Five symptoms of one omission

Over three earlier ladders this collection recorded five things about its knitted fabric that it could not explain. They were written in four different ladders, by four different lines of argument, and each was recorded honestly as a shortfall with a note about what it would take to close.

They are one defect.

Finding that out is worth more than closing any of them would have been, and this rung exists to put the five side by side, because side by side they say something none of them says alone.

The sentence #

The defect is one modelling decision, made when this collection first let its knitted loop out of the fabric’s plane.

A course’s crest is a needle loop’s head and its trough is the feet resting on the head below. The head lies half a yarn diameter behind the fabric’s mid-surface and the feet half a diameter in front. So a half period climbs a whole diameter as it descends a course spacing, and the interlacing is a point where two centre lines pass one diameter apart.

Every part of that is right about distances. It is what makes the fabric two diameters thick, which is this collection’s most falsifiable knitted result and the first prediction it made with nothing fitted.

And it places the yarn beside the yarn below rather than through it.

Symptom one: the extension ceiling is three times too high #

The oldest of the five, and the first to be recorded.

At a fixed loop length the yarn has to reach from one interlacing to the next, so widening a wale costs course height, and the model’s course-wise extension ceiling is where the course spacing reaches a yarn diameter. It comes out at three hundred and twenty-two per cent for an ordinary jersey.

Measured jerseys stop at about a hundred, which is what a knit gives when it is pulled.

The recorded diagnosis was that the model does not stop adjacent courses passing through one another. That is true, and it is nearly this — but only nearly, and the difference matters. Stopping the courses overlapping turns out to bring the ceiling down by seven per cent, which is nothing. Threading them through one another is a much stronger constraint, because two threaded loops cannot separate beyond the length of yarn joining them however much room there is.

Symptom two: the course has no writhe #

A thread whose ends are held can shed twist only by acquiring writhe. Writhe is the mechanism the standard account of spirality is made of, and this collection had priced spirality at nine degrees per unit of twist factor with the nine fitted, because it had no torsional compliance to derive it from.

With the compliance in hand, the writhe of the model’s own solved course is minus six parts in a million per wale, and it is not small: it is exactly nought, by a symmetry. Successive half periods are exact mirror images, so the course is carried to itself by a reflection, and writhe changes sign under a reflection.

The symmetry exists because a near miss has no handedness. A real knitted fabric is one of the most obviously handed structures in the subject.

Symptom three: the linking number is nought #

The same integral applied to two courses rather than one gives the sharper version, because here the right answer is known rather than merely absent.

A knitted tube of twelve wales has a linking number of twelve between adjacent courses, and a point cannot link. The model’s is minus thirteen parts in a million.

That is not a quantitative error. It is a statement that the model’s fabric is not knitted, in the one respect that distinguishes a knitted fabric from any other, and it holds at every size, every tightness and every state.

Symptom four: the two half periods at a crest touch #

Measured rather than argued, and found while asking a different question.

Two half periods meet at a crest as mirror images, and they approach one another to 0.018 of a yarn diameter — eighteen thousandths — and stay within that for more than a millimetre of arc. Two strands of the same yarn, a fiftieth of their own thickness apart, running alongside.

In a fabric what holds them apart is the needle loop of the next course, drawn between them — which is what holds a crest apart. That is precisely the loop the model does not thread. The strands have nothing between them because nothing was put between them.

Symptom five: the wale-direction curl cannot be computed #

The most expensive of the five, and the one that consumed most of a ladder.

Bending the fabric about its wale direction turns a thread’s end tangents out of the fabric’s plane. A thread bent hard in a plane is only conditionally stable in that plane, and with its ends turned it is not: the solve leaves the fabric altogether, by more than three times the fabric’s own thickness, for thirty-seven per cent off the energy.

The recorded diagnosis was that the buckled configuration is one the thread’s neighbours would not allow. It named the cause correctly and did not connect it to anything: the neighbours in question are the loops that ought to be threaded through it.

Worse, the failure does not announce itself. Both signs of curvature find the same buckled branch, their energies are identical, and the moment comes back at nought — which is exactly what the correct branch gives. A wrong calculation agreeing with a right one is the worst kind, and what separated them was the shape rather than the number — the same trap a branch jump always sets.

Why none of them located it #

The interesting question is not what the defect is. It is why five encounters with it produced five diagnoses.

Each symptom was found by a different instrument, and each instrument reported in its own vocabulary. The ceiling was found by comparing a computed extension with a measured one, and reported as a factor of three. The curl was found by a shape check on a solve, and reported as a branch jump. The crest was found by a distance measurement, and reported as an overlap. The writhe and the linking number were found by an integral, and reported as zeros.

None of those vocabularies contains the word the others need. A factor of three, a branch jump, an overlap and a zero do not obviously belong to one another, and the only thing that made them belong was computing a quantity — the linking number — that is about the arrangement rather than about any of the things the four instruments measure.

A defect in an arrangement shows up in every measurement and in no measurement’s own language.

And why the fifth was the one that named it #

The linking number is the only one of the five instruments whose output is an integer with a known correct value.

The ceiling’s correct value is “about a hundred per cent”, which is a measurement with a spread. The curl’s is a moment nobody has published. The crest’s is a distance nobody has measured. The writhe’s correct value is unknown, because nobody has computed the writhe of a real jersey either.

The linking number’s correct value is twelve, for a twelve-wale tube, and it is twelve because that is what the machine does. So it is the only one of the five where the model’s answer could be compared with the truth rather than with another model or another estimate, and it is therefore the only one that could say how wrong rather than that something is wrong.

That is worth generalising: the instrument that locates a defect is usually the one with the least tolerance, not the one closest to the symptom.

What a repair would have to do #

The repair is not a term and not a constant. It is a geometry, and it can be stated precisely.

At each interlacing the arriving yarn must pass round the yarn already there: leave the mid-surface on one side, travel past, and return on the other, so that the two centre lines wind about one another once rather than approaching and separating.

Three consequences follow immediately.

It costs about half a diameter of extra yarn at each interlacing — around two and a half per cent of the loop length at ordinary counts — and every dimension the fabric has is computed at a fixed loop length, so every one of them moves.

It costs bending, at a radius of about one diameter, which is the tightest bend anywhere in the fabric.

And it turns the problem from one with three endpoint equalities into one with a non-penetration inequality, which is a different class of problem: the constraint is active on part of the domain and inactive on the rest, and the part it is active on is an unknown of the solve.

What survives the defect #

This is the practical output of the rung and it is worth being precise, because a model with a known structural omission is not a model to discard.

Every quantity that is an integral over one segment survives. The bending energy of a stitch, the contact force, the fabric’s thickness, its bending rigidity, its thermal resistance, its fibre fraction, the pressure a cuff applies. Each is decided by where a half period’s two ends are and how much yarn runs between them, and threading the curve through another one changes none of that to first order.

Every comparison at fixed topology survives. A rib against a jersey, a tight fabric against a slack one, one relaxation state against another: the omission is the same in both members and cancels.

Nothing that depends on the fabric holding together survives. The extension ceiling, the run, the curl about the wale axis, the spirality.

That division is clean, it is easy to apply, and it means a reader can use the collection’s knitted results without having to decide case by case.

The one that is not on the list #

There is a sixth zero in this collection’s knitted work and it is not a symptom of this, which is worth saying because it would be easy to sweep it in.

The curling moment about the course direction is exactly nought, and it is nought because the fabric’s mid-surface bisects every segment, so bending gains nothing at first order either way. A real jersey’s mid-surface very nearly does bisect its segments, so that zero is a genuine statement about the fabric, and the small eccentricity that produces the observed curl — about five per cent of a yarn diameter — is a correction to it, which is how little asymmetry a curl needs.

Two null results that look identical on the page. One is physics and one is an omission, and telling them apart needed a symmetry argument in one case and a threading argument in the other.

The rule that follows #

A model’s null results are where its omissions live. An omitted mechanism contributes nothing, and contributing nothing looks exactly like a converged calculation of something negligible.

So the question to ask of any quantity that comes back at nought is not whether it is small enough to ignore. It is:

Is it zero for a reason, and does the subject share the reason?

Here the reason was a mirror symmetry, and the subject does not share it. There the reason was a bisected mid-surface, and the subject does share it. Same output, opposite meanings, and no gate anywhere can tell them apart because a gate reads a number.

Why the decision was defensible when it was made #

It is worth defending the sentence that caused all this, because the point of the rung is not that somebody was careless.

When the interlacing was declared a point, the model had just acquired a third dimension and the question in front of it was how thick a knitted fabric is. For that question, “two centre lines a diameter apart” is exactly right, and the answer it produced — 0.334 millimetres, with nothing fitted and no dependence on the gauge — is the most falsifiable thing this collection has said about a knit, and a state does not move it.

Threading the crossing properly would have cost a much harder solve and would have moved the thickness by a few per cent at most. Nobody trades a working answer for a few per cent.

The cost was not in the thickness. It was in five other questions, none of which was being asked at the time, and four of which were not askable because the quantities they need had not been built.

What was counted, and how #

Nothing here is new arithmetic. All five numbers are the collection’s own and are recomputed rather than quoted: the ceiling from the jamming geometry, the writhe and the linking number from the Gauss integral on the solved course, the crest approach from a point-to-point minimum over the sampled curves, and the curl’s branch jump from the shape check that found it.

What is new is the claim that they are one thing, and the check on that claim is a counterfactual: each of the five is a consequence of the crossing being a point, and each of the five would be changed by threading it. That is stated for each in turn above, and it is the whole of the argument.

Where the model stops #

Nothing here solves the threaded problem. The repair is priced and not made.

And the list may not be complete. Five is what has been recorded; there is no reason to think a sixth is not sitting in some ladder waiting for somebody to ask a question that needs the fabric to hold together.

Nor does the rung say the repair is worth making. Four of the five symptoms are about fabric failure and deformation, which is a smaller part of this collection’s subject than the geometry and the forces, and the geometry and the forces are unaffected. A reader who wants to know how warm a rib is does not need any of this.

The cost of a shortfall queue, paid back #

There is a process finding here as well as a modelling one, and it is the reason this rung exists in the form it does.

Every one of the five was written into a shortfall list at the end of the ladder that found it: not fixed, not hidden, and not left as a rough edge somebody meant to return to. Each entry named what was wrong, what it would take to close, and what the collection was doing meanwhile.

That habit costs something. Writing a careful account of a thing that is not working is slower than moving on, and none of the five entries paid for itself at the time.

All five paid together. The joining above is a piece of reading rather than a piece of computation: the linking number supplied one new fact, and the other four were already written down in enough detail to be recognised. Had any of the four been recorded as “the curl calculation was unstable” or “the ceiling seems high”, none of it would have connected.

The rule this supports is worth stating for whoever writes the next ladder: a shortfall is recorded with its diagnosis, not with its symptom. A symptom is a note to self. A diagnosis is a sentence that can be matched against another ladder’s sentence three ladders later by somebody who was not there.

Who found it, and when #

Each symptom was found by this collection and recorded in its own ladder, over three earlier ladders. The joining of them dates from the arrival of the linking number, which is the only instrument here that could compare the model against a known integer.

The habit of recording a shortfall rather than quietly dropping it is what made the joining possible at all. Five diagnoses written down in five places, in four vocabularies, over three earlier ladders — and none of them would have been available to be joined if any of them had been left as a rough edge somebody meant to come back to.

Where the ladder goes next #

The topology ladder closes here, and the collection returns to a mechanism that is available in the model as it stands: a twisted thread that nothing is holding straight, which stops being straight. Why a slack yarn snarls needs only the two stiffnesses and their ratio, and it produces a number anybody can check by letting go of a piece of sewing thread.

The contact side of the same story runs in parallel and starts from a measurement that ought to have been made years ago: whether the fabric this collection has been computing with actually fits together. It does not, and what it does about that turns out to predict a parameter this site has swept since its second phase.