A rib's relaxation is not its bending either

The most useful negative result on this ladder is that a relaxed jersey is not at a minimum of anything. Its bending energy falls as the fabric is widened and falls as it is lengthened, all the way to the geometric limit where its yarn runs straight between interlacings, and the fabric everybody measures sits nowhere near there. What holds it is friction and the setting of its yarn, and the model computes forces because it cannot compute dimensions.

A rib has a dimension of its own that a jersey does not — the distance between the two beds’ loop planes — and it is the obvious place to ask whether the same is true. It is, and the arithmetic is cleaner because the dimension is one number.

The question #

A rib comes off a machine with its two beds a set distance apart. Take it off, wash it, tumble it, dry it flat — the treatment this collection calls full relaxation — and it settles at some gap of its own.

At what gap? A fabric that settled where its bending energy was least would settle where the curve above stops falling.

The question is worth asking in that form because it has a definite wrong answer. A jersey’s relaxed dimensions are famously predictable — the courses and wales go as one over the loop length, with constants Munden measured once and everyone has used since — and the reason that works is that a jersey has one length in it and everything else follows. A rib has two: the loop length and the gap between the beds, and only the first of them is set at the machine in a way that survives being taken off it. So the second has to come from somewhere, and the natural place to look is the energy.

It is also the question a knitter asks in commercial terms without recognising it as this one. A rib knitted at a wide setting and relaxed to a narrow one has shrunk in thickness, which is a fault; a rib knitted narrow and relaxing wider has grown, which is a different fault. Either way the number that decides is the relaxed gap, and it is not on any specification because nobody knows how to compute it. The rest of this essay is an account of why nobody does, which is a more useful thing than a wrong formula.

The answer, which is the same as the jersey’s #

The curve does not stop falling. Bending energy falls monotonically as the climb grows, from a jersey’s one diameter to five and beyond, and there is no stationary point anywhere in the range.

So the model’s answer is that a rib should open its beds until something stops it, and the something is not energetic.

That is exactly the jersey’s situation, one dimension over. The jersey’s bending pushes it wider and longer; a rib’s pushes it thicker as well; and in every one of the three directions the fabric sits on a slope rather than in a well.

Where the geometric ceiling is #

Something does stop it, and it is a length rather than an energy: the yarn between two interlacings cannot be shorter than the straight line joining them.

A half period has half a loop length of yarn — 1.75 millimetres for the 20 tex cotton at a 3.5 mm loop. It has to span half a wale spacing along the fabric, a course spacing and a diameter across it, and its climb through it. Solve for the largest climb that keeps the chord under the thread:

1.50 millimetres, which is nine yarn diameters.

Since a crossing’s climb is a diameter plus half the gap, that is a bed gap of

16 yarn diameters — 2.67 millimetres for this yarn.

Which is nowhere near where a rib is #

Machines are set to gaps of a few yarn diameters, not sixteen. At sixteen the two beds’ loops would be nearly three millimetres apart with nothing between them but a nearly straight thread, and the fabric would be a very open spacer rather than a rib.

So the geometric ceiling is a long way above anything anybody knits, in exactly the way the jersey’s extension ceiling — three hundred per cent, against a jersey that jams near a hundred — is a long way above anything anybody stretches.

Both ceilings are real, both are computed exactly, and neither is reached. That similarity is the point of this rung rather than an aside: it says the same thing is missing in both cases.

What is missing, in both cases #

The model stops a thread reaching further than its own length and stops nothing else. Adjacent courses may pass through one another; the two beds’ loops may approach without noticing; nothing in the arithmetic knows that a fabric is made of solid yarn.

Add that and both ceilings come down. A jersey stops extending when its courses meet, well short of its yarn running out. A rib stops opening when — when what? That is the question this rung cannot answer, and it is worth being exact about why.

Opening a rib does not bring anything into contact. It takes things apart. So the constraint that would stop it is not contact at all, and the geometric ceiling really is the only hard limit the fabric has.

Which makes the rib case the cleaner one #

A jersey’s relaxed width has two candidate explanations — friction holding it where it was, and courses coming into contact — and the model can only rule out one of them.

A rib’s bed gap has only one. Nothing comes into contact as the beds open, so the gap a relaxed rib settles at is set entirely by what the yarn has been set into and what friction holds, with no geometric competitor at all.

That is a stronger negative result than the jersey’s, and it points at the same missing quantity: the set fraction, which this collection has established cannot be measured from a fabric’s dimensions because it cancels out of every balance the dimensions can be put into.

The jersey’s version, for comparison #

The parallel is close enough to be worth setting out in full, because the two cases together are stronger than either.

A jersey’s bending pushes it wider and longer, and its geometric ceiling on extension is around three hundred per cent — where the straight line between two interlacings reaches the yarn between them. A real jersey jams near a hundred, so the ceiling is three times beyond anything observed and the difference is courses coming into contact, which the model does not have.

A rib’s bending pushes its beds apart, and its geometric ceiling is sixteen yarn diameters of gap. A real rib is knitted at two or three, so the ceiling is five times beyond anything observed — and here the difference is not contact, because opening a rib separates things.

Two ceilings, both computed exactly, both unreached, and only one of them with a candidate explanation. That asymmetry is what makes the rib the cleaner test.

What relaxation actually does to a rib #

The trade’s observation is that a rib relaxes narrower and thicker: it pulls in across its width, its wales fold further past one another, and the fabric gains bulk.

Neither half of that is predicted here. The pull-in is the fold closing, which is geometry once the wales’ positions are known and is not a force calculation. The thickening is the gap doing something, and the model’s only statement about the gap is that its energy wants it larger.

So the model and the observation are not in conflict — they are about different things. The model says which way the bending pushes; the observation says where friction and setting leave it. The whole content of several rungs on this ladder is that those are different questions and that the second one is not answerable from the first.

What the model does say about relaxation #

Two things, and they are worth having.

The direction of the forces at any stated gap. At a three-diameter gap a rib’s loops hold 23,086 nanojoules a stitch and press their interlacings with 34.8 millinewtons along the wales and 10.1 through the thickness. Those are computable at every gap and they are what a relaxation would have to overcome or be helped by.

And that relaxation makes it worse, not better. A knit’s bending energy rises through the three relaxation states — dry, wet, fully relaxed — because relaxation is the yarn moving to where friction lets it stop rather than to where its energy is least. There is no reason for a rib to be different, and every reason to expect the same.

The false hope, named #

There is a feature in the gap sweep that looks like an answer and is not, and it is worth disposing of explicitly.

The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters. A quantity with a minimum in a sweep is exactly the kind of thing that gets pressed into service as an explanation of where a fabric settles.

It cannot be. A fabric settles where its energy is stationary, not where a particular force component is least — and the energy is monotone through that gap and everywhere else. The dip is a feature of how the through-thickness component is shared between two half periods, and a fabric does not minimise a component.

This collection has retired a headline result for exactly this shape of error before: a predicted crimp ratio that turned out to be the minimum of an energy over the wrong set of states. Naming the set a minimum is taken over is the whole of the work, and here the set the fabric can move over does not have a minimum in it at all.

What would settle the gap #

Three things, in increasing order of how much work they are.

Measure it. A relaxed rib’s bed gap has never been published in a form this collection could quote, and a section photomicrograph of a relaxed rib would supply it directly. That is the cheapest and the model would then have a number to be checked against rather than swept over.

Add contact. Give the model solid yarn that cannot pass through itself and the jersey’s extension ceiling comes down to something realistic. It does not fix the rib’s gap, because opening a rib brings nothing into contact — but a model with contact in it could be trusted about the cases where contact does bite.

And measure the set fraction. Two experiments would separate it — cutting a wale and watching what happens, and boiling an unloaded fabric — and neither has been run. Everything on this ladder that cannot be pinned down is waiting on that one quantity.

What this rung is for #

It would be reasonable to ask why a negative result needs its own rung when the jersey’s version has one already.

Because the two-bed account is new and somebody will read its energies as predictions. Every table in it moves monotonically with the bed gap, and a reader who takes the energy column at face value would conclude that the machine should be set as wide as possible.

The energy column is not advice. It is what the bending does, in a fabric where the bending does not decide the dimensions. Saying so in the same voice as the results is the only way to stop the results being misread, and it is the reason this ladder’s negative rungs are written at the same length as its positive ones.

What a set yarn does to the picture #

Everything above is at the free end of the setting bracket — an unset yarn, a straight rod bent into a loop and pressing to get out of it. It is worth asking what changes at the other end, because that is where a real fabric is.

Nothing about the shape of the curve. Setting scales every force and every energy by one minus the set fraction, uniformly, so a curve that falls monotonically goes on falling monotonically at a smaller amplitude. A fully set fabric’s bending energy is flat in the gap because it is zero everywhere.

Which makes the refusal stronger rather than weaker. If the model’s energy had a minimum, a reader could argue that setting moved it. It does not have one at any degree of setting, so the conclusion — that the gap is not decided by bending — holds across the whole bracket rather than at one end of it.

That is the same structure as the friction balance, where the set fraction cancels out of the ratio exactly and the balance is therefore consistent with any degree of setting whatever. A quantity that cancels is a quantity a measurement cannot reach, and this is the second place on the ladder where that has been the answer.

What a machine setting is, as against a fabric dimension #

There is a distinction running through this whole two-bed account and this is the right place to state it plainly.

A machine setting is imposed. The bed gap on a knitting machine is where the beds are, and the fabric on the needles has no say in it.

A fabric dimension is what is left after the fabric has been taken off and allowed to move. It is what this collection means by a relaxed dimension and it is the thing every relaxation constant is about.

The two are different numbers and the account here uses the first everywhere, because the second has never been published. So every figure in the two-bed ladder should be read as what a fabric would do at a stated gap, not as what a fabric taken off a machine set to that gap actually does — and the difference between them is exactly the relaxation this rung says nothing can predict.

What is genuinely new here #

Two numbers and a caution.

A rib’s geometric ceiling is a bed gap of sixteen yarn diameters, computed exactly from the loop length and the fabric’s plan, and it is five times anything a machine is set to.

And nothing brings that ceiling down, because opening a rib separates things rather than bringing them together — which makes the rib’s relaxed gap a purer test of setting and friction than the jersey’s relaxed width is.

The caution is that the model’s monotone energy is not a recommendation and the through-thickness dip is not an equilibrium.

What the pictures cannot show #

Neither curve here has a stationary point on it, and a reader looking for where a fabric settles will not find it. That absence is the content of the page, and an absence is hard to draw.

The gap sweep has a visible minimum, which is exactly the feature this rung warns against reading as an equilibrium. It is drawn because it is real and because it is checkable; it is not drawn as an answer.

What is genuinely new here, restated as a list #

A geometric ceiling on the bed gap, at sixteen yarn diameters, computed exactly from the loop length and the fabric’s plan.

A demonstration that nothing brings it down. Opening a rib separates rather than compresses, so the contact this collection’s other ceilings are waiting for does not apply.

And a warning about the one feature in the sweep that looks like an equilibrium and is not. The through-thickness force has a minimum at two diameters, and a minimum in a force is not a minimum in an energy — it is a stationary point of the derivative, which is an inflection in the thing that would actually be minimised. A fabric does not settle there and nothing in the model says it should. The two quantities sit in adjacent columns of the same table and are one differentiation apart, which is exactly the sort of adjacency that gets misread.

What the list amounts to is a negative result with a boundary on it, which is the most a model of this kind can honestly produce here. The ceiling is real and computed; the mechanism that would pick a point below it is absent; and the one feature that looks like that mechanism belongs to a different quantity. A rib’s relaxed gap is set by friction and by setting — by what the yarn has been persuaded to remember — and this collection holds no model of either.

The three quantities that do not wait on the gap #

It would be easy to read this rung as saying the two-bed account is held up, and most of it is not.

The crossing share is a count over a grid and needs no gap at all. The flatness balance is a second count and needs none either. And every ratio between two structures at one gap is a comparison the missing number cancels out of.

What waits on the gap is the absolute value of a two-bed fabric’s thickness, its warmth, its density and its forces — and those are exactly the quantities reported as sweeps rather than as values throughout. The discipline of refusing a default is what keeps the missing measurement visible instead of buried in a table.

What this leans on #

That the energy falls monotonically with the climb, which is the single-bed result extended by one parameter and follows from the chord lengthening.

That the geometric ceiling is a chord condition, which is the same arithmetic as the jersey’s extension ceiling and needs no force at all.

And that a relaxed knit is not at a minimum, which is the finding this whole ladder’s account of relaxation rests on and is supported by the three relaxation states coming out in the wrong order for a springing yarn.

If any of the three were wrong the conclusion would change. The third is the one with a measurement behind it.

Where the ladder goes next #

The gap that is not predicted is the one every two-bed number depends on, so it is the standing gap in the whole account: where a two-bed fabric’s yarn is sweeps it rather than assuming it, and how dense a knitted fabric is shows what a factor of three in it does to a fabric’s bulk.