A knit's dimensions come from its loop

A woven cloth’s dimensions come from its sett, and its sett is decided by the reed — or rather is not, since the cloth contracts off it. Ask how many threads per inch a fabric has and the answer was determined when someone chose a reed, before a single pick was inserted.

A knit has no reed and no sett. Its dimensions are not set by the machine at all — a plain knit relaxes to whatever size it wants, and the machine’s gauge merely decides how many needles were involved in producing that much fabric.

So what does decide them? The answer is one measurement, and it is a remarkably clean result.

The result #

For a plain weft-knitted fabric in a relaxed state, with ℓ the length of yarn in one loop:

courses per unit length × ℓ = k_c wales per unit length × ℓ = k_w stitch density × ℓ² = k_s courses ÷ wales = k_r

and the four k’s are constants. Not constants for a particular yarn — constants across everything anyone has tried, within the scatter of the measurement.

The dimensional structure is worth pausing on. Courses per unit length has dimensions of one over length, ℓ has dimensions of length, so their product is dimensionless and can be a pure number. Stitch density is one over length squared, so it takes ℓ². The relations are the only ones that could be dimensionally consistent, which is a hint that the result is geometric rather than material — and that is exactly what it turns out to be.

Why the yarn count cancels #

The mechanism is that a relaxed knit’s shape is decided by the loop’s own geometry, and the loop’s geometry is decided by the length of yarn in it.

A loop of yarn, released and allowed to settle, takes up a shape that minimises its bending energy subject to being interlocked with its neighbours. That shape is similar — in the geometric sense — for any loop, scaled by the length of yarn in it. A long loop makes a large loop of the same shape; a short one makes a small loop of the same shape.

So the fabric built from those loops is a scaled copy of itself, and every length in it scales with ℓ while every count per unit length scales as 1/ℓ. The same cancellation is what makes a loop’s occupancy a ratio of two lengths with no fabric dimension in it.

The yarn’s thickness enters only as a second-order correction, because it decides how much the loop’s own bulk interferes with the shape — and for ordinary knitting, where the loop is many times the yarn diameter, the correction is small enough to sit inside the scatter. That is the whole reason the constants are constants.

This makes a plain knit a genuinely different object from a woven cloth. A woven cloth’s geometry is dominated by yarn diameter: the jamming sett, the cover, the crimp and the thickness are all functions of it. A knit’s is dominated by a length nobody specifies directly and everybody controls, since the loop length is set by the machine’s cam settings and yarn feed.

The constants and their state #

The values differ by relaxation state, and quoting them without the state is the mistake this whole field exists to name.

These are measurements. Nothing on this site derives them, and this essay is one of very few here that quotes a number without computing it. What justifies quoting them is that they are the inputs to a geometric argument rather than its outputs, and the argument’s content is what follows from them rather than what they are.

The shape constant k_r barely moves across the three states, which says something: relaxation makes a knit smaller in both directions at nearly the same rate, so the fabric’s aspect ratio is close to a genuine invariant while its size is not.

The check the constants do not quite pass #

Here is what this site can contribute to a set of measured constants, and it is the kind of contribution available whenever a published set is redundant.

Stitch density is courses times wales. That is not a measurement — it is what stitch density means. So

k_s = (c·ℓ)(w·ℓ) = k_c × k_w

is an identity, and likewise k_r = k_c ÷ k_w. The published sets have four numbers where two would do, and the other two must follow.

They very nearly do and they do not exactly:

What the gap means #

The gap is not an error and reading it as one would be the wrong conclusion.

It is what separately fitted regression constants look like. Each k is obtained by fitting a line through measurements of a different quantity — courses against 1/ℓ, wales against 1/ℓ, stitch density against 1/ℓ² — over a set of fabrics with scatter in all of them. Three independent fits to noisy data do not satisfy an exact algebraic relation, and a published set that did would be evidence that somebody had computed two of them from the third rather than measuring them.

So the right reading is: the constants are three measurements of one fabric family, and their mutual disagreement is a measure of the scatter.

That makes the pattern across the states interesting. The dry-relaxed set is by far the worst — 5.3 per cent, against 0.6 and 0.2 — and there is a good reason. Dry-relaxed is defined by the fabric having been left alone: it is the state a knit reaches with no wetting and no mechanical action, which means it is defined by an absence rather than by a procedure. Two laboratories’ dry-relaxed specimens have had different amounts of nothing done to them, and the scatter is correspondingly larger.

Wet-relaxed and fully-relaxed are both defined by positive procedures with stopping conditions, and their constants agree with their own arithmetic an order of magnitude better. A state defined by a procedure is more reproducible than a state defined by neglect, and here that shows up as a number.

What was counted, and how #

mundenConsistency computes k_c × k_w and k_c ÷ k_w for each published set and compares them against the published k_s and k_r, asserting each within a tolerance — and it asserts something else that is easy to miss the point of: that no set satisfies the identity exactly.

That second assertion is the unusual one. It would fail if somebody tidied the table by computing k_s from the other two, which would be a small, well-intentioned change that destroyed the evidence about scatter. Asserting the inexactness protects a finding that only exists as a discrepancy.

The tolerance is a documented decision. It was first written at three per cent and failed on the dry-relaxed set, and the temptation was to treat the failure as a defect in the constants. Raising it to eight per cent — with the reasoning recorded in the source — admits the published sets and still refuses a mistyped one, which is a different and achievable job. A bound written to admit what is known and refuse what is wrong is worth more than a bound chosen for its roundness.

knitRelaxation then computes what a knit does between two states, and asserts that the answer is identical for every loop length, because ℓ cancels out of the ratio. That check has teeth: it compares three fabrics of very different loop length and requires their area shrinkages to agree to twelve decimal places.

The practical half #

That last result is the one that matters in a knitting mill and it is worth stating on its own.

A plain knit’s shrinkage from one relaxation state to another does not depend on the fabric. Dry-relaxed to fully relaxed is a 15.4 per cent area change for every plain knit, whatever its yarn, its gauge or its loop length, because it is the ratio k_s(dry)/k_s(full) and nothing else.

That is a much stronger statement than anything available for woven cloth, where the shrinkage depends on the crimp, which depends on the weave and the sett and the yarn diameter. A knitter can predict relaxation shrinkage from a table with three rows; a weaver needs the construction.

It is also the reason knitted garments are specified by loop length rather than by dimensions. Control the loop length and the finished dimensions follow, which is why a knitting machine’s most carefully controlled parameter is the yarn feed rather than anything about the fabric.

What a knitter controls, and what a weaver does #

The comparison at the end of this essay is worth one more turn, because it changes what each maker has to watch.

A weaver controls the spacing directly, through the reed, and the fabric’s crimp follows from the spacing and the yarn. So a weaver’s dimensional problem is that the crimp is a consequence and the relaxed dimensions follow from it.

A knitter controls the loop length directly, through the yarn feed, and the fabric’s dimensions follow from it by Munden’s constants. So a knitter’s dimensional problem is that the loop length is hard to measure in-process — it is inferred from yarn consumption per course — and easy to specify.

Both are one measurement away from control and it is a different measurement. That is the sharpest way to put the difference between the two structures, and it explains why a knitting mill’s quality system is built around yarn metering and a weaving mill’s around loom setting.

The second redundancy, which points the same way #

The published sets carry four constants where two would do, so there are two independent checks and the essay above runs one of them. Running the other is free and it corroborates the finding rather than merely repeating it.

The first identity is that stitch density is courses times wales. The second is that their ratio is the shape constant:

k_r = k_c ÷ k_w.

The dry-relaxed set is the worst on this check too, and by a similar margin — three per cent against a half and one and a half. Two redundancies, computed from different combinations of the same four numbers, both singling out the same row.

That is a considerably stronger statement than either alone. A single discrepancy in a single identity is what any three separate regressions would produce, and it could sit anywhere; two identities agreeing about which set is loosest is evidence that one of the three measurements really was noisier than the others, and the essay’s explanation — that a state defined by neglect is less reproducible than one defined by a procedure — now has two independent measurements behind it rather than one.

Two further remarks the pair makes available.

The two checks disagree about the size and agree about the ordering, which is the ordinary situation when several fitted quantities are being reconciled: the products and the ratios weight the individual errors differently, so no single per-cent figure is the inconsistency of a set. Quoting both, and noticing that they rank the three states identically, is more honest than averaging them.

And four constants with two degrees of freedom is a two-dimensional over-determination, so a least-squares reconciliation of each set would give a best pair of k’s and a residual. Doing that would be tidier and would destroy exactly what the essay is reading — the scatter between three independent fits — so it is not done here. The redundancy is the measurement, and reducing it away would leave four numbers that satisfy their own arithmetic and say nothing about how well they were measured.

Where the model stops #

The constants are for plain single jersey. Rib, interlock, purl and every patterned structure have their own, and some of them are not well described by a single set at all.

The scaling assumes the loop is large compared with the yarn. At very tight settings — a short loop in a thick yarn — the yarn’s own bulk stops being a correction and starts being the constraint, and the fabric jams in a way this argument does not describe.

Nothing here derives the constants, and this essay is unusual on this site for that reason. What it does is check them against arithmetic they must satisfy, which is a weaker and still worthwhile thing.

And the relaxation states are three named points on a continuum, with the procedures behind them varying between laboratories more than the tidy names suggest.

What the woven equivalent would look like #

The obvious question for this site is whether a woven cloth has a Munden result, and the answer is instructive: it does not, and the reason is structural rather than historical.

A knit’s dimensions depend on one length because a relaxed loop’s shape is scale-invariant — it is decided by the yarn’s own bending, and a longer loop is a bigger loop of the same shape.

A woven cloth’s dimensions depend on two lengths that are set independently: the yarn’s diameter, and the spacing the reed imposed. Those are not related by anything, so no single parameter can carry the fabric’s geometry, and this site’s woven arithmetic accordingly takes both.

The knit’s simplicity comes from the machine not setting a dimension. A knitting machine’s needles decide how many wales there are in the piece and not how far apart they sit; the fabric decides that when it relaxes. A loom’s reed decides the spacing directly, and the cloth is not free to choose.

So the two structures are opposite in exactly the way that matters here: the knit has one free geometry and one measured constant, and the weave has two imposed geometries and no constants at all. That is why the woven side of this site solves equations and the knitted side quotes a table, and neither is a failure of the other.

The rib and the interlock do not obey it #

The constants are for plain single jersey, and the structures beside it are worth a paragraph because their departures are informative.

A rib has wales alternating between the two faces of the fabric, so the loops are not all in one plane and the fabric contracts widthwise by pulling the two sets of wales together. Its widthwise dimension depends on how far it has been allowed to contract, which is not decided by the loop length alone — and a rib’s characteristic large extensibility is exactly the room left in that contraction.

An interlock is two ribs interlocked, and it is dimensionally more stable than either, because each rib restrains the other.

So the further a structure gets from plain jersey, the more of its geometry is decided by interactions between loops rather than by one loop’s shape, and the less a single-parameter description can carry. Munden’s result is cleanest for the simplest structure, which is the usual way with results of this kind.

Who found it, and when #

D. L. Munden published the result in 1959, from work at the Wool Industries Research Association, and it has been the basis of knitted-fabric dimensional control ever since. The scope of the claim was larger than the data strictly supported at the time and it has held up: the constants have been re-measured for many fibres and the values move very little.

The result’s real significance is what it removed. Before it, a knitter’s dimensional control was empirical per yarn and per machine, and a change of yarn meant re-establishing everything. Afterwards it was one measurement — the loop length — and a table.

That is the same kind of simplification this site’s own arguments look for, and it is worth noticing that it came from measurement rather than from geometry. The geometric argument for why the yarn count cancels is a rationalisation offered after the fact; what Munden had was fabrics, a tape measure and the observation that the numbers collapsed onto one curve.

Where the ladder goes next #

Every essay in this field has turned on the same unstated thing: a quantity quoted without the state it was measured in. The last essay collects them, and finds that the woven side of this site has been doing something the knitted side stopped doing sixty years ago.