The shog is the anisotropy

Two bars cannot share a beam established what an underlap is, for an arithmetic about beams: a straight run from a needle on one course to a needle s spaces away on the next, of length

$$u(s) = \sqrt{(s\,w)^2 + c^2}$$

with w the wale spacing and c the course spacing. That was used as a quantity of yarn.

It is also a constraint. The underlap’s length is fixed by the thread; the two spacings are not. So a warp knit’s underlaps put it on a locus, exactly as a woven cloth’s crimp puts it on one — and this one is a circle.

A quarter circle, because an underlap is straight #

Pull the fabric wider and the sideways span s·w grows. The underlap cannot grow, so the course spacing must shrink, and the relation is Pythagoras:

$$(s\,w)^2 + c^2 = u^2, \qquad u \text{ fixed.}$$

That is the same accounting as crimp interchange — pull a cloth one way and it narrows, because the extra length is taken out of one direction and put into the other — and it is simpler, because a crimped thread’s path is Peirce’s locus and an underlap’s is a straight line. A straight line between two points is a hypotenuse, and a hypotenuse of fixed length traces a circle.

The two extremes are the two axes. Laid flat, the underlap lies along the course, the fabric is at its widest and its course spacing is nothing. Stood upright, it lies along the wale, the fabric is at its longest and has no width at all. The as-knitted state sits between them.

And where it sits is decided by the shog. A short shog puts the underlap nearly along the wale to begin with, so most of its length is already spent on length and there is a great deal of width to gain. A long shog puts it nearly along the course, and the reverse.

The numbers, and they are not a tendency #

At 28 needles an inch and 14 courses a centimetre — an ordinary tricot machine — the wale spacing is 0.907 millimetres and the course spacing 0.714:

A tricot can be pulled 27.3 per cent wider and 61.6 per cent longer. A cord can be pulled 7.5 per cent wider and 173 per cent longer.

The last column is the ratio of the two ranges, and it spans a factor of 96 across four lappings a machine actually runs. That is not a material property, a yarn, a gauge or a finish. It is one integer.

The intuition it inverts #

The obvious expectation is that a longer underlap is more thread, and more thread is more stretch. It is more stretch — a cord’s 173 per cent against a tricot’s 62 — and it is more stretch in the wrong direction.

The reason is that an underlap’s length is spent on whichever direction it is already pointing. A cord’s underlap runs two needle spaces across and one course down, so it is already mostly horizontal and has almost nothing horizontal left to give. A tricot’s runs one across and one down, so it is at forty degrees and has room both ways.

So the width-way extension is decided by how far the underlap already leans, and leaning further costs width rather than buying it. A designer wanting a fabric that gives across its width wants the shortest underlap that still makes cloth — which, by the coprimality condition, is a shog of one.

That is the tricot, and it is the commonest warp-knit fabric there is.

The two drawings are the same machine at two settings of one integer, and the difference between them is not what a reader would guess from looking. The cord’s threads travel further, its picture is busier, and it is the construction that gives less. What separates them is an angle rather than a length: the cord’s underlap has already leaned most of the way over and has nothing left to lean with, and the tricot’s is halfway and can go either way. That is the whole of the anisotropy, and it is visible in the two pictures once the thing to look at is which way the diagonals point rather than how far they reach.

The machine moves the point and not the circle #

The as-knitted state sits at (1, 1) by construction, so the interesting question is where on its own circle it sits — and that is what the gauge and the course density decide.

The ordering by shog is the same on every machine — a longer shog always gives less width and more length — and the numbers move by a quarter across the range.

What moves them is the ratio of the two spacings, which is the angle the underlap lies at. A machine whose courses are close relative to its wales has its underlaps nearly horizontal and little width to give; one whose courses are open has them steep. So a course density is an anisotropy setting, continuously, alongside the shog’s discrete one — and it is the knob a mill turns when it wants a tricot to give a little more or a little less across.

That is a second lever nobody names as one. Course density is set for weight and for cover; it is also setting how the fabric moves, by up to a quarter of the width-way range, with no change of lapping and no change of yarn.

Which is why a warp knit is stabilised by a second bar #

What a second guide bar is for answered the question structurally: two shogs join the wales into $\gcd(s_1, s_2, \text{width})$ groups, so a pair that each fail alone can succeed together. This essay adds the mechanical half and it is the one a mill cares about.

A fabric with two bars lies on two loci at once, and it can only move where both allow. A tricot bar admits 27.3 per cent of width and a cord bar 7.5, so the pair admits 7.5 — the tighter of the two, because the cord’s underlaps come taut first and stop the fabric.

So a second bar at a longer shog is a stabiliser, and the amount of stabilising is computed rather than described: it is the ratio of the two widest factors, which for a tricot-and-cord pair is 1.273 against 1.075 and is a reduction of 78 per cent in the width-way give.

And it costs what the essay before it priced: a cord bar wants 111 metres more yarn per wale over a hundred-metre piece. That is the exchange rate between stability and yarn on a warp-knitting machine, and both halves of it are now numbers.

The same argument runs in the other direction for the length. A tricot bar admits 61.6 per cent of length and a cord bar 173, so the pair admits 61.6 — the tricot is the stabiliser lengthways. Neither bar is the stabiliser; each stabilises the direction the other is loose in, which is a better account of why two bars are standard than “more threads hold it together”.

Where the circle meets the woven locus, and where it does not #

Setting this beside the woven side is the comparison the account exists to make, and the two agree in shape and differ in one respect that matters.

Both are constant-thread-length loci. Crimp and interchange has a woven cloth narrowing when it is pulled because the extra length moves from one system to the other, with no fibre stretching; this has a warp knit narrowing lengthways when it is pulled across, for the same reason and with the same absence of strain.

Both are free and both are opposed by friction. Nothing on either locus costs a fibre anything, which is why a cloth gives back less than it took applies to both — the hysteresis is at the crossings.

And both have one equation too few. A woven cloth’s locus does not say where on it the cloth sits; that is the crimp ratio, which the geometry cannot supply. A warp knit’s does, and this is the difference: the as-knitted point is fixed by the machine, because the wale spacing is the gauge and the course spacing is a gear. Both are set rather than relaxed to.

So the warp knit has, for free, the thing the woven side has spent an account failing to get. It is not a better model; it is a simpler machine. A loom lets its cloth choose where on its locus to sit and a warp-knitting machine does not, and the whole of the difference is that a warp knit’s two spacings are needle positions rather than the outcome of a tension balance.

What is not on the locus #

The circle is a mechanism and not a total, and the distinction is the same one crimp interchange needs on the woven side.

The loops are not in it. A warp knit’s overlaps are loops, and a loop straightens under load like any other; the extension the loops contribute is additional to everything above and is not computed here, because this account has no loop model that survives being loaded — A knit’s dimensions come from its loop.

So the locus is a floor on the extension and not a value. A tricot gives at least 27.3 per cent across before anything in it is stretched, and gives more once the loops start to go.

And nothing on the circle costs anything. Every state on it keeps every thread’s length, so moving along it is free in the sense the woven locus is free: no fibre is strained, and what resists is friction at the crossings rather than elasticity in the yarn. That is why a warp knit recovers from a width-way pull and why it recovers slowly — the mechanism is a shape change opposed by friction, which is what makes a knit’s dimensions depend on how it has been treated.

What the shog cannot be chosen for on its own #

The shog is now carrying four quantities in this account and they do not all want the same value.

Connectivity wants it coprime with the width, which rules out very little. Yarn wants it short: an underlap is what a bar consumes beyond what every bar consumes, and a cord costs 111 metres a wale more than a tricot. Width-way give wants it short — 27.3 per cent at one space against 7.5 at two. Length-way give wants it long — 62 per cent against 173.

Three of the four agree on a short shog and the fourth disagrees, which is why tricot is the default and why the exceptions are fabrics wanted for their length-way stretch. The disagreement is exactly one column wide, and the arithmetic says so rather than the trade having to.

The same argument in a weft knit, where it does not work #

It is worth saying why this is a warp-knit result and not a knitting result, because the weft-knit side of this account has no such locus and the reason is structural.

A weft knit’s extension is the loop straightening, and a loop is a curve. What stops a knit extending found that nothing elastic opposes it at all — the mechanism is the loop changing shape against friction, and the loop’s shape is a solved elastica rather than a triangle. So a weft knit has a locus, and its locus needs a loop model this account does not have.

A warp knit has both. Its overlaps are loops and its underlaps are straight lines, and the two mechanisms are in series: the underlaps come taut first, because they are straight and short, and then the loops begin to give. So the locus here is the first part of a two-part extension curve, and it is the part that is exactly computable.

That ordering is worth testing and it makes a prediction with a shape. A warp knit pulled across its width should show a knee — a first region where the fabric gives freely at almost no load, ending at the locus’s own limit, and a second, much stiffer region where the loops are being worked. For a tricot the knee is at 27.3 per cent and for a cord at 7.5, and both are well inside the strain a fabric meets in wear.

Nothing in the trade’s own extension curves is reported as a knee, and the extension figures quoted for tricot — commonly thirty to forty per cent across — bracket the 27.3 computed here in a way that is either a confirmation or a coincidence, and only the shape of the curve would say which.

What was counted, and how #

The locus is Pythagoras and the check is that it is. Every state on it is required to keep the underlap’s own length, to nine figures — which is what the locus is, so the check is a check on the sweep rather than a result.

The spacings are the machine’s, 28 needles an inch and 14 courses a centimetre, and both appear only as the two legs of the triangle. A different gauge moves the as-knitted point along the same circle and changes both ranges; the ratio between them moves with it, so the last column is a property of a machine and a lapping together rather than of the lapping alone.

The monotonicity is required in both directions: a longer shog must give away less width and more length. Either failing would mean the triangle had been mis-assembled, which is the kind of error that produces a plausible table.

And the extremes are the axes rather than a chosen range. The widest state is the underlap flat and the longest is it upright; neither is a load, a strain or a measurement, and both are where the geometry stops.

What this cannot say #

Nothing here is a force. The locus says which states exist and says nothing about what it takes to reach them, which needs the loops’ own resistance and the friction at the underlap crossings — neither of which this account has for a warp knit.

The underlap is taken as straight, and a loaded one is not. An underlap lies over the overlaps of its own course and is pushed out by them, so it is slightly longer than the hypotenuse and slightly curved; straightening it is therefore a two-stage business, the curvature first and then the lean. That makes the width range computed here a floor a second time.

And the fabric is taken as free. A warp knit in a garment is seamed, and a seam pins the course spacing at its edge — so the locus describes the middle of a panel and not its border, which is where every extension test grips it.

Who found it, and when #

Warp-knit extensibility is described in every account of the trade as a consequence of the lapping, and the descriptions are qualitative: a short underlap gives a more extensible fabric across its width, a long one a more stable one. The direction has been known as long as the machine has existed.

What is not anywhere is the circle. The relation is Pythagoras applied to a quantity the trade computes for a different purpose — run-in — and turning a yarn quantity into a geometric constraint is a step nobody has had a reason to take, because the run-in tables and the extensibility descriptions live in different chapters.

The part worth carrying is that the shog is an integer. Every other lever on a warp-knitting machine is continuous: the course density, the run-in, the tension, the gauge. The anisotropy is set by a whole number of needle spaces, and it moves by a factor of ten between consecutive values of it. A design decision with a step of ten in it is a decision that is made once, which is why a fabric is a tricot or a cord and never anything between.

Still open: what an atlas does, which is on no single circle #

Every lapping here has one shog, and an atlas does not: it walks several spaces one way and returns, so its underlaps are all of one space and its thread ranges across four wales.

The locus argument applies to each underlap and the underlaps are all identical, so an atlas’s circle is a tricot’s — which says its width-way give should be a tricot’s 27.3 per cent, whatever the thread’s range.

That is a prediction and it is testable, and it is the kind that is worth writing down because it sounds wrong. An atlas fabric looks and behaves quite differently from a tricot; the arithmetic says the difference is not in this mechanism. If a measured atlas gives less than 27 per cent across, the missing constraint is the thread’s own path over several courses — which is a longer-range object than an underlap and would be the first thing on this account that a single crossing does not explain.