How little asymmetry a curl needs

Knit a rectangle of plain jersey and it will not lie flat. That is the first thing anybody notices about the fabric and the last thing this collection has been able to compute.

Why stockinette curls explains it structurally: the loop is not symmetric front to back, work every course the same way and the asymmetries add along the edges, alternate them and they cancel. It counts face changes across a repeat, gets zero for stockinette and one per course for garter, and says plainly that the count computes no bending moment and predicts no radius.

A moment is a force at an offset. An offset needs a thickness to be measured across. So the reason that rung stopped where it did was not caution; it was that the model had no third dimension in it. It has one now, and this is what happens when the question is finally asked.

How the question is asked #

A fabric curls because the energy it holds is not symmetric in the curvature imposed on it. Write the energy per unit area as a constant plus a moment times the curvature plus half a rigidity times the curvature squared, and the free fabric settles where the derivative vanishes: at a curvature equal to the moment over the rigidity. So the curl radius is the rigidity divided by the moment, and nothing else.

The rigidity is already known. It is the yarn’s bending stiffness times the length of yarn per unit area times a factor accounting for which way each piece of thread is pointing, and it comes out at 1.57 micronewton-metres per unit width about the course direction and 3.40 about the wale direction — an anisotropy of 2.17 that is a structural constant of the loop rather than a property of any yarn.

The moment is what had to be computed. And the way to compute it is not to reason about which parts of the loop are on which face, which is where an argument of this kind usually goes wrong. It is to bend the fabric and re-solve.

Both numbers in that ratio deserve a word about their status. They are the free end of a bracket a hundred and thirty wide, because a spun yarn’s bending stiffness depends on whether its fibres slide. But the ratio of a moment to a rigidity has the stiffness in both places, so the curl radius carries no bracket at all — which is the first sign that this is a question worth asking of a model that cannot pin a force down.

Bending the fabric, literally #

Curling the fabric to a cylinder moves the only things the loop is anchored to — its interlacings — and turns the surface directions the thread has to leave and arrive along. Both are exact statements about where the yarn must run.

So: take the two interlacings that bound one half period, map them onto a cylinder of a given radius, work out the new separation and the new end tangents in the segment’s own frame, and solve the thread again. Do it at a small positive curvature and a small negative one, and the difference is the moment.

There is nothing approximate in that procedure and nothing fitted. It is the same solve the rest of the ladder uses, asked about a fabric that is not flat.

One thing about it is worth flagging before the answer, because it is the part that is easy to get wrong and hard to notice. The separation of two points on a bent surface depends on where the frame is put: measure it from one end and a rigid rotation of the whole picture creeps into the answer. Taking the frame at the segment’s own mid-point removes it, and the removal is not a convenience — an answer that changes when the origin moves is not an answer.

The answer #

Zero, to the precision of the arithmetic.

Not small: zero. The energy at a small positive fabric curvature and the energy at the same negative one come back identical, so their difference is nothing at all.

A number that comes out at machine zero rather than at something tiny is a statement about a symmetry rather than about a magnitude. And the symmetry is not hard to find once it is looked for.

The symmetry that does it #

The crest of a course is a needle loop’s head, half a yarn diameter behind the fabric’s mid-surface. The trough is the feet, half a diameter in front. The mid-surface bisects the half period exactly.

Bending the fabric stretches the material on the outside of the curve and compresses the material on the inside, in proportion to how far it is from the neutral surface. If the loop’s two anchors sit symmetrically about that surface, one is stretched by exactly what the other is compressed by, and the segment between them spans the same distance it did before. To first order, nothing has happened.

That is why the moment is zero, and it is why no amount of care with the solve would have produced anything else. The mid-surface of a fabric of these loops is, by construction, the plane that bisects every one of them.

Which makes the model wrong, usefully #

A jersey curls. The model says it does not. That is not a small discrepancy to be absorbed into a bracket; it is a qualitative failure, and the right response is to find out exactly what is missing rather than to add something until the answer comes out.

What is missing is the difference between a head and a foot.

In the model the crest and the trough are mirror images — two extreme points of one wave, related by a half-period translation and a reflection. In a real fabric they are not remotely alike. The head is an arc with two legs threaded through it, holding them; the sinker loop is an arc with nothing through it at all, running free between two neighbours. They wrap different things, they carry different contacts, and there is no reason whatever for the fabric’s neutral surface to fall exactly between them.

The wave model cannot represent that difference, because a wave has one crest and one trough and they are the same shape upside down.

So the question is inverted #

If the moment is proportional to the eccentricity — how far the neutral surface fails to bisect the loop — and the model’s eccentricity is zero, then the useful thing to compute is not the moment. It is the exchange rate: how much moment an eccentricity buys, and therefore how much eccentricity a measured radius implies.

That is computable with the same machinery, by displacing the loop’s anchors off the neutral surface by a stated amount and re-solving. The moment comes out strictly proportional to the displacement — checked at two, five and ten per cent of a yarn diameter, agreeing to within three per cent of a straight line through the origin — so one solve settles the whole curve.

The number #

A fine jersey rolls to a radius of two or three millimetres. Take three, and:

an eccentricity of 9.0 micrometres accounts for it — 5.4 per cent of a yarn diameter.

Nine micrometres, on a yarn 167 micrometres across. A twentieth of the thickness of the thread.

That is the whole of stockinette curl, expressed as the quantity a better loop model would have to get right. It is also, immediately, an explanation of why nobody has computed a curl radius before: the answer lives inside a rounding error on the geometry, and any model that idealises the loop at all is likely to idealise it away.

Only one of the two edges can be asked #

The rung this one follows says that a jersey’s four edges do not behave alike: the top and bottom roll toward the back and the two sides roll toward the front. Two imbalances, along perpendicular axes, with opposite senses.

Everything above is about the first of those. Bending the fabric about the course direction — the bending whose edges are the top and the bottom — moves the interlacings and nothing else, because the thread’s own heading along the courses is unchanged by that bending. The solve is the same solve with different endpoints.

Bending it about the wale direction is a different problem, and this model cannot do it.

Why the other axis is refused #

Bending a fabric about its wale direction turns the fabric’s own along-the-course direction with it. A thread spanning half a wale therefore has to leave one interlacing and arrive at the next with its tangent tilted out of the fabric, by half a wale times the curvature at each end.

That is an exact statement about where the yarn runs, and it is fatal to the solve for a reason that is physics rather than arithmetic.

A thread bent hard in a plane is only conditionally stable in that plane. Turn its end tangents out and the flat configuration stops being a minimum: the segment can lower its energy a great deal by swinging out of the fabric altogether. At half a hundredth of a radian of end tilt — a third of a degree — the solved thread’s out-of-plane excursion is 3.2 times the fabric’s own thickness, and its energy is 37 per cent lower.

That configuration is a correct solution of the free thread’s problem. It is not available to a thread in a fabric, because the neighbouring courses are in the way, and this model has nothing in it that says so.

Which is worse than not being able to compute it #

There is a specific trap here and it is the reason the refusal is written into the machinery rather than into a paragraph.

The buckled branch is found for both signs of curvature, and by symmetry the two are the same configuration mirrored. Their energies are identical. So a solver asked for the wale-direction moment does not fail, does not warn and does not produce nonsense: it produces zero — which is exactly the answer the correct branch gives.

A wrong calculation that agrees with a right one is the worst kind, because nothing distinguishes them downstream. What distinguishes them here is the shape: the correct segment stays within a yarn diameter of the fabric and the buckled one is three times outside it. So every bent solve in this account is checked against the flat one’s out-of-plane excursion, and one that has left the fabric is refused rather than differenced.

That check is what stopped this rung reporting a second moment. It found the branch jump in the wale-direction calculation and in nothing else.

What cancels out, and what it explains #

The rung this follows makes a scaling argument without computing anything: the yarn’s stiffness appears in the moment and in the rigidity, so it cancels, and the curl radius is a length set by the loop’s geometry rather than by the material.

The computation confirms it exactly. The moment is a contact force times an offset, and every contact force here is linear in the yarn’s bending stiffness. The rigidity is the same stiffness times a length of yarn per unit area. Divide and the stiffness is gone.

So a stiff yarn does not curl less. It resists more and pushes harder in the same proportion. That was an inference and it is now an identity, and it explains the observation that started it: blocking a jersey in a stiff yarn is no easier than blocking one in a soft yarn, and knitters have always known it.

What sets the radius, then #

Two things, and neither is the fibre.

The gauge, through the loop length. Every length in the problem scales with the loop, so if the eccentricity scales with the yarn diameter — which it must, since it is a feature of how two threads sit at a crossing — the radius scales with the diameter too. A coarse knit curls into a wider tube than a fine one, in proportion.

And the structure, which is the other rung’s subject. A rib and a garter do not curl less than a jersey; they do not curl at all, because their sums are zero rather than small. Alternate the facing of successive courses and the eccentricities alternate with them; sum them over a repeat and they cancel; the fabric lies flat. That is garter stitch, and which knitted fabrics lie flat is where the same sum is done for every structure a two-bed machine can make.

Why the earlier count was right for the right reason #

It is worth being explicit that nothing here overturns the counting argument. It underwrites it.

The count said: a fabric curls when the signed sum of its stitches’ facings is not zero, and lies flat when it is. The computation says that each stitch’s contribution to the fabric’s moment is a single constant times a signed eccentricity — the same constant for every loop in the fabric, since every loop is the same loop.

A constant times a signed sum is zero exactly when the signed sum is. So the count and the moment agree about which fabrics lie flat, and the count was answering the question it could answer with the tool it had.

What it could not do is put a radius on the ones that do not, and the reason is now stateable: the radius needs the constant, the constant needs a thickness, and the count had none.

The residual shortfall, stated precisely #

The list of things this collection’s loop model cannot say had the curl radius on it, and it is still on it. What has changed is what is owed.

Before: the model computes no bending moment and predicts no radius.

Now: the model’s course-direction moment is exactly zero by a symmetry it should not have, and a radius of three millimetres about that axis needs an eccentricity of 5.4 per cent of a yarn diameter. The wale-direction moment is not computable at all, because a free thread with turned end tangents buckles out of the fabric.

That is a much smaller debt in one direction and a sharper one in the other. What a model would have to do to close the first is distinguish a threaded head from a free sinker loop. What it would have to do to close the second is stop its threads passing through one another — which is the same absence that puts this collection’s extension ceiling three times beyond any jersey.

What would measure it #

Two experiments would settle the eccentricity without a better model, and both are within reach of anybody with a swatch.

Measure the radius of a free edge, at two gauges of the same yarn. The model says the radius scales with the loop length; a departure from proportionality would say the eccentricity is not simply a diameter’s worth of geometry.

And measure the radius at two gauges of the same yarn again after wet relaxation. Setting moves every force on this ladder by the same factor and moves the fabric’s rigidity by the same factor too, so the curl radius should be unchanged by setting. That is a strong prediction and an easy one to falsify: if a blocked and re-washed swatch curls to a different radius than an unblocked one of the same dimensions, the moment and the rigidity are not scaling together and something outside this account is carrying part of one of them.

Neither has been run here. The prediction is offered as a prediction, and it belongs with the two experiments the set fraction rung asked for and did not get: a curl radius is one of very few things about a relaxed knit that is easy to measure with a ruler and hard to get out of a model, which is exactly the wrong way round from most of this collection.

Which makes curl the best measurement on the ladder #

That deserves its own paragraph, because it inverts the usual situation here.

Almost every quantity in the knitted half of this collection is easy to compute and hard to measure: a contact force of 38 millinewtons a stitch has never been measured on any fabric, and the load–extension curve that could be compared with a measurement has not been. The curl radius is the reverse. It is a length, it is a few millimetres, it is visible from across a room, and it needs no apparatus at all.

So it is the one number on this ladder where an experiment could embarrass the model quickly and cheaply — and the model, as it stands, predicts infinity. That is not a comfortable position and it is a clear one.

What is genuinely new here #

Three things.

A theorem where there was a gap. A knitted fabric whose neutral surface bisects its loops carries no curling moment at all, in either direction, however hard its yarn is bent. That is exact and it is why no simple model of a loop will ever curl.

An exchange rate. Moment per unit of eccentricity, in both directions, so that a measured radius converts to a geometric asymmetry and back.

And a number small enough to explain the difficulty. Five per cent of a yarn diameter. The whole of the most obvious mechanical behaviour of the most common knitted fabric in the world lives in nine micrometres.

A fourth thing is a refusal rather than a result and belongs on the list anyway: the side edges’ moment is not computable here, because a free thread with turned end tangents leaves the fabric and a threaded one cannot.

What the pictures cannot show #

The curve of radius against eccentricity rises steeply toward the left and passes through no point at zero, because a radius of infinity does not fit on an axis. The model’s own fabric is at that missing point.

A reader who takes the curve as the model’s prediction has it backwards. The curve is the model’s sensitivity; the model’s prediction is the empty left-hand edge. Every point drawn is a fabric with something in it that the model does not have.

Where the ladder goes next #

The eccentricity that curl needs is a feature of a single loop. Everything else on this ladder from here is about what happens when the loops are put on two beds, where the asymmetry stops being a rounding error and becomes the bed gap.

A rib climbs a gap is where that begins, and which knitted fabrics lie flat is where this rung’s signed sum is finally done over every structure a machine can knit rather than over the two that everybody already knows.