Where a torsion model stops

A ladder that ends should say what it bought. This one was built because an earlier one named the absence of a torsional rigidity as the first item to disbelieve in this collection’s account of a knitted fabric, and said in as many words that a three-dimensional solve with torsion in it was the obvious next thing to build.

The rigidity now exists. What it settled and what it did not are unevenly distributed, and the honest summary is that it settled several things it was not built for and left the thing it was built for untouched.

What it settled #

The ratio is knowable when neither stiffness is. The bracket that afflicts a yarn’s bending rigidity — the fibre count over the square of the packing factor, three hundred and twenty-seven fold for an ordinary cotton — afflicts its torsional rigidity by exactly the same factor, so C/B is 2G/E at both ends of it and everywhere between. That is the ladder’s foundation and it is an identity rather than an estimate.

A fibre’s shear modulus can be tabled with a control in it. Glass is an isotropic solid, so its 2G/E is one over one plus Poisson’s ratio and is known before anybody measures anything. Every other fibre sits below it, and the ordering runs with molecular orientation — which turns a column of numbers into a physical claim.

A slack yarn’s snarl is a threshold, and the threshold reads the bracket. The tension a twisted thread needs to stay straight is 1.9 metres of its own weight at the free bound and 616 at the coherent one. Anybody who has handled thread knows which, so an everyday observation closes a factor of three hundred that no laboratory measurement had.

And a snarl’s size is bracket-free. The coil radius is two over the stiffness ratio times the twist rate, with the bending rigidity cancelling completely — 1.6 millimetres for a cotton at ordinary twist, at any count and either bound.

What it refuted #

One trade explanation, and the refutation came with its replacement attached.

The rule that a fold is twisted at about two thirds of its singles is explained everywhere as a torque balance. The balance is now computable and gives a fifth, not two thirds; and reaching two thirds would need a torsional rigidity twice the bending one, which needs a shear modulus equal to a tensile one, which no solid has.

So the rule is not a torque balance. It is a surface-angle condition — one over the square root of the number of folds — which reproduces all three of the trade’s brackets exactly, depends on the fold count as the practice does, and depends on the fibre as the practice does not.

That is the ladder’s best result, and it is worth noticing that it came from the direction nobody expected: a rigidity added to settle a question about knitting settled a question about spinning instead.

What it did not touch #

Spirality. The ladder was built to derive the lean of a hard-twisted jersey, which this collection had priced at nine degrees per unit of twist factor with the nine fitted.

It cannot. The mechanism requires the fabric to relieve the yarn of twist by writhing, and this collection’s own solved course has a writhe of nought — exactly nought, by a mirror symmetry, at every size and tightness and state. There is nothing for the twist to be converted into.

The nine remains fitted. The ladder’s contribution to it is to say precisely why it cannot be derived here, which is worth something and is not what was wanted.

The share of a loop’s energy that torsion carries. The ratio is known, and how much twist a knitted loop’s yarn actually carries is not — that depends on how the yarn was fed and how much the fabric has been allowed to rotate at its interlacings, and neither is a question about a rod.

The wale-direction curl. Untouched, and for a reason that turns out to belong to the omission the topology found rather than to any material constant.

And torsion inside a woven cloth. A thread at a crossing is twisted as well as bent, and nothing here computes it. The ladder went to the yarn and to the fabric’s topology and never to the crossing.

Why the target moved #

The pattern is worth recording because it is not an accident of this ladder.

The rigidity was built to answer a question about a knitted fabric. The obstacle turned out not to be the rigidity at all: it was that the fabric’s own model has no chirality, because its interlacing is a point where two centre lines pass a diameter apart, and a near miss has no handedness.

So the missing piece was topological rather than material, and adding a material constant could not supply it. That is the single most useful thing the ladder found, and it was found by computing a quantity — the writhe — that nobody would have computed without a torsional rigidity to motivate it.

A model’s missing piece is often not in the same category as the symptom. The symptom was a fitted constant in a mechanical account, and the cause is an omitted arrangement.

Which results a reader can use, and how far #

A ladder’s output is only usable if somebody can tell which parts of it carry a bracket, so it is worth sorting the results by that rather than by subject.

No bracket at all. The stiffness ratio, the balanced fold ratio, the surface-angle folding rule, the snarl radius, and every linking number. Each of these is either a ratio of two things that share the unknown, or a count over an arrangement. They can be quoted flatly.

A bracket that the observation closes. The snarling threshold. Two calculations three hundred-fold apart, and one of them contradicted by common experience.

A bracket carried in full. Every absolute torque, every energy, and any force computed from them. These are quoted at the free bound and said to be so, exactly as the rest of this collection’s forces are.

Three categories, and the first is much larger than anybody expected at the start of the ladder — which is the practical reason the ladder was worth building even though its target survived it.

What the ratio is worth elsewhere #

The ladder’s one genuinely portable result is the bracket-cancellation, and it is worth saying where else to look for it.

The rule is: a quantity that depends only on the shape of the section survives the bracket; a quantity that depends on how the fibres are behaving does not. The polar second moment of a circle is exactly twice its flexural one, and no arrangement of fibres inside the circle changes that.

Three results in this work used it. The stiffness ratio itself. The snarl radius, which is a ratio of a bending resistance to a torsional drive. And the balanced fold ratio, which is where two moments cancel.

Two more candidates are sitting unexamined in this collection. A cloth’s crimp interchange is a ratio of two systems’ responses to the same imposed strain, both of them linear in the same bracketed stiffness. A knit’s two bending directions likewise. Both are more robust than the numbers they are made of and neither has been described that way.

What the ratio is not worth #

One correction to the ladder’s own headline, and it comes from the other half of this work.

C/B is 2G/E for a circular section, and a yarn in a fabric is not circular. A flattened section has two different flexural moments — easy about the long axis, hard about the short one — and the polar moment is still their sum, so a flattened thread has three constants rather than two.

At the flattening a knitted fabric’s own geometry demands, the multiplier on C/B in the easy bending direction is about 1.32. A knitted loop bends in the easy direction.

So the headline number is a lower bound for a yarn in cloth, and the ladder’s cleanest result carries a correction from a ladder it did not know about when it was written. That is the section that changes both stiffnesses.

The ledger, as a list #

Set out plainly, because a reckoning that is only prose is a reckoning nobody can check against later.

Settled: the stiffness ratio and its independence from the bracket; the shear table and its control; the snarling threshold and what it says about which bound a yarn sits at; the snarl radius and its independence from both stiffnesses; the folding rule’s true condition and its extension to cables.

Refuted: the torque-balance explanation of the folding rule, with a replacement.

Found while looking for something else: that this collection’s knitted fabric has no writhe and no linking, that the two facts have one cause, and that four earlier unexplained findings share it.

Not touched: spirality, the torsional share of a loop’s energy, the wale-direction curl, torsion at a woven crossing, and the behaviour of a set yarn.

Five settled, one refuted, one found, five untouched. That is a fair ladder and it is not the ladder that was planned, which is the ordinary condition of a piece of work worth doing.

Two things a reader should not take from this #

Two over-readings are available and both are worth heading off.

The ratio is not a licence to compute torques. Knowing C/B does not give C. Every absolute torque on this ladder is quoted at the free bound and carries the same three-hundred-fold bracket every other force in this collection carries. What escapes are ratios, thresholds and lengths — and the essays say which is which each time.

And the topological results are not about torsion. They were found on this ladder because a torsional rigidity gave somebody a reason to compute a writhe, and they belong to the fabric’s geometry rather than to its material. A collection that had never acquired a second stiffness could have computed the same linking numbers on the same day, and did not, for four ladders.

That second point is the more useful. A new constant is often worth less than the questions it makes somebody ask, and the questions here were free.

What was counted, and how #

Every number on this rung is recomputed from the ladder’s own functions rather than quoted from its essays, so a change anywhere would show here.

The stiffness ratio, the identical brackets and the bracket-free ratio are checked across four counts, five fibres and three packing factors, at both ends of each bracket, to a part in ten thousand million million.

The snarl threshold is checked at the degenerate case — no twist, no tension needed — and as an exponent across four twist levels, because a relation that holds at one pair of points holds for any monotone function.

The fold balance is checked against its own closed form at low twist and required to depart from it monotonically as the helix steepens, because a limit that is approached is a different object from a formula that nearly works.

The linking integral is checked against four arrangements whose answers are known by inspection, and the writhe against its own reflection.

Where the model stops #

The fibres are straight and parallel to the axis in both bounds. They are not, and the same idealisation runs through the bending bracket this collection has used from its earliest work. It biases both bounds in the same direction, which is part of why the ratio is more robust than either.

The shear moduli are working values with wide ranges. Cotton’s runs over a factor of two, so every result that is linear in the ratio carries a factor of two, and only the results that are thresholds or comparisons escape it.

A spun yarn’s torque relaxes and a set yarn has none. Every result here treats the yarn as elastic in torsion for ever, and steaming is the trade’s whole answer to torque. That is the largest single omission and it makes the snarl results statements about fresh yarn.

And there is no torsion in any fabric solve. The loop is still minimised over a bending energy alone. Nothing in this collection’s knitted arithmetic has changed.

What it cost #

A ladder should say what it consumed as well as what it produced, and this one was cheap by the standards of the collection.

It added one material table — a shear modulus and a range for each of the ten fibres — and one control row inside it.

It added no new solver. Every result above comes from arithmetic on quantities the collection already computed, plus two integrals that are definitions rather than approximations.

And it added two integrals: the Gauss writhe and the Gauss linking number, each a double sum over a few hundred segments, each checked against arrangements whose answers are known by inspection.

Against that, the ladder that preceded it — the one that let the loop out of the fabric’s plane — needed a new three-dimensional solver, a new basis, a new convergence argument and a branch check. This one needed a table and two sums.

The reason is worth carrying: the expensive part of modelling is usually the configuration space, not the physics. Adding a second material constant to an existing solve is nearly free. Adding a second coordinate to what the solve is over is not, and adding a topological constraint to it is harder still — which is exactly why the four items in the untouched column are untouched.

The generalisation #

The most transferable thing the ladder produced is a method rather than a number, and it has two halves.

Convert every threshold into the unit somebody would notice it in. A critical tension of 0.37 millinewtons is a number nobody can disagree with. The same number as 1.9 metres of the yarn’s own weight is a claim about something everybody has seen, and it settled a bracket by being disagreeable.

And ask of every null result whether it is zero for a reason the subject shares. The writhe came back at six parts in a million, which reads as a converged calculation of something negligible and is a calculation of something forbidden. Two of this collection’s knitted results are exactly nought and only one of them is nought for a reason a real fabric has.

Neither of those needed a torsional rigidity. Both were learnt while building one.

What would close the untouched column #

Each of the five untouched items has a named next step, and listing them is more useful than restating that they are open.

Spirality needs the interlacing threaded rather than crossed, so that the fabric has a chirality to trade against. That is a contact problem and it is the same repair four other findings need.

The torsional share of a loop’s energy needs a solve whose configuration space includes the material’s rotation as well as the centre line’s shape — a genuine Kirchhoff rod solve rather than an elastica. The machinery is a step up from what this collection has and the ratio it would multiply is now known.

The wale-direction curl needs the same threading, for the same reason: the solve runs away to a configuration the neighbours would block.

Torsion at a woven crossing needs nothing new at all. A thread wraps its neighbour through a known angle at a known radius, so the twist it acquires is computable from geometry this collection already has, and nobody has done it. That is the cheapest of the five by a long way.

And a set yarn’s torque needs a model of setting, which this collection has wanted for several ladders and has approached only as a fraction of a natural curvature that cancels out of every balance the dimensions can be put into.

One of those five is an afternoon. Three are the same piece of work. One is a research programme.

Who found it, and when #

The rod mechanics is Kirchhoff’s, from 1859, with the stability criterion Greenhill’s from 1883 and the twist–writhe identity Călugăreanu’s from around 1960. None of the physics on this ladder is new.

What is this collection’s own is the bracket pairing, the observation that the two brackets are identical and the ratio therefore survives, the reading of an everyday observation as a measurement of which end of the bracket a yarn sits at, and the refutation of the folding rule’s usual explanation together with its replacement.

Where the ladder goes next #

This collection’s other half starts from a question this collection should have asked years ago and did not: whether the fabric it has been computing with actually fits together.

It does not. Two adjacent courses of the relaxed solved fabric approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere, before anything is loaded. The fabric that does not fit is where the contact ladder begins, and what the fabric does about it turns out to predict a parameter this site has swept since its second phase.