What holds a crest apart

The measurement that found this collection’s knitted fabric overlapping itself reported three numbers, and only one of them has been used so far.

Between two adjacent courses, the closest approach is 0.780 of a yarn diameter, and the fabric meets that by flattening its yarn.

Within one course, at a whole loop length of separation along the yarn, the closest approach is 3.02 diameters, which is comfortable.

And at a crest, where two half periods meet, the two strands come to 0.018 diameters. Eighteen thousandths.

Why this is not the same defect #

A fiftieth of a diameter is not a flattening. It is worth being emphatic about that, because the two overlaps were found by one measurement and it would be natural to treat them as one problem.

A flattening can accommodate a fifth of a diameter. Squash a yarn to four fifths through the fabric’s thickness and two centre lines four fifths of a diameter apart fit perfectly well. That is what the fabric does and it costs nothing.

Nothing accommodates a fiftieth. A yarn squashed to a fiftieth of its diameter is not a yarn; it is a film. And even if it were, the two strands are the same yarn — two parts of one thread, both of which would have to be squashed simultaneously against one another with nothing between them.

So the crest overlap is a different kind of failure. The section is not wrong; the arrangement is.

What the arrangement should be #

In a real fabric the two half periods that meet at a crest do not meet. They pass on either side of the needle loop of the next course, which is drawn through the space between them.

That loop is a yarn diameter thick. It is what holds the two strands a diameter apart, and it is why a needle loop’s head is a loop rather than a fold.

This model has no such loop. Its interlacing is a point where two centre lines pass a diameter apart, and a point holds nothing open. So the two half periods meet at a mathematical crest and there is nothing between them.

The same omission, seen again #

That is the fifth appearance of one modelling decision, and by now the shape of it is familiar.

The course has no writhe, because a near miss has no handedness.

Adjacent courses have no linking number, because a point cannot link.

The wale-direction curl cannot be computed, because the solve runs away to a configuration the missing loops would block.

The extension ceiling is three times too high, because two loops threaded through one another cannot separate and two loops passing beside one another can.

And the two half periods at a crest run within a fiftieth of a diameter, because the loop that ought to be between them is not there.

Five symptoms, five ladders, one sentence.

How to tell the two overlaps apart #

Since one measurement produced both, it is worth extracting the rule that separates them, because it will be needed again.

An overlap that a plausible section change accommodates is a section problem. A fifth of a diameter is within what a fibre bundle can rearrange to, so the model’s geometry is right and its section assumption is wrong.

An overlap that no section change accommodates is an arrangement problem. A fiftieth of a diameter cannot be reached by any yarn that is still a yarn, so the model’s geometry is wrong.

The test is quantitative and cheap: compare the overlap to what the material can supply. A yarn can be flattened to perhaps a half of its round diameter before it stops being a bundle at all; anything inside that is a section question and anything outside it is not.

That rule would have separated the two on the day the measurement was made, and it did not, because nobody had stated it.

Why the free length between them matters #

There is a second reason the crest overlap is worse than its number suggests, and it is about extent rather than depth.

The two strands are not merely close at one point. They run within a fiftieth of a diameter of one another for more than a millimetre of arc, which at a three and a half millimetre loop length is nearly a third of the yarn in a stitch.

An overlap at a point is a contact. An overlap over a third of the yarn is two strands lying in the same place, and no local repair fixes it.

That extent is a consequence of the mirror symmetry: the two half periods either side of a crest are exact reflections, so they approach and depart symmetrically and run parallel for as long as the curvature is low. A model that broke the symmetry would also break the parallel run.

What it costs the numbers #

An honest accounting of what depends on it, because a defect that changes nothing is not worth a rung.

Nothing local is affected. The bending energy of a half period, the contact force, the fabric’s thickness, its rigidity: each is an integral over one segment, and a segment does not know what is beside it.

The crest’s own curvature is affected, because a strand that has to pass round a loop rather than fold back on itself takes a different path there. That path is longer, so at a fixed loop length it costs the rest of the stitch some yarn.

And every quantity that depends on the fabric holding together is affected, which is the same list as before.

So the crest overlap is not an additional defect with its own consequences; it is the same defect with an additional symptom. What makes it worth its own rung is that it is the cheapest to see of the five: no integral, no invariant, no branch check — just a distance measured between two points on a curve.

The measurement that separates the model’s two problems #

The rung suggests a measurement, and it is one this collection could make on its own machinery rather than in a laboratory.

Build the threaded crossing — the arriving yarn passing round the standing one rather than beside it — and re-measure all three approaches.

The prediction is specific. The crest approach should go from a fiftieth of a diameter to about one diameter, because the loop between the two strands is what sets it. The adjacent-course approach should stay at about four fifths, because it is a different geometry that threading does not change. And the control should stay where it is.

If all three move, the threading has changed more than it should have. If only the crest moves, the two problems really are separate and the flattening result survives the repair.

That is a check on a future model rather than a result, and writing it down now is what makes it a check rather than a hope.

Why the loop between them is not free #

It is worth pricing the thing that ought to be there, because the price is what makes the repair a real piece of work rather than a decoration.

A strand that passes round the standing yarn rather than beside it travels further. The extra is about half the circumference of a yarn — half of π times a diameter, so about 0.26 millimetres for a twenty tex cotton — at each of the two interlacings a stitch has.

That is 0.52 millimetres against a loop length of 3.5, which is fifteen per cent.

Fifteen per cent of the loop length is not a correction. Every dimension this collection computes for a knitted fabric is computed at a fixed loop length, so taking fifteen per cent of it away for threading changes the wale spacing, the course spacing, the thickness, the areal weight and every force.

Some of that is already accounted for, because the model’s loop length is the measured one and a real fabric’s yarn is already doing the threading. So what the repair changes is not the fabric but the model’s accounting of where its yarn goes: fifteen per cent of it is currently spent on a free run and should be spent on a wrap.

That is a large enough reallocation to change results, and it is the strongest argument that the repair is worth making.

What the wrap would cost in bending #

The second half of the price is curvature, and it is worse than the length.

A strand wrapping round another at contact follows a path whose radius is about one yarn diameter — the standing yarn’s radius plus its own. That is the same radius as the loop’s own tightest bend, so the wrap is as hard a bend as anything already in the fabric.

Bending energy goes as the square of curvature times the length bent, so a wrap of half a circumference at one diameter of radius costs about as much as a substantial fraction of the loop’s whole bending.

Adding it would therefore raise the fabric’s computed energy, raise its contact force, and change the balance the fabric’s dimensions come from. Whether the fabric would end up denser or slacker is not obvious from the sign of one term.

What was counted, and how #

The crest approach is a self-approach on one sampled course, excluding pairs closer together than a stated arc separation, because two points a few samples apart on a smooth curve are close for reasons that have nothing to do with an overlap.

Two separations are reported. At half a half period of arc, the answer is the crest: 0.018 diameters. At a whole loop length, the answer is the genuine self-approach: 3.02 diameters.

Choosing the exclusion is the delicate part and it was got wrong first. A separation of one half period picks up the crest, which is what it is for; a separation of a few samples picks up the curve being continuous, which is meaningless; and reporting the second as a self-approach would have said the fabric overlaps itself everywhere.

The check asserts both: the whole-loop-length approach exceeds one diameter, so the between-courses overlap is not an artefact of a folded course; and the crest approach is under a fifth of a diameter, which is the finding stated so that a repaired model fails it.

Why a fold is not a loop, and what the difference is worth #

The distinction the rung turns on can be put in one sentence and it is worth putting there, because it is the difference between two crafts.

A fold is a thread doubled back on itself. Nothing is between the two sides. Pull the two ends and the fold straightens out, and nothing was holding it.

A loop is a thread doubled back round something. The something is between the two sides. Pull the two ends and the fold cannot straighten until whatever is inside it comes out.

This collection’s model builds a knitted fabric out of folds. A knitting machine builds one out of loops, and the difference is a needle.

That is why every symptom on the list of five is a symptom of holding rather than of shape. A fabric of folds has all the geometry of a knitted fabric — the same paths, the same spacings, the same thicknesses, the same forces — and none of its integrity, because the geometry does not include what the folds go round.

What a fabric of folds would actually be #

It is worth taking the idea seriously for a moment, because such a fabric exists and is instructive.

A sheet of yarn laid in a serpentine path, course by course, with each course’s crests resting against the previous course’s troughs and nothing threaded through anything, is a real arrangement. It is what a fabric looks like the instant before the needles pull the new loops through, and it is what a fabric becomes when it runs.

It has no integrity at all. Lift one course and it comes away, taking nothing with it.

So the model this collection has been computing with is a picture of a knitted fabric at the moment before it becomes one, and every number it has produced is a number about that arrangement. The numbers are right because the geometry is right; the fabric is not a fabric because the geometry is not what makes it one.

Where the model stops #

Nothing here builds the threaded crossing. The rung measures what is wrong and prices what a repair would have to do; it does not do it.

The extent is measured at one sampling. The parallel run’s length depends on how close “within” is taken to mean, and the millimetre quoted is for a fiftieth of a diameter.

And the crest is the only place looked at. A trough is the mirror image of a crest in this model and should behave identically, which is checked; in a real fabric a crest is a needle loop’s head and a trough is a sinker loop, and they are not the same object at all.

That last is a real gap. The model’s crest-trough symmetry is itself a modelling choice, and it is very likely the thing that would have to break first for any of the five symptoms to be repaired.

The generalisation #

The rule this rung extracts is the transferable part, and it is short.

When a model’s output is impossible, ask by how much. An impossibility that a plausible material change absorbs is a statement about the material. An impossibility that nothing absorbs is a statement about the arrangement.

Those two need different repairs, they live in different parts of a model, and they are indistinguishable from the fact of the impossibility alone. What separates them is a comparison against what the material can actually do, and that comparison is usually available and usually not made.

This collection has one other place where the same question is open and has not been asked. Peirce’s woven geometry joins circular arcs to straight lines and the curvature jumps at every join, which is impossible for a real thread. Whether that impossibility is one a plausible material change absorbs, or one that says the geometry is wrong, has never been settled here — and the site’s own elastica exists because somebody suspected the second.

What this says about the model’s other symmetry #

There is one more thing the crest overlap points at, and it is the piece of the model most likely to break first when somebody repairs it.

The two half periods that meet at a crest are exact mirror images in this model: the second is the first reflected through the fabric’s mid-surface. That symmetry is what produces the parallel run, and it is also what makes the course achiral and its writhe nought.

In a real fabric the two are not mirror images at all. One arrives at a needle loop’s head and the other leaves it, and the head is a loop with something through it while the space between two heads is not. A knitted loop is asymmetric and this collection has measured that asymmetry in another context without connecting it to this one.

So the cheapest available repair may not be threading at all. Breaking the crest–trough symmetry would give the course a handedness, give it a writhe, separate the two half periods at a crest, and cost nothing but a longer solve — and it would leave the linking number at nought, so it fixes three of the five symptoms and not the other two.

That is a partial repair with a known cost and a known scope, which is a much more attractive proposition than the full one.

Who found it, and when #

That a needle loop’s head is a loop rather than a fold is the definition of knitting and is not a discovery.

What is this collection’s own is measuring the consequence in its own model: finding that the two half periods meet, quantifying by how much, and separating that failure from the other one the same measurement produced.

The date is the same afternoon as the between-courses overlap, and the two were reported together and treated as one thing for several hours before the difference in magnitude made the distinction obvious.

Where the ladder goes next #

The contact ladder has one question left about the fabric and it is the one it has been avoiding: what a model would have to do to fix any of this, and whether it is worth doing.

What a contact model would have to do prices the repair in yarn, in bending and in the class of problem it becomes — and comes to a conclusion about whether this collection should attempt it.