A tube can only be shaped by its loop

A circular knitting machine has its needles arranged round a cylinder, and a fabric knitted on all of them comes off as a tube with no seam in it. The tube’s circumference is a fixed number of wales, because the needles are a fixed number of needles.

So how does a seamless garment get narrower at the waist?

There are exactly two answers and they are not two versions of one thing. One of them changes the fabric and reaches a fifth of the way; the other does not change the fabric at all and reaches the whole way. Which is which is not what the arithmetic first suggests.

The circumference is the loop #

A relaxed plain knit has k_w / ℓ wales per centimetre, so the wale spacing is ℓ / k_w — the loop length over a constant. A tube of N needles therefore has a circumference of

C = N · ℓ / k_w

and the loop length is the only quantity on the right that anybody can change without stopping the machine. Not the yarn, not the fibre, not the gauge, not the fabric structure: the needle count is the cylinder and the constant is a constant.

That makes the circumference exactly proportional to the loop length, which is a much stronger relation than the usual “a tighter fabric is narrower”. It also makes the density’s response exact: stitch density goes as k_s / ℓ², so the ratio of densities across any taper is the square of the ratio of circumferences.

Both of those are checked rather than asserted, to twelve decimal places, because the whole essay is arithmetic on two proportionalities and a proportionality that had been implemented as something slightly different would produce a plausible answer.

What bounds the loop #

A loop cannot be arbitrarily small. The needle has to draw it round a yarn with a diameter, and below some length the fabric is a board and the machine is dropping stitches. It cannot be arbitrarily large either, because past some length the fabric is a net that will not hold its shape.

The trade’s number for where a fabric sits between those is the tightness factor, √tex over the loop length in millimetres, and plain jersey is knitted between about 1.3 and 1.6 of it. That is an empirical range and it is stated as one; what matters below is not its endpoints but that it is a bounded interval whose ratio is a little over a fifth.

For a 20 tex yarn that gives a loop between 2.795 mm and 3.440 mm — a ratio of 1.231. So:

• the circumference can be reduced by at most 18.8 per cent;
• the stitch density at the narrow end is 1.51 times the density at the wide end;
• and on a 240-needle machine the tube goes from 192 cm round to 156.

The taper is real and it is not nothing. It is also a fifth of what a garment needs between a chest and a waist, and it arrives with a fabric change nobody asked for.

The other route, and why it wins #

The second way to narrow a tube is to stop using some of the needles — transfer their loops onto neighbours and knit the rest.

That is the same operation as a fashioned edge, applied round a cylinder instead of at a panel’s side, and it inherits the same quantisation: the circumference changes by whole wales. On a 240-needle machine one wale is 0.42 per cent of the circumference, which is a step of eight millimetres on a 192 cm tube — fine enough to call continuous for any garment purpose.

And it costs nothing. The wale spacing has not changed, the loop has not changed, the stitch density has not changed. What was a tube of 240 wales is a tube of 236 wales of the same fabric.

So the comparison is stark. One route is bounded at a fifth and changes the cloth; the other is unbounded and does not. A reader who expected the loop route to be the fine adjustment and the needle route to be the coarse one has it exactly backwards, and the reason is that the loop is bounded by the yarn while the needle count is bounded only by zero.

The cost of the loop route is the inverse square of what is left #

The taper and the density move together, and the relation between them is exact, so it is worth writing down as a curve rather than as the one point the tightness range happens to land on.

Circumference is proportional to the loop and stitch density to its reciprocal square, so a tube narrowed to a fraction f of its wide circumference has a density of one over f squared. The 18.8 per cent taper is f = 0.8125 and 1.51 is one over 0.8125 squared, exactly.

That relation is the whole reason the route is bounded in practice rather than merely in the tightness table. A thirty per cent taper would need a fabric twice as dense; a halving of the circumference would need one four times as dense. Neither of those is a fabric anybody would put in the same garment as the wide end, whatever a machine could be persuaded to knit — so even if the tightness range were wider than the trade’s, the density cost would close the route at very nearly the same place.

It also explains why the bound does not move with the yarn. The taper is the ratio of the two ends of the tightness range and nothing else, because the count cancels out of a ratio of tightness factors, so a coarse yarn and a fine one reach exactly 18.8 per cent. The only way to widen the route is to widen the range of fabrics a knitter is willing to accept, which is a decision about the garment rather than about the machine or the material.

Two controls, two outputs, and they are independent #

Put the two routes together and they stop competing, because they change different pairs of things.

The needle route moves the circumference at constant fabric. The loop route moves the circumference and the density together, along the fixed curve above. So the two are independent controls over a two-dimensional output — how wide the tube is, and how dense the cloth is — and a machine with both can reach any combination inside a band rather than any point on a line.

That is what a seamless garment machine is doing when it knits a welt. It is not choosing between the two routes; it is using the loop route to get the density a welt needs, and then the needle route to put the circumference wherever the garment wants it, independently. The 18.8 per cent is not a shaping budget being spent — it is the width of the band the fabric decision drags the circumference across, and the needle route puts it back.

The needle route’s own step is one over the needle count, which means its resolution improves exactly as the machine gets finer: a 240-needle cylinder steps by 0.42 per cent and a 480-needle one by 0.21, so the finer gauge is better at shaping as well as at fabric. The loop route’s reach does not move at all with the gauge. Two controls whose relative merit widens as machines improve is not a balance anybody has to manage; it is a decision that was made once and will not need remaking.

Read that way the earlier comparison is not that one route beats the other. It is that only one of them is a shaping control at all, and the other is a fabric control whose unwanted side effect the first one exists to cancel.

Where the taper is spent instead #

If the loop route reaches only a fifth and costs a fabric change, why is it used at all? Because it is doing something else at the same time.

A knitted garment’s welt, cuff and body are routinely knitted at different loop lengths, and the reason usually given is fit — a tighter welt grips, and a rib grips harder still. That is true, and the arithmetic above says the grip and the narrowing are the same operation: a welt knitted at the tight end of the range is 18.8 per cent narrower and 1.51 times denser than the same needles at the loose end, and both of those are what a welt is for.

So the loop route is not a poor shaping method. It is a fabric method whose side effect is a taper, and it happens to be used where both are wanted at once. The needle route is the shaping method, and it is used where a taper is wanted and a fabric change is not.

The needle route has a quantisation of its own, and it is easiest to see where a panel’s edge makes it visible.

What was counted, and how #

Everything is computed from the two proportionalities and the stated tightness range, and the proportionalities are checked rather than assumed.

The circumference at each end of the loop range is computed by building the fabric from its loop length — courses, wales and stitch density all from Munden’s constants — and dividing the needle count by the wales per centimetre. The ratio of the two circumferences is then compared with the ratio of the two loop lengths and required to agree to twelve decimal places, and the ratio of the two densities with the square of it, to the same tolerance. If either had come out different, the relation would be something other than proportional and the essay’s arithmetic would be wrong rather than the numbers being slightly off.

The needle route’s step is the wale spacing over the circumference, which is one over the needle count, and the assertion attached to it is that it is under two per cent — the claim that it can be called continuous.

Two refusals guard the range. A tightness range whose loose end is tighter than its tight end is refused, which is the failure mode of getting the inequality the wrong way round. And a taper that reached more than half the circumference is refused, because that would mean the loop had been left effectively unbounded and the finding would be an artefact of the bound rather than a consequence of it.

The third route, which is not shaping #

There is a way of narrowing a knitted tube that this essay has left out, and leaving it out deliberately is worth a paragraph.

An elastic yarn — a covered elastane, a bare spandex laid in — pulls the fabric in without changing either the needle count or the loop length. A sock’s leg is narrower than its foot largely for that reason, and it is by far the commonest way a knitted tube changes width.

It is not shaping in the sense of this essay, because the fabric’s relaxed dimensions have not moved: the wale count is the wale count and the loop is the loop, and what has changed is that the structure is now held below its relaxed state by a second yarn. Everything Munden’s constants describe is a statement about a fabric that has been allowed to reach its own size, and a fabric under a permanent internal tension has not.

So the elastic route is outside every number here, and it is outside them in a way that matters: it does not merely add a term, it removes the condition the whole geometry is stated under. What it buys in exchange is a tube whose circumference is a range rather than a value, which is the property a sock is actually sold on and which neither of the two routes above can produce at all.

Where the model stops #

The tightness range is empirical and is an argument. 1.3 to 1.6 is ordinary for plain jersey; a particular yarn, machine and end use will differ. What the essay depends on is that the range exists and is narrow, not on its endpoints, and the bound moves linearly with the ratio of the endpoints rather than dramatically.

Munden’s constants are a relaxed plain knit’s, so the whole calculation is for plain jersey. A rib is a different structure with different constants, and this site has already found that the constants do not compose — so a rib welt’s taper is a separate calculation from constants nobody has published for every structure.

Neither route says anything about the shape of the transition. Changing the loop over a few courses produces a gradual taper; dropping needles produces a step. What a wearer sees at the join, and whether a needle-dropped taper shows a line — the same question a fashioning mark raises, is a question about the fabric’s appearance that this site has no model for.

And the machine’s own constraints are absent. A circular machine’s needle selection is a mechanism with its own limits — how many needles can be taken out per revolution, whether they can be put back — and the arithmetic here treats the needle count as freely settable. The real constraint on seamless shaping is very often that mechanism rather than the geometry.

The generalisation #

Two control routes to the same output, one of which is bounded by a material property and one of which is bounded by nothing, and the bounded one is also the one that changes something else.

That is a recognisable situation and the useful part is the diagnostic. Ask what bounds each route and what each route changes besides the target. Here the loop is bounded by the yarn and changes the density; the needle count is bounded by the machine’s mechanism and changes nothing. The answer is not close, and it does not depend on any of the numbers being right to better than a factor.

The second half generalises differently and is the more interesting one. When a control has a side effect, and the side effect is also wanted somewhere, the control stops being a shaping method and becomes a compound operation. A welt is not a narrow piece of the same fabric; it is a denser fabric that is narrower as a consequence. Reading it as shaping-with-a-side-effect gets the design backwards.

Who found it, and when #

Circular knitting is Victorian and seamless garment machines are recent. The practice — shape by needle selection, not by stitch length; knit the welt tighter for grip — is universal and is taught as a rule, and the rule is correct.

The arithmetic that says how far stitch length alone would get is not usually done, presumably because nobody wants to shape that way. Doing it anyway is what produces the two numbers worth having: the 18.8 per cent, which says the route is not merely inferior but bounded, and the 1.51, which says the cost is a fabric a customer would notice.

The relation between the tightness factor and the loop length is standard knitting technology; the relation between the loop length and the fabric’s dimensions is Munden’s. Putting the two together to bound a taper is this site’s, and it is one line once both are present.

Where the ladder goes next #

The shaping ladder has two rungs and both are about a discrete operation approximating a continuous shape. What neither reaches is what a transfer does to the fabric — a needle carrying two loops is a force balance, the mark it leaves is the visible sign of the whole method, and this site has no model of a loop’s shape under load. That is the same gap the knit-geometry ladder records: the yarn accounting composes exactly and the size the structure relaxes to does not.

Sideways, the same tube in a woven cloth is a tubular double cloth, where the circumference is fixed by the loom’s width and the shaping question does not arise at all; the geometry both rungs use is the loop’s own dimensions; and the operation that shapes by cutting instead is the bias cut, which reaches any angle at all and pays for it in cloth.