The constants do not compose

A knitted fabric is bought by its weight. The number on the specification is grams per square metre, and it is arrived at in a mill by knitting the fabric and weighing a disc of it, which works and explains nothing.

Two quantities stand behind that number. One of them is a count and it is exact. The other is a measurement and there is no route to it that does not involve knitting the fabric — which is the whole subject of this essay, because the boundary between them is sharper than anybody states and it falls in an unexpected place.

The half that composes, and the assertion behind it #

The yarn a structure uses is a count over its array. A knitted loop is k_w wale spacings of yarn, a tuck is a stated multiple of a loop, and a run of yarn past a missed needle is however far it has to go. Add them up.

The claim in that sentence is not the arithmetic, it is the word add. The total is the sum of what each element takes and there is no interaction term — no correction for a tuck standing next to a miss, nothing that depends on the arrangement rather than on the counts. That is what composing means, and a claim of that shape needs a test it could fail.

The test is to compute the total twice by routes that share nothing but the structure. One route counts elements and groups them by kind: so many knits, so many tucks, and the segments the float classification found, added up class by class. The other walks each course from needle to needle, accumulating distance and loop as it goes, and never asks what kind of segment it is on. The two are required to agree to twelve decimal places, and they do, on every structure here — including composites the site has not measured and nobody has published anything about, like a rib that both tucks and floats.

That assertion has teeth. A segment counted in two classes, or a float the classification missed, changes one total and not the other, and the failure would be a plausible-looking number rather than a crash.

The exact half shrinks when the second bed arrives #

Here is the thing that was not expected and is the more useful half of the result.

On one bed the yarn arithmetic is a pure count. Every element sits in one plane, the distance from any loop to the next is one wale spacing, and the only geometry in the whole calculation is already inside k_w. A structure’s yarn is knits plus tucks plus misses, weighted, and nothing else enters.

On two beds it is not. Every course that works both beds sends its yarn across the gap between them and back, and how much yarn that takes depends on how far apart the beds hold their planes of loops. That distance is a machine dimension — real, knowable, and not a fitted constant — but it is one more input than the single-bed arithmetic needed, and it is not a count.

The size of it is worth quoting because it is small and it is not negligible. Knits and tucks alone are 97.3 per cent of a one-by-one rib’s yarn, 97.4 of a half-cardigan’s, 98.6 of a full Milano’s — and only 91.2 per cent of an interlock’s, which has more crossings than anything else here. The remaining few per cent is where the bed gap lives, and moving that gap from three quarters of a needle pitch to one and a half moves a rib’s total yarn by 16.2 per cent while leaving single jersey’s at exactly zero change, which is the check that none of this has leaked into the old arithmetic.

So the honest statement of what composes is narrower than the site had it. The counts compose exactly. The geometry composes exactly given the machine’s own dimensions. Neither of those is the dimension the fabric relaxes to.

The half that does not compose #

Munden’s constants are the site’s route from a loop length to a fabric’s size: courses per unit length times the loop length is k_c, wales per unit length times it is k_w, and stitch density times its square is k_s. They are measurements, they are quoted with their relaxation state, and they are for plain weft-knitted fabric and for nothing else.

The knitting literature does have tables for named structures — rib, interlock, half-cardigan — obtained the way Munden obtained his, by knitting fabrics over a range of loop lengths and fitting a line. This site quotes none of them, because quoting a constant for a structure means having a source for that structure, and inventing a plausible one is precisely how a fitted number gets into a calculation that claims to have none.

What can be said without any of them is why they will not compose.

A structure’s relaxed dimensions are the outcome of a force balance. The loops settle where the yarn’s bending stiffness, the friction between loops and the tension each loop is pulled by leave them, and every loop’s position depends on its neighbours’. Take half a fabric’s positions from one structure and half from another and the composite settles somewhere the two do not average to, because the neighbours have changed and the neighbours are the whole mechanism. A held loop pushes its wales apart; a float pulls them together; a held loop next to a float does neither of those by half.

A force balance does not average, and that sentence is the whole of why the tables stop where they do. It is also why nothing on this site will predict the composite: predicting it needs the loop’s shape under load, which needs the yarn’s bending stiffness, and the site’s knitted geometry stops at the loop length.

The result with no free parameter #

Now the useful part, because the position is much better than “the dimensions are unknown” suggests.

Substitute Munden’s relations into the weight identity and something collapses. Areal weight is yarn per repeat times linear density divided by the area of the repeat; the yarn per repeat is measured in wale spacings and the area carries a wale spacing of its own, so the wale spacing cancels. What is left is

$$\text{g/m}^2 = \frac{\text{tex} \times Y_\ell \times k_s}{10 \times n_w \times n_c \times \ell}$$

where Y_ℓ is the yarn per repeat in loop lengths, n_w and n_c are the wales and courses in the repeat, and ℓ is the loop length.

Everything in it is known except one thing. The linear density is the yarn’s specification, the repeat size is the array, the loop length is a machine setting, and Y_ℓ is a count. The entire unknown is k_s, one number per structure, and the weight is linear in it — so a ten per cent error in the stitch-density constant is a ten per cent error in the weight and nothing else moves at all.

That is a far more precise statement of the shortfall than the site had. It is not that the weight cannot be computed; it is that the weight is a straight line through the origin whose horizontal position nobody has measured. Every line in the figure at the top of this essay is exact. Where to read it is the question.

The check that the identity is right is the one that costs nothing to run and would have caught a slip anywhere in it: computed for plain single jersey with Munden’s own constants, the two-bed machinery returns 135.142857 g/m² at 20 tex on a 3.5 mm loop, and the single-bed arithmetic returns the same figure to the last digit it has. Two files, two models of the fabric, one number.

What a naive composition would predict, and what checks it #

The obvious move, once the yarn composes, is to compose the constants too — take the structure’s fractions and average. The machinery here will do it, and it returns the prediction with a flag saying checkable: false, because there is nothing to check it against.

The size of the guess is worth putting a number on, and it can be done using only the published plain-jersey sets. Munden’s k_s for plain runs from 19.0 dry-relaxed to 23.6 fully relaxed. That is a 24.2 per cent spread, for one structure, from nothing but the state it was measured in. A one-by-one rib’s predicted weight moves from 223 to 277 g/m² across the same span. Any naive composition is choosing a value inside a range at least that wide and is not choosing it from anything.

The function refuses to help. Asked to compose without being given a constant per part, it fails rather than defaulting — because a default here would be a fitted constant wearing a sensible-looking value, and the whole discipline of this site is that a number says where it came from.

What would have to be measured, exactly #

This is the valuable half and it is a list rather than a lament.

Settling one structure means obtaining k_c and k_w for it, by Munden’s own procedure and there is no other: knit fabrics over a range of loop lengths, in more than one yarn so the collapse can be seen to be real, relax them to a defined state, and fit a line through each relation. Five loop lengths and three yarns — thin by Munden’s standards and enough to fit — is fifteen fabrics per relaxation state, and the state is not optional, so three states is forty-five fabrics per structure.

Two constants come off the same forty-five fabrics, because both are measured on each fabric. So the cost is per structure and not per constant, and that is the whole difficulty: the cost is fixed and the structures multiply.

The second row is the one that settles the question. Eleven hundred structures is the smallest two-bed repeat there is — two needles and two courses — and every real double jersey is written on something larger. Forty-nine thousand fabrics is more than the whole knitted-dimensional literature is likely to hold, for the smallest corner of the space.

So the table can be extended and it cannot be completed, and that is a statement about the subject rather than about anybody’s diligence. It is also why the trade does what it does: a mill settling a new structure knits it, weighs it, and writes the number down, because for one structure that is forty-five times cheaper than the general answer and the general answer does not exist.

How far the shortfall is closed, and how far it is not #

An earlier essay here recorded this in writing: “Nothing computes the dimension change a tuck or a miss produces. The yarn count composes exactly and the dimensional constants do not.” It is worth saying precisely what has changed and what has not.

Closed. The composition is now an identity with a test rather than an assertion — computed twice, over composites nobody has published anything about, agreeing to twelve decimal places. The boundary is located: the counts compose, the machine geometry composes given the machine’s dimensions, and the relaxed dimensions compose not at all. The weight is reduced to exactly one unknown per structure, linearly, which is a much smaller hole than “the dimensions are unknown”. And the cost of filling it is priced.

Not closed, and not nearly. Nothing here computes the dimension change a tuck or a miss produces. Not for a two-bed structure, not for a single-bed one, not approximately. The essay has replaced a vague shortfall with a precise one, which is progress of a kind the site values and is not the same as a result.

And one thing got worse. The single-bed arithmetic had no machine dimension in it anywhere; a missed needle saved exactly 76.7 per cent of a loop, always. On two beds a missed needle on the far bed saves 105.5 per cent of a loop at a bed gap of one needle pitch — and 95.4 per cent at three quarters, and 127.0 at one and a half. A quantity that was an identity is now a quantity with a stated machine dimension in it. That is honest arithmetic rather than a regression, and it is a real loss of cleanliness.

Where the model stops #

The conversion to centimetres hides the same gap twice. Yarn here is measured in needle pitches and area in relaxed wale spacings, and treating those as one unit is exactly what the float essay flagged as the float’s slack: a knit is laid at the machine’s pitch and lives at a smaller one. On a rib the two differ by more than on single jersey, because a rib contracts widthwise by folding, and the fold is not in this arithmetic.

The tuck factor is a stated calibration and it is unchanged. A tuck is taken as 1.15 of a loop, which is the trade’s figure and not this site’s, and every number involving a tuck moves with it.

The bed gap is stated and not measured. It is quoted with every figure that depends on it, it never enters the float classification at all, and it is not obtained from any fabric.

And the relaxation state is doing more work than it looks. Every constant on this site is quoted with its state because that is the field’s standing error, and the 24.2 per cent spread above is the size of the error a specification makes by leaving it off — for the one structure anybody has measured. Two beds make that worse rather than better, since a rib’s relaxation has a fold in it that plain jersey has not.

Who found it, and when #

Munden’s 1959 paper is the foundation and its scope was narrower than its influence. It measured relaxed plain weft-knitted fabrics and found that their dimensions depend on the loop length and on nothing else — not the yarn count, not the fibre, not the machine gauge — which was a much stronger claim than the data strictly supported at the time and has held up.

He did not extend it to tuck and float structures and nobody has extended it cleanly since. What exists instead is a scattered set of tables, structure by structure, each obtained the same way and each answering only for its own fabric. The gap has been visible for sixty-five years and it has never been posed as a composition question, which is the small thing this essay adds: the reason the tables do not join up is not that nobody has done the arithmetic, it is that there is no arithmetic to do.

The nearest thing to a general result is the woven side of the same question, and the contrast is the sharpest way to see what is missing. A woven cloth’s weight comes out of sett, count and crimp, and the crimp comes out of Peirce’s geometry — so the whole thing computes from a specification with no fitted constant. The weave’s difficult quantity was the crimp and it was solved by geometry; the knit’s difficult quantity is the relaxed dimension and it is settled by a force balance nobody has written down. The difference is structural rather than historical: a loom’s reed puts the threads where they are, and a knitted fabric decides for itself.

Where the ladder goes next #

The next rung on this anchor is the one nothing here can do without: a model of the relaxed loop that predicts a dimension rather than reading one off a table. That needs the yarn’s bending stiffness and the friction between loops, which would be the first mechanical model of a knit on this site and the first thing in this field that is not a property of the array.

Failing that — and it should be said plainly that failing that is likely — the useful rung is narrower and worth having: a sensitivity ladder rather than a prediction. The weight is linear in one constant, the extension and the cover are not, and knowing which quoted property of a knitted fabric moves how far with an unmeasured dimension would tell a specifier which numbers on a data sheet are counts and which are guesses.