A knit relaxes for as long as it is allowed to

A knitted fabric’s dimensions are a loop length times a dimensionless constant, and there are three sets of constants because there are three states a specimen can be measured in: dry relaxed, wet relaxed and fully relaxed. Every standard for knitted fabric dimensions says which of the three it means, and the reason it has to is that they differ by several per cent.

The three are usually presented as a table — three pairs of numbers, one pair per state — and read as three answers to the question “how big is this fabric?”.

Read as a path, they say something the table does not.

The claim #

A knit’s relaxation is two steps, and the anisotropy reverses between them.

The first step, from dry relaxed to wet relaxed, is 5.66 per cent along the courses and 2.44 across the wales — a ratio of 2.3 to one, strongly in the course direction. The second step, from wet relaxed to fully relaxed, is 3.64 and 4.65 — a ratio of 0.78, slightly the other way.

So a knit does not shrink in one direction and then shrink less in the same direction. It shrinks lengthwise and then widthwise, and a specimen stopped after the first step has a different shape from one taken to the end, not merely a different size.

That has a consequence for testing which is not usually stated: the difference between a fabric measured wet-relaxed and one measured fully relaxed is not a scale factor, so a correction applied as a percentage to both dimensions is wrong in both.

Why the steps compose, and why that is a check #

The three states are three independently measured sets of constants. Nothing about the way they were obtained forces them to be consistent with one another, and a table of three pairs of numbers is exactly the sort of thing that acquires a transcription error nobody notices.

Composing the two steps is a check on all three. If the fabric goes from dry relaxed to wet relaxed losing a fraction s₁ of its length, and from wet relaxed to fully relaxed losing s₂ of what is left, then the whole change from dry relaxed to fully relaxed must be 1 − (1 − s₁)(1 − s₂). It is, to machine precision.

The composition is multiplicative rather than additive, which is worth stating because getting it wrong is easy and the error is small enough to look like rounding. Adding 5.66 and 3.64 gives 9.30 where the correct answer is 9.09 — four tenths of a point, on numbers quoted to two decimal places. This collection has recorded that shape of mistake before, in the arithmetic of pre-shrinking: a shrinkage composes on what is left rather than on what has gone.

The same shape as a laundering series #

Along the courses the two steps fall in the way a laundering series falls: the second is 0.64 of the first. That is not a coincidence and it is not a separate mechanism.

A wash takes a woven cloth through a sequence of states, each one a little further along its own locus, and the steps get smaller because the cloth’s remaining excess gets smaller while the distribution of frictional barriers holding it does not. A knit’s three states are the same process, sampled at three points that somebody chose to name.

The names are the interesting part. Dry relaxed means the fabric has been laid flat and left, so the only thing that has moved is what the yarn’s own bending could push past friction unaided. Wet relaxed means it has been soaked, which lowers friction and lets more move. Fully relaxed means it has been tumbled, which lowers the effective barrier further still.

None of the three is a duration. Each is a quantity of agitation, and the standard specifies the machine, the temperature and the number of cycles precisely because those are what the state is defined by.

That is the sentence the table of constants hides. A fabric left wet in a bucket for a week does not become wet relaxed; a fabric tumbled for twenty minutes does. The variable is not time.

Why the anisotropy reverses #

The reversal is the finding that a table of constants cannot show, and this collection can say what it is not rather than what it is.

It is not the fibre swelling. A swelling enters both of a loop’s dimensions through the same loop length, so it cannot produce an anisotropy at all, let alone one that changes sign — which this collection established when it asked whether Munden’s states were the fibre taking up water, and found that they are not.

It is not a change in the loop length. The loop length is fixed by the knitting machine and does not change in relaxation; every constant here is dimensionless and the loop length divides out.

What is left is the loop’s shape, and the shape has two independent ways to move. A loop can get taller and narrower or shorter and wider at constant thread length, exactly as a woven crossing can, and it can also change how much of its thread is in the legs versus in the heads. Two mechanisms, two directions, and no reason for them to be released in the same order.

So the honest statement is that the reversal is evidence for two distinct rearrangements with different frictional barriers, released in sequence, with the lower-barrier one acting mostly along the courses and the higher-barrier one mostly across the wales. This collection has no model of a loop detailed enough to say which is which, and says so rather than guessing.

What the reversal costs a specification #

A knitted garment is cut from a fabric measured in one state and worn in another, and the reversal decides what that costs.

If the whole relaxation were a single anisotropic shrinkage, a maker could allow for it with two percentages and be done. It is not: a fabric that has been wet-relaxed but not fully relaxed has taken most of its course-direction change and less than half of its wale-direction change, so what remains is mostly across the wales — 4.65 per cent of width against 3.64 of length.

The residual is a different shape from the total. A garment cut from wet-relaxed fabric and then washed and tumbled loses proportionally more width than length, which shows up as a garment that got shorter and much wider, or rather as one that got shorter and did not get narrower nearly as much as expected. A maker allowing a single percentage on both dimensions gets one of them wrong by a factor of nearly two.

This is why the standards are as insistent as they are about which state a measurement was taken in, and it is a stronger reason than the one usually given. The usual reason is that the numbers differ. The better reason is that the numbers differ differently in the two directions, so no single correction converts one state’s measurement into another’s.

The two steps are equal in area #

The two steps differ in shape and the essay stops there. Composing each step’s two dimensions into an area gives a result the shape comparison hides.

Step one, dry relaxed to wet relaxed: 1 − (1 − 0.0566)(1 − 0.0244) = 7.96 per cent of area.

Step two, wet relaxed to fully relaxed: 1 − (1 − 0.0364)(1 − 0.0465) = 8.12 per cent.

The two steps take almost exactly the same area — within two per cent of each other — while being completely different in how they take it. One is course-dominant at 2.3 to one and the other is wale-dominant at 0.78, and both remove an eighth of a twelfth of the fabric.

So the area is the invariant of the path and the shape is not. That is a stronger statement than the reversal on its own, and it is a second consistency check on three independently measured sets of constants: the composition test says they belong to one fabric, and the equality of the two area steps says something about what the relaxation is doing.

Which points at what is being conserved #

An equal-area pair of steps with opposite anisotropies is what a fixed budget spent two ways looks like.

The loop length does not change, so the yarn per unit area does not change, so an area contraction is the loop’s plan footprint shrinking — the same quantity this collection calls the occupancy, which is the yarn’s plan area over the cell’s. A given amount of occupancy is recovered in each step, and how it divides between the wale and the course is what the two mechanisms differ in.

That is a much more specific account than two rearrangements with different barriers. The two mechanisms are not two ways of shrinking; they are two ways of dividing one shrinkage, and the quantity being divided is the same size both times.

It also says what a fuller model would have to produce. Any loop model that explained the reversal would have to explain it at constant area per step, which is a strong constraint and rules out any account in which one mechanism is simply larger than the other.

And it gives a maker one number that works #

The practical consequence inverts the essay’s own advice in a useful direction.

A maker cannot allow for relaxation with one percentage on both dimensions, because the residual’s shape differs from the total’s by nearly a factor of two. But they can allow for it in area with one number per step — eight per cent, twice — and the area is what decides fabric consumption.

So the two quantities separate cleanly. A garment’s cloth consumption is predictable from a single number and its finished dimensions are not. A cutting room ordering fabric for a run needs the area and gets it; a pattern grader needing the length and width gets two numbers whose ratio depends on which state the fabric was in when it was measured.

That is a real division of labour and it explains why the two departments have historically had different complaints about knitted fabric. The buyer’s allowance works and the grader’s does not, and the reason is that one of them is asking about the invariant and the other about the thing that moves.

Whether three states are the right three #

It is worth asking whether the three named states are natural or conventional, because the argument here treats them as samples of a continuous path.

They are conventional, and the convention is a good one. Dry relaxed is the state a fabric reaches on its own, and it is the only one of the three that needs no equipment. Fully relaxed is the state a fabric reaches when agitation has taken it as far as agitation takes it, and it is the one a garment eventually arrives at in use. Wet relaxed sits between them and is the one that is hardest to justify on its own terms — it is a state defined by a soak, and a soak is a weak and variable amount of agitation.

The path picture says what the middle state is for. It is a sample at an intermediate barrier, and its value is that it separates the two rearrangements: without it, the reversal of the anisotropy would be invisible, because the whole change from dry to fully relaxed is 9.09 per cent along the courses and 6.98 across, which is a single anisotropy of 1.3 to one and looks like one mechanism.

The middle state is what makes the second mechanism visible, which is a good reason to keep measuring it even though it is the least well defined of the three.

What was counted, and how #

The three sets of constants are measured and are quoted as such; every number in this rung is arithmetic on them.

The composition is asserted rather than checked by eye, to a tolerance of a billionth, and it is asserted in the multiplicative form. That assertion is doing real work: it is the only test available that the three sets belong to one fabric.

The ordering of the two steps is asserted in the course direction — the first must be the larger — because that is the claim being made about the mechanism, and a table in which it failed would mean the states had been mislabelled or the sequence misunderstood.

The loop length is a parameter throughout and every result is checked to be independent of it, which it must be, since every quantity in the calculation is a ratio of constants.

Where the model stops #

Three points is not a series. A woven cloth’s laundering test gives five numbers and lets a distribution be fitted to them; a knit gives three states with two steps between them, which is enough to see the shape and not enough to measure anything about the distribution. What this rung can say is that the shape is the right shape.

The states are plain knit only. Munden’s constants are for single jersey. A rib, an interlock or a purl has its own constants and its own path, and there is no reason the two steps should have the same ratio or the same reversal — a rib’s two beds are doing different things and its wale spacing is set by the gating rather than by the loop alone.

The mechanism is named and not modelled. Nothing here computes a frictional barrier for a loop, so the account of why the anisotropy reverses is a statement about what is left after two candidates are eliminated. That is weaker than a derivation and stronger than a guess, and it is where this collection’s knitted mechanics currently stops: the interlock force between two loops is a quantity that has been recorded as missing for several fields.

And nothing is time-dependent, on purpose. Whether a fabric reaches a state in ten minutes or ten hours is not a question this ladder asks, and the fact that the standards specify agitation rather than duration is the evidence that it is the right thing not to ask.

Why a knit has states and a woven cloth has washes #

The two fabrics are doing the same thing and the vocabulary is completely different, which is worth accounting for.

A woven cloth’s relaxation is quoted as a series of laundering cycles because that is how it is measured and because the cloth arrives at the test carrying an excess put in by the loom — the beam tension, the take-up, the finishing route. How much excess is a property of the mill rather than of the construction, so the useful quantity is what comes out per cycle rather than where the cloth ends up.

A knit arrives with much less. A knitting machine does not hold its fabric under anything like the tension a loom holds a warp under, and a loop’s geometry is not fixed by a reed. So a knit’s destination is well defined enough to be worth naming, and naming states is more useful than counting cycles.

The two conventions therefore encode a real difference: a woven cloth’s relaxation is mostly about where it started and a knit’s is mostly about where it ends. That is why a woven specification quotes a residual shrinkage after N washes and a knitted one quotes dimensions in a state.

It also says which of the two is more likely to surprise a wearer. A woven cloth’s residual is bounded by what the loom put in and is mostly gone after five washes. A knit’s is bounded by the distance to the fully relaxed state, and a garment sold in a state short of that will keep moving toward it for as long as it is worn and washed — with the width doing most of the moving.

The generalisation #

A sequence of named states is usually a sampled path, and the samples were chosen for convenience rather than for the physics. Reading them as a path rather than as a set of answers asks two questions the table does not: do the steps compose, and are the later steps the same shape as the earlier ones?

Both questions are cheap and both are informative. Composition tests the internal consistency of independently measured sets, and a table that fails it has an error in it. Shape tests whether one mechanism is acting or several, and a change of anisotropy between steps is about as clear a signal of two mechanisms as a sequence of three points can carry.

The second lesson is about what a standard’s procedure is telling the reader. When a specification insists on a machine, a temperature and a number of cycles rather than a duration, the underlying variable is not time, and any model built on a rate is modelling the wrong thing. The knitted fabric standards have been saying so, in their own vocabulary, for as long as they have existed.

Who found it, and when #

Munden’s constants are from 1959 and are the foundation of knitted fabric geometry: a plain knit’s course and wale spacings are proportional to its loop length, with constants that depend only on the state. The three states and their definitions are from the same tradition and are embodied in the testing standards.

That the states differ, and by how much, is thoroughly documented. Reading them as two composing steps and noticing that the anisotropy reverses between them appears not to be standard, which is unsurprising: the constants are used to predict a fabric’s dimensions in a stated state, and the path between states is not a quantity anybody has needed.

Where the ladder goes next #

The same yarn that relaxes into these states also arrives carrying a torque it was never allowed to release, and what that does is a question the structure answers rather than the fibre: a jersey leans and a rib does not.

Sideways, the woven version of this series is a laundering test read as a measurement, and the frictional band that makes both of them progressive is what decides how much load a held fabric keeps.