The second binder buys half of everything

A bouclé wears from the loops down found that a bouclé wears in the opposite order from a woven cloth. A woven cloth’s crowns are the threads that carry its tension, so a rub takes strength before it takes mass; a bouclé’s surface is a forest of loops hanging off a core that carries everything, so the same rub takes three tenths of a millimetre of thread that does nothing at all before it reaches anything that does.

It also priced a snag. A caught loop drags thread past its own binder point and then past the next, and every wrap multiplies the pull the capstan equation demands. Starting from the four millinewtons a loop is already pushing with in a hundred-gram cloth, the pull reaches the effect thread’s breaking load of 4,366 millinewtons after 7.43 binder points — so a snag draws about twenty-one millimetres of yarn and then breaks rather than running on.

That whole bound rests on one wrap per binder point, which is what a single binder laid in one direction gives. Ordinary practice for a yarn expected to be handled is two, and the essay before it said in a sentence that a second binder would square the grip. It does. What it does not say is that the consequence has a closed form.

Why it is exactly inverse, with nothing fitted #

The capstan equation gives the ratio of tensions across a wrapped contact as the exponential of the friction coefficient times the wrap angle. A binder laid in one direction wraps the effect thread through half a turn at each binder point, so

$$\text{grip} = e^{\mu\pi} = 2.566 \quad \text{at } \mu = 0.3.$$

Two binders, one laid each way, wrap it through half a turn each — a full turn in total — so the grip is $e^{2\mu\pi} = 6.586$, which is the first one squared. Three binders give the cube, and n binders give the n-th power.

Now the reach. Thread dragged from k binder points away arrives through k such wraps, so the pull needed is the starting force times the grip to the k-th, and the thread breaks when that reaches its breaking load:

$$k = \frac{\ln(F_\text{break} / F_\text{start})}{\ln(\text{grip})} = \frac{\ln(F_\text{break} / F_\text{start})}{n\,\mu\pi}.$$

The binder count is in the denominator, linearly, and nowhere else. The numerator has the friction in it nowhere at all. So the reach is $k_1 / n$ exactly, for every yarn, every fibre, every cloth weight and every friction coefficient — the only thing any of those decide is $k_1$.

The steps are what matter, and they fall as one over n squared #

An inverse law’s differences fall much faster than the law itself, and that is where the practical answer is.

Going from n binders to n + 1 removes

$$\frac{k_1}{n} - \frac{k_1}{n+1} = \frac{k_1}{n(n+1)}$$

binder points. At n = 1 that is $k_1/2$ — half of everything, in one step. At n = 2 it is $k_1/6$, a sixth. At n = 3, a twelfth.

So the second binder takes 3.72 binder points off the reach, the third a further 1.24, and the fourth 0.62. The second binder is three times as effective as the third and six times as effective as the fourth, and each of them costs the same fifteen tex of thread.

Which is the arithmetic behind a practice nobody has had to justify. A fancy yarn that will be handled carries two binders. Not one, because one leaves a snag running through seven loops; not three, because the third costs as much as the second and buys a third as much. The trade’s number is the number the arithmetic picks, and this is the first place in this account where a fancy-yarn convention turns out to be an optimum rather than a habit.

The halving survives everything the cloth can do #

The reach itself is not one number — it depends on where the cloth sits in its bracket, because the starting force is the force the loops are already being pressed with and a denser cloth presses them harder.

The reach rises as the cloth gets heavier, which is the opposite of the obvious expectation and is the maximum in the loop’s own force curve showing through: a heavier cloth presses its loops past the peak, the force falls again, and a snag therefore starts from less pull and takes more binder points to reach the breaking load.

And the halving is exact in every row. 6.48 and 3.24; 7.00 and 3.50; 7.43 and 3.72; 8.03 and 4.02. The cloth’s weight moves the numerator and the binders divide it, and the two never interact — which is what “exactly inverse” buys and is why the factor is quotable where the reach is not.

The same is true across the other free parameter. At a loop spacing of 1.5 millimetres the reach is 5.31 points and at 6 millimetres it is 9.61, because a wider spacing means a bulkier yarn and a more open cloth; both halve to 2.66 and 4.80. The reach spans a factor of two across everything a designer can choose, and the binder count divides all of it.

The half the sentence left out #

The essay before it priced the second binder as “another fifteen tex on a resultant of eighty-nine, which is a seventh more yarn for half the snag”. That is true and it is not the whole cost, because a bouclé’s count arithmetic charges for thread in three places.

The resultant count rises from 89 tex to 104, which is 16.9 per cent more yarn — tex being mass per length, so the rise is the thread itself and not a change of convention — and moves the count diameter from 352 to 381 micrometres.

The jamming bracket narrows from 12.56 to 11.73. A bouclé’s bracket is wide because its two diameters are far apart — the one its count implies and the one its loops occupy — and adding binder thread raises the first without touching the second, so the bracket closes from below. At a hundred grams a square metre the cloth’s sett falls from 5.62 ends a centimetre to 4.81.

And the share of the yarn that is on no load path falls from 60.7 per cent to 51.9. That column looks like the best news in the table and it is the one to be careful about. The sacrificial share is not a defect; it is the whole reason a bouclé wears the way it does. The essay before it’s finding was that a rub takes six tenths of the yarn’s mass before it reaches anything the cloth’s strength depends on. A second binder spends nearly nine points of that.

So the second binder trades a snag that runs half as far against a fifth less sacrificial material and an eighth less bracket, and which of those a fabric wants is a question about how it will be damaged rather than about how well it is made.

Where the two effects meet #

Setting the two together gives the design rule the account has been working towards, and it is a comparison rather than a number.

A snag and a rub are different damage. A snag is a single event that takes a length of thread out of the surface and leaves a visible run; its size is the reach, and the reach is what the binders decide. A rub is a slow removal of whatever is on top; its budget is the sacrificial share, and the binders spend it.

A fabric that will be caught on things — an upholstery, a coat sleeve, anything near a fastening — is a fabric whose damage is snags, and two binders halve them. A fabric that will be rubbed — a chair seat, a cushion face — is a fabric whose damage is abrasion, and two binders take nine points off the depth it can lose for nothing.

These are opposite recommendations from one change, and the count arithmetic prices both of them in the same units. That is what the account’s whole apparatus is for: an upholstery bouclé and a throw bouclé are not two qualities of one yarn, they are two points on a trade with a computed slope.

The slope is steep in one direction and shallow in the other. Doubling the binders halves the snag; tripling them reduces it by only a further third. Doubling the binders costs nine points of the sacrificial share; tripling them costs six more. The trade is worst at three binders and best at two, in both quantities at once, which is why two is the answer and why nothing beyond two appears in any yarn anybody sells.

What the second binder does not change #

Three things a reader might expect to move do not, and saying which is most of the discipline here.

The loop’s shape is untouched. A bouclé loop is an elastica whose shape is fixed by the overfeed and the binder spacing — it is a length of thread leaving the core along the core at two binder points, and nothing about it knows how many threads are doing the binding. Two binders at the same points give the same two boundary conditions as one, so the free height, the overhang and the neck are the same to the last figure.

The force a loop presses with is untouched, for the same reason: it is the elastica solved with its apex pushed down, and the binders enter only as the points the thread leaves from. That is why the starting force in the reach arithmetic is the same in both columns, and it is what makes the inverse law clean rather than approximately clean.

And the surface is untouched. A bouclé’s surface is its loops exactly when the loop envelope stands clear of the count diameter, and the envelope is 4,426 micrometres against a count diameter that rises only from 352 to 381. The ratio falls from 12.56 to 11.73 and the surface is loops either way, by a factor of more than eleven.

So the second binder changes exactly two things — how far a snag runs, and how much of the yarn is expendable — and leaves the yarn’s geometry, its pressing force and its appearance where they were. A change that moves two quantities and no others is rare enough in this subject to be worth naming, and it is why the binder count is a design variable in the way the overfeed is not: the overfeed moves everything at once.

The loop that comes free is the loop that pills #

There is a third consequence and it belongs to a different account, so it is stated here and computed there.

A snag that runs seven binder points does not remove the thread; it draws it out and leaves it standing proud of the surface as a loop several millimetres long. That is precisely the object a pill is anchored by — a length of fibre free at the surface but held at one end — and the pilling arithmetic’s whole question is how much free length there is and how firmly it is held.

So halving the snag reach halves the length of drawn thread each snag leaves, from twenty-one millimetres to nine, and the material that would have become a pill is still in the yarn. That is a larger effect than it sounds, because the drawn length is what distinguishes a bouclé’s surface damage from an ordinary cloth’s: abrasion takes the hairs first on a spun-yarn fabric and takes whole loops here, and a loop is three orders of magnitude more material than a fibre end.

It also runs the other way from the sacrificial share. Two binders leave less material that may be lost for nothing and leave less material standing where it can be worked into a pill, so on this quantity the two effects agree rather than trading — which makes the pilling consequence the cleanest argument for a second binder and the one nobody makes.

What was counted, and how #

The capstan ratio is this account’s own capstan calculation, the same function a seam’s grip on its threads is computed with. A binder point is taken as half a turn per binder, which is what one binder laid in one direction gives and what the essay before it assumed; two binders laid in opposite directions give a full turn between them.

The starting force is the loop’s own pressing force at the cloth’s position in its bracket, from the elastica solve, and it does not depend on the binders at all. That is why the reach’s numerator is unchanged and the whole effect is in the denominator.

The breaking load is the effect thread’s, from this account’s yarn tensile model at the same count and fibre. A snag that reaches the breaking load breaks; it does not go on pulling.

The inverse law is required rather than plotted. Every row of the reach figure is checked against the first row divided by its own binder count, to within a part in a million, because the relation is exact algebra and a figure that merely looked inverse would be a figure nobody had checked.

And the count consequences are the account’s own arithmetic run again, not a new model: the resultant count, the two diameters, the bracket and the sett all come from count arithmetic with the binder count changed and nothing else touched.

What the arithmetic cannot say #

The friction coefficient is assumed and every grip is exponential in it. At μ = 0.2 the single-binder reach is 11.15 points and at μ = 0.4 it is 5.57 — a factor of two across a range of coefficient that is entirely ordinary for cotton on cotton. What survives that is the inverse law, because μ cancels out of the ratio between one binder and two. So the reach is a bracket and the halving is not, and only the second of those is worth quoting without a range attached.

The two binders are taken as laid in opposite directions. Two laid the same way wrap the effect thread through a full turn at the same place rather than at two places, which is the same total angle and the same grip — so the arithmetic does not distinguish them. It should: two binders laid the same way put twice the torque into the yarn, and a yarn with torque in it snarls. A yarn that has been set has no torque is the collection’s account of what to do about that, and nothing here computes it.

And nothing here is about the binder breaking. The reach assumes the binder holds while the effect thread is dragged through it. A binder at fifteen tex is thinner than the effect thread at thirty, so it breaks first at a lower load — and once a binder has broken, the loops it held are free and the snag runs unopposed to the next binder. That failure is not in this arithmetic and it would cap the reach from the other end.

Who found it, and when #

That fancy yarns are doubly bound is universal practice and is in every account of fancy twisting, always as a construction to be described and never as a quantity to be chosen. The capstan equation is Euler’s, from 1762, and its use for thread held in a fabric is standard in every account of seam slippage and pull-out.

The inverse law is this account’s, and it is elementary once the two halves are set beside one another: a grip exponential in a wrap angle that is linear in the binder count, inverted by a logarithm, gives a reach inverse in the count. Nothing about that needed to be discovered; it needed the reach to have been computed in the first place, which is the essay before it’s contribution and is not a thing the trade has ever had a number for.

What is genuinely new is the cost column, and it is new because it is uncomfortable. A second binder improves the yarn by one measure and makes it worse by another, and the second measure — the depth of material a rub may take for nothing — is a quantity nobody has named because nobody has computed it. It is the thing a bouclé is for.

Still open: whether a real doubly bound bouclé snags at half the reach #

The prediction is a factor and not a direction, which makes it the cleanest test this account offers. Take one effect thread, one core and one binder count; make two yarns differing in exactly one thing, the number of binders; weave both at the same weight; snag both with the same hook under the same pull.

The singly bound one should give up about twenty-one millimetres of yarn and the doubly bound one about nine. If the friction coefficient is wrong, both numbers move together and the ratio does not; if the half-turn-per-binder geometry is wrong, the ratio moves and says by how much.

The measurement would also settle the thing the arithmetic is least sure of. A reach of 3.72 binder points is between three and four loops, and a loop is a discrete object — so the prediction is really that the snag stops at the fourth loop rather than at 3.72 of them, and whether the distribution of stopping points sits where the continuous calculation puts it is a question the calculation cannot ask itself.