The singles inside a ply are not the singles

Here is a puzzle that the previous two rungs create and neither of them answers.

Folding untwists: a single spun at 800 turns a metre and folded at the trade’s ratio is left with 260 of its own, and its surface fibres lie at 7.8° instead of 22.8°. And what grips the end of a fibre falls very steeply as that angle falls — at 7.8° the critical length is longer than the staple, which is the regime in which a yarn can be pulled apart in the fingers.

So a folded yarn ought to be nearly as weak as a roving. It is not. Folded yarns are the strong ones; sewing thread is folded precisely because it has to be strong.

Something is holding the fibres, and it is not the singles’ own twist.

The claim #

A folded yarn’s strength is almost independent of its folding twist, because the grip removed from inside the singles is supplied from outside them.

• The ply’s own helix presses on the singles by the same closed form that a single’s helix uses to press on its fibres, one level up.
• At the trade’s folding ratio, sixty-five per cent of the pressure holding a fibre comes from the ply and thirty-five from its own single. At a ratio of 0.9 it is ninety-eight per cent from the ply.
• Across the whole range the realisation varies by 7.5 per cent, with a shallow minimum near 0.70 rather than a maximum. So the spinner is free to fold for balance, which is exactly what the trade does — and this is the arithmetic that says the freedom is real rather than a lucky accident.

The pressure, one level up #

The radial pressure in a twisted assembly was derived for fibres in a single, and nothing in the derivation was about fibres.

It said: members under tension, lying at a helix angle around a common axis, pull inward with sin²θ of their tension per unit length divided by the radius; integrate outward to the surface, where the pressure is zero because nothing lies outside. The members were fibres because that was the assembly in view. In a folded yarn the members are singles, the axis is the ply’s, and the same integral gives the same closed form.

Converting the ply’s pressure into the units the fibre comparison is in takes one factor — the packing, which turns a single’s axial stress into the fibre stress the critical length is measured against — and after that the two pressures simply add. A fibre’s critical length is its diameter over four times the friction times the total pressure, and both terms are in it.

Why they nearly cancel #

The two contributions move in opposite directions as the folding twist rises, and they move at similar rates, which is the whole result.

Raising the folding ratio takes twist out of the singles: their residual angle falls, their own pressure falls as roughly the square of it, and their grip collapses. In the same act it raises the ply’s helix angle: the ply’s pressure rises as roughly the square of that, and the grip it supplies climbs.

The transfer is nearly complete and nearly free. At a ratio of 0.5 the two angles are equal — the exact cancellation — and the pressures are not, because the ply’s helix radius is half the single’s diameter while the single’s fibres are distributed across its whole section. Past 0.7 the ply is doing almost all of the holding.

The minimum, not a maximum. The curve has a shallow trough near 0.70 and rises slightly on either side, which is the opposite of the shape everybody expects. It is worth being precise about how shallow: the difference between the best and the worst folding ratio in the whole range is 7.5 per cent of the yarn’s realisation, against the 34 per cent that ten degrees of singles twist costs. The folding twist is an order of magnitude less consequential for strength than the spinning twist, and that is the practical statement.

What it leaves the spinner free to do #

A quantity that barely varies is a quantity that can be spent on something else, and folding twist is spent on balance.

That is not a small thing. A single yarn is lively; a loop of it kinks; in a knitting machine the fabric leans and in a needle the thread snarls. The folding ratio that cancels the residual moment is a torque condition this collection declines to compute, and it lands where the trade puts it, between 0.6 and 0.75 for a two-fold.

Look at that band against the curve above. It sits at the bottom of the trough, which means the trade’s folding ratio is very close to the worst one for strength — and costs about one per cent against the flat region on either side. Spending one per cent of strength to remove the liveliness entirely is an obviously good trade, and it is available only because the curve is flat.

There is a second freedom in the same place and it is the one that makes folded yarns useful for knitting. A knitting yarn wants to be soft — a soft yarn bends more easily into a loop and gives a fuller, warmer fabric — and softness is exactly what a low twist gives. A single spun soft enough for that is too weak to knit; a folded yarn whose singles are soft inside it is not, because the ply is holding them. So a two-fold knitting yarn delivers the surface and the bending of a soft single with the strength of a hard one, and the arithmetic above is why the combination is available at all.

What moves the answer, and what does not #

The obliquity bracket moves the level and not the shape.

The contact efficiency moves the level too, and flattens the curve further: at η = 0.2 the whole range spans 4.7 per cent rather than 7.5, because the grip is ample everywhere and the transfer between the two sources hardly registers. Every value of the fitted number makes the conclusion stronger, which is the useful direction for a fit to be uncertain in.

The same transfer, at every level #

Once the mechanism is named it is visible in three places, and the third is the one that makes it a principle rather than a coincidence.

In a folded yarn, the ply holds the fibres its singles have stopped holding. That is this essay.

In a cabled yarn — folded yarns folded again, which is how the heaviest sewing threads and all rope-like constructions are made — the same thing happens once more: the cabling twist untwists the folds, which untwists the singles further still, and the cable’s own helix supplies pressure to the whole assembly. Each level takes grip from the one below and supplies it from outside. That is why a cable can be built to almost any size without the innermost members losing their hold.

In a cloth, the same argument is the crimp. A thread in a woven fabric is gripped by the crossings it passes under, and the pressure comes from the cloth’s geometry rather than from the thread’s own twist. A soft-twisted weft in a firmly set cloth is held perfectly well; the same weft loose on a bobbin is not.

So the general form is: an assembly holds its members, and a member that is itself an assembly is held from outside as well as from within. A structure with several levels can afford to be soft at any one of them, provided the level above is doing its work — which is the same statement as the criterion that decides whether a cloth is one cloth, read as a question about force rather than about connection.

The recursion, and what is left at the bottom of a cable #

Each level of folding untwists the level below by the same rule, so a construction of several levels leaves its innermost members with a residual that decays geometrically — and the decay is fast enough to be worth a number.

A single spun at 800 turns a metre and folded at the trade’s ratio keeps 260, which is about a third of what it had. Cable that folded yarn and the same operation runs again on the fold, which runs again on the singles inside it: after two levels an innermost single retains something near a tenth of its spun twist, and after three it retains a few per cent.

At a tenth of 800 turns a metre a single’s surface fibres lie at two or three degrees, which is well inside the region where the critical length exceeds any staple. Taken on its own that strand would fall apart in the fingers. Inside the cable it is carrying its share, and every bit of what is holding it comes from the two helices above it.

That is why a cut cable behaves the way it does. Cut one and the outer members can be unlaid, and as each level is opened the level below loses the pressure that was holding it — so a rope’s inner yarns can be teased apart by hand a few centimetres from a cut end and are immovable a metre back. The grip is a property of the assembly and disappears with it, which is the same statement as the essay’s title one level further down.

It also says what limits the number of levels. Nothing in the pressure arithmetic does: each level supplies what the one below lost, so the recursion could in principle go on. What stops it is that every level costs retraction — the assembly is shorter than the strands that went into it — and the retractions compound, so a four-level construction is spending a great deal of fibre on being a construction.

The two twists are one grip, so they trade #

Because the two pressures add, the critical length depends on their sum and not on where the sum came from — which means there is a whole family of yarns with the same grip.

A hard single folded softly and a soft single folded hard can arrive at the same total pressure, the same critical length and the same realisation. They are not the same yarn in any other respect: they differ in liveliness, in surface, in how the fibres lie against the light, and in what happens when the yarn is bent. They are the same yarn only in the one quantity this rung computes.

So the spinner is not choosing a point but a point on a curve, and choosing where on it by criteria the strength arithmetic cannot see. The trade’s answer — spin hard, fold to balance — is one end of that curve, and it is chosen because the singles have to survive winding and doubling before the folding twist exists to protect them. That is a manufacturing constraint rather than a property of the finished yarn, and it is the reason the curve is walked from one end rather than optimised over.

Why nobody would have guessed the flatness #

It is worth saying why this is not the expected answer, because the expected answer is a maximum.

Every trade rule about folding twist is stated as though there were an optimum: fold too little and the yarn is not balanced, fold too much and it “loses strength”. The second half of that is the part this arithmetic contradicts. Folding too much does not lose strength; it moves the source of the grip and loses about one per cent.

The reason the wrong expectation is so durable is that the singles’ contribution behaves exactly as the rule says. Measure a single taken out of an over-folded yarn and it is feeble, because it is at a fraction of its twist. The mistake is in supposing that the yarn is the sum of its singles — and a folded yarn is precisely not the sum of its singles, because the folding is doing something to them that a measurement made on them separately cannot see.

That is the general lesson and it is one this collection keeps meeting. A property measured on a component removed from an assembly is a property of a different object. It is the same mistake as reading a crimp ratio off a thread taken out of a cloth, and it fails in the same direction: the component looks worse than it was, because what was holding it has been taken away in order to measure it.

What was counted, and how #

The ply’s pressure is the same function as the single’s, called at the ply’s angle rather than re-derived, so a fix to one is a fix to both and the two cannot drift apart. That is deliberate: writing a second body for the same closed form is exactly how two levels of an argument come to disagree.

The critical length takes the two pressures as a sum, with the packing factor appearing only in the term that crosses between levels — it cancels within a level and does not cancel between two, which is stated in the code rather than left as an inference.

The relation between the strength optimum and the balance band is asserted to be consistent across the fitted number rather than asserted to hold: the check is that the optimum is inside the band at every contact efficiency or outside it at every one, so a conclusion that depended on the fit would fail rather than pass quietly.

And the range is asserted to be small. A folded yarn’s realisation must vary by under twelve points across the whole span of folding ratios, which is the claim the essay is about — a claim about a quantity being flat, which is unusual enough to be worth a gate of its own.

Where the model stops #

The two pressures are added and they may not simply add. The ply’s pressure acts on the single as a body; how it is transmitted to a fibre in the middle of that single is a question about the single’s own internal stress distribution, and treating it as a uniform addition is the crudest step here. The direction of the error is not obvious: the core of a single may be shielded, in which case the transfer is less complete and the curve less flat.

The singles are treated as circular and uniform inside the ply. They are not — folding flattens them against one another, which changes both the helix radius and the pressure, in the same way that a thread flattens where it crosses another.

The contact efficiency is the same fitted number as before, and every absolute realisation quoted here carries it.

Migration is a bracket at both levels, and a folded yarn has two helices for a fibre to migrate along. Nothing here says whether a fibre in a folded yarn migrates between singles, which it plainly cannot, or only within its own — the second is what is assumed.

And nothing here is about abrasion or about fatigue. A sewing thread’s life is a hundred thousand passes through a needle eye rather than one pull, and the folding twist matters far more for that than for the single pull this curve describes.

Where the ladder goes next #

Sideways, into where a folded yarn is actually used and why. A sewing thread is folded, balanced, lubricated and finished, and every one of those is a response to a requirement no weaving yarn has.

And back down into the single, where the same radial pressure decides a quite different question: not whether a fibre can be pulled out, but whether the fibres can slide past one another when the yarn is bent. That settles which end of the stiffness bracket a thread in cloth is at, and the answer is not the one the bracket’s width suggests.