How far a knit could go if its yarn were the limit

A knitted stitch has a fixed length of yarn between one interlacing and the next, so there is a hard limit on how far the fabric can be pulled: the straight line between the two cannot exceed the thread spanning it. That is a ceiling with nothing elastic in it, and no force whatever reaches beyond it.

For an ordinary jersey the ceiling is at three hundred and twenty per cent course-wise extension. Jerseys jam at about a hundred.

The arithmetic #

Half a loop length of yarn joins two interlacings that are half a wale apart across the fabric and one loop height apart along it, and the loop height is the course spacing plus one yarn diameter.

So the two spacings must satisfy a Pythagorean inequality, and the reachable states are the inside of an ellipse quadrant. Widening the wale therefore costs course height whether the fabric likes it or not, and the largest wale spacing available is reached when the course spacing is as small as it can be — which is a yarn diameter, since the courses cannot pass through one another.

Everything in that is a length. There is no material in it, no force, no rigidity and no relaxation state.

The numbers #

For a 20 tex cotton at loop lengths from two point eight to four and a half millimetres, the course-wise ceiling runs from three hundred and eighteen per cent to three hundred and twenty-five, and the wale-wise ceiling from a hundred and forty-two to a hundred and fifty-four.

The near-constancy is the same effect as the slack barely moving: both spacings scale with the loop length, so the ratios that decide the ceiling are almost independent of it. A knit’s geometric extensibility is a property of the structure, not of the construction.

What jerseys actually do #

Somewhere between eighty and a hundred and fifty per cent course-wise, depending on the construction and on who is measuring and how hard. Nobody reports three hundred.

That is a discrepancy of a factor of three in a quantity computed from four lengths, and the four lengths are not in doubt. So the ceiling is right and it is the wrong ceiling: something stops a jersey long before its yarn runs out.

What stops it #

Loops meeting sideways, and the model does not carry the constraint that would say so.

A knit’s occupancy is over one at rest — the yarn overlaps itself seen from above — so there is no room to spare in the plane to begin with. As the fabric widens, the loops flatten towards each other, and the yarn of one course meets the yarn of the next long before either runs straight.

Why the model was built without it #

Because adding it turns a boundary-value problem into a contact problem, and the two are different orders of difficulty. A boundary-value problem is a dozen Newton steps; a contact problem needs the constraint set to be discovered as part of the solve.

That is a legitimate reason to leave something out and it is not a reason to leave the consequence unstated. The consequence is that every extension quoted on this ladder past about a hundred per cent is a region the model reaches and a real fabric does not, and the load–extension curve’s stiffening — which the model puts at two hundred per cent — happens earlier in a real fabric and for a different reason.

What the gap is worth knowing #

Three things, and the first corrects something this collection has said.

What stops a knit extending argued that a knit’s extension ends when the loop runs out of reconfiguration rather than when the yarn runs out of stretch. The first half of that is now measurable and the argument is wrong in its detail: the loop does not run out of reconfiguration at a hundred per cent, it runs out at three hundred, and what happens at a hundred is loops touching.

The distinction matters because the two have different levers. Running out of reconfiguration is a function of the loop length; running into a neighbouring course is a function of the loop length and the yarn diameter, so it moves with the count at a fixed loop length and the other does not.

Where the fabric goes on the way #

The path across the state space is as informative as the endpoint, and it is not a straight line.

That descent is what makes the ceiling reachable at all. A fabric held to constant height while being pulled would run into the boundary far sooner, because the boundary is a relation between both spacings; letting the height collapse is what buys the last two hundred per cent.

Which is worth saying because a garment does not let it collapse. A sleeve stretched round an arm is under biaxial extension with neither dimension free, and its available extension is much less than the uniaxial ceiling — a point on the edge of the ellipse in one direction only. The ceiling quoted here is the most generous case there is.

A prediction that follows #

If the real limit is loops meeting, then a fabric knitted from a finer yarn at the same loop length should stretch further before jamming, because there is more room in the plane between the loops.

If the real limit were the yarn running straight, a finer yarn would make no difference at all — the ceiling depends on the diameter only through the loop height, which is a small correction.

Those two predictions differ by a large factor and are separated by an experiment anybody with a knitting machine can run: three fabrics, one loop length, three counts, measured to jamming. Nothing here has run it.

The two ceilings are not one number #

A caution about how a single figure gets quoted, because “a knit stretches three hundred per cent” would be a bad sentence to take away.

The ceiling is a curve rather than a point: every pair of spacings on the edge of the ellipse is a state at which the yarn is straight, and which one a fabric reaches depends entirely on what the second dimension was allowed to do. Free transversely, a jersey reaches three hundred and twenty per cent by giving up nearly all its height. Held to constant height, it reaches under a hundred.

So the number attaches to a test condition rather than to a fabric, and the condition is almost never stated. That is the same complaint this collection makes about a dimension quoted without its relaxation state, applied to an extensibility instead of a length, and it is worth making twice because the mechanism is identical: a quantity with two free variables has been reported as though it had one.

The other direction #

The wale-wise ceiling is about a hundred and fifty per cent, which is half the course-wise one, and it is reached differently.

Pulling along the wales lengthens the course spacing and narrows the wale, and the limit is the wale spacing reaching a yarn diameter — the loops of adjacent wales in the same course touching. So the wale-wise limit already is a contact limit in the model, because the wale spacing appears in the geometry directly, while the course-wise one is not.

That asymmetry is an artefact of how the model is set up rather than a fact about knitting, and it is worth flagging so that the two numbers are not read as comparable.

What the picture cannot show #

Contact. Every drawing here is of centre lines with a stroke width, and two strokes overlapping in the plan view of a planar model is not a collision — the real fabric has a third dimension in which one of them is in front.

So a reader looking at the stretched panels cannot tell, from the picture, whether the crowding is real or an artefact of projection. The honest answer is that some of it is real and the model does not distinguish, which is the same admission in a different form.

The ceiling as a specification #

For anybody choosing a construction, the useful form is not the ceiling but its insensitivity.

So tightness is a strong lever on how hard a knit is to stretch and almost no lever at all on how far it can go. A knitter wanting more extension cannot get it by loosening the fabric — the loosening changes the force at every extension and leaves the ceiling where it was.

What does move the reachable extension is the structure: a rib opens by folding before it extends its loops at all, which is why a cuff is ribbed and not merely knitted slackly. The rung on rib puts the two mechanisms side by side.

What is solid in it #

The ceiling itself, as a bound. Whatever stops a jersey first, nothing gets past three hundred and twenty per cent, and a claimed extension beyond that for a plain jersey at that loop length is a measurement of something else — the yarn stretching, or the loops slipping, or the specimen slipping in the jaws.

That is the useful form of a ceiling that is not tight: it is not the answer, and it is a fact that rules things out.

Where a tight ceiling would come from #

Adding a non-penetration constraint between the sampled points of neighbouring courses. That is the missing piece and it is the same missing piece flagged in the list of what the model cannot say, arriving here as a number rather than as an item on a list.

The measurement that would confirm the account before the machinery is built is optical: photograph a jersey at increasing extension and record where adjacent courses first touch. If that extension is where the load–extension curve turns up, the account is right and the constraint is worth adding.

What a woven cloth’s ceiling looks like #

The comparison sharpens what kind of quantity this is, because a woven cloth has exactly the same sort of bound and it is tight rather than loose.

A woven cloth’s ceiling is where one system goes straight or the other jams, and a real cloth gets very close to it, because its threads have almost no slack to begin with. The geometric bound and the observed limit are within a few per cent of one another.

A knit’s are a factor of three apart. That is the difference between a structure whose behaviour is decided by its own thread lengths and a structure with so much room that something else finds the limit first — and it is one more consequence of the same slack that runs through this whole ladder.

What the number is good for anyway #

Three uses survive the bound being loose, and they are worth listing because a loose bound looks useless.

It rules things out: a plain jersey reported at four hundred per cent is measuring something other than loop reconfiguration.

It bounds the model’s own validity: everything computed on this ladder past a hundred per cent describes a fabric that is not there, and the sentence saying so belongs on every curve.

And it isolates the missing constraint: knowing the yarn-length bound exactly means that whatever gap remains is entirely the contact between courses, with nothing else left to attribute it to. A loose bound with a known cause is worth much more than a tight one with an unknown one.

The same limit in a knit that has been finished #

One place the ceiling does bind is a fabric that has been shrunk deliberately, and it is worth naming because the trade already exploits it.

A knit that has been milled or felted has a shorter effective loop length in the fabric plane, because fibre migration has locked the loops and taken slack out of them. That moves the whole state space, and it moves the ceiling with it — a felted knit stretches far less than an unfelted one of the same original loop length, and the reason is not that its yarn is stiffer.

Conversely a fabric stretched on a stenter and dried in that state has been given a longer effective loop and a larger ceiling, at the cost of everything else on this ladder shifting with it. That is a real and familiar trade-off with an arithmetic under it now: a finisher moving the fabric’s dimensions is moving its position on the state space, and every mechanical property this ladder computes moves along the same axis.

The ceiling depends on the count only at second order #

The near-constancy of the ceiling is reported above as an observation across a sweep of loop lengths. It has a reason, the reason is one line, and having it makes the proposed experiment far sharper than the essay claims.

The largest wale spacing available is twice the square root of the half loop length squared less the minimum loop height squared, and the relaxed wale spacing is the loop length over a shape constant. Dividing one by the other, the extension ratio is

k × √(1 − 4(d/ℓ)²),

so the whole of the yarn’s contribution enters as the square of the diameter over the loop length — which is the tightness factor, and which is a small number.

For a 20 tex cotton at a 3.5 millimetre loop, d/ℓ is 0.048, so the correction is 2(d/ℓ)² = 0.45 per cent. Across the essay’s own sweep of loop lengths the correction runs from 0.28 per cent to 0.71, a difference of four tenths of a point — which is the reason the ceiling moves from 318 to 325 and no further.

So the count cannot move the ceiling. Doubling the tex at a fixed loop length raises d by 41 per cent and the correction by a factor of two, which is half a per cent of the answer. No change of yarn available to a knitter moves the geometric ceiling by more than about one per cent in either direction.

Which makes the discriminating experiment much easier #

That turns the essay’s proposed test from a comparison of magnitudes into a comparison against zero, and a test against zero needs far less precision.

The essay’s version is that a contact limit should move with the count and a yarn-length limit should not, and it notes the two predictions “differ by a large factor”. The stronger statement is available: the yarn-length limit cannot move by more than one per cent, at all, for any count.

So the experiment does not have to establish how far the jam moves. It has to establish that it moves at all by more than a per cent or two — and three fabrics at one loop length in 15, 30 and 60 tex, measured to the nearest five per cent of extension, would settle it. If the jamming extension differs between them by anything a tape measure can see, the limit is not the yarn’s length, and the argument is closed without any model of contact at all.

That is worth having because the alternative — building the non-penetration constraint — is the expensive route the essay names as the missing piece. The cheap measurement can rule out the whole class of explanations before anybody writes the solver, and it can do it with an instrument that measures to a centimetre.

The same argument disposes of a second hypothesis for free. Yarn extension cannot be the limit either, because a cotton yarn stretches a few per cent and the discrepancy to be explained is a factor of three; and slippage at the loops would move with friction rather than with count, which the same three-fabric series separates by using one fibre at three counts.

So the series has three outcomes and each identifies a mechanism. No dependence on count: the yarn’s own length, and the model’s ceiling is somehow being reached after all. A strong dependence: contact between courses, as the essay argues. A dependence in the wrong direction — a coarser yarn stretching further — would mean something is happening that neither account contains, and would be the most interesting result of the three.

What was known before #

That a knit’s extension is limited by its geometry rather than by its yarn is not new; it is the first thing anybody says about knitted fabric. What has not been available is either number — the geometric ceiling, or the extension at which loops meet — so the sentence has been true and unquantified.

Half of it is quantified now, and the half that is quantified turns out to be the half that does not bind. That is a common and useful shape for a result: the easy bound is loose, and knowing how loose says which hard bound is worth computing.

How the two candidate limits differ in shape #

Beyond the experiment, the two accounts predict different curves rather than different points, which is a stronger discriminator.

A limit set by the yarn running straight arrives as a stiffening that is smooth and accelerating: the geometry degrades gradually as the chord approaches the arc, and the force rises as a power. A limit set by loops meeting arrives as a corner: nothing is touching, then something is, and the fabric acquires a whole new set of contacts within a small extension.

A measured load–extension curve on a jersey has a knee rather than a smooth acceleration, and it has one at about a hundred per cent rather than at two hundred. Both of those favour contact over yarn-length, and neither is decisive on its own because friction and yarn extension both blur a corner.

What would be decisive is the count dependence predicted above, because the two accounts differ in direction rather than in degree.

Where the ladder goes next #

Away from extension and towards the single dimensionless group that has been doing all of the work: the yarn diameter over the loop length, which the trade already quotes as its tightness factor and which turns out to be the model’s only variable.