A second bed changes what a float is

A float is a length of yarn lying on a surface with nothing holding it down. Six rungs of one anchor rest on that sentence, and the knitted rung rests on it too: a missed needle takes no yarn, so the yarn passes straight across the back of the fabric, and the trade and this site agree where it ends up.

Both halves of the sentence have been true because of an assumption nobody wrote down. Every knitted structure on this site has been single jersey — one bed of needles, one plane of loops, and a fabric with a face and a back and nothing between them. There is nowhere else for the yarn to be.

A rib machine has two beds facing each other, and the yarn of every course runs through the gap between them.

The notation, extended by exactly one thing #

The array survives the second bed intact and needs one addition.

A needle position now exists twice — once on the front bed and once on the back — and each may knit, tuck or miss on each course. So a two-bed structure is a pair of arrays over the same three symbols, on the same number of courses and the same number of needle positions, and every symbol means exactly what it meant on one bed.

The addition is not a symbol. It is the gating: how the two beds’ needles line up, which is a property of the machine rather than of the structure written on it. Rib gating sets the back needles half a pitch along from the front ones, so a back wale sits in the gap between two front wales; interlock gating sets them opposite, sharing a needle space. That single number — half a pitch, or none — is what rib and interlock differ by at the machine, and it is the whole of what the pair of arrays does not carry.

It has one consequence that is a machine constraint rather than a convention. Under interlock gating the two needles at a position occupy the same space, so at most one of them may be in action on any course. An array asking for both is refused, which is the two-bed version of the tuck limit — a condition on the machine rather than on the cloth.

A needle that misses for ever is not a needle that fails #

One condition had to change and the change is the reason a second bed can be written down at all.

On one bed the site’s rule is that every wale must knit at least once in the repeat, or its needle never casts off and the held loop piles up until something breaks. A column with no K in it is a failure.

On two beds that rule makes single jersey inexpressible, because a bed out of work is a bed whose needles miss every course. A needle that misses on every course of the repeat takes no yarn at all. It is not holding a loop; it has none, and nothing accumulates on it. It is simply not in the machine’s action, which is exactly what an operator does when a rib machine is set to knit plain.

So a needle here is in work if it takes yarn at least once — by knitting or by tucking — and a needle in work must knit. A needle that never takes yarn carries no wale and is asked nothing. That is a real disagreement with the single-bed rule and not a bug in either: one is about a needle holding a loop, the other about a needle holding nothing, and the difference is whether the needle was ever put into work.

The practical payoff is that everything the single-bed family already draws is now a two-bed structure with one bed idle, and the arithmetic agrees to the last bit — checked rather than assumed, on all nine of the site’s named single-jersey structures that are fabrics.

Three places a float can be, where there used to be one #

Every miss floats the yarn past a needle, and the yarn of a course runs in the gap between the beds because that is where the needle hooks meet it. So the question of where a float lies has an answer with three cases and no others.

Both ends on the front bed. The yarn lies against the gap side of the front fabric — its technical back. If any back-bed wale is spanned, the back fabric closes over it when the cloth relaxes and the float is interior. If there is no back wale across it, it is exposed on the back of the cloth. That second case is single jersey, and it is the only one the site has ever drawn.

Both ends on the back bed. The mirror image: interior if a front wale is spanned, otherwise exposed on the face.

One end on each bed. The yarn crosses the gap, so it is inside the cloth by construction. Interior, always.

The rule is short enough to be suspicious of, so it is worth saying what makes it exact. Interior does not mean buried by the yarn’s own thickness or hard to see. It means a wale of the opposite bed lies across the float, so that when the two sets of wales close on each other the float has fabric on both sides of it — a statement about the array, decidable from the array.

What was counted, and how #

Three symbols on two beds over two needles and two courses is 3⁸ arrays, which is 6,561, and every one of them is walked.

1,135 of them are fabrics — 17.30 per cent. The failures sort into two heaps and they are wildly unequal: 5,265 arrays have a wale that takes yarn and never knits, and 161 have a course that never knits at all. The first heap dwarfs the second for the same reason it does on one bed — a wale is a column of two symbols and a course is a row of four, so a course has twice as many chances to contain a K.

Those 1,135 fabrics carry 1,272 floats between them, and 1,248 of them are interior. That is 98.11 per cent. Of the 910 fabrics that have any float at all, 890 have nothing but interior floats.

The twenty-four exposed floats are worth naming, because they are what the rule does not say. They are the structures where two wales of one bed sit next to each other with no wale of the other bed between them — a wide rib, or a bed with needles taken out — so a float between them has nothing behind it. They are rare here because a two-needle repeat has very few ways to leave a gap that wide.

Four claims that are void inside the cloth #

This is the part that matters, and the ladder is explicit about which rungs it takes out.

Lustre is void. A float shines because it is straight — an uninterrupted run of parallel yarn reflects specularly where an interlaced surface scatters. The argument needs a straight length of yarn and an eye, and an interior float supplies the first and is not available to the second. A rib jacquard’s floats contribute nothing whatever to the fabric’s appearance, and the sheen a float jersey gets on its back has no two-bed equivalent.

Abrasion is void. Wear happens at the high points, and a float is a high point. An interior float is on no surface, so it is not a high point of anything and takes no rubbing at all. This one inverts rather than merely vanishing: on single jersey the float is the first thing to wear, and on a rib the float is the last, because everything that touches the abradant is a loop.

Snagging is void. A float catches because something can get under it over its unsupported length, and nothing gets under a length of yarn with fabric on both sides. A rib can carry floats far longer than a single jersey before it snags — a genuine engineering freedom, and the reason two-bed jacquards float across colour repeats that would be unusable on one bed.

Raising is void. Only a float can be raised, because a raising wire has to catch a length of yarn that is not tied down. An interior float is not tied down and cannot be caught, so a two-bed structure cannot be raised on the strength of its floats however long they are. That is the sharpest of the four, because it is the one where the trade’s rule of thumb — long floats raise well — is not merely weakened but reversed in its precondition.

What survives is everything about length and weight. An interior float still removes a loop and still takes only its own run of yarn, so the yarn arithmetic is untouched by where the float lies. The split is the useful one: the optical claims about a float need it to be on a surface and the arithmetical ones do not.

What a miss costs, once there are two beds #

There is a second consequence and it lands in the arithmetic rather than in the optics.

The single-bed result is one of the cleanest on this site: a missed needle saves exactly (k_w − 1)/k_w of a loop — 76.7 per cent — every time, on every structure, because it is one subtraction repeated. Nothing about the arrangement enters it.

On two beds it is not one subtraction. Missing a back-bed needle in a rib removes its loop and replaces two sinker loops crossing the gap with one straight run along the front bed, so the yarn saved is the loop plus the detour that is no longer taken. Differencing a full one-by-one rib against the same rib with one back needle missing gives 4.536 wale spacings, which is 105.5 per cent of a loop — more than the whole loop that was removed.

And it moves. The saving depends on how far apart the beds hold their planes of loops, which the single-bed number never did: 95.4 per cent of a loop at three quarters of a needle pitch, 105.5 at one, 127.0 at one and a half. Every figure here states the separation it was drawn at. A number that was an identity on one bed is a number with a machine dimension in it on two, and that is the honest cost of the second bed.

Where the model stops #

The section is drawn at the machine. The upper panel of every section figure here has the beds held apart, which is where the needles are and is not where the fabric is a moment later. The lower panel closes them, and it closes them by shrinking a gap rather than by folding a concertina — which is what a relaxed rib actually does and what the rung below models with a cosine. The two models are not joined here.

Interior is a topological statement, not an optical one. A float with a wale across it is covered; how well it is covered depends on how tightly the fabric relaxed, how thick the yarn is and how far the beds were apart, and none of that is here. A slack rib knitted very open has floats that are interior by this classification and perfectly visible in the hand.

The bed gap is a stated machine dimension. It is not measured on any fabric, it is quoted with every figure, and it enters the yarn arithmetic and nothing else. The float classification does not depend on it at all, which is worth saying: whether a float is interior is decided by which wales lie across it, and no distance enters that question.

And nothing here computes the dimensions. A two-bed structure’s relaxed width and length are a measurement this site does not have, for any structure, and the essay after next is about exactly how far that shortfall reaches. The rung below draws the same fabric a second way — a rib in section as a folded plan, where the extension is the fold opening out — and the two pictures cannot answer each other’s questions: one asks how wide the fabric is and the other asks where the yarn is. Putting the fold angle and the bed gap into one model is not done here.

The generalisation, which is about backed cloths #

The woven side of this site has met the interior float already and did not call it that.

A backed or stitched cloth has two thread systems on the reverse and a binding system tying them to the face, and the long floats of the backing yarn lie between the two layers where nothing touches them. Everything said above about a rib’s floats is true there, for the same reason, and the woven trade has always known it: a backing weft is chosen for warmth and weight rather than for handle, because its handle never reaches a surface.

So the general statement is not about knitting at all. A float is exposed when it is on the outside of the outermost layer, and a structure with more than one layer has floats that are on the outside of none of them. Any construction with a thickness has them — backed cloths, double cloths, pile grounds, and every two-bed knit — and the site’s float arguments have been written as though a cloth were a surface because until this rung every cloth it drew was.

That is the connection worth taking away, and it is uncomfortable rather than tidy: the float ladder’s optical rungs are not wrong, they are conditional on a hypothesis about layers that no draft and no array records.

Who found it, and when #

The knitting trade has never needed the word. A rib jacquard designer knows that the float is on the inside and says so in those terms, and the technical vocabulary has a phrase for the effect rather than for the object — a two-bed jacquard’s reverse is described as clean, which is a way of saying the floats are somewhere else without saying where.

The oldest machine that exploits it is not a rib machine. Tubular knitting on two beds — one course on the front, one on the back — puts every float inside the tube, and hosiery has been made that way since the frame had two beds to put together. Nobody wrote it down as a fact about floats, because there was no arithmetic of floats to write it into.

What is new here is only the bookkeeping: that the case is decidable from the array, that it is the usual case once there are two beds, and that four of the site’s own arguments have a precondition they never stated. The measurement — ninety-eight floats in a hundred — turns a remark into a rung. Before it this was a caveat somebody might add to a paragraph; after it the caveat is the general case and the single-bed float is the special one.

The shortfall this closes was recorded in writing at the end of the float in a knit, which said that the two-bed structures were absent and that a rib’s float lying inside the fabric breaks the lustre and abrasion arguments entirely. It does, and now the site says which ones and by how much. What it still does not say is what an interior float does — whether it stiffens the cloth, whether it carries load, whether it wears against the wales it lies between — and that is the same missing force the float ladder has been short of for six rungs.

Where the ladder goes next #

The array now describes a fabric with two beds in it, and the site’s central question has not yet been asked of one. The next rung asks whether a two-bed structure is one fabric or two, finds that the answer is a connectivity question exactly as it is for a weave, and finds by enumeration that the criterion anybody would state at a machine is wrong on sixty-two arrays out of eleven hundred.