A thread between two crossings is an elastica

Every force in this collection so far has come from a tension. A warp end held at T arrives at a crossing at the weave angle and leaves at its negative, so the transverse components add and the thread beneath carries 2·T·sin θ. That is exact, it needs nothing fitted, and it is where the first force here came from.

It also has a restriction that has been said in the same breath every time it has been used. It is the contact force in a cloth that is under tension — on the loom, in a seam, in a fabric being pulled. A cloth lying on a table is under none, and its threads still press on one another hard enough to decide whether a seam slips, a cut edge frays, or a tuft comes away in the hand.

Two places where this collection stopped #

The admission has been made twice, in nearly the same words, on two different ladders.

The first is on the woven side. The essay that recovered the relaxed contact force got it by inverting a thickness measurement through a compression energy, and opened by saying why it had to: Peirce’s geometry joins arcs of constant curvature to straight lines, its curvature jumps at every join, and the forces are not recoverable from it. That route needed an elastica, and there was none.

The second is on the knitted side, three times over. A knit’s dimensions come from its loop ends by conceding that the woven half of this site solves equations and the knitted half quotes a table. Why stockinette curls states plainly that it computes no bending moment and predicts no radius. Why a knit recovers quotes a lower bound and says it is one.

Four ladders, one missing object.

What is actually missing #

Not a number and not a measurement. What is missing is a shape with a derivative.

A force is what an energy does when something moves. To ask how hard one thread presses on another, the honest question is: if that crossing were displaced by a hair’s breadth, how much would the bending energy of the thread change? Divide the change by the displacement and there is the force.

That question has an answer only if the shape is the one the thread would actually take. A path assembled by hand out of an arc and a straight line has a perfectly well-defined energy — integrate the square of the curvature along it and multiply by half the bending rigidity. What it does not have is the property that displacing its endpoint gives back a force, because the assembled path was never in equilibrium in the first place.

The jump that gives it away #

The tell is visible without any calculus. Along Peirce’s path the curvature is 2/D inside the wrap and exactly zero along the straight, and it changes between the two in no distance at all.

The bending moment a thread carries is its rigidity times its curvature. So Peirce’s thread carries a moment on one side of the join and none on the other, with nothing in between. A step in bending moment along a rod is a concentrated couple applied at that point, and there is nothing there to apply it: the join is in mid-air, between one crossing and the next, with no thread touching and no load acting.

The construction is a good description of where the thread goes. It is not a description of a thread in equilibrium, and only a thread in equilibrium has forces.

The elastica #

The shape a thread does take has been known since Euler, and it is the one that minimises

$$E \;=\; \tfrac{1}{2} B \int \kappa^2 \, \mathrm{d}s$$

over all inextensible curves of the right length running between the right two points. B is the thread’s bending rigidity, κ its curvature, s its arc length. There is nothing in it to choose. No arcs, no straight portions, no assumption about where the bending goes: the bending distributes itself, and where it goes is the answer rather than the input.

Such a curve is an elastica, and the whole of the machinery under this ladder is a way of finding one.

Why the unknown is an angle #

There is an obvious way to set the problem up and it is the wrong one. Take the curve’s coordinates as the unknowns, sample them, and minimise. Inextensibility — the statement that the thread has a fixed length and does not stretch — is then a constraint at every single sample point, and the whole difficulty of the problem is spent enforcing it.

Take instead the tangent angle φ as the unknown, as a function of the fraction of the way along. Integrating cos φ and sin φ recovers the curve, and a curve recovered that way has exactly the arc length it was integrated over, whatever φ does. Inextensibility is free.

Better still, both ends of a thread’s span between crossings are extreme points of its wave — the top of one bend and the bottom of the next — so the tangent lies along the fabric at both. Expand φ in sines of whole multiples of π and every term vanishes at both ends by construction. The boundary conditions cannot be violated by rounding, because there is nothing to round.

What that buys #

The energy becomes exactly quadratic in the coefficients, with no quadrature at all: it is a weighted sum of their squares. The only nonlinear part of the whole problem is the pair of statements that the curve arrives where it is supposed to, and those are two equations rather than a few hundred.

A Newton step on that system is a small matrix solve, and from a straight-line start it converges in about a dozen of them. One solved thread costs under a millisecond, which is what makes it possible to sweep a fabric’s whole state space rather than solve one configuration and argue about it.

What the rigidity is worth #

B is the one material number in the whole statement, and it is worth seeing how badly it is known before any force is quoted.

For a 20 tex cotton the two bounds are a hundred and thirty apart. That is not a defect in the measurement; it is a statement about what twist does, and this collection does not pretend to solve it. What follows from it is a rule about how every number on this ladder has to be quoted: a single force is given at the free bound and called one, and every claim that matters is a ratio of two forces computed the same way, in which the bracket cancels exactly.

The check with a closed form #

A solver that returns plausible numbers is a solver nobody should believe. The cleanest available test is a case where the answer is known exactly.

Ask for a thread of length πR joining two points 2R apart with its tangent reversed between them. Exactly one curve of that length does that: the semicircle of radius R, whose curvature is 1/R everywhere and whose energy is therefore πB/2R. Nothing in the machinery knows that. It returns the closed-form energy to two parts in ten thousand million, and a curvature within two per cent of one over the radius at every station.

The check the solve never used #

The stronger test is a different kind. A rod carrying no load between its ends transmits a constant internal force, and its bending moment then satisfies

$$B\kappa(s) + n_y\,x(s) - n_x\,y(s) = \text{constant}$$

along the whole length. That is a statement of equilibrium. The minimisation never saw it: all it ever knew was a quadratic form and two endpoint conditions.

So fitting that straight line to the solved curvature recovers the internal force from the shape alone, by a route with nothing in common with the one the solver used. The two agree to a tenth of a per cent, and the residual of the fit halves every time the basis doubles — which is the signature of a solution converging rather than of a relation that only nearly holds.

What must still be refused #

An assertion that has quietly stopped failing is worse than none, so the machinery is fed things it has to reject. A thread asked to reach further than its own length is refused, because an inextensible thread cannot. A thread of negative length is refused. Both throw rather than returning something confident and wrong, and the checks that they still throw run every time the site is built.

The first of those two is the one that earns its keep, and the reason is the shape of the solver rather than the plausibility of the input. Finding an elastica is finding the value of one parameter that makes the curve reach a stated end point, and the reaching is monotone in that parameter right up to the point where the thread is straight — after which nothing further can be reached and the solver, left to itself, will happily return the straight thread and call it a solution. The refusal is the difference between a model that says no such thread exists and one that answers a question it was not asked. Every span in every figure in this essay is checked against it before it is drawn.

The second is cheaper and is here for a different reason. A negative length cannot arise from anything the site computes; it can only arise from an argument passed in the wrong order, which is the mistake that produces the most convincing wrong pictures. A guard against an impossible input is a guard against a caller, and callers are what change.

What the model does not contain #

Three things, and they are named here once so that no result downstream has to be read as claiming more than it does.

Torsion. The curve is planar. A real thread twists as it goes round a crossing, and twisting carries energy. Nothing here counts it.

A contact with any width. The thread is held at its ends and free in between. A real thread wrapped round the one it crosses touches over an arc rather than at a point, and where the free run wants to bend harder than that arc, the point description has stopped being admissible. The measurement of whether it has is made at every use.

Any resistance but bending. A spun yarn’s bending is itself partly frictional — its fibres slide over one another and do not slide back — so its rigidity is a bracket rather than a number, a hundred and thirty wide for an ordinary cotton. Every force computed from it inherits that bracket, and the comparisons below are all ratios, in which it cancels.

What the picture cannot show #

The figure at the head of this essay draws a curvature, which is not a thing anybody can see. It is drawn as a spine standing off the curve because there is no other way to put a scalar field on a line, and the length of a spine is a number rather than a distance in the fabric.

What the drawing does show honestly is the absence of a jump, and that is the whole content. A figure of Peirce’s path drawn the same way would have spines of one length all through the wrap, no spines at all along the straight, and nothing between: the step would be at a single point, and a single point is exactly where a picture cannot say whether something is smooth.

What a construction could already do #

None of this displaces the geometry. Peirce’s path puts a thread where a thread is, to within the accuracy anybody has ever measured, and its bending energy is perfectly well defined — which is why the energy route has been usable here all along.

The distinction is worth keeping sharp, because it is easy to read this ladder as saying the old construction was wrong. It was not wrong. It was silent on one particular question, and silent in a way that looked like an answer: integrate the square of its curvature, get a number in joules, and there is nothing in the arithmetic to warn anybody that differentiating it would be meaningless.

The first thing it was pointed at #

The natural first test was an ordinary shirting, and the solver would not converge on it.

That looked like a defect for about an hour. It is not. A cloth’s threads sit a whole pick spacing apart with a crimp height a fifth of it, so the yarn between two crossings is longer than the straight line between them by a few parts in ten thousand. A curve with that little slack, which must nonetheless turn from lying along the cloth, down across it, and back, has to do all of its turning within a hair’s breadth of each end — and the thread it is turning round is in the way.

A woven thread has no free run to solve. The rung after next is about what that means, and it turns out to explain why the woven side of this collection has spent its whole existence solving geometric constructions rather than mechanical ones.

The second thing, which worked #

A knitted loop carries about twenty-one yarn diameters of thread in a cell five diameters wide and four tall. The straight line between two interlacings is half the yarn available. There is nothing tight about it at all, and the same solver converges in a dozen steps.

That is the division this whole ladder rests on, and it was not anticipated. A woven cloth and a knitted one are not two arrangements of the same mechanical object. One of them has a thread with room to find a shape and the other does not, and every difference in how the two fabrics behave has that underneath it.

The shallow limit has a closed form, and it says what the span is worth #

There is one case where the solve can be done on paper, and it is worth doing because it shows which quantity the answer depends on.

Take a thread whose departure from the straight line is small compared with the span — a shallow wave of amplitude a over a span L. Its curvature is then the second derivative of its offset, its energy is the integral of the square of that, and its excess length over the chord is half the integral of the square of its slope. Both integrals are elementary for a single half-wave, and dividing one by the other eliminates the amplitude entirely:

E = 4π²·B·Δ ÷ L², where Δ is the excess length the span is carrying.

Two things follow, and neither is obvious from the variational statement.

The energy is linear in the slack, not quadratic. So the tension the thread pulls back with is the derivative of that, which is 4π²B/L² — a constant, independent of how much slack has been fed in. A shallow thread resists the first hair of crimp exactly as hard as the last.

And the span enters squared. Halving the distance between crossings — which is what doubling the sett does — makes the thread four times as reluctant to take up crimp. That is a sett dependence arriving from bending alone, with no contact force and no friction in it anywhere.

The expression is also honest about where it stops. It assumed a shallow slope, and a knitted loop’s excess is comparable to its own span rather than small against it. So the paper case covers the woven thread and fails on the knitted one — the same division the rest of this ladder rests on, reached here by arithmetic instead of by watching a solver refuse.

What comes of having one #

Once a thread’s shape between two crossings is an equilibrium rather than a construction, the derivative exists, and the derivative is a force. The next rung is about where it comes from and why it arrives without being differenced.

After that the forces are available for use: what a loop presses with, what friction has to hold, what a knit gives when it is pulled, why a fabric everybody can measure sits nowhere near where its own bending would put it.

Where the idea comes from #

The variational statement is Euler’s, from 1744, in the appendix on curves of maximum and minimum properties — the same paper that gave the calculus of variations its name and the buckling load its formula. The application to yarns is much later and belongs mostly to the knitted side: Peirce built a geometric loop in 1947 and the minimum-energy treatments came in the fifties and sixties.

What is worth noticing is which way round the history runs. The woven side got its geometry first and has kept it for ninety years, because the geometry is nearly forced. The knitted side got a geometry too and it never fitted as well, and the reason is the slack.

Where the ladder goes next #

Three directions, and this collection takes all of them. The immediate one is the force, which the next rung supplies. The second is the comparison between a woven thread and a knitted one, which turns into a statement about which fabric grips its own yarn and by how much. The third is the awkward one: run the model honestly on a relaxed knit and it says the fabric should be somewhere else entirely, and the disagreement turns out to be the most useful thing on the ladder.