The fabric that does not fit

This collection has computed a great deal about a knitted fabric: its thickness, its contact force, its bending rigidity in both directions, its warmth, its fibre fraction, the pressure a cuff applies to a wrist. All of it came from minimising a bending energy over a centre line — a loop solved as an elastica rather than drawn.

A centre line has no thickness. So nothing in any of those solves forbids one part of the yarn from occupying the same place as another, and nobody had asked whether the fabric that came out could be built.

It cannot.

The measurement #

Take the collection’s own relaxed jersey — a twenty tex cotton at a three and a half millimetre loop, fully relaxed, solved as a three-dimensional elastica. Lay two adjacent courses one course spacing apart, which is what the model says a fabric is. Then ask, over the whole of both curves, how close they come.

The answer is 0.780 of a yarn diameter.

Not at one point. That is the minimum over the whole sampled length, and the region where the two curves are inside a diameter of one another runs for a fifth of the course.

So the fabric this collection has been computing with has its yarn passing through itself by a fifth of its own thickness, everywhere, at rest, before anything is loaded.

Where the closest approach is #

The immediately suspicious thing is that the model placed the two curves one diameter apart, deliberately, at the interlacing. That is what the climb is: a half period rises one whole yarn diameter as it descends a course spacing, because the head lies half a diameter behind the fabric’s mid-surface and the feet half a diameter in front.

That placement is exactly right, and it is the reason the fabric comes out two diameters thick with nothing fitted.

What nobody asked is whether one diameter at the interlacing is the closest the two curves come. It is not. The interlacing is a single point on two curves that are still approaching either side of it, and the minimum sits a little way along — at a place the model never places anything, and therefore never checked.

That is the whole of the defect and it is one sentence: the model got the distance right at the point where it set it, and nowhere else.

Why it is a finding rather than a bug #

The temptation is to record this as an error and move on, and that would waste it.

A round yarn cannot occupy the arrangement above. A flattened yarn can — and flattened is what a yarn in a fabric measurably is. Every microscope section of every cloth ever cut shows threads that are not round, and this collection has carried a flattening ratio as a free parameter since it first jammed a warp precisely because nothing predicted one.

So the overlap reads as a prediction rather than a defect:

The fabric’s own geometry requires the yarn to be flattened to about four fifths of its round diameter, and it requires it at rest.

That number was not fitted to anything. It falls out of a solve that knew nothing about flattening, made no allowance for it, and would have been written the same way if flattening had never been heard of.

What was already known and not connected #

Three facts sat in this collection separately and this measurement joins them.

A yarn in cloth is flattened. Everybody knows it, every micrograph shows it, and the site’s own racetrack section exists because of it.

Nothing predicts the flattening. The site’s own transverse-rigidity bracket has no floor at all — a bundle of fibres free to slide resists a change of section not at all — so the flattening cannot be estimated from the fibre, and it has been swept rather than computed.

And the knitted solve produced a fabric nobody checked for self-consistency, even while the model’s own shortfalls were being carefully listed. Every gate on this site reads a fabric and asks whether something about it is right; not one of them asks whether two parts of it are in the same place.

Put together, they say the flattening is decided by the fabric’s own geometry rather than by the yarn’s mechanics, which is a shift in where the number comes from.

Two other approaches, and what they say #

The measurement asks three questions rather than one, and the other two are controls that make the first mean something.

Within one course, a whole loop length apart, the yarn comes to 3.02 diameters of itself. So the overlap is between courses rather than inside one, and it is not an artefact of a course being folded on itself.

Between courses once removed, the minimum is 2.87 diameters. Those two never touch in a fabric and they do not touch here, which is the control on the layout: if that number had come out small, the arrangement would be wrong rather than the model.

And at a crest, where two half periods meet as mirror images, the two strands come to 0.018 diameters — a fiftieth — and run within that of one another for over a millimetre of arc.

That third number is not a flattening. Two strands a fiftieth of a diameter apart are not a squashed yarn; they are two strands in the same place. What holds them apart in a real fabric is the needle loop of the next course drawn between them, and that is the loop the model does not thread.

So there are two defects, not one #

Separating them matters because they have different repairs.

The between-courses overlap is a section problem. The centre lines are where they should be and the yarn is drawn round when it is not. Flattening fixes it, and the fabric flattens.

The crest overlap is a topology problem. No amount of flattening puts two strands a fiftieth of a diameter apart into separate places; the arrangement is wrong rather than the section. It is the same omission that gives the model’s course no writhe and its adjacent courses no linking number.

Two numbers from one measurement, and they are symptoms of two different things. It took drawing both to see it.

What the overlap is worth against the transverse bracket #

There is a reason this particular prediction is worth more than most, and it is about which quantity it replaces.

This collection’s bending rigidity is bracketed over a factor of three hundred, which is bad. Its transverse rigidity — how hard a yarn is to squash out of round — is worse than bracketed: its lower bound is exactly nought, because a bundle of fibres free to slide resists a change of shape at constant area not at all. A quantity bounded below by zero cannot be estimated from its bounds at all, and this collection said so when it computed them.

That is why the flattening has been a free parameter. It is not that nobody tried; it is that the mechanical route to it is closed.

So a geometric prediction of the flattening is not a marginal improvement on a poor estimate. It is the first prediction of a quantity that had none, arriving from a direction — the fabric’s own arrangement — that nobody had looked in.

Whether it is right is a separate matter and the next two rungs are about testing it. What makes it worth testing is that the alternative is not a worse prediction but no prediction.

Why nothing failed #

A fair question is how a fabric that cannot exist passed thirty-eight gates, and the answer is instructive rather than embarrassing.

Every figure on this site is checked for whether its labels fit, whether its elements paint, whether anything overflows its viewBox, whether two labels collide, whether the options a caption names are options the generator reads. Every essay is checked for its length, its links, its voice and its internals. Every weave is checked for whether it is one cloth.

None of that is a check on physical consistency, and none of it could be. A drawing of two overlapping yarns is a perfectly good drawing: its labels fit, its strokes paint, nothing overflows. The gates read pictures and prose, and an impossible fabric drawn correctly passes every one of them.

The check that would have caught it is the one written for this ladder, and it did not exist because the quantity it measures — the distance between two parts of one fabric — had never been computed by anything.

What was counted, and how #

The two courses are the same solved curve, one course spacing apart, sampled at a hundred and twenty points per half period over two wales — about five hundred points a course, so a quarter of a million pairs, which takes a moment.

The approach is measured point to point rather than segment to segment. The difference is at most half a sample spacing, which at this sampling is a part in a thousand of a diameter — the same reasoning the site uses for its own sampled curves elsewhere, and saying so is cheaper than writing a segment-pair routine whose own correctness would then need checking.

Three separations are reported and they are three different questions: between adjacent courses, within one course at a whole loop length of arc, and between courses once removed. The last is the control.

The check on all of it is written so that fixing the model breaks it: it asserts that the approach lies between 0.70 and 0.88 diameters, so a later solver that keeps its threads apart will report one diameter and fail this check. That is the signal that the thing this ladder is about has been dealt with, and it is why the assertion is a band rather than “is less than one”.

What the fabric does about it, mechanically #

Saying that the fabric flattens is a geometric statement and it is worth asking what it costs, because a fabric that has a choice will take the cheap one.

Flattening a yarn is a change of section at constant area, and what resists it is the yarn’s transverse rigidity. At the free end of that bracket — the fibres sliding freely — the resistance is exactly nought: a bundle in cross-section is a fluid, and nothing at all opposes it changing shape.

At the coherent end, the section is a solid rod of fibre material and the energy of squashing it to four fifths is thirty-six times the whole bending energy of a stitch. A yarn like that could not be knitted into this fabric at all.

So the fabric has a choice between flattening, which is free at one end of the bracket and impossible at the other, and the alternatives — a longer loop, a wider course spacing — which Munden measured and which the fabric did not take.

It flattens. That is a fourth everyday observation landing at the free end of this collection’s stiffness bracket, and it is the cheapest of the four.

The alternative the fabric did not take #

It is worth spelling out what the fabric would have had to do instead, because it shows how strong the constraint is.

The overlap is a fifth of a diameter. To remove it without changing the section, the two courses would have to sit further apart — a course spacing larger by about a fifth of a diameter, which is five per cent on the spacing.

That is not a small change. A five per cent change in course spacing moves the fabric’s stitch density by five per cent, its areal weight by five per cent, and every quantity computed from either. And it is a change away from what Munden measured on real fabrics, which is the input this collection’s whole knitted arithmetic rests on.

So the measured fabric is denser than a round-yarn geometry allows, by an amount that a flattening of four fifths exactly accounts for. Which is a slightly stronger statement than the rung has made so far: the flattening is not merely permitted, it is required by the measurements the collection was already using.

Where the model stops #

Nothing here re-solves anything. The measurement diagnoses a solved configuration; it does not minimise the energy subject to a non-penetration constraint, which is a different and much larger piece of machinery. Every number on this ladder is a diagnosis rather than a correction.

The yarn is treated as a circular tube of constant diameter along its length. It is not — that is the whole finding — and it is not constant either, which is where a yarn is thinnest.

And the fabric is plain. A rib, an interlock or a purl fabric has courses arranged differently and none of them is measured here.

A note on what “at rest” means here #

One clarification, because the phrase does a lot of work above.

The fabric measured is fully relaxed: wetted, tumbled, dried, and left alone. No tension, no load, no gauge pressing on it. It is the state a garment ends up in after a few washes and the state this collection quotes most of its knitted numbers for.

That matters because a fabric under load is expected to press its threads together, and nobody would be surprised by an overlap in a stretched or compressed one. The finding is that the overlap is there when nothing is doing anything to the fabric, which means it is a property of the structure rather than of a loading.

It is also worth saying that the overlap is a little larger in the dry-relaxed state and a little smaller in the fully relaxed one — 0.796 against 0.780 — because relaxation brings the courses closer together. So the flattening the fabric requires increases as it relaxes, which is a small and specific prediction: a washed fabric’s yarn is flatter than the same fabric’s off the machine, at an unchanged loop length.

That is checkable on a micrograph and nobody has looked.

The generalisation #

The habit worth taking is short and this collection had not been keeping it.

Ask whether a model’s output can be built.

Every gate on this site reads a fabric and asks whether something about it is right: whether a label fits, whether a weave is one cloth, whether a figure’s options are read, whether a caption’s number matches its picture. All of those presuppose that the object exists.

Not one asks whether the object is realisable — whether the thing described occupies a consistent amount of space. That is a different question, it is cheap to ask, and asking it once found a defect that four ladders of careful work had left in place.

The reason it went unasked is worth knowing too: the model’s outputs were all local. A bending energy, a contact force, a thickness, a rigidity — each is an integral over one segment, and one segment cannot overlap itself. Realisability is a property of an assembly, and nobody had assembled anything.

The same question, asked of a woven cloth #

The obvious next thing to ask is whether a woven cloth has the same problem, and it is worth flagging here because the answer is not obviously no.

A woven cloth’s geometry in this collection comes from Peirce, which joins circular arcs to straight lines and places the two thread systems in contact at the crossing. That is the same kind of construction as the knitted one: a distance set correctly at one point, with the rest of the curve free.

If the closest approach of two Peirce threads is somewhere other than the crossing, a woven cloth has the same defect and the same reading — and the flattening the trade has always assumed would be predicted rather than swept, for cloth as well as for knit.

The measurement is cheap and this collection has the geometry, so it is asked directly two rungs along.

Who found it, and when #

That yarn in a fabric is flattened is old and universal knowledge, and Peirce’s 1937 geometry already replaced the circular section with a racetrack for exactly that reason.

What is not in the literature, as far as this collection can tell, is a prediction of how flat. The flattening has always been a measured or assumed input, and this collection has swept it for the same reason everybody else has.

What is this collection’s own is asking its own solved fabric whether it fits, finding that it does not, and reading the shortfall as the flattening the fabric requires rather than as an error in the solve.

Where the ladder goes next #

The number is 0.780 for one fabric, and one number is an anecdote. The next rung asks whether it is a property of that arithmetic or of knitted fabric, by computing it across five loop lengths, three relaxation states and three counts.

They fall on one curve against the tightness factor, which is the model’s only dimensionless group — so it is a structural requirement rather than an accident, and it is the flattening nobody fitted.