolting cloth for flour, screen-printing mesh, curtain scrim, agricultural netting, surgical gauze, and the open fabrics used for insect screening are leno or leno-derived almost without exception. The trade explanation is that leno stops the threads sliding, which is correct and is usually offered as a fact about the construction rather than as a consequence of anything.

It is a consequence of the wrap angle, and the arithmetic above is how much.

What it costs is the matrix. A leno has no fixed column order — the doup end is on one side of its partner and then the other — so it cannot be written as a binary matrix at all, which is why leno is not a matrix and why every count, census and enumeration on this site excludes it. That is the price of the stability: the construction that keeps its holes is the one construction whose holes cannot be enumerated by the machinery that enumerates everything else here.

And it costs a loom. A doup needs a half-heald or an easer, the shed is formed differently, and the rate is lower. Nothing in this collection prices that, and it is the reason plain-woven filter cloths exist at all.

Why the advantage grows exactly where it is needed #

The ratio column runs from 3.3 at a close sett to 18.6 at an open one, and the shape of that is worth naming because it is the reason the construction exists rather than a curiosity about the table.

The leno’s grip is a constant and the plain weave’s is a function of the cover, so the ratio between them is one over the other, and the plain weave’s own curve is what supplies the shape. As a cloth opens its weave angle falls, its normal force falls with it, and its grip — the capstan less one — falls faster than linearly because an exponential near one is very nearly its own argument. So the plain weave’s hold is roughly proportional to its weave angle, and the weave angle is roughly proportional to the cover at the open end.

The ratio is therefore roughly one over the cover, which is a hyperbola: it grows without bound as the cloth opens, and it flattens as the cloth closes. Nothing about the leno enters the shape at all.

That says two useful things. There is no open sett at which a plain weave becomes adequate, because the deficiency grows as the cloth opens and never turns round — so a designer who finds a plain-woven mesh unstable cannot fix it by opening the sett further, and cannot fix it by closing it either without abandoning the mesh. And the leno’s advantage is not a fixed multiple to be traded against its cost: it is a multiple that depends on the fabric, so the loom’s extra cost buys eighteenfold on a bolting cloth and three on a shirting.

Which is why the boundary between the two constructions is so sharp in practice. A trade decision between two options whose relative value varies by a factor of six across the range does not produce a gradual transition; it produces a threshold, with every cloth on one side of it made one way and every cloth on the other made the other. The threshold sits at whatever cover makes the leno’s extra cost worth three times a plain weave’s grip, and above it nobody bothers and below it nobody dares.

The hyperbola has a second consequence for how a mesh should be specified. Because the plain weave’s grip depends on the cover and the leno’s does not, the two constructions’ stability is comparable only at one sett — so a trial that establishes a plain weave is adequate at one construction says nothing about the same weave at a more open one, and the direction of the error is the unhelpful one. A cloth opened a little to raise its flow has lost grip faster than it has gained openness, because the grip falls with the weave angle while the openness rises only with the gap.

The cloth that is held by neither #

There is a third case and it is the one a filter cloth actually spends its life in, so it is worth naming.

A cake builds on the upstream face of any working filter within minutes, and thereafter the cake is doing the filtering. That does two things at once. It makes the cloth’s own rating less important, which is the usual argument for not worrying about any of this — and it presses on the cloth unevenly, which is a lateral disturbance applied continuously for the whole of the cloth’s service life.

So the arithmetic above is about a new cloth’s rating, and it is about the mechanism that decides whether that rating survives being used. A cloth that leaves the loom at 264 µm and is asked to hold a soil at 300 has thirty-six micrometres of margin; thirty-six micrometres is a fifth of a thread’s diameter, and it is the whole of what stands between the specification and the cake.

And the same reasoning applies to the seam. A woven filter is sewn into a sleeve or a belt, and a sewn seam pulls the ends beside it out of place by far more than any service load does — which is the point at which what holds a thread in a seam stops being a strength question and becomes a rating question.

Where the model stops #

The capstan is a rope on a fixed drum and neither thread here is fixed. Both move, both are compliant, and the contact is a finite patch rather than a line. This collection has already noted that a real grip is friction along a gripped length rather than at a point, which is a better model and is not the one used here.

Nothing computes how large a disturbance to expect. The arithmetic prices the consequence of a displacement and the force needed to cause one; what actually pushes an end sideways in service — handling, tensioning, a cake building unevenly on a filter — is not modelled, so there is no prediction of drift in micrometres.

The wrap is π and a real doup’s is less. The half turn is the idealisation; a real crossing has the doup end leaving at an angle, and the effective wrap is somewhat under π. That reduces the leno’s advantage and does not touch the ordering, since the ordering needs only that the wrap exceed twice any achievable weave angle.

And a leno’s own hole is not a rectangle. Everything in this collection’s opening arithmetic assumes four threads bounding a cell in a fixed order. A leno’s ends cross, so its opening is bounded by two threads that swap sides, and the shape is a triangle or a lens rather than a rectangle. Nothing here computes it — which means the stability of a leno’s hole is priced and its size is not.

The generalisation #

A constraint enforced by friction degrades continuously and a constraint enforced by topology does not, and a design that needs the constraint at its weakest should not use the first.

The distinction is sharper than strong against weak. A frictional hold has a magnitude that depends on everything — the load, the surface, the humidity, the angle — and it goes to zero smoothly as those change. A topological hold is either present or absent: the two ends are threaded through one another or they are not, and there is no continuum in between.

The diagnostic is to ask what happens as the load goes to zero. A frictional constraint that depends on a normal force supplied by the same tension it resists has no floor: reduce the tension and the constraint goes with it. A geometric interlock does not care.

The examples are everywhere once the shape is named: a knot against a clamp, a splice against a friction grip, a bayonet against a taper fit, a dovetail against a glued butt. In each case one member is available at any preload and the other is not, and in each case the frictional one is chosen because it is cheaper to make — which is exactly the trade a plain-woven filter cloth is.

Who found it, and when #

The capstan equation is Euler’s, from 1762. Peirce’s geometry is 1937. The leno’s half-turn wrap is a description of the construction rather than a result.

That leno constructions hold their spacing is universal trade knowledge and is the reason the construction exists.

What appears to belong to this collection is the pairing with a rating: that a filtration specification is a maximum over holes, that a maximum over holes moves one-for-one with a displacement, that a displacement is resisted by a grip which vanishes as the cloth opens, and that the ratio between the two constructions’ grips is therefore largest exactly where the specification is tightest. Each of those four is known separately and the chain does not seem to have been drawn.

Where the ladder goes next #

Everything in this ladder has assumed a cloth with holes between its threads, which a woven cloth has and a knitted one does not: a knit has no hole to lose, because at any ordinary tightness its loops are already over-full, and the whole planar apparatus refuses it.

Sideways, the leno’s stability is bought by giving up the matrix, and what a construction gives up when it leaves the encoding is a question this collection has asked before under what the matrix cannot say.