Where a two-bed fabric's yarn is

A structure written on two beds is a pair of grids of letters, and it is entirely determined by them. Everything mechanical about it in this account follows from one derived quantity: how much of its yarn crosses from one bed to the other, and therefore how far that yarn climbs through the fabric.

That quantity has never been counted here. This rung counts it, for every named structure the collection holds, and the answer sorts them into three groups with two surprises in it.

What is being counted #

One course of yarn is one pass of one carrier across both beds. It meets the wales in work in a definite order — that order is what makes a course a single continuous length of yarn, and it is the fact every connectivity argument on this ladder rests on.

Walk that order and note where the bed changes. Each change is a crossing: the sinker loop between those two wales has to travel the whole distance between the beds. Each half period of the wave either sits either side of a crossing or it does not, and the crossing share is the fraction that does.

Nothing in that count is a measurement or a convention. It is a walk over a grid of letters.

It is worth saying what the count is not reading, because a grid of letters supports several different walks. It is not counting knits, or tucks, or floats, or needles, or courses. Those are all counted elsewhere on this ladder and each answers a different question — how much yarn a repeat holds, whether the thing can be knitted at all, whether it hangs together. This walk asks one thing: between one wale and the next, did the yarn change beds?

The three groups #

Zero. Single jersey, and a single jersey with a float in it. Both have their back bed out of work, so there is no other bed to cross to. And one more, which is the first surprise and is dealt with below.

A half to a whole. Every rib, every cardigan, both milanos, interlock, and the composites. A one-by-one rib crosses at every sinker loop, so its share is exactly one. A two-by-two crosses at every second one, so its share is a half. A half-milano is at two thirds; a fabric with three front wales to one back wale is at four sevenths; a rib that both tucks and floats is at six sevenths.

And nothing in between zero and a half, in the thirteen structures held here. That is a fact about the sample rather than a theorem, but it says something: a fabric either uses its second bed or it does not, and the structures that use it use it a great deal.

The first surprise #

A tubular fabric — one course on the front bed, the next on the back — has a crossing share of zero.

That is not a defect in the count; it is exactly right, and it is what a tube is. The two faces are made on separate courses, so no course of yarn ever crosses from one bed to the other. The two layers are joined at the selvedges and nowhere else, which is precisely why a circular machine knitting this structure produces a tube rather than a double fabric.

So the count separates a tube from a rib without being told which is which, on a criterion that never mentions either.

The same count also puts a single jersey with a float in it at zero, which is worth a moment. A float is a length of yarn running across the face of the fabric past needles that missed, and it is the longest unsupported piece of yarn in any weft-knitted structure. It does not cross beds, so it does not climb, and in this account it is mechanically identical to the fabric it is written on. Everything that makes a float matter — snagging, the distance it spans, what happens when it catches — is a fabric-scale question that lives on a different ladder.

Which makes its thickness undefined #

The count’s consequence for a tube is sharper than the count itself, and it is the reason this rung reports one structure’s thickness as unavailable rather than as a number.

Every other structure here has a thickness that follows from its geometry: two loop planes a bed gap apart, with a radius on the outside of each. A tube has two loop planes too — but nothing in any course of its yarn decides how far apart they lie. The layers rest on one another because of gravity and a finishing process, not because a thread runs between them.

Reporting a jersey’s thickness for a tubular fabric would be reporting a number about one of its two layers as though it were the fabric. Reporting twice that would be assuming the layers touch. So the honest answer is that this model does not have one, and it is returned as an absence with the reason attached rather than as a plausible figure.

That is the same discipline the counting rungs already apply to a two-bed fabric’s relaxed spacings. An absence with a reason is worth more than a number nobody can defend.

What the share predicts #

The crossing share, the bed gap and the yarn are between them enough for every mechanical quantity on this ladder. At a gap of three yarn diameters, with a 20 tex cotton at a 3.5 mm loop:

Every column is monotone in the share, which is not a coincidence and is not a result either: the share is a weighted average over two kinds of half period, and each quantity is an average of the same two values, so a monotone relation is arithmetic rather than physics.

What the table does say is that a structure’s mechanical position is fixed by one number. Two structures with the same crossing share have the same energy, the same forces and the same thickness in this account, whatever else is different about them.

The second surprise: milano and a two-by-two rib #

Those two fabrics are nothing alike. A two-by-two rib is a rib: every course knits on both beds, and the wales alternate in pairs. A milano is a sequence — a rib course, then a course on the front bed alone, then a course on the back bed alone — and it is used precisely because it behaves unlike a rib.

They come out with the same crossing share, the same energies and the same forces.

That is not the model failing to distinguish them; it is the model saying something specific. The two fabrics put the same fraction of their yarn across the gap, and everything this account computes is a function of that fraction. What separates them in practice — a milano’s much lower extensibility, its different appearance, its stability — comes from which courses cross and where they sit relative to one another, and that is a fabric-scale question the per-stitch arithmetic does not reach.

Where the two do differ in this account is in the number of crossings per course rather than the share: a two-by-two rib crosses twice a course, a milano one and a third. The difference is that a milano has courses which do not cross at all, and a rib does not. That is exactly the distinction a reader would draw, and it is available; it is just not the quantity that drives the mechanics.

Interlock, which is the awkward case #

One structure in the list sits uncomfortably and it is worth saying so rather than letting the table imply otherwise.

Interlock is written here as two one-by-one ribs on interlock gating, each using half the needles of each bed. Its crossing share comes out at one, like a plain rib’s, and its energies and forces come out identical to a rib’s.

That is right as far as it goes and it misses what interlock is for. The reason interlock is used where stability matters is that its two rib fabrics are interknitted and cannot fold past one another, and folding is a fabric-scale behaviour that a per-stitch climb does not see. The count says the two fabrics put the same fraction of their yarn across the gap; it does not say they behave alike, and anybody reading the table as though it did would conclude that a rib and an interlock are the same fabric, which they are conspicuously not.

The connectivity argument is where interlock’s own peculiarity is actually adjudicated, and it is a different question asked of the same grid.

Where the count can go wrong, and how it is stopped #

The profile is assembled from a traverse order and a table of which bed each wale is on. Both can be read wrongly without the answer looking odd, so there is a condition it has to meet.

A course’s climbs must add to zero over the repeat. A course of yarn is a closed traverse: 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 through its own thickness, which is not a fabric.

Every structure here satisfies it exactly rather than to a tolerance, because the climbs are made from two lengths added and subtracted in an order that has to balance. It is a weak condition in that many wrong profiles would pass it, and a strong one in that the commonest ways of getting a bed table wrong fail it at once.

What a needle out of work is, and why it matters here #

The count depends on a distinction the counting rungs had to make and that is easy to miss: a needle in work takes yarn at least once in the repeat, and a needle out of work takes yarn never and holds nothing.

That is how a single-bed fabric is written on a two-bed machine — the back bed’s needles miss on every course, so they hold no loops and carry no wales. If they were counted as wales the traverse would visit them, the count would find crossings, and single jersey would come out with a crossing share of one.

It does not, because a needle that takes no yarn is not in the machine’s action. The distinction was made for the connectivity argument and it turns out to be load-bearing here too, which is a small piece of evidence that it was the right distinction rather than a convenience.

The composites, which are the point #

Three of the structures in this list exist for no other reason than to have something with no published constants: a rib that both tucks and floats, a fabric with three front wales to one back, a rib with a miss on the back bed.

Those are the structures a census is for. Anybody can quote a figure for a one-by-one rib. Nobody has ever published one for a four-course repeat with a tuck on the second course and a miss on the fourth, and there is no reason they should — there are more such structures than anyone could tabulate.

A census computed from the grid handles them at the same cost as it handles a plain rib: a tenth of a second, no lookup, no exception. The census is the answer to a question the tables cannot be asked.

What it costs to compute #

A tenth of a second for the whole family, which is worth stating because it decides what kind of tool this is.

A lookup table of published constants is a reference: it answers questions about the fabrics somebody measured and nothing else, and extending it means measuring. A count over a grid is a function: it answers the same question about any structure that can be written, at the same cost, including ones nobody has made.

That difference is the whole reason this collection counts rather than quotes wherever it can. It is the same reason its weave arithmetic enumerates every four-by-four draft rather than listing the named weaves — and it has the same limitation, which is that a function can only answer the question it was written for.

What this census does not say #

Three things, and each of them is the kind of claim a table like this invites.

It does not say which structure is better at anything. A crossing share is not a merit; it is a position.

It does not carry the fabric’s own dimensions. Every number above is computed at single jersey’s relaxed spacings, because a rib’s have never been published in a form this collection could use. So the columns are a comparison of topologies at a fixed geometry, and a real rib is much narrower than a jersey.

And it treats every stitch as knitted. The cardigans in the list are full of tucks, where the yarn is held over an old loop rather than pulled through it and therefore arrives at a crossing at an angle rather than at an extreme point of its wave. That breaks the condition under which the third dimension is a rotation, and it is the first thing to fix before any cardigan’s numbers are taken seriously.

The thing to disbelieve first #

If one of these numbers is going to turn out wrong, it is the assumption that a crossing’s climb is shared equally between the two half periods either side of its sinker loop.

There is no measurement behind that. It is the simplest thing that satisfies the closure condition — the climbs have to add up over the repeat, and splitting a crossing down the middle is the symmetric way to make them. A real sinker loop crossing a gap might do most of its climbing on one side, and if it does, the two half periods either side of it are further apart in tilt than the figures here say, and every average over them shifts.

What would settle it is a fabric-scale solve rather than a per-segment one: let the crossing find its own division of the climb by minimising the pair’s energy together. That has not been done, and it is the most obviously available next piece of arithmetic on this ladder.

Why a fraction rather than a count #

The census reports a share of half periods rather than a number of crossings, and the choice is worth a line because the two say different things.

A count is a property of a repeat, and repeats are written at whatever width the author found convenient — a one-by-one rib could be written across two needles or across twenty, and the count would change tenfold while the fabric did not.

A share is invariant under that. It is what fraction of the yarn is doing the crossing, which is a property of the fabric rather than of how somebody chose to write it down. The same rule is why the collection’s weave counts are done over minimal repeats rather than over drafts.

What is genuinely new here #

Two things.

A structural number for two-bed fabrics that was not available. Thirteen structures, each with a crossing share computed from its own grid, and every mechanical quantity on this ladder a function of it.

And an absence, named. A tubular fabric’s thickness is not a small number or a large one; it is not determined by anything in the fabric’s own course, and the model returns that rather than a plausible figure. Being able to refuse is what makes the other twelve numbers worth having.

What the pictures cannot show #

The bar chart puts thirteen structures in one order and invites a reader to see a spectrum. There is no spectrum. There is a group at zero, a group between a half and one, and nothing in between — and the ordering within the second group is an artefact of which structures somebody chose to name.

A census of every structure a four-course repeat can hold would fill the gap or would not, and it has not been run. What is drawn here is a list of fabrics with names, which is a biased sample of fabrics.

Where the ladder goes next #

The same grid of letters supports a second sum, over which bed each stitch’s yarn goes to rather than whether it crosses. That one decides whether the fabric lies flat, and which knitted fabrics lie flat is where it is done.

And the crossing share’s consequence for the through-thickness force is not the monotone thing the table suggests once the gap is allowed to move: a rib is quietest at two diameters.