A loop has no closure condition

A woven cloth’s two thread systems are tied together by one equation. The crimp height the warp takes and the crimp height the weft takes must add to the cloth’s thickness, so thickness one system has is thickness the other cannot have. Swell the threads and the demand rises in proportion while the supply does not, and for three of this collection’s eight cloths there is a swelling past which no state exists at all.

A knit does not do that. Wet a jersey and nothing refuses; the geometry has no equation to fail.

The obvious explanation is that a knit is an open fabric with room to spare, so a swollen thread has somewhere to go. That explanation is wrong, and the arithmetic says so before any water is added.

The occupancy, and the surprise #

A woven cloth’s cover factor is a diameter over a spacing. A loop has no such spacing, so the comparable question has to be asked on area: how much of one stitch’s rectangle — one wale wide by one course tall — is thread, seen from above?

The thread in one stitch has length ℓ and width d, so its plan area is ℓd. The rectangle is (k_w ℓ) × (k_c ℓ), with the two shape constants this collection has used since the finishing field. So the occupancy is

$$\frac{\ell d}{k_w k_c \ell^2} = \frac{d/\ell}{k_w k_c}$$

A ratio of the yarn diameter to the loop length, and nothing else. No fabric dimension survives; the loop length cancels once and appears once.

For a 20 tex cotton at a 3.5 millimetre loop it comes out at 1.129. Above one, before any water.

That is not an error. A jersey is opaque, its loop legs lie over one another, and the plan area of the thread in one stitch really does exceed the stitch’s footprint. Wetting takes it to 1.355, which is further above one, and nothing happens.

It is the knitter’s own number #

The occupancy has a shape a knitter will recognise. A yarn diameter goes as the square root of its count, so d/ℓ is √tex/ℓ times a constant made of the fibre density and the packing factor.

√tex/ℓ, with the loop length in centimetres, is the tightness factor — the trade’s own index of how tightly a fabric is knitted, quoted in the range 13 to 17 for an ordinary cotton jersey. It is normally presented as an empirical index: a number that correlates with handle, cover and dimensional stability and is not derived from anything.

Divide the occupancy through and

$$\text{occupancy} = 0.0884 \cdot K$$

for cotton at a packing of 0.6. The constant is not a fit — it is √(4/(π ρ V)) over k_wk_c with the units carried through — and it changes with the fibre exactly as a density does.

The tightness factor is the fraction of a stitch’s own footprint that its thread fills, times twelve. That identity is asserted across four counts and five loop lengths to machine precision, because it is two ways of writing the same quotient and not a correlation.

The boundary where a loop is exactly full sits at K = 11.87, which is slacker than anything commercially knitted. Every jersey the trade makes is over-full before it is wetted: 1.095 at a tightness factor of 13, and 1.432 at 17.

What an occupancy above one says #

A number above one where a fraction was expected is a signal, and it is worth reading rather than filing.

The plan area of the thread exceeds the cell, so the thread must be somewhere other than in the plane of the cell — and it is: a stitch’s two legs pass in front of the head of the loop below and behind its own neighbours, so a jersey has a real thickness that is between two and three thread diameters rather than one. A knit is a fabric whose thread occupies a third dimension as a matter of course, and a woven cloth is one whose thread does so only as much as its crimp requires.

That is the same fact the knits ladder has been circling from several directions. It is why a jersey of a given yarn weighs more per unit area than a woven cloth of the same yarn at a comparable hand; why a knit is warmer, since it traps more air; why it recovers from extension, since it has thread stored out of the plane to give; and why it curls at a cut edge, since the two faces are not the same.

Reading the occupancy as a cover factor would lose all of that. A cover factor near one means a fabric with no holes; an occupancy above one means a fabric whose thread does not lie flat, which is a different statement about a different fabric.

So what does save a knit #

Not room. What saves it is that there is no equation to fail.

A woven cloth’s two spacings are tied to its thickness by a closure condition, because the two systems physically interleave: each must bend over and under the other in the same thickness of fabric. That is a geometric constraint with no slack in it, and it is what runs out.

A knit’s dimensions are a loop length times a dimensionless constant. That is the whole of it. Swell the thread and d changes; the wale spacing is k_w ℓ and the course spacing is k_c ℓ, and neither has a d in it, so neither moves. There is nothing to satisfy and therefore nothing to fail.

This is asserted in the only form a negative takes: the knit’s predicted spacings are computed at a swollen diameter and required to be unchanged to the last bit at three counts, while the woven cloth’s closure margin at the same swelling is required to have moved. One fabric’s geometry knows about the swelling and the other’s does not, and that is the whole comparison.

Which is a statement about the model, not only about the fabric #

It is worth being careful here, because “there is no closure condition” could mean two very different things.

It could mean that a knit really has no constraint of that kind. It does not quite: at some swelling a loop’s own legs must press against each other, and a loop cannot shrink its head below the thread’s own diameter. Those are real constraints and they are far away — the loop’s free dimensions are several thread diameters, not a fraction of one.

Or it could mean that the model this collection uses for a knit has no such condition in it, which is certainly true and is the weaker claim. Munden’s constants are an empirical result: wales and courses per unit length are a constant over the loop length, fitted across yarns, counts and loop lengths, and found not to move with any of them. A model built on a fitted constant cannot predict what happens when the fitted range is left.

The honest position combines the two. The constants were fitted over ordinary constructions and ordinary yarns, including wet-relaxed states, so the empirical claim already covers the swollen case within its range — and the mechanism that would produce a failure is visibly far away. The absence is real and the model’s silence about it is not evidence.

The consequence, which runs the other way from the reputation #

Knitted fabrics have a bad reputation for dimensional stability. They shrink more in the wash than wovens, they grow when they are worn, and a knitted garment’s size is a much softer number than a woven one’s.

So it is worth stating plainly that the mechanism that ruins a woven cloth in water does not touch a knit. A close cotton sheeting generates megapascals against itself when it is wetted and is permanently changed by it. A jersey of the same yarn generates nothing.

The knit’s instability is a different mechanism entirely and this collection already has it: the loops are held where the machine left them by friction, and water and agitation let them move to where the yarn’s own bending wanted them. That is relaxation, it is frictional, and it is large precisely because a knitted loop is a much longer lever than a woven crossing — a knit’s yarn is far more bent, so it has far more stored bending to spend and far more distance to spend it over.

A knit is unstable because it is under-relaxed, and a woven cloth is damaged because it is over-constrained. Those are opposite failures and they are both called shrinkage.

The one place a knit does meet a limit #

There is a construction where a knitted fabric does run out of room, and it is worth naming so that the claim above is not read as broader than it is.

A rib or an interlock has two beds, and a yarn crossing between them has to fit in the gap. That gap is a machine setting, and a second bed changes what a float is. Swell the thread and the gap does not change: it is set by the needle beds’ separation, which is steel.

So a two-bed structure does have a constraint of the woven kind — a fixed dimension that a swollen thread has to fit into — and it is imposed by the machine rather than by the fabric. Whether it bites is a question about the bed gap, which is a millimetre or two against a thread of a fifth of a millimetre, so the answer at ordinary constructions is comfortably not.

But the shape of the argument matters. The constraint that could fail comes from outside the fabric, and a single-bed knit has nothing of the sort. That is a cleaner statement of what this rung found than “a knit has no closure condition”, which was true of the fabric and needed the qualification.

What was counted, and how #

The occupancy is a ratio. Asserted to 1e-15 against the same quantity computed from the two spacings, because the closed form is what this essay argues with and the spacings are what the figure draws.

And it is the tightness factor. Asserted across counts from 10 to 60 tex and loop lengths from 2.5 to 5 millimetres, requiring the occupancy per unit of K to be constant to 1e-12. A later change to the diameter route or to the shape constants would break the proportionality — which would mean the trade’s index and this occupancy had stopped being the same number, and the claim here would be wrong.

A jersey at trade tightness is over-full. Asserted over the trade’s own range of K rather than over a grid of counts chosen here, which matters: a very fine yarn at a long loop is under-full — a 15 tex at 3.5 millimetres fills 0.978 of its cell, at a tightness factor of 11.1 — and that is not a commercial jersey, it is a fabric a knitter would call slack.

And the knit’s spacings do not move when its thread swells, while the woven cloth’s margin does. Both halves of the comparison, in one assertion, because either alone would be uninformative.

One tightness factor is not one occupancy #

The identity is written for cotton at a packing of 0.6, and the constant carries the fibre in it — a diameter comes from a count by conservation of volume, so the occupancy per unit of tightness factor scales as one over the square root of the fibre density times the packing.

That is not a small dependence across the fibres a knitter actually uses.

A twenty-seven per cent spread, top to bottom, in what a single number means — and the tightness factor is quoted, specified and compared across fibres as though it were fibre-neutral.

The two ends of the table are the ones that matter commercially. A nylon jersey at a tightness factor of thirteen has an occupancy of 1.33; a filament polyester jersey at the same thirteen has 1.04. One is a third over-full and the other is barely over at all, and every number on both specifications is identical.

That is a real caution about a real index, and it points the same way as this collection’s standing complaint about thread count. The tightness factor is an honest measurement of √tex over a loop length; it is simply not the quantity anyone is reaching for, which is how much of the stitch is thread. Divide by the square root of the fibre’s density times its packing and the comparison becomes a comparison.

Which explains a piece of trade practice #

Knitting manuals give different recommended tightness ranges for different fibres, and the reason usually offered is handle: a wool at a cotton’s tightness is said to feel harsh, a filament at a spun yarn’s tightness to feel thin.

The arithmetic gives the same recommendations without mentioning handle. The recommended ranges are the ranges that put each fibre at a comparable occupancy. A fibre whose constant is high needs a lower tightness factor to reach the same fullness, and a filament yarn — packed harder, so a smaller diameter at the same count — needs a higher one.

Run the table backwards from a target occupancy of 1.25, which is roughly a mid-range cotton jersey: cotton needs K = 14.1, polyester 13.5, wool 13.1, nylon 12.2, and filament polyester 15.6. Those are recognisably the ranges the trade uses, and the spread between them is the spread the manuals give.

So the fibre-specific tightness ranges are not a handle judgement wearing arithmetic’s clothes; they are arithmetic wearing a handle judgement’s. The occupancy is the quantity that was being held constant all along, and the tightness factor is the coordinate it was being held constant in.

The caveat is the packing factor, which is the least secure number in the whole calculation and the one the constant is most sensitive to. Ring-spun cotton at 0.6 is well established; wool at 0.6 is a borrowing; and a filament yarn’s packing depends on how it was textured and can run from 0.7 to nearly 0.9, which alone moves its constant by ten per cent. The ordering is robust and the third figures are not, which is the same statement this collection makes about every diameter it computes.

Where the model stops #

The occupancy is a plan area and a loop is three-dimensional. The legs of a stitch lie over one another rather than beside one another, which is exactly why an occupancy above one is possible and is also why the number cannot be read as a cover factor. It is a ratio worth having because it is the trade’s index in disguise, not because it is a fraction of anything.

The shape constants are for a plain knit at one relaxation state. A rib, an interlock or a tuck structure has its own constants, and this collection carries some of them; the tightness-factor identity holds for each with its own numerical constant, and the numbers here are the jersey’s.

Nothing here says what a knit does when it is genuinely jammed. There is presumably a tightness factor at which a loop cannot be knitted at all, and it is set by the needle and the machine rather than by the fabric. That is a machine limit and this collection has no machine model for a knitting frame.

And the whole comparison is at one swelling. A viscose knit swells thirty-five per cent, which takes the occupancy of an ordinary jersey to 1.45, and the same argument applies with more room to spare — because there is no condition for the extra swelling to violate.

The generalisation #

Before explaining why something does not fail, check that the explanation is not about a quantity that has already failed.

The plausible account here — a knit has room — is not merely unsupported; it is contradicted by the very quantity it appeals to, which was above one before the question was asked. Believing it would have meant carrying a mental picture of a knit as an airy fabric with gaps, which is what a stretched jersey looks like and not what a relaxed one is.

The correct account is structural and is nearly the opposite in flavour: a knit is safe not because it has slack but because it has no simultaneous equation. Slack is a quantitative property that can be used up; the absence of a constraint is not.

Look for the equation before looking for the room.

Who found it, and when #

The tightness factor is the knitting trade’s, and it is usually attributed to the same body of work as the dimensional constants — Munden’s at the Hosiery and Allied Trades Research Association in the 1950s and 60s, where the loop length was established as the master variable of a knitted fabric.

That the tightness factor is dimensionally a diameter over a loop length is implicit in its definition and is presumably known to everybody who has looked at the units.

What is this collection’s is the identification of it with a plan occupancy, the constant that connects them, the observation that a commercial jersey is over-full at every tightness the trade uses, and the argument that a knit’s immunity to swelling is the absence of a constraint rather than the presence of room.

Where the ladder goes next #

To the change a knit does undergo in water, which looks exactly like a swelling and turns out not to be one — and the form of the constants is what rules it out.