A loop is nine tenths free run

A jersey knitted from a 20 tex cotton at a three-and-a-half millimetre loop puts twenty-one yarn diameters of thread into a cell four point nine diameters wide and three point eight tall. The straight line from one interlacing to the next is a little over half of the yarn available to span it.

Forty-nine per cent slack, against four parts in a thousand for a poplin. That is the whole of this rung, and everything the rest of this ladder does follows from it.

What the number is #

The slack is one minus the chord over the arc. The arc is half a loop length — the yarn from the crown of a needle loop’s head to the bottom of the next sinker loop. The chord is the straight line between the same two points, which is half a wale across and a loop height down.

The loop height is the course spacing plus one yarn diameter, because the loops interlock: a course’s feet rest on the head of the course below with their centre lines one diameter apart. Everything in the calculation is a measurement — the loop length a knitter sets, the two spacings that follow from it, and the yarn’s own diameter — and nothing is fitted.

Why the answer barely moves #

Sweep the loop length from two and a half millimetres to five, which is the whole range a knitter of that count can work in, and the slack goes from forty-six per cent to fifty-one.

That is a five-point range over a doubling of the loop length, and it is worth pausing on. The reason is that both the chord and the arc scale with the loop length in almost the same way: the two spacings are proportional to it by Munden’s constants, and the only thing that does not scale is the yarn’s own diameter, which enters the loop height and nothing else.

So the slack is nearly a property of the structure rather than of the fabric, and a knit is a loose object at every tightness anybody can knit.

Why the slack has a ceiling and the fabric never reaches it #

Five points of movement over a doubling of the loop length is a small range, and the expression says exactly why — and what the number would be if a knitter could go on loosening indefinitely.

The arc is half a loop length. The chord is built from half a wale spacing across and a course spacing plus a diameter down, and the two spacings are both proportional to the loop length by the shape constants. So dividing chord by arc leaves everything proportional to the loop except the yarn’s own diameter, which sits in the loop height alone and does not scale.

So the slack is a function of the diameter over the loop length and of the two constants, and of nothing else. As the loop grows, the diameter term shrinks towards nothing and the slack approaches a limit set by the two constants by themselves — a number with no yarn in it at all, and the value the fifty-one per cent at the loose end is climbing towards.

The fabric never gets there, and it is worth saying why the ceiling is not merely far away but unreachable. A larger loop at a fixed count is a looser fabric, and below the bottom of the trade’s tightness band the structure stops holding its shape: the loops slide, the fabric distorts under its own weight, and it is a net rather than a knit. The approach to the ceiling is cut off by a fabric property rather than by a geometric one, which is the same shape of bound as the tightness range that limits a tube’s taper.

Slack is not extension #

The one thing this number invites and does not license is reading forty-nine per cent of slack as forty-nine per cent of stretch.

Slack is a statement about one span: the thread between two interlacings is about twice as long as the straight line between them. Extension is a statement about the fabric: how far the whole thing moves before something resists. The two are related and the relation is not an equality, because straightening a leg does not lengthen the fabric by the length recovered.

Pull a jersey across its wales and the loops do not straighten in place. They reconfigure — the heads narrow, the legs rotate towards the direction of pull, and yarn is drawn from the loops above and below through interlacings that resist it by friction. The fabric gets longer one way and narrower the other, and how much of the slack is available depends on what the neighbouring loops will give up, which is a question about the whole fabric rather than about one span.

So the slack is best read as an upper bound with the right order of magnitude. A jersey stretches by tens of per cent across its width and a woven cloth by a few, and the ratio between those is roughly the ratio between the two slacks — which is the useful statement, and it is a great deal weaker than the arithmetic looks.

The tightest bend, which is a different question #

Slack says the thread has room. It does not say the thread uses it gently, and the second measurement is the one that decides whether the contact is a point or an arc.

A yarn wrapped hard round another of the same size follows a centre line of radius one diameter, so one over the diameter is the curvature of the tightest wrap any fabric ever asks for. The question is whether the solved loop’s own tightest bend gets there.

What the crossing at one means #

Below a tightness factor of about thirteen the loop’s own tightest bend is looser than a wrap, and the contact between two loops is genuinely a point: the thread arrives, touches, and leaves, and everything in between is free.

Above it the peak creeps past the wrap’s curvature, and the description is no longer exactly right — the contact spreads a little, the way a woven thread’s does. The departure is small over the band a knitter uses, reaching about a quarter at the tightest end, and it is stated here rather than smoothed over because it is the model’s own boundary.

The trade’s band begins where the loop stops being loose. That is not a claim about causation, and it may be a coincidence of two numbers with the same units. It is worth recording because a coincidence recorded is a coincidence somebody can later refute.

The comparison, at one scale #

The two fabrics are easier to believe side by side than in a table.

The picture is the argument. Nothing about the woven panel suggests a thread with choices; nothing about the knitted one suggests a thread without them.

Drawing them at one scale is what makes it an argument rather than an illustration, and it is the reason the two panels are different sizes on the page. The obvious way to draw a comparison is to fill two equal boxes, and it would put a poplin’s pick spacing and a jersey’s course spacing at the same width — which is precisely the fact the figure exists to show, thrown away in the drawing. A loop is large. Its free run is several times the whole repeat of the woven thread beside it, and a reader who is shown the two normalised will conclude that a knitted thread is a woven one with more curvature in it.

The twenty-nine per cent is the number to carry away, and it is worth saying what the complement of it is. Just under a third of the woven thread is inside a wrap where its shape is dictated by the yarn it is going round; the rest is a free span between two wraps, and that span is short, shallow and under tension from both ends. The knitted thread has no wrap at all — nothing in a plain knit’s loop is bent round another thread — so the whole of it is free run, and its shape is decided by nothing but its own stiffness and where its two ends have to be.

Where the room comes from #

A loop is long because a knitting machine puts a whole loop length of yarn onto one needle before casting the old loop off. The yarn per interlacing is set by the machine’s take-up rather than by the geometry, and it is set generously — a loop length is several times the distance between the interlacings it joins.

A weaver has no such freedom. The reed is not the sett but it sets the bounds on it, the loom’s own construction fixes the picks, and the thread length between two crossings is then whatever the crimp works out to be. There is no knob that puts spare yarn between two intersections without changing the cloth into a different cloth.

That is a difference in the machines rather than in the fabrics — what comes off a loom is a cloth whose thread lengths were decided by its geometry, and what comes off a knitting machine is a fabric whose geometry was decided by its thread length — and it produces a difference in mechanics that neither machine’s designer had in view.

What the slack is spent on #

Not on being loose. A relaxed jersey’s plan occupancy is over one — the thread’s own plan area exceeds the cell it sits in, which is why a jersey is opaque and why swelling it does not jam it.

The yarn is crowded and the path is free, and those are compatible because the crowding is in the fabric plane and the freedom is along the thread. A loop is a long piece of yarn taking a leisurely route through a small box, and it overlaps itself doing so.

Why the solver converges #

Mechanically the difference is between a well-conditioned problem and a nearly infeasible one.

With forty-nine per cent slack, the constraint that the thread arrive at the right point is a loose one: many shapes satisfy it, they differ by a lot, and the minimum among them is a broad and unambiguous thing. With four parts in a thousand, the feasible set is a sliver, every admissible shape is nearly the same shape, and the numerical problem of picking the best one is the problem of resolving differences below the tolerance of the arithmetic.

That is the practical face of the same fact, and it is what made the refusal on the shirting informative rather than annoying.

What the picture cannot show #

The slack. Drawn at any scale that fits on a page, a forty-nine per cent slack and a nought-point-four per cent slack both look like a curve between two points — one more curved than the other, with nothing to say that the difference is two orders of magnitude in a quantity nobody drew.

Nor can any of these figures show which strand is in front. The interlacing is a place where two centre lines pass one diameter apart across the fabric, and the model treats that ride as free; the drawings are plan views of a planar curve, and the casing that makes the crossings legible is an engraver’s convention rather than a computed depth.

Where the bending actually sits #

Having room does not mean the bending is spread evenly, and the solved curvature says where it goes.

Two thirds of the bending energy is in the two turns and a third is spread along the legs. That distribution is an output rather than an assumption, and it is why a loop is nothing like a chain of rigid links joined at hinges: a hinge would put all of the compliance at the joint, and a loop puts a third of it in the middle of the legs, which is where a knit’s softness comes from.

It also explains a thing every knitter knows and nobody derives. Pull a jersey and it is the legs that visibly straighten first, not the heads. The heads are where the yarn is already bent hardest and where the marginal cost of bending further is highest, so the fabric spends its extension where the yarn is cheapest to move.

The one dimensionless group #

There is exactly one ratio in the whole model: the yarn’s diameter over the loop length.

Everything else in it is proportional to the loop length, so two knits that share that ratio have the same loop shape, in units of their own loop length, whatever they are made of and whatever count they are spun to. That is asserted rather than asserted-and-hoped: a 12 tex cotton and a 30 tex wool matched on the ratio give coefficients that agree to fifteen figures.

And that ratio is, up to a constant made of the fibre’s density and its packing factor, the tightness factor the trade already quotes. The rung that pursues it finds that the trade’s empirical index is the model’s only variable, which is a reason for it working that nobody had.

What the model does not carry #

Three things, and the third is the one that bites here.

The curve is planar; a real loop twists, and torsion is not counted. The interlacing is a point, and the out-of-plane ride the topology needs is priced at about three per cent of the loop’s bending and then ignored. And the loops of adjacent courses are not stopped from occupying the same space in the fabric plane, which is fine at the relaxed state and stops being fine when the fabric is stretched a long way.

The third one has a consequence later, and it is flagged where it appears rather than here.

A knit and a chain #

The right mental picture is a chain rather than a cloth, and it explains several things at once.

Links in a chain are held to each other and free in between; a chain has enormous freedom of configuration at almost no load; and what stops a chain moving is friction at its contacts rather than the stiffness of its links. All three are true of a jersey and none is true of a poplin.

What follows immediately #

Two forces, both from the same solve. Across the fabric there is a contact force — how hard a loop presses on the loop it hangs from — and along it there is the force the loop’s bending exerts on the fabric’s own spacings.

The first is what the next rung computes and what the second half of this ladder spends. The second is what makes the ladder interesting, because it does not come out to zero at the fabric everybody measures.

A number for the crowding #

The occupancy is the other half of the picture and pulls the opposite way, so it is worth quoting rather than gesturing at.

The thread in one stitch has a plan area of its length times its width, and the cell it belongs to is one wale by one course. The ratio is the yarn diameter over the loop length, divided by the product of the two shape constants — with no fabric dimension in it at all. For an ordinary jersey it is one point one three, and wetting the yarn takes it higher.

Above one means the thread overlaps itself seen from above, which is exactly what the drawings show and exactly why a jersey is opaque at a cover the same yarn could not reach woven. The rung that found it was after something else — why a knit has no swelling it cannot take — and the number is worth re-reading here for what it says about room: a knit is crowded in the plane and loose along the thread, and no single word covers both.

What was known before #

The slack itself is not a quantity anybody has quoted, as far as this collection can tell, and it is not hard to compute — every ingredient has been in the literature since Munden. What has been known for a long time is the consequence: knitted-fabric mechanics has been treated as a bending problem since the 1950s and woven-fabric mechanics as a geometric one since Peirce, and each field kept its own habit without anybody writing down the number that separates them.

That is a common shape for a missing measurement. Everyone acts on it correctly and nobody states it.

The slack across the whole table #

One column, computed rather than argued, so the two fabrics can be read against each other in one place.

Read the other way, the free share runs from ninety-three per cent down to forty-five for the woven cloths and is a hundred for the knit — but “free” there means not inside a wrap, which is not the same as having room. A cheesecloth has ninety-three per cent of its warp outside the wrap and four parts in ten thousand of slack, because the free part is a straight run whose length and span agree.

That distinction is the one thing in this rung most easily lost. Free run and slack are different quantities, and a woven thread has one of them and not the other.

Where the ladder goes next #

The forces. A loop presses with tens of millinewtons where a woven crossing presses with hundreds, and the ratio is what decides which fabric gives up a fibre, which pills, which frays and which runs.

Then the awkward result: run the model on a relaxed knit and it says the fabric ought to be somewhere else. What is wrong with it turns out to be the most useful thing on the ladder.