A snarl comes in one size

The tension at which a twisted thread stops being straight depends on how stiff it is, and this collection does not know how stiff a yarn is to better than a factor of three hundred. So the threshold is a bracket, and the argument that closes it is an observation rather than an arithmetic.

The size of the coil it takes is different, and it is different in the best possible way.

The cancellation #

The wavelength the writhing instability selects sets a coil radius of 2B/M: twice the bending rigidity over the torque.

The torque is the torsional rigidity times the twist rate. The torsional rigidity is the bending rigidity times the stiffness ratio. So

R = 2B/(C·ω) = 2B/(r·B·ω) = 2/(r·ω)

and the bending rigidity has gone.

What is left is the stiffness ratio and the twist rate. The ratio is 2G/E — a property of the fibre, the same at both ends of the bracket, and known to whatever the fibre’s shear modulus is known to. The twist rate is a number a spinner sets and a twist tester reads.

Nothing about the yarn’s count, its diameter, its packing factor, its fibre density or how freely its fibres slide appears anywhere in it.

The number #

For a cotton, whose ratio is a quarter, at eight hundred turns a metre — five point zero three radians a millimetre —

R = 2 / (0.25 × 5.03) = 1.59 mm

so the coil is about three and a quarter millimetres across.

Take the same cotton at ten tex or at a hundred: 1.59 millimetres. Take it at the coherent bound rather than the free one: 1.59 millimetres. Change the packing factor from 0.5 to 0.7: 1.59 millimetres.

That is an unusually clean prediction for this subject, and it can be checked with a ruler.

Why the cancellation happens #

It is worth understanding rather than merely noticing, because the reason says which other quantities will behave the same way.

The coil radius is set by a competition between bending and the work the torque does. Bending resists at a rate set by B and by the curvature; the torque pays at a rate set by M and by the rate of writhing. Both sides of the competition are linear in a rigidity, and the two rigidities are proportional to one another with a factor that does not depend on the section’s size.

So the size of the section cancels and its shape does not. That is the same rule this collection arrived at when the ratio itself was derived: a quantity that depends only on the shape of the section survives the bracket, and a quantity that depends on how the fibres are behaving does not — which is why flattening the section moves it.

Here it is one level further along. The ratio survives because the section is circular; the radius survives because the ratio does, and because the radius is a ratio of two energies rather than a value of either.

What it does depend on #

Two things, and both are easy to vary.

The twist rate, inversely. Double the twist and the coil halves. A crepe yarn at fourteen hundred turns a metre coils at 0.91 millimetres; a soft weft at four hundred coils at 3.2. That range of twists is the range the twist curve is drawn over. That is a factor of three and a half across the range of twists anybody spins, and it is visible.

The fibre, through the ratio. A wool at 0.8 coils at 0.50 millimetres for the same twist — a snarl a millimetre across. A polyester at 0.167 coils at 2.4 millimetres, and an aramid at 0.044 at nine.

So the prediction is not a single number but a two-parameter family with no free constants in it, which is the most testable form a prediction can take.

The test #

Take a fresh reel of sewing thread of known count and known twist. Produce a snarl by the usual method — hold both ends and bring the hands together — and measure the coil across with a ruler or, better, photograph it against a scale.

The prediction for a twenty tex cotton at eight hundred turns is a coil three and a quarter millimetres across.

If the measured coil is around three millimetres, the whole chain holds: the torsional rigidity is the bending one times 2G/E, the shear modulus in the table is about right, and the writhing instability is the mechanism.

If it is around thirty millimetres, the shear modulus is ten times too large.

If it depends on the count, something in the derivation is wrong at a level that would be worth knowing about, because the count is supposed to have cancelled exactly.

That last is the most valuable outcome and it is the reason the test is worth describing: a prediction of independence is stronger than a prediction of a value. A value can be right by accident and a cancellation cannot.

Why tight snarls are worse #

The size has a practical consequence that follows immediately and is worth stating, because it explains a piece of trade lore.

A snarl is held closed by friction where the two strands press together. The pressing force comes from the bending of the coil, which goes as the bending rigidity over the square of the radius — so a tighter coil presses harder, by the square.

A crepe yarn’s snarl at 0.91 millimetres presses about three times harder than a soft weft’s at 1.59, and a wool’s at 0.5 millimetres presses ten times harder than a polyester’s at 2.4.

That is why hard-twisted yarns snarl in a way that will not comb out, and soft-twisted ones produce loose kinks that shake free. The threshold governs how often it happens; the radius governs how bad it is when it does.

The parameter that is absent and should not be #

There is a term missing from the radius and its absence is the honest weakness of the result.

A coil of radius R made from a yarn of diameter d has a curvature of about one over R at its tightest, and the yarn cannot be bent to a radius below its own. For a twenty tex cotton the diameter is 0.167 millimetres, so the coil at 1.59 millimetres is nineteen yarn radii — comfortable.

For an aramid at nine millimetres it is even more comfortable. For a wool at 0.5 millimetres and a coarse count it is not: a forty tex wool has a diameter of about 0.26 millimetres, and a coil of radius 0.5 millimetres is under four yarn radii, which is getting close to the tightest bend a thread can take.

At that point the linearised criterion stops being the right instrument, because the coil is no longer a shallow perturbation of a straight thread. Nothing here says where the crossover is, and it is a real limitation on the coarse, lively yarns where the effect matters most.

Why this is this collection’s cheapest experiment #

This work names several experiments and none of the others is as cheap as this one.

The knitted thickness prediction needs a swatch, a gauge, a wash and a tumble dryer, and it measures a structural thickness through a hair layer that moves in the opposite direction. The contact force prediction needs a compression tester. The knot efficiency prediction needs a tensile tester and a great many specimens, because knot strength scatters badly.

This one needs a reel of thread and a ruler.

More than that, it needs no control. Most of the collection’s proposed measurements are comparisons — this fabric against that one, before and after a treatment — because the absolute numbers depend on constants the model brackets. Here the absolute number is predicted with no bracket at all, so a single measurement of a single snarl is informative.

That is rare enough in this subject to be worth saying twice, and it is entirely a consequence of the cancellation. A prediction that survives a factor of three hundred of ignorance is a prediction that can be tested by somebody who owns nothing.

What was counted, and how #

The radius is the wavelength the linearised instability selects, which is 2B/M, and the substitution that removes B is one line of algebra.

The numbers are the collection’s own: a bending rigidity at both bounds, a torsional rigidity that is the bending one times the fibre’s 2G/E, and a twist rate in radians per millimetre. The check that the cancellation is real is that the two bounds return the same radius to every figure printed, which they do — the function computes the radius from B and M separately and the equality is an output rather than an assumption.

The counts, packing factors and fibres were varied across the collection’s whole table and the radius depends on none of them except through the ratio.

The same cancellation, looked for elsewhere in this work #

A cancellation this clean invites the question of where else it happens, and this work has an answer in each of its two halves.

On the torsion side, the balanced fold ratio cancels the bracket for the same reason: it is where a torsional moment and a bending moment are equal, both of them linear in a rigidity, and the answer comes out as C/(B+C) with no diameter, no count and no number of folds in it.

On the contact side, the flattening a knitted fabric demands cancels differently and more surprisingly. It is a purely geometric quantity — the closest two solved curves come, in diameters — so no rigidity enters at all, and it comes out the same for every count at a given tightness factor. That is a flattening that follows the tightness factor, and it is a stronger cancellation than this one because it removes the fibre as well.

Three results in one phase that survive the collection’s worst uncertainty. That is not luck: it is what happens when a ladder spends its time on ratios, thresholds and geometry rather than on values.

Where the model stops #

It is a linear stability result. The radius quoted is the size of the perturbation that first grows, not the size of the coil that finally settles. A fully formed snarl is a self-contacting plectoneme whose geometry is a contact problem: the two strands press on one another, friction resists sliding, and the equilibrium is somewhere the linear analysis does not reach. The number is the right scale and should not be trusted to a per cent.

It says nothing about how many turns the snarl takes up. That depends on how much slack there was and how much twist had to go somewhere, and it is what the trade’s own liveliness test measures, which is the threshold rather than the size.

And the fibres are treated as straight and parallel. They are at a helix angle, and the same caveat applies here as everywhere else on this ladder — with the mitigation that a helix correction affects both rigidities and the radius depends only on their ratio.

The generalisation #

The rule this rung is an instance of is one of the more useful ones in the collection and it deserves stating in general form.

A ratio of two quantities that share an unknown is knowable when neither quantity is.

Three results in this work have that shape. The stiffness ratio survives a bracket of three hundred because both rigidities carry the same bracket. The snarl radius survives it because it is a ratio of a bending resistance to a torsional drive. And the balanced fold ratio survives it because it is where two moments cancel.

The instruction that follows is practical: when a quantity in this collection is bracketed, look for the questions whose answers are ratios, because those are the questions that can be answered anyway. Several of the collection’s standing unknowns have that structure and have not been mined for it. The lateral rigidity has no floor at all, which is worse than a bracket — but the ratio of the flattening in two directions might well survive it, and nobody has asked.

What the trade already knows in this shape #

Trade knowledge about snarling is mostly about how to stop it, and the size of the coil is not something anybody specifies. But the size appears in practice in two places, and both are consistent.

The first is combing out. A snarl in a hard-twisted crepe yarn is described as impossible to pick apart and one in a soft weft as shaking out. The radius arithmetic says the first is half the size of the second and presses three times harder, which is the right ordering and about the right magnitude.

The second is kink formation on the machine. A yarn running slack over a guide produces kinks whose size is characteristic of the yarn and is used by machine operators as a diagnostic: a fine tight kink means the twist is too high for the tension being run. That is the same measurement made without a ruler.

Neither of those is data. What they are is a check that the prediction is not absurd, which is worth having before anybody spends an afternoon on the measurement itself.

Why the ratio is the weakest link #

If the test is run and the answer is wrong, the first thing to disbelieve is the shear modulus.

The radius is two over the ratio times the twist rate, and the twist rate is measured to a per cent by an ordinary twist tester. So all the uncertainty is in the ratio, and the ratio’s uncertainty is the shear modulus’s: cotton’s runs over a factor of two, from 0.7 to 1.5 gigapascals.

That puts the predicted radius for a twenty tex cotton at eight hundred turns somewhere between 1.06 and 2.27 millimetres — a coil between two and four and a half millimetres across.

A measurement landing in that band confirms the mechanism and not much else. A measurement landing at ten millimetres or at half a millimetre says something is wrong with the chain, and the ratio is the term to look at first because it is the only one with a factor of two on it.

That is a less satisfying test than it first appears, and saying so is better than pretending otherwise. What saves it is the independence claim: the band is wide, and the prediction that the radius does not move with the count is sharp, and a measurement at three counts settles that regardless of where the band sits.

Who found it, and when #

The instability and its wavelength are Greenhill’s, from 1883, and Love’s treatise gives the analysis in the form used here. The observation that plectonemes have a characteristic size independent of the rod’s stiffness is standard in the rod-mechanics literature and is used constantly in the study of supercoiled DNA, where the same cancellation makes the coil size a readout of the molecule’s stiffness ratio.

What is this collection’s own is applying it to a spun yarn with its own numbers, noticing that the cancellation makes the prediction bracket-free, and pointing out that it therefore constitutes a test anybody can run with a ruler on something they were going to do by accident anyway.

One more thing the size decides #

A last consequence, and it is the one that connects this rung to the fabric rather than to the reel.

A yarn that snarls on its way to the needle or the shuttle produces a fault in the cloth: a slub, a loop, a place where the yarn doubled on itself and was carried in doubled. The size of that fault is the size of the coil, so it is a millimetre or two for an ordinary yarn — which is exactly the scale at which a fault is visible in a cloth and not at which it is catastrophic.

That is why snarl faults are a quality problem rather than a strength problem. A three-millimetre doubled length in a warp end shows as a slub and does not weaken the cloth measurably. A three-centimetre one would be a different matter, and the arithmetic says it cannot happen at any twist anybody spins.

So the trade’s tolerance for snarling is calibrated to a length nobody has computed, and the length comes out of a cancellation between two rigidities neither of which is known.

Where the ladder goes next #

The snarl group closes here, and the ladder turns from what a single twisted thread does to what happens when two of them are twisted together on purpose.

The trade has a rule for that — fold at about two thirds of the singles twist — which this collection has carried as a bracket quoted from practice and derived nowhere. A torque balance is a ratio of two moments and is therefore exactly the kind of quantity this rung says should be computable. It is, and the answer is wrong by a factor of three, which turns out to be the most useful result on the whole ladder.