A loop is a plane curve in another plane

A knitted loop has to pass through another knitted loop. That is not a detail of the fabric; it is the definition of it, and it is the one thing a plane curve cannot do. Two curves in a plane meet or they do not, and no amount of care with a drawing makes one of them go behind the other.

So the account of the loop that this collection solves has a hole in it, and the hole was declared rather than hidden: the interlacing is a point where two centre lines pass one diameter apart across the fabric, and the excursion that arranges is treated as free. What a loop model still cannot say lists it first among five omissions and prices it at a few per cent by a curvature ratio.

This rung stops declaring it and solves the thread in three dimensions instead. The answer is stranger and much simpler than the correction anybody would have budgeted for.

What the third coordinate actually is #

Begin with where the yarn has to be, which is a question about the fabric rather than about mechanics.

A course of yarn runs as a wave along the fabric: one crest and one trough to a wale. The crest is a needle loop’s head. The trough is the feet of that loop, and the feet are what were drawn through the head of the course below — so they are on the far side of it. Head and feet are therefore on opposite faces of the fabric, one yarn diameter apart through the thickness, and a half period of the wave climbs that diameter while it descends a course spacing and a diameter.

That is the whole of the geometry. It has no free constant in it: the climb is the yarn’s own diameter because an interlacing is two centre lines passing at a diameter, which is what a crossing of two threads of the same size is.

For a 20 tex cotton at a 3.5 mm loop, relaxed, the numbers are a diameter of 0.167 mm and a drop of 0.803 mm — the course spacing of 0.636 plus the diameter. So the half period travels 0.803 mm across the courses and 0.167 mm through the fabric.

Why the departure is a tilt rather than a wobble #

The obvious picture of an out-of-plane excursion is a ripple: the thread leaves the fabric plane, goes round something, and comes back. That picture is wrong here, and it is wrong in a way that decides everything downstream.

The climb is monotone across the half period. The thread starts at the back, ends at the front and never returns, because the two ends of the half period are the two ends of the interlacing. So the thread’s out-of-plane position is not oscillating about the fabric’s plane; it is walking steadily across it, in step with its progress across the courses.

A curve whose third coordinate is proportional to its second is a curve in a tilted plane. That is what the picture above is showing, and the tilt is

twelve degrees, for an ordinary jersey.

More exactly, 11.75° — the arctangent of 0.167 over 0.803.

The identity that makes it exact #

The tilt is not an approximation to the solved shape. It is the solved shape, and the reason is a symmetry rather than a smallness.

A half period leaves its crest along the course direction and arrives at its trough along the course direction, because both are extreme points of the wave. So both end tangents lie on one axis. Rotating the entire problem about that axis maps the constraint set to itself and leaves the bending energy alone, since a rotation changes no curvature. The energy can therefore depend on the drop and the climb only through the square root of their squares — and the solved curve is the flat solution rotated.

That is an exact statement about the continuous problem. Nothing in it is a series, a small angle or a leading order.

How the identity was checked, which is the interesting part #

An identity that a solver was built to satisfy is not evidence of anything. This one is checkable precisely because the solver was not built to satisfy it.

The unknown in the three-dimensional solve is a pair of angles — the heading in the fabric’s plane and the elevation out of it — each expanded in a series that vanishes at both ends. Rotating a flat solution about the course direction gives an elevation whose expression is an arcsine of a product of sines, and that is not a sine series in the arc length. No finite basis can represent the rotated answer exactly.

So the discrete solve cannot match the flat one by construction. What it can do is converge on it, and it does: six terms leave the two apart by nine parts in a hundred thousand and thirty-two terms leave them apart by five parts in a hundred million. Four decades of agreement across a basis that cannot express the thing being agreed on is what an exact identity looks like from the inside of a numerical method, and it looks nothing like a relation that merely nearly holds.

What the tilt depends on #

The angle is the diameter over the drop, so it is a statement about how tightly a fabric is knitted and about nothing else. Loosen the loop and the course spacing grows while the diameter does not, and the tilt falls:

Across the whole range a knitter can reach, the tilt runs between nine degrees and fifteen. It is never zero and it is never large, and both halves of that matter: a model that ignored it is not badly wrong, and a model that treats the fabric as flat has nowhere to put a thickness.

Coarsen the yarn instead and the angle rises, because the diameter is the numerator: a 40 tex cotton at the same tightness sits at 15.2°, a 10 tex at 8.9°. The tightness factor is the group that collapses all of it, exactly as it collapses everything else in the loop’s geometry.

What it costs, which is nothing #

If the third dimension were an extra bend added to a curve that was otherwise unchanged, it would cost energy. It does not, and the sign is the point.

Letting the thread climb lengthens the straight line between its two interlacings. A thread of fixed length spanning a longer chord has less slack, and less slack means less curvature to store. The solved loop’s bending energy falls from 25,074 to 24,395 nanojoules a stitch — two and seven tenths per cent, downwards.

An earlier estimate on this ladder put the ride at about three per cent of the loop’s bending and called it a cost. The magnitude was right within a factor and the sign was wrong, and the reason is worth naming because it is a general trap: the estimate asked what curvature the excursion would have on its own, and added it. The excursion is not on its own. It replaces part of the drop.

The one thing that does change #

Every quantity the flat model produced moves by under four per cent, which is the honest summary and a dull one. One quantity is not on that list at all, because the flat model has nowhere to put it.

The contact force is the derivative of the loop’s energy with respect to where its interlacings are. Where they are is now a question with three answers, so the force has three components — and the third of them acts through the fabric’s thickness. It comes out at 7.81 millinewtons a stitch, a fifth of the whole contact force, and it is what holds a knitted fabric’s two faces apart. The force that holds a knit open is what that number turns out to be for.

Why the force turns by the same angle #

The contact force and the loop’s own plane are turned by the same rotation, and they have to be: a rotation of the problem rotates its answer, and the force is part of the answer.

That is a check with real teeth, because the two are computed by routes with nothing in common. The plane’s tilt is arithmetic on the fabric’s dimensions — a diameter over a drop, no solve involved. The force’s direction comes out of the Lagrange multipliers of a constrained minimisation over a couple of dozen coefficients. They agree to a part in ten thousand at a basis of twelve terms, and closer as the basis grows.

A sign error in a multiplier would be invisible any other way. A contact force pointing the wrong way through a fabric is still a plausible number of millinewtons.

What the tilt says about the fabric’s thickness #

Two centre lines a diameter apart, each with a radius on either side of it, make a fabric two yarn diameters thick. For the 20 tex cotton that is 0.334 mm.

Nothing was fitted to get that, and it does not depend on the gauge: knit the same yarn loosely and the fabric is not thicker, because the interlacing still passes at a diameter. It depends only on the yarn. That is a strong prediction and a checkable one, and it lands inside the band of the woven cloths in this collection’s own table, which run from 0.16 mm for a voile to 0.44 for a duck.

It is also, as a prediction about a real jersey, a low one, and how thick a knit is is where the difference between a model’s thickness and a gauge’s reading is taken apart.

The independence from gauge is worth dwelling on, because it is the kind of claim that is easy to make and easy to test. Everything else about a knitted fabric moves when the loop length moves: its weight, its openness, its extension, its bending rigidity, the force at its crossings. The thickness does not, because the only thing setting it is that two threads of one diameter cross at one diameter. A knitter who wants a thicker fabric out of the same yarn cannot get it by knitting looser; the fabric gets more open and stays as thick as it was. What does change it is a second bed, which is a different lever entirely and moves the thickness by a factor rather than a fraction.

Where the angle sits in a fabric’s other angles #

A dozen degrees is a small angle, and this collection has several others to measure it against.

A woven thread’s crimp angle — the angle its path makes with the cloth at the steepest point — runs from about ten degrees in an open cheesecloth to over fifty in a close poplin. A twill’s own line runs at forty-five degrees when the cloth is square-set. A jersey’s spirality, the lean its wales take because the yarn is still trying to untwist, is a few degrees to a dozen.

The loop’s tilt is in that last company: small enough that it was reasonable to leave out of a drawing, large enough that leaving it out of a force is not the same decision at all. A twelfth of a right angle removed from a vector removes a fifth of it in the perpendicular direction, and a fifth of a contact force is the entire mechanism of the next rung.

What is not a tilt #

The identity holds because both end tangents lie on one axis. Take that away and it fails, and it is worth being exact about when it is taken away, because that is where the third dimension stops being free.

A thread that arrives at a crossing rather than at an extreme point of its wave does not leave along the course direction. A tuck holds an old loop while a new one forms over it, so the yarn arrives at an angle. And a fabric bent to a curve turns its own surface directions, so a loop in a curled edge has its two ends pointing different ways.

In every one of those the rotation is not available and the solve is genuinely three-dimensional. That is not a failure of the identity; it is what the identity is for. It says exactly which questions need the extra coordinate and which do not, and most of the questions on this ladder do not.

Which retires an entry from a list of omissions #

The list this rung began from has five items on it. The first was that the curve is planar and a loop is not, priced at about three per cent.

That item is now wrong in both halves. The curve is planar, and the price is negative. What replaces it is not a correction to a number but a different statement: the loop’s plane is not the fabric’s, the angle between them is a dozen degrees, and the consequence is a force the flat model could not have had.

The second item on that list — that adjacent courses may pass through one another — is untouched by any of this and remains exactly as recorded. A tilt does not stop two courses overlapping; it says where they overlap.

The same picture for a fabric with two beds #

Everything above is about a fabric knitted on one bed, where the only reason to leave the plane is the interlacing. Put a second bed in and the yarn has a much bigger reason: alternate wales are on opposite faces, and the sinker loop between them has to cross the whole gap between the beds.

The model does not change at all. The climb is a parameter, and a rib is the same solve with a bigger one — three to five diameters rather than one. That is the whole mechanical difference between a jersey and a rib in this account, and a rib climbs a gap is where it is followed through.

A rib’s course alternates between a steep half period and a shallow one — 27.5° and 5.9° at a three-diameter gap — and its through-thickness force rises from a fifth of the contact force to over a quarter, and to nearly a half on an open gap.

The two scales, and why every picture here has both #

A jersey is a third of a millimetre thick and its wale is four fifths of one. A section drawn at a single scale is therefore a line with some marks on it, and every figure in this ladder that shows the thickness expands it.

That is a hazard rather than a convenience, and the hazard is specific: an expansion of three makes a twelve-degree tilt look like thirty-five. So every figure here that expands says so in its own note, and the one that is about the angle draws the fabric twice — once expanded, so the arrangement can be seen, and once at one scale, so the angle is true. Neither picture alone is honest.

What the identity does not license #

It says the energy and the shape are a rotation of the flat problem’s. It does not say the fabric is a rotation of anything.

A fabric is a lattice of these curves, and the lattice is not rotated: successive courses are displaced by the course spacing, which lies in the fabric’s plane and not in the loop’s. So the fabric is genuinely three-dimensional even though each of its threads is flat, in the same way that a stack of tilted cards is a solid.

That distinction is what makes the next two rungs possible. A force through the thickness exists because the loop is tilted; a fabric has a thickness for it to act across because the lattice is not.

What is genuinely new here #

Three things, and the third is the one to keep.

The climb has a value rather than an adjective. It is a yarn diameter, from the interlacing, with nothing fitted.

Its cost is negative and was estimated positive. A curvature added to a curve is not the same object as a curvature that replaces part of it, and the estimate that got the sign wrong is the kind that looks like arithmetic and is a modelling choice.

And the departure from planarity is a rotation, exactly. That is a theorem about a boundary-value problem rather than a fact about jerseys, and it says which of this collection’s knitted results were safe all along: all of the ones about shape, and none of the ones that would have needed a third component of a force.

What the picture cannot show #

The tilt is drawn as a straight line because the projection is a straight line, and a reader could take that to mean the yarn runs straight. It does not. Seen from the face of the fabric it is the same curved loop it always was, bending hardest at its crowns to a curvature within a third of a per cent of one over the yarn’s diameter — which the tilt leaves almost exactly where the flat model put it.

What the projection removes is the curvature, not the curve. Looking along a plane curve’s own plane is exactly the view in which its bending is invisible.

Where the ladder goes next #

Three questions follow, and each has its own rung.

The energy fell and the force turned; what leaving the plane costs puts numbers on the whole set and works out which of them a reader should now quote.

The turned force has a component nothing here has had before; the force that holds a knit open is that component, as a force, as a pressure and as a correction to a friction balance.

And a fabric with a thickness can finally be asked whether it curls, which is a question about a moment and a moment is a force at an offset. The answer is not the one anybody would guess, and how little asymmetry a curl needs is why.