The folding rule is not a torque balance

There is a rule in the trade for how hard to twist a folded yarn. Take two singles at a given twist and fold them, in the opposite hand, at about two thirds of it. For three singles the figure is nearer a half to two thirds, and for four nearer a half.

This collection has carried those three figures for a long time as a bracket copied out of practice, sitting in a table with a note saying that they are what folders use. Nothing derived them, and the reason nothing derived them was written down at the time: deriving them needs a torsional rigidity, and there was none.

There is one now, and the derivation is short, and it produces the wrong answer by a factor of three.

What a balanced yarn is supposed to be #

The stated purpose of the rule is balance. A folded yarn is folded so that it has no residual torque: hang a loop of it and it should stay a loop rather than kinking back on itself.

That is a well posed condition and it has an obvious form. The fold has a net moment about its own axis; set the moment to nought; solve for the fold twist.

So the derivation is a matter of writing down the moments.

The two moments #

Each single arrives at the folding frame with a residual twist rate of its own — two π times its turns per millimetre — and a residual torque of the torsional rigidity times that rate.

Folding does two things to it.

It untwists the single. One turn of fold takes one turn of twist out of each single, on a length of single that is longer than the corresponding length of fold by one over the cosine of the helix angle. So after folding, the single’s residual twist rate is two π times the singles twist minus the fold twist over that cosine.

And it bends the single onto a helix. The single now follows a helix of radius half a single diameter — for a two-fold, where the two singles touch — with a curvature of the cosine squared of the helix angle over that radius.

Both produce a moment about the fold’s axis, and both have to be resolved onto it. The single’s own torque points along the single, which is at the helix angle, so its axial component is multiplied by the cosine. The helix’s bending moment points along the binormal, whose axial component is the sine.

Setting them equal:

C·2π(T_s − T_p/cos α)·cos α = B·(cos²α/R)·sin α

The closed form #

In the small-angle limit — which the helix angles here are, at four or five degrees — the cosines go to one, the sine goes to the tangent, and the tangent of the helix angle is two π times the fold twist times the radius. Substitute and the radius cancels:

T_p / T_s = C / (B + C)

which is the ratio of the two stiffnesses and nothing else. Written in terms of the fibre’s own constants it is 2G / (E + 2G).

The count has gone. The single’s diameter has gone. The twist level has gone. The number of folds has gone. And the stiffness bracket has gone, because both terms carry it identically.

That is the third bracket-free result on this ladder and it is the cleanest of them.

The number #

For a cotton, whose stiffness ratio is a quarter, the balance is at

0.25 / 1.25 = 0.200

Two tenths. The trade folds at between 0.60 and 0.75.

For a wool, whose ratio is the highest in the table at 0.8, the balance is at 0.444. For a polyester, at 0.167, it is 0.143. For a viscose it is 0.231.

None of those is anywhere near two thirds, and the fibre that comes closest is wool — which is not what a two-fold cotton sewing thread is made of. The full spread across the table is in what a balanced yarn is balanced about.

The size of the disagreement #

A factor of three is not a refinement, and it is worth saying precisely how bad it is before asking what went wrong.

To get a balance at two thirds, the closed form needs C/(B+C) = 0.667, which needs C = 2B: a torsional rigidity twice the bending rigidity.

For a circular section, C/B is 2G/E, so that needs G = E. No solid has a shear modulus equal to its tensile modulus — isotropy alone caps 2G/E at one over one plus Poisson’s ratio, which for any real material is under one — and an oriented fibre is far below the isotropic value rather than above it.

So the balance point cannot be moved to two thirds by any choice of fibre, any refinement of the shear table, or any correction to the helix geometry. The trade’s rule is not a torque balance. It is a rule about something else, and finding out what took the next rung.

What the picture says #

The visual check is worth making because it is immediate.

A yarn folded at a fifth of its singles twist has a surface helix angle of about a degree and a half. The two singles lie almost parallel with a slow wrap, and the structure has none of the qualities a folded yarn is folded for: it does not present a smooth surface, the singles are not held together, and it will separate under handling.

A yarn folded at seven tenths — which is what the trade does — has a surface angle near the singles’ own, the two components wrap visibly, and the result is the round, smooth, coherent thing anybody recognises as a folded yarn.

So the disagreement is not subtle and it is not in a regime nobody looks at. The two predictions produce visibly different objects and the trade’s is obviously the one that gets made.

Which half of the derivation to suspect #

Given a factor of three, the derivation has to be examined rather than the answer.

The untwisting assumption is the largest candidate. The derivation assumes one turn of fold removes one turn of twist from each single, which is the trade’s own statement of what folding does and is what this collection derived several ladders ago, where it was also noted that the singles inside a ply are not the singles. It is right about the direction and it is an idealisation about the amount.

The alternative bookkeeping — that the fold inserts writhe rather than removing twist, so the twist falls by the fold’s writhe rather than by its turns — gives a smaller untwisting for the same fold, which pushes the balance point higher. That is the right direction. How much higher is a calculation nobody has done, and it is bounded above: even complete failure to untwist would not reach two thirds, because at that point the singles’ torque never falls and there is no balance at all.

The bending term is the other candidate. The derivation prices the helix’s bending against a straight natural shape, which is right for a single that was straight before folding. A single that has been on a package is not straight, and one that has been steamed is not elastic. Both reduce the bending term, which pushes the balance point lower — the wrong direction.

So the plausible corrections move the answer partway at most, and the honest conclusion is that the rule is about something else.

What a refutation is worth here #

This collection keeps an index of trade claims it has tested, and most entries in it are claims that turned out to be roughly right for a reason nobody had stated. This one is different: the claim is right as a practice and wrong as an explanation, and separating those is the useful part.

Folders do fold at two thirds. Yarns folded that way are good yarns. What is wrong is the account of why, and the account matters because it is what a person reasons from when the situation changes — a new fibre, an unusual count, a filament rather than a staple.

Anybody reasoning from “two thirds balances the torque” will predict that a wool should be folded at the same ratio as a polyester, since both are being balanced. Anybody reasoning from the correct account will predict that the ratio does not depend on the fibre at all but does depend on the number of folds, in a specific way.

The second prediction is right and the first is wrong, and the two are only distinguishable once somebody has done the arithmetic that shows the first is not what is happening.

What the disagreement is not #

Three explanations suggest themselves and none of them survives, which is worth working through because each is the first thing a reader will reach for.

It is not the small-angle limit. The exact solve and the closed form agree to a part in a thousand at ordinary twists, and where they part company at high twist the exact answer is lower rather than higher.

It is not the fold count. The closed form has no fold count in it and the exact solve confirms that to four figures: the radius on which the singles sit multiplies the bending term and divides the helix angle, and the two cancel. So the fact that the trade’s rule does depend on the fold count — 0.67, 0.58, 0.50 for two, three and four — is itself evidence that the trade’s rule is not this condition, because this condition is fold-count-free.

That last point is worth more than the factor of three. A wrong number can be a wrong constant. A wrong dependence is a wrong mechanism, and the trade’s rule varies with something the torque balance is blind to.

And it is not the fibre. The trade quotes one set of ratios for all fibres. The torque balance varies by a factor of three across the collection’s table. If the rule were a balance, a wool would be folded very differently from a polyester, and it is not.

Three independent mismatches — the magnitude, the fold-count dependence and the fibre-independence — and only the first of them is a number. The other two are structural, and structural mismatches are what refute a mechanism.

What was counted, and how #

The two moments are computed exactly rather than in the small-angle limit, and the crossing is found by bisection on the fold twist between nothing and twice the singles twist.

The closed form is checked against the exact solve rather than substituted for it: they agree to a part in a thousand at two hundred turns a metre and drift to five per cent at sixteen hundred, which is the right behaviour for a small-angle limit and is why both are kept. The check asserts the agreement at low twist and asserts that the departure grows with the twist rather than scattering, because a limit that is approached is a different thing from a formula that nearly works.

The stiffness ratio comes from the collection’s own fibre tables. The single’s diameter comes from its count and the packing factor by the site’s own volume arithmetic. The radius on which the singles sit is half a diameter for a two-fold and the circumradius of a regular polygon of touching circles for more.

Every one of those was varied — four fibres, three twist levels, three fold counts — and the ratio moved only with the fibre.

Where the model stops #

The singles are treated as elastic in torsion. A spun yarn’s torque relaxes over hours and disappears entirely when it is steamed, and steaming after folding is standard practice. So the balance the derivation computes is a balance for a fresh, unset yarn, and a folded yarn as delivered is balanced by having been set rather than by having been folded correctly.

That is not a small caveat and it may be most of the answer to why nobody in the trade is troubled by any of this.

The fold is treated as a regular helix of touching singles. It is not: the singles migrate, flatten against one another and share a common surface, and this collection has a rung on how much a folded yarn is not two singles.

And the whole derivation is about a two-component idealisation. A cabled yarn — a fold of folds — has three levels of twist and two balance conditions, and none of that is here.

The generalisation #

The pattern is one this collection has now met three times and is worth naming.

A rule of thumb with a number in it usually encodes a real constraint, and the constraint is often not the one the rule is explained by.

The first instance was the thread count as a measure of quality, where the number is real, the correlation is real, and the stated mechanism is not what produces it. The second was the racetrack section, where a shape used for a century turned out to be justified by a different argument from the one usually given.

This is the third, and the shape of the resolution is the same in all three: compute the stated mechanism, find it gives the wrong number, and then ask what else in the situation has that number in it.

Here the answer is waiting and it is geometric rather than mechanical: 1/√n, for n folds, which is 0.707, 0.577 and 0.500 — and every one of the trade’s three brackets contains it.

What a folder would say #

It is worth imagining the objection somebody who folds yarn for a living would make, because it is a good one and it sharpens what the rung is claiming.

They would say that nobody folds to zero torque, that the ratio is chosen for the yarn’s appearance, hand, evenness and strength, and that balance is one consideration among several and not the governing one.

That is exactly right, and it is the conclusion this rung arrives at from the other direction. The disagreement is not with practice; it is with the explanation attached to practice, which appears in the textbooks as a balance and is repeated as one.

What the arithmetic adds is that the explanation is not merely incomplete but impossible: no fibre can be balanced at two thirds, so whatever the trade is optimising, balance is not it. That converts a vague “several considerations” into a definite “not this one”, and it leaves exactly one candidate standing.

The candidate is geometric and it reproduces all three numbers exactly rather than roughly, which is a stronger claim than the one it replaces.

A note on what balance costs #

If the trade’s fold is not balanced, folded yarn carries residual torque, and it is worth asking how much.

At the trade’s ratio the singles have been untwisted by seventy per cent rather than by eighty, so a residual remains and the fold is torsionally live. It is much less live than a single — the threshold tension for snarling goes as the square of the residual twist, so removing seventy per cent of it reduces the tension needed by a factor of eleven — but it is not nought.

That is consistent with practice in a way that the balance account is not. Folded yarns are slightly lively, they are steamed, and a folded sewing thread left in a loop does eventually take up a small twist. If folding balanced the torque exactly, none of that would happen and steaming would be unnecessary.

So the residual is real, it is small, and the trade deals with it by setting rather than by folding. Which means the fold ratio was free to be chosen for something else, and it was.

Who found it, and when #

The folding rule is old trade practice and appears in every spinning text with the same numbers and the same one-line explanation.

The torque balance of a plied yarn has been analysed several times in the textile literature, mostly with more machinery than is used here and mostly to compute residual torque rather than to test the rule. The conclusion that the balance point is well below the practical fold ratio is not new; what is unusual is having it fall out as a closed form in the ratio of two stiffnesses, with everything else cancelling.

What is this collection’s own is the cancellation, the closed form, and the observation that the required ratio exceeds what any solid can supply — which is what turns “the numbers disagree” into “the mechanism is impossible”.

Where the ladder goes next #

If the rule is not a torque balance it is something, and the something has to reproduce three numbers rather than one: two thirds for a two-fold, a little under six tenths for a three-fold, and a half for a four-fold.

One ratio does that exactly, it involves no mechanics at all, and it falls out of arithmetic this collection has had since its second phase. The folding rule is a surface angle.