A float presses on nothing

The float is this site’s most useful single number and its ladder is the longest here. One length behind lustre, drape, snagging, abrasion, tear strength and the sett a cloth can be woven at. The rung that priced a thread’s hold was the one place the float was doing work that had not been checked, and that rung said so in its own list of what it had not done:

The crossover arithmetic counts every intersection as a grip, which is right for a plain weave and overstates a satin’s hold by something like the ratio of their interlacing rates.

Half of that is right. A float presses on nothing, the grips are the interlacings, and counting crossings overstates a satin’s grip count by exactly two and a half. The other half — that the correction changes how a satin compares with a plain weave — is wrong, and the arithmetic says so in closed form.

Where the interlacing rate sits in the answer #

The crossover under a capstan grip is

$$L^* = \frac{1}{\mu\rho\beta}\ln\!\left(1 + \frac{T_{\text{break}}\,\beta}{n_0}\right)$$

with ρ the turns per unit length and β the angle of each turn. The floor n₀ — the resistance per unit length the cloth supplies on its own — is itself μ·N·ρ, so writing it out,

$$L^* = \frac{1}{\mu\rho\beta}\ln\!\left(1 + \frac{T_{\text{break}}\,\beta}{N}\right)$$

and ρ has left the logarithm. It appears once, as a factor outside, and nothing else in the expression knows the weave.

So the ratio of two weaves’ crossovers at the same cloth is the ratio of 1/ρ, which is the ratio of their interlacing rates, which is exactly what a sum of independent contacts gives. The two models disagree about every absolute number and agree about every comparison.

Why the expectation was reasonable #

It was reasonable because the exponent contains ρ, and an exponential is where small differences become large ones. A satin’s grip builds far more slowly per millimetre than a plain weave’s, so over a fixed length its pull-out force is much less than two and a half times smaller — and that part is true.

The mistake is in what the crossover is. It is not a force at a fixed length; it is the length at which the force reaches a fixed value, and the two questions have different answers. At a fixed length the exponent matters and the satin loses more than its grip count. At a fixed force the exponent is what is being solved for, so ρ moves to the outside and the extra loss disappears exactly.

Both statements are true of the same model. Which one is the useful one depends on the question, and the questions that matter here — how far a cut edge frays, how wide a seam allowance has to be, how long a tuft has to be bound for — are all fixed-force questions.

So the correction is a single number #

ln(1 + z)/z, with z the thread’s breaking load times the wrap angle, over the contact force. It contains no friction coefficient — that cancelled — and no interlacing rate.

For a sheeting it is 0.269. Every weave’s crossover is multiplied by 0.269, so a plain weave frays 2.0 mm instead of 7.4 and an eight-end satin 8.0 instead of 29.8, and the eight-end satin still frays four times as far as the plain weave in both accounts.

Which is the useful form of the answer #

A correction that rescales everything is invisible in every comparison and decisive in every absolute number, and that is the opposite of the failure mode most corrections have.

Everything the float ladder says about relative behaviour survives untouched. A satin frays further than a twill, a twill further than a plain weave, and by the ratios the drafts give. Why satin shines is unaffected, and so is where the abrasion happens. Whether a loose weave tears better is unaffected. The float ladder’s whole architecture — one number behind six behaviours — is unaffected, because it is an architecture of comparisons.

What changes is every place a length in millimetres was quoted against something in the world: a seam allowance, a fray width, a tuft’s bound length. Those were all four times too long.

And it makes a known fact into an arithmetic #

Every weaver knows a satin frays and a plain weave does not. It is a rule of thumb, it is taught as one, and it is one of the clearest cases of trade knowledge that is exactly right.

The rule now has a number under it that comes from the draft and nothing else, in two steps. Count the interlacings per thread per repeat — a property of the matrix, decidable, no tolerance to choose. Divide by the repeat’s length in cloth. That is the turns per millimetre, and the crossover is inversely proportional to it.

A trade rule about a fabric’s behaviour turns into a count on a piece of point paper, which is what this collection is for.

The count itself, weave by weave #

Every one of these four weaves turns exactly twice per thread per repeat, and that is not a coincidence to be noticed and moved past — it is the whole ordering.

A plain weave’s end runs over one pick and under the next, so over a repeat of two picks it goes down once and up once: two turns. A 2/2 twill on a repeat of four has two floats of two, so again down once and up once: two turns. A five-end satin has one float of four and one of one, so two turns. An eight-end satin has one float of seven and one of one: two turns.

The count is the same and the repeat is not, so the turns per millimetre go as 2/n, with n the repeat. Against a plain weave’s 2/2, the rate is exactly 2/n — a half for the twill, two fifths for the five-end satin, a quarter for the eight-end — and the crossover is the reciprocal of that.

That is why a satin’s fraying width is set by its order rather than by its longest float, which is a real distinction and an easy one to lose. A weave with two floats of four in a repeat of eight has twice the turns of an eight-end satin and the same longest float, so it frays half as far. What decides fraying is the number of turns per unit length, and a longest float is a poor proxy for it — except among the regular satins, where each repeat holds one float and the two quantities move together.

One more consequence follows and is worth having, because it pairs two weaves nobody would put together. A 2/2 basket has floats of two on a repeat of four, and its end goes down once and up once in that repeat — two turns, exactly as a 2/2 twill does on the same repeat. So a 2/2 basket and a 2/2 twill fray by identical amounts, twice as far as a plain weave, though one reads as a diagonal and the other as a chequer and they have nothing else in common. Fraying does not see the arrangement of the turns at all. It sees how many there are and how far apart, and both of those are the same in the two drafts.

What the count really is, once the four weaves are left behind #

The four weaves above all turn twice per thread per repeat, which makes the rate 2/n and makes the ordering look like an ordering by repeat size. It is not, and the general statement is worth having because the exceptions are ordinary weaves rather than curiosities.

A thread’s turns per repeat is twice the number of runs in its own sequence — one turn at each end of each run, counted cyclically. So the interlacing rate is

2 × (runs per repeat) ÷ (repeat length),

and the crossover, and therefore the fraying width, is its reciprocal.

The four weaves on this page each have two runs, which is why the count of runs disappeared and the repeat did all the work. Across the twills a repeat of eight actually admits, the number of runs varies from two to eight — a 4/4 twill has two, a 1/2/3/2 has four, and the alternating sequence has eight — so the fraying widths of the twills on one repeat span a factor of four, and nothing about the repeat says which one a given draft is.

Two consequences, and the first contradicts a proxy this ladder has used comfortably elsewhere.

The longest float does not order fraying. A 4/4 twill and an eight-end satin both have two runs on a repeat of eight, so they fray by identical amounts — and their longest floats are four and seven. A 1/2/3/2 twill has a longest float of three, shorter than either, and frays half as far as both, because it has twice the turns. So a designer choosing a weave for a cut edge should count runs and not floats, and the two orderings disagree on ordinary cloths rather than on contrived ones.

Among the regular satins the two agree, which is why the confusion survives. A regular satin has exactly one float per repeat and therefore exactly two runs, so its float is n − 1 and its rate is 2/n: the longest float and the turn count are two readings of the same number. Every satin behaves as the float proxy says, and every satin is where the proxy was learnt.

The practical form is the one worth carrying into a specification. A float limit does not bound a fraying width, because a weave can satisfy any float limit and still have only two runs in a long repeat. What bounds a fraying width is a minimum run count, which is a different constraint on the same draft and is not, as far as this collection has found, ever written down.

What was counted, and how #

The ordering claim is asserted directly and in the strongest available form. For four weaves at one cloth, the ratio of crossovers is required to equal the ratio of resistances per millimetre to one part in a thousand million — not approximately, not within the model’s uncertainty, but as an identity, because it is one.

That assertion is worth its place precisely because it is a negative result. If a later change to the geometry gave the wrap angle a weave dependence — which it should, and which is the largest thing this rung does not do — the identity would break, and it should break, and the assertion is what would say so.

The shortening factor is separately asserted to be identical across the four weaves, and to be identical across six friction coefficients, and to equal ln(1 + z)/z exactly. Three assertions on one closed form, because it could fail three ways.

The interlacing counts are enumerated from the matrix by the same function the site has used since its first essays, and the repeat lengths come from the cloth’s own setts. Nothing here is quoted.

The two questions this rung keeps apart #

It is worth ending on the distinction the whole result turns on, because it is easy to collapse and collapsing it is how the shortfall’s expectation was formed.

At a fixed gripped length, how much force does it take to withdraw the thread? There the exponent matters, a satin loses far more than its grip count suggests, and the comparison between weaves is not the ratio of their interlacing rates. That question is the one a tuft asks, because a tuft’s bound length is fixed by its fastening.

At a fixed force, how much gripped length is needed? There the exponent is what is being solved for, the interlacing rate moves outside the logarithm, and the comparison is exactly the ratio. That question is the one a fraying width and a seam allowance ask, because both are lengths chosen against a thread’s breaking load.

Both are true of the same model and neither is a correction to the other. What decides which applies is whether the length or the force is the thing held fixed — and in this collection the force is held fixed far more often, because a thread’s breaking load is a property of the yarn and appears in every applied question here.

So the ordering survives in most of what this site computes and fails in the one place it does not, and knowing which is which is the difference between quoting a ratio correctly and quoting it in the wrong régime.

Where the model stops #

The wrap angle is a plain weave’s, in every weave, and this is where it hurts most. A satin’s crimp really is smaller than a plain weave’s at the same construction — that is the float ladder’s own result, the reason a satin can be set denser — so its wrap angle is smaller too, and a smaller β lowers the grip further. The satin is therefore worse than this rung says, and the correction would enter through β, which sits inside the logarithm as well as outside it.

That last clause matters. A weave-dependent wrap angle would break the exact ordering, because β appears in two places and ρ in one. So the finding of this rung is conditional on a limitation the rung itself states, and the honest form of it is: within a model that gives every weave the same wrap angle, the interlacing ordering is exact. Supplying a per-weave geometry is the obvious next thing and is recorded as not done.

The float is treated as touching nothing. A long float on the face of a cloth does rest on the threads it passes over, and there is a small normal force from the thread’s own path and from anything pressing the cloth. Treating it as zero is right to first order and is why a satin frays at all rather than falling apart.

And the contact force is a tensioned cloth’s, so every length here is a length in cloth under load.

The generalisation #

Where a parameter enters an expression decides which questions it changes the answer to.

The interlacing rate enters this one exactly once, outside a logarithm, and that single structural fact settles what looked like a modelling question: whether a better account of friction changes how weaves compare. It does not, and no amount of computation was needed to find out — reading the closed form was enough.

The wider point is about corrections and expectations. The rung below expected this correction to change an ordering and it changes a scale, and the difference between those two is the difference between a result that invalidates a ladder and one that rescales it. Before computing a correction, it is worth asking which of the two it can possibly be, and the answer is usually visible in where the parameter sits.

There is a converse worth carrying. A correction that changes only the scale is the hardest kind to notice from inside a body of work built on comparisons, because every internal check passes. It shows up only when something is quoted against the world — a millimetre against a cut edge, a seam allowance against a garment — and this site had quoted very few.

Who found it, and when #

That a satin frays and a plain weave does not is weaving practice and is as old as satin. That the difference is the interlacing count is equally old and equally uncontroversial.

The capstan treatment of yarn pull-out is standard in the fibre-mechanics literature and the exponential build-up with embedded length is measured routinely, most often for a fibre in a matrix rather than a yarn in a fabric.

What is this site’s is the observation that the two combine to leave the ordering exact — which nobody would look for, because nobody was worried about the ordering. It was found by asserting the ordering and watching the assertion pass at a tolerance of one part in a thousand million, which is not the tolerance an approximate agreement passes at.

Where the ladder goes next #

Sideways, this is spent in the applied field, where a fray width and a seam allowance are numbers somebody has to choose and were four times too long.

Along the float ladder, the missing piece is a per-weave geometry: every crimp on this site is a plain-weave crimp, and a satin’s is genuinely different. That is a setting-field problem rather than a weaves-field one, and it is the largest single thing standing between this ladder and an answer it could quote in the trade.