What holds a pick in

The previous rung ended with an uncomfortable result. A leno gauze and an open plain weave at the same sett are both, by this site’s central criterion, one cloth — and one of them holds together in the hand and the other does not.

The missing quantity is the same one the pile fastenings needed, and it is asked here of the weft rather than of a tuft: how hard is a pick held in?

Where a plain weave’s grip comes from #

A weft in a plain weave is held by friction at its crossings, and the friction available at a crossing is governed by the same capstan relation the pile rung introduced: the holding force is exp(μθ), where θ is the angle through which the thread turns as it passes over the pick.

That angle is the weave angle, and this site has a solver for it. Peirce’s geometry takes two yarn diameters and two thread spacings and returns the crimp heights and weave angles that satisfy its closure condition, and the numbers here come from that solver rather than from a separate model — so the grip curve agrees with everything else on the site by construction.

The end passes over the pick and back, so the total wrap is 2θ.

Now open the cloth out. As the spacing increases, the thread has more room to lie straight and less reason to bend, and the weave angle falls. At a sett of thirty threads to the inch in a sixtieth-inch yarn the angle is about 37°; at ten to the inch it is under 10°. So the wrap falls, and the grip with it, smoothly, towards nothing.

There is no threshold. A cloth does not become slippery at some sett; it becomes gradually less grippy at every sett, and a designer opening out a fabric receives no warning of any kind. That gradualness is worth noticing because it is the shape of failure that most reliably escapes notice.

Where a leno’s comes from #

A leno crossing is not a crimp. The doup end passes beneath its standard end and comes up on the other side, which is half a turn around the pick trapped in the crook, and it is half a turn regardless of how far apart the threads are.

The sett does not enter. Open the cloth to any degree whatever and the doup end still goes under and comes up the other side, because that is what the mechanism does — it is a permutation, not a bend, and a permutation has no magnitude to lose.

So the leno’s wrap is π and its grip is exp(μπ), a constant.

The comparison has an exact answer #

Set the two side by side:

plain weave — grip = exp(μ·2θ), where θ is the weave angle.

leno — grip = exp(μ·π), a constant.

The plain weave reaches the leno exactly when 2θ ≥ π, which is θ ≥ 90°.

A weave angle cannot reach a right angle. A thread turning through 90° as it crosses another has doubled back on itself; Peirce’s geometry has no such solution at any spacing, and the assertion that every solved angle is under 90° runs while the figure draws.

So 2θ < π always, the capstan is monotone in the angle, and:

No plain weave, in any yarn, at any sett it can be woven at, and at any friction coefficient, grips its pick as hard as one leno crossing.

The μ cancels. That is the part worth dwelling on. Every other frictional comparison in this field — V against W, corduroy’s binding, tuft withdrawal — is a statement at a stated coefficient, with a range attached and a warning that the range matters. This one is not. Whatever μ is, it multiplies both exponents equally, and the inequality between the angles decides the inequality between the grips.

How close it gets, and where #

The inequality is strict, so the interesting number is how close the plain weave comes at its best.

Its best is its tightest sett, because grip rises with closeness — and the tightest sett is where the geometry runs out. Peirce’s closure condition has no solution once the threads have no room, which is the jam, and this site has computed jammed setts since its foundation.

At a sixtieth-inch yarn the geometry solves up to a sett of thirty and refuses beyond it. At that limit the weave angle is 37°, the wrap is 74°, and the grip is 1.47 against the leno’s 2.57 — 57 per cent.

So the tightest plain weave that can be woven in that yarn is still well short of a gauze that is mostly holes. Stated that way the result stops being a technicality and becomes the reason the construction exists: the leno’s grip advantage is largest exactly where a plain weave has least, and it does not go away even where the plain weave has most.

Why the plain weave’s grip rises with the sett, which is not obvious #

The direction of the plain-weave curve is worth an argument, because the naive expectation runs the other way and the naive expectation is not stupid.

A closer sett means more crossings per inch of weft, so more places where friction acts — that much is obvious and points the right way. But it also means less room, and one might expect a crowded thread to be straighter rather than more bent, since it has neighbours pressing on it from both sides.

Peirce’s geometry settles it and the answer is the opposite. The closure condition says the two thread systems between them fill the thickness of the cloth: the warp’s crimp height plus the weft’s equals the sum of the diameters. Crowd the cloth and the crimp height each system must supply is unchanged, while the horizontal distance available to supply it in has shrunk — so the thread must rise the same amount over a shorter run, which is a steeper angle.

The crimp is set by the yarn’s thickness and the room to achieve it is set by the sett, so the angle rises as the room falls. That is the mechanism behind the curve, and it is the same mechanism behind crimp rising with sett, which this site computed at foundation from the same solver.

The consequence for this essay is that the plain weave’s grip and its openness are directly opposed by geometry rather than by accident. There is no clever plain weave that is both open and grippy, because the two are the same parameter read in opposite directions.

Why an open cloth is wanted at all #

It is worth saying what the openness is for, because otherwise the obvious response is to weave the thing tighter.

Air. Curtain gauze, mosquito netting, shade cloth, agricultural fleece: the fabric’s function is to be mostly hole, and closing it defeats the point.

Weight. A fabric that must cover an area for as little mass as possible — surgical gauze, bandage, some geotextiles — buys its area with openness.

Drainage. Filter media, sacking, the leno bands in industrial belting: the holes are the product.

And stability at the edge. The commonest industrial use of leno is not an open cloth at all: it is a leno selvedge on an otherwise ordinary fabric, where a pair of crossed ends at each edge locks the weft ends of a shuttleless loom’s picks that would otherwise pull straight out. That is this essay’s result put to work at the one place in a normal cloth where a weft has a free end.

What was counted, and how #

For each sett, the thread spacing is its reciprocal and the pair of spacings goes into this site’s own Peirce solver with a crimp ratio of one, giving the weave angle. Setts at which the solver finds no solution are dropped rather than extrapolated into — the cloth is jammed there and a number obtained by continuing the curve past its domain would be exactly the kind of over-reach this site’s fourth invariant forbids. Which setts were dropped is reported.

The capstan is then applied: twice the weave angle for the plain weave, π for the leno.

Four assertions run. A plain weave grips less the more openly it is set, which is the monotonicity. A leno’s grip does not move with the sett at all, which would fail if the leno’s wrap had accidentally been made to depend on the geometry. Every solved weave angle is under a right angle, which is the fact the theorem rests on. And every plain-weave grip is below the leno’s, which is the theorem itself — asserted rather than observed, because if it ever failed the argument in this essay would be wrong rather than the figure being interesting.

Where the model stops #

This is the most heavily modelled result in this field and the qualifications are correspondingly heavy.

The capstan is an idealisation applied to yarns. It assumes a flexible line on a fixed rigid cylinder. A pick is neither fixed nor rigid, it is under tension itself, and it flattens where it is gripped. The wrap angles are real; the equation relating them to force is a model with a name on it.

Peirce’s geometry is one model of many. Kemp’s racetrack gives different thicknesses for the same cloth, and this site has been careful about saying which is in use. The weave angles here are Peirce’s. The theorem survives any model that keeps the weave angle below a right angle, which every sane one does — so the result is more robust than the numbers on the curve.

“Grip” is not the same as “will not slip in use”. A fabric fails at its weakest crossing under a real load applied at an angle, repeatedly, in a piece that also has a selvedge and a seam. The capstan gives a per-crossing ratio, not a fabric property.

And the crossing itself has been idealised. A leno’s half-turn is taken as a clean half-turn. In a real gauze the doup end is under different tension from its partner, the crossing is not symmetric, and the wrap is somewhat less than π on one side and somewhat more on the other.

What a result with no free parameter is worth #

It is worth separating this result from the ones around it, because the difference in kind is easy to lose when everything is presented in the same voice.

Most of what this field computes takes the shape: given a friction coefficient μ, quantity A exceeds quantity B by a factor f(μ). That is a useful shape and it is honest, and it comes with an obligation — quote the coefficient, quote the range, and say what survives the range. The V-and-W comparison of the pile ladder is exactly this, and its ratio moves by a factor of five across the values people report.

This result has a different shape: quantity A is less than quantity B, full stop. The parameter is present in both sides and cancels. What is left is an inequality between two angles, one bounded by a geometry and one fixed by a mechanism, and neither of them is measured.

That is worth more than a large factor at a plausible coefficient, for a reason about how claims fail. A claim of the first kind fails if the coefficient turns out to be wrong, or to vary with humidity, or to differ between the two cases being compared — and the last is the dangerous one, because a leno’s crossing and a plain weave’s crimp are not obviously the same contact and might well have different coefficients. A claim of the second kind survives all of that: even if the two coefficients differ, the plain weave would need one substantially larger than the leno’s to close a gap that geometry has already opened, and the direction of that requirement is itself implausible.

So the value of doing the arithmetic was not the numbers on the curve. It was discovering that the numbers on the curve were not what the argument needed.

Where the leno’s grip does run out #

The result is an inequality and not a licence, and it has a limit of its own that the trade knows well.

A leno crossing grips the pick it traps. It grips that pick, at that pair of ends, and nothing else. So a gauze’s resistance to a weft being pulled out is a product of the number of crossings that weft passes through, and an open gauze has few pairs per inch by construction — which is the whole point of it.

The comparison this essay makes is therefore per-crossing, and the fabric-level comparison needs the crossing count too. A plain weave at thirty ends per inch grips its pick at thirty places, weakly; a leno at ten pairs per inch grips it at ten places, strongly. Which fabric holds its weft better overall depends on both numbers, and at very open setts the leno wins decisively because the plain weave’s per-crossing grip has collapsed towards one while its crossing count has fallen too.

Where the leno’s advantage narrows is at close setts, where a plain weave has both a good crossing count and its best angle. That is exactly where nobody uses leno, and the reason is now visible as arithmetic rather than as convention.

The fabric-level comparison, which the per-crossing one leaves open #

The essay closes by saying that which fabric holds its weft better overall depends on the per-crossing grip and the crossing count, and leaves the product unevaluated. It can be evaluated, and the answer is one-sided.

The capstan compounds along a thread: a pick passing m crossings, each of wrap θ, is held by e^(μ·mθ). So the comparison is between the two total wraps, and the friction coefficient cancels there exactly as it did per crossing.

A plain weave at n ends to the inch delivers 2θn. A leno at p pairs delivers πp, and a leno’s pairs are two ends each, so at the same end count n = 2p the leno delivers πn/2 = 1.571n.

At the plain weave’s tightest sett, where θ is 37 degrees, its total is 1.29n.

So at equal end counts the leno wins even at the plain weave’s best, by 22 per cent in the exponent — which at a coefficient of three tenths and thirty ends to the inch is a factor of thirteen in holding force. Open the cloth to ten ends and the angle falls to ten degrees, the plain weave’s total collapses to 0.35n, and the factor is forty.

The per-crossing inequality survives being multiplied by the crossing count, and it survives it because the factor of 2.44 in wrap per crossing more than covers the factor of two in crossings that pairing the ends costs.

How many more ends a plain weave would need #

The same expression inverts into a number a designer can use, and it is the honest form of the comparison because it prices the two constructions in the currency that matters.

Setting the two totals equal, a plain weave needs

n ÷ 2p = π ÷ (4θ)

times the leno’s end count. At the plain weave’s densest, θ = 37°, that is 1.22 — twenty-two per cent more ends. At ten ends to the inch, where θ is ten degrees, it is 4.5 — four and a half times as many.

So the trade-off is stated: to hold a pick as firmly as a leno does, a plain weave needs a fifth more threads at its very densest and four times as many at the openness a gauze is actually used at.

And the second half of that sentence is why the construction exists. The comparison is worst for the plain weave exactly where the fabric is wanted, because both terms move against it at once — the per-crossing wrap collapses and the crossings thin out, while the leno loses only the second.

Which explains the selvedge #

The same arithmetic settles the industrial case the essay names and does not price: the leno selvedge, where a pair of crossed ends locks the cut weft ends of a shuttleless loom.

There the pick has a free end, so the hold is not shared over the fabric’s width — it is whatever the last few crossings supply. A plain-weave selvedge would offer 2θ per end at whatever the cloth’s own sett is; a single leno pair offers π, which is 2.44 times as much from one pair as from one end and more than the last two ordinary ends together at any sett.

So one leno pair at each edge does the work of about two and a half ordinary ends, at every sett, and the advantage does not decline as the cloth is opened. That is why the device is a pair rather than a band, why it is used on cloths that are otherwise entirely plain, and why nobody bothers to specify a denser selvedge instead: two ends of leno beat any number of ordinary ends that would fit in the same width, because the ordinary ends are competing on a wrap the geometry keeps under a right angle.

Who found it, and when #

The capstan is Euler’s, 1762. Peirce’s cloth geometry is from 1937, and the site has worked out what its constant assumes elsewhere. Putting them together to compare a leno with a plain weave is not, as far as this collection knows, a standard textbook derivation, and the reason may simply be that nobody needed it: leno’s superiority at open setts has never been in doubt, and a result confirming something obvious rarely gets written down.

What makes it worth writing down here is its form. Almost every claim in this field is a claim at a stated friction coefficient with a range attached, and this one is not — the coefficient cancels, and what remains is an inequality between two angles, one of which is bounded by a geometry and the other of which is fixed by a mechanism. That is a much stronger kind of statement than the ones around it, and the value of computing it is discovering that it is available.

Where the ladder goes next #

Two constructions have now demanded the same missing quantity and got it from the same equation. The essay that puts them together states the boundary in general: what this site’s central check decides exactly, what it structurally cannot see, and why the second is where the fancy weaves live.