A rib climbs a gap

This collection has counted two-bed fabrics for several rungs. It knows which of them hang together, how much yarn a repeat holds, which needle takes yarn on which course, and what a float is once there are two beds for it to run between. It has never said a word about what any of them costs in force, and it said so in as many words: no dimensional constants for a two-bed structure, because none are published that could honestly be quoted.

The missing piece was never the constants. It was that a knitted loop was solved as a flat curve, and a rib is the one fabric whose whole character is that its loops are not in one plane.

The whole of the model #

A half period of a course of yarn runs from a crest to a trough. It travels half a wale spacing along the fabric, a course spacing and a diameter across it, and some distance through it.

In a single-bed fabric that last distance is one yarn diameter, because a loop’s feet were drawn through the head below and are on the far side of it. That is not a choice; it is what an interlacing of two threads of one diameter is.

In a two-bed fabric the yarn also has to get from one bed to the other, and the sinker loop that does it travels the gap between the beds. The two half periods either side of that sinker share the journey, and each of them is also carrying the interlacing’s own diameter — one climbing with it and one against.

So the climb of a half period is a diameter plus or minus half a bed gap, and everything else about the solve is unchanged. A rib is the jersey model with a bigger number in one place.

Which is a stronger claim than it sounds #

It would be easy to read that as a simplification, and it is the opposite: it is a claim that no new mechanism is needed.

A rib is usually described as a fabric that folds, and folding sounds like a different kind of behaviour from anything a jersey does. It is not. The wales on the two beds sit on opposite faces because the yarn crossed between the beds; the fabric corrugates because those wales are on opposite faces; and the yarn crossed because a sinker loop had a gap to span. Every one of those is a consequence of the climb, and the climb is one length.

Nothing here says a rib is simple. It says the complication is topological rather than mechanical, and the topology was already counted.

Where the climbs come from #

They are read off each structure’s own traverse rather than assumed.

One course of yarn is one pass across both beds, and it meets the wales in a definite order — front bed, back bed, front bed, and so on for a one-by-one rib; two of one then two of the other for a two-by-two. A bed change between consecutive wales is a crossing, and the climb profile of any structure is the list of those changes with the interlacing’s diameter added to each half period.

That works for structures nobody has a rule for as well as for the ones everybody knows, and the whole census is a tenth of a second’s arithmetic. A one-by-one rib comes out with every sinker loop crossing; a two-by-two with half of them; a half-milano with two thirds; a fabric with three front wales to one back wale with four sevenths.

And it comes out right where the answer is not in doubt. Single jersey has no crossings at all, because its back bed is out of work. A tubular fabric — one course knitted on the front bed, the next on the back — also has none, because its two faces are made on separate courses and the yarn never crosses between them within a course. That is exactly what a tube is, and a criterion that got it wrong would be worth nothing.

The one number that is not available #

The bed gap is a machine setting. On a machine it is set by how far apart the two needle beds are and how the cams are timed; in a relaxed fabric taken off the machine it is whatever the yarn and the finishing have left it at, and nobody has published that measurement in a form this collection could quote.

So the gap is an argument here with no default that means anything, and every figure of a rib either sweeps it or says in its own caption what it was drawn at. That is the same discipline the counting rungs already used for a two-bed fabric’s relaxed spacings, and for the same reason: a number computed at an invented measurement is a number about no fabric.

What is available is the comparison. Hold the dimensions fixed at a jersey’s, change only the climb, and the answer is a statement about what the topology does — which is the comparison worth having and the only one that can be made honestly.

What the topology does #

At a bed gap of three yarn diameters, for the 20 tex cotton at a 3.5 mm loop this collection quotes throughout:

The energy falls by five per cent, for the reason the single-bed ladder established: a longer climb lengthens the straight line the thread has to span, so it leaves less slack to be spent on curvature. The force along the wales falls with it. The force through the thickness rises by nearly a third — the component that holds a knit open — and the fabric it is holding open has doubled, which is how thick a knit is’s other case.

None of that needed a new mechanism, a fitted constant or a measurement.

The steep half period and the shallow one #

There is a detail in the arithmetic that turns out to matter, and it is the reason a rib does not behave like a jersey with everything scaled up.

The two half periods either side of a crossing do not climb equally. One of them climbs half the gap plus the interlacing’s diameter and the other climbs half the gap minus it. At a gap of three diameters that is 0.418 mm against 0.084 mm — a factor of five — and the tilts of the two are 27.5° and 5.9°.

So a course of yarn in a rib alternates between a steep half period and a shallow one, twice per repeat. The fabric’s average tilt of about sixteen degrees is not a description of any piece of yarn in it; it is an average over two very different pieces.

That is not a curiosity. It is why the through-thickness force does something unexpected as the gap is opened, and a rib is quietest at two diameters is what.

Why the dimensions are a jersey’s, said plainly #

Every number in the table above uses single jersey’s relaxed spacings, and a real rib’s are not those. A one-by-one rib relaxed off the machine is much narrower than a jersey of the same yarn and loop length, because alternate wales fold to opposite faces and each hides the one beside it.

That is a real objection and there are only two answers to it. One is to invent a rib’s spacings, which is not available. The other is to hold the spacings fixed and say so, which makes every comparison a statement about the climb alone.

The second is what is done here, and it is worth being clear about what it buys and what it does not. It buys the direction and size of the effect of a second bed at a fixed geometry. It does not buy a prediction about a particular rib, and any number here read as one would be wrong by however much a real rib’s spacings differ from a jersey’s.

The check that the profile has to pass #

A climb profile is assembled from a traverse order and a table of which bed each wale is on, and either could be read wrongly without the result looking odd. So there is a condition it has to meet and it is not a tolerance.

A course’s climbs must add to zero over the repeat. One course of yarn is a closed traverse across the width: it starts on some bed, visits every wale in work, and comes back to where it began. A repeat whose climbs did not cancel would describe a fabric drifting steadily through its own thickness, which is not a fabric.

Every structure in the list satisfies it exactly — to the last bit of a double-precision sum, since the climbs are made of the same two lengths added and subtracted. That is a weak check in the sense that a great many wrong profiles would also pass it, and a strong one in the sense that the commonest way to get a bed table wrong fails it immediately.

The thickness, which is the exception #

One row of that table is not a comparison at fixed dimensions. It is a prediction.

A single-bed fabric’s thickness is two yarn diameters, because its interlacing passes at a diameter. A two-bed fabric’s is the gap plus a diameter — the two loop planes, plus a radius on the outside of each.

That does not depend on the fabric’s spacings at all, so the objection above does not touch it. It says a rib on a three-diameter gap is twice as thick as a jersey of the same yarn, a rib on a five-diameter gap three times, and a rib is thicker than a jersey by a factor set entirely by the machine rather than by the yarn or the gauge.

Anybody who has handled both fabrics knows the conclusion. What is new is that the factor is a machine setting rather than a property of the structure.

What a crossing costs in yarn #

The counting rungs already know how much yarn a two-bed repeat holds, because a traverse is a walk and its length is a sum. What the climb adds is where that yarn is.

A crossing’s half period spans a longer straight line than a jersey’s — at a three-diameter gap, 0.993 mm against 0.916, out of the 1.750 mm of yarn each of them holds — so it uses up more of its slack going somewhere and less of it bending. That is the same statement as the energy falling, seen as a length rather than as an energy, and it is the more useful form for anybody thinking about what a machine has to feed.

It also says which way a rib’s yarn consumption moves when the beds are opened: not at all, at a fixed loop length. The loop length is set by the cams and the yarn feed; the gap decides what the yarn does with itself, not how much of it there is. A rib knitted on a wider gap at the same stitch length is the same weight of fabric arranged differently.

What happens to the fold #

The folding that gives a rib its width behaviour is not in this model, and the reason is the one that has been true of every relaxed dimension on this ladder.

A rib relaxes to about half the width it is knitted at because each wale hides the one beside it, and that is geometry rather than force. What force would be needed for is to say where the fabric stops, and this collection established some rungs ago that a relaxed knit does not stop where its bending energy is least — it stops where friction and the yarn’s setting leave it, on a slope in both directions.

That is as true of a rib as of a jersey. The energy of a rib falls monotonically as its bed gap opens, exactly as a jersey’s falls as its wales spread, so the model has no more to say about a rib’s relaxed gap than it has about a jersey’s relaxed width. A rib’s relaxation is not its bending either is where that is followed through rather than waved at.

What the second bed does to the friction balance #

The condition for friction to hold a relaxed knit along its wales came out at the friction coefficient exceeding a half, with everything else cancelling — and then, once the force was allowed out of the fabric’s plane, at a half times the cosine of the loop’s tilt.

A rib’s tilt is much larger, so the cosine is much smaller: at a gap of three diameters the average tilt of about sixteen degrees puts the requirement at about 0.48, and at five diameters nearer 0.44. That is still above what any fibre in this collection’s table supplies, so the conclusion is unchanged — friction alone cannot hold a relaxed knit along its wales, on one bed or two.

But the direction is worth having. A rib is measurably closer to holding itself than a jersey is, and the reason is that more of its contact force is pointing through the fabric where it is not being asked to hold anything.

Why the same solve works for both #

There is a reason to be slightly suspicious when one model covers two fabrics that feel unalike, so it is worth saying what the two have in common that makes it legitimate.

Both are made of one continuous yarn laid in courses. Both have the yarn free between interlacings and constrained at them. Both have interlacings that pass at a yarn diameter, because that is what two threads in contact are. And in both, the two ends of a half period are extreme points of the wave, so the thread leaves and arrives along the course direction.

That last one is what licenses the solve. It is the condition under which the third dimension is a rotation, and it holds on a rib for exactly the reason it holds on a jersey: a crest is a crest whichever bed it is on.

What would break it is a tuck — a stitch where the yarn is held over an old loop rather than knitted through it, so it arrives at an angle rather than at an extreme point. The cardigans in the list above are full of tucks, and their numbers here treat every stitch as a knitted one. That is a stated simplification rather than an oversight, and it is the first thing to correct if any of those fabrics is taken seriously.

The structures the counting could not reach #

The value of reading the climbs off a traverse rather than quoting them shows up on the fabrics nobody writes rules for.

A half-cardigan — a rib with the back bed tucking on alternate courses — has every half period crossing, like a plain rib. A milano has half of them. A composite that both tucks and floats, written for no other reason than to have something with no published constants, has six sevenths.

Those are computed from the structure matrix in a tenth of a second, and each of them then has an energy, a pair of contact forces and a thickness. Whether they are right is a separate question and one this collection cannot settle without measurements it does not hold. What it can say is that the arithmetic no longer stops at the fabrics somebody happened to publish constants for.

What this does not settle #

Four things, listed because each is easy to assume from the numbers above.

A rib’s relaxed width. Not here, not anywhere on this ladder, and for the reason relaxed dimensions have never been available: the model has no interior minimum to stop at.

Its recovery force. The force a cuff pulls back with is a fabric-scale quantity computed from an extension curve, and it is unchanged by anything here.

Which gap a machine should be set to. The energy falls monotonically as the gap opens, so the model would recommend opening it for ever, which is plainly not advice. What stops it is the yarn running out — a chord cannot exceed the thread that spans it — and that limit is geometric rather than energetic.

And anything about a tuck. Every stitch here is a knitted one.

What is genuinely new here #

Three things.

A rib has forces. Until now this collection could count two-bed structures and could not price them.

The bed gap is the parameter. One length separates a jersey from a rib mechanically, it is read off the machine rather than the fabric, and every quantity here is a function of it rather than of a structure’s name.

And the thickness is predicted rather than compared. A two-bed fabric is the gap plus a diameter thick, whatever its spacings, which makes it the one number here that a measurement could contradict outright.

What the pictures cannot show #

The rib section is drawn at a bed gap of three diameters because a picture has to be drawn at something, and a reader could take that as the gap of a rib. It is not the gap of any rib; it is a plausible middle of a range that has not been measured.

The section is also drawn with the yarn at its true width and the wale spacing at a jersey’s, so the crossings look tight. On a fabric relaxed to half its knitted width they would be tighter still, and the picture would be much harder to read — which is the usual trade between a drawing that is legible and a drawing that is to scale.

Where the ladder goes next #

Three questions follow the climb.

The two half periods either side of a crossing climb by very different amounts, and at one particular gap the shallower of them climbs nothing at all: a rib is quietest at two diameters.

The crossings can be counted for any structure, which turns the question of which fabrics lie flat from a description into a sum: which knitted fabrics lie flat.

And a fabric with a doubled thickness and a doubled through-thickness force is a different object under a hand, which is where how thick a knit is picks the question up.