What comes off the loom is not the cloth

Sixty-one essays on this site have quoted a sett, a cover factor, a crimp, a thickness or a mass per unit area. Every one of those numbers describes a fabric in a particular condition, and not one of them names it.

The condition is the loom state: the warp under tension from the moment it was beamed, the cloth held out to the reed width by the temples as it is formed, the fabric wound onto a roller under a further tension of its own, and nothing at any point wetted. It is a real, reproducible, perfectly measurable condition. It is also one that no fabric is ever in by the time anybody looks at it, because the first thing done to a piece of cloth is to take it off the machine.

The reference condition nobody states #

A measurement without a reference condition is not a measurement. Everyone knows this about temperature and about pressure, and the textile trade knows it too — the standards for fabric testing specify a conditioned atmosphere, twenty degrees and sixty-five per cent relative humidity, and a specimen left in it long enough to come to equilibrium. What the standards specify much less loudly is the mechanical state, and it is the one that moves the numbers furthest.

Here is the difficulty in one sentence. The quantities this site computes exactly are quantities of a cloth in equilibrium with itself, and a cloth on a loom is not in equilibrium with itself.

Peirce’s geometry, which supplies almost every thread-geometry number here, closes on the condition h₁ + h₂ = D: the two thread systems between them fill the thickness of the cloth. That is a statement about a fabric with no external load on it. A cloth being woven has a large external load on it in one direction and a smaller one in the other, and the closure condition does not hold.

This matters more than a footnote, and the way it matters is worth being precise about. It is not that the loom-state numbers are wrong. They are right, about the loom state. It is that they are quoted as though they were properties of the fabric.

What actually changes, and by how much #

The whole of the change is one quantity moving: crimp.

A thread in a cloth is longer than the cloth it crosses, because it goes over and under rather than straight. That excess is the crimp, this site has computed it from the foundation, and the arithmetic that connects it to a dimension is one line: a thread of length l lying at crimp c spans a cloth length of l/(1 + c).

On the loom the warp is held straight by the tension. It has less crimp than the fabric’s own geometry would give it. Take the load off and the yarn takes up the crimp it prefers — and the thread cannot get longer, so the only thing that can change is the length of cloth it spans.

Nothing is lost. Nothing is destroyed or consumed. The length moves from the cloth into the crimp, which is exactly the bookkeeping this site’s assertCrimpInterchange has enforced since the first commit, run in one direction until it stops.

Why this is a whole field and not a correction #

It would be possible to treat all of this as an erratum — a paragraph appended to the earlier essays saying these numbers are loom-state numbers — and that would be honest and would miss the interesting part.

The interesting part is that every operation done to cloth after the loom moves a quantity this site already computes, and computes exactly. There is no new physics in the finishing works. There is a great deal of chemistry, and a great deal of very sophisticated machinery, and what the machinery does is move parameters that are already in the yarn and cloth models:

• Relaxation and washing move the crimp, and the crimp decides the dimensions.
• Calendering flattens the yarn’s section, which is the flatten parameter of Kemp’s racetrack — a model this site has kept beside Peirce’s circle for a long time without noticing that the two are the same yarn on either side of a machine.
• Mercerising swells the fibre, which is the packing factor in diameterFromTex, and every cover and jamming consequence follows from that single number.
• Raising pulls fibre ends out of the floats, so which cloths can be raised at all is a question about the float map and is decidable over the whole four-by-four census.
• Milling entangles the fibres until the cloth holds together for a second reason that has nothing to do with the weave — which is the nonwoven’s mechanism arriving inside a woven fabric.

That last one is the only place in this field where something genuinely new is needed, and what is needed is a ratchet.

The size of the effect #

A correction worth a field ought to be large enough to matter, so here it is at the numbers this site’s own solver produces.

Take a balanced plain weave in a yarn of diameter 0.2 in the site’s usual units, set at a spacing of 0.42 — an ordinary, comfortably unjammed cloth. Solve its relaxed geometry with Peirce’s equations and it wants about 12.8 per cent crimp in each direction. Suppose the loom took seventy per cent of the warp crimp out and a quarter of the weft’s, which is an unremarkable pair of numbers for a shuttleless loom with temples.

Then the cloth loses 7.9 per cent of its length and 2.8 per cent of its width the first time it is properly relaxed. Its area falls by 10.6 per cent, and its mass per unit area rises by 11.8 per cent without a gram of anything being added to it.

Eight per cent of a length is not a rounding error. On a two-metre trouser leg it is sixteen centimetres. On a shirt sleeve it is enough to change the size on the label, and it is the reason the label exists.

What was counted, and how #

Everything above comes from one function and one input that is a model rather than a measurement, and both are worth stating plainly.

The relaxed cloth is solved with peirce() at the given spacings — the same solver used since the foundation, unchanged, with its own consistency check re-run forwards on every solution. That gives the crimps a fabric in equilibrium wants.

The loom state is then that same thread with a stated fraction of each crimp pulled out of it, and the shrinkage in each direction is (c_relaxed − c_loom)/(1 + c_relaxed). It is computed twice — once from the crimps and once from the spacings — and the two are asserted equal, because agreeing is not automatic and a mistake in either bookkeeping is invisible in a picture.

The fraction of crimp the loom removes is an input, not a computation. It depends on the loom, the yarn, the let-off and the weaver, and nothing on this site derives it. Every shrinkage quoted here is therefore a shrinkage at a stated loom tension, and the essays in this field quote ranges wherever one number would be misleading.

The one thing that is not an input is the ceiling, which is the subject of its own rung.

The order the argument has to go in #

This field inverts something. Everywhere else on this site the matrix comes first and the yarn comes second: the draft decides the floats and the interlacings exactly, and then a diameter is introduced and a smaller set of quantities becomes available with a model attached.

Here the yarn comes first. Finishing barely touches the matrix at all. A milled cloth, a mercerised cloth, a raised cloth and a calendered cloth all have exactly the draft they were woven with; every float is where it was, every interlacing is where it was, and the integrity criterion returns precisely the same answer before and after. Run the check on a piece of melton and it reports, truthfully, that the threads form one cloth, and it will be describing a fabric whose weave can no longer be seen with a lens.

That is not a failure of the criterion. It is the criterion answering the question it was asked. But it does mean that a field about finishing is a field in which this site’s sharpest instrument is largely silent, and the honest response is to say so at the start rather than to stretch the instrument.

Why none of this shows up in a gate #

There is a reason this went sixty-one essays without being said, and it is the same reason several other findings on this site went unnoticed for a long time: nothing that runs here has any way to ask the question.

Consider what the checks actually test. clothcheck re-runs the integrity assertions and requires them still to reject — and the integrity of a relaxed cloth is identical to the integrity of a loom-state cloth, because relaxation moves no thread from above another to below it. svg_check asks whether a label fits and contrasts. labelcheck asks whether any label carries a stroke. figbox asks whether the drawing fits its viewBox. optcheck asks whether an option a placement passes reaches a parameter. Not one of them can distinguish a number that is right from a number that is right about a different fabric.

That is the general shape and it is worth naming, because it recurs. A check tests the relation between a figure and its own generator, and the state a quantity was measured in is not in either of them. The caption says crimp 12.8 per cent and the generator computed 12.8 per cent, and the machinery is entirely correct and entirely silent about which cloth it is 12.8 per cent of.

The repair is not a gate. It is a habit: any quantity whose value depends on the mechanical state is quoted with the state, in the same sentence, and where a figure draws two states it draws them at one scale so that a reader can see the difference before reading the number.

Which quantities need a state, and the rule that decides #

The repair is proposed as a habit rather than a gate, on the ground that no check can distinguish a number that is right from a number that is right about a different fabric. That is true of the checks as they stand and it is not true in principle, because the quantities that need a state can be told apart from the ones that do not by a rule with no judgement in it.

A quantity needs a state if and only if its computation touches a spacing.

Everything this collection computes falls into three classes on that test, and the classes are the collection’s own architecture seen from a new angle.

Matrix quantities are state-free. Longest float, interlacing count, layer count, shaft count, balance, plane group, blind intersections, raisability. Every one is a function of a binary matrix and of nothing else, so relaxing a cloth, washing it, calendering it or milling it leaves all of them exactly where they were. The integrity criterion returning the same answer before and after a milling is not a blind spot; it is this class behaving correctly.

Geometric quantities are state-dependent, all of them. Sett, cover factor, crimp, thickness, mass per unit area, hole size, air permeability, jamming sett, locking angle, bending length, crown line. Each computation takes a thread spacing or a diameter, and both move with the state.

And fibre quantities are state-free but conditioned. A count, a density, a tenacity, a swelling: none moves when a fabric relaxes, and every one is quoted at a standard atmosphere that is a state of a different kind.

Which makes it a gate after all #

That rule is mechanical, and it is checkable from the code rather than from the prose.

A generator whose inputs include a length produces a state-dependent number. One whose inputs are a matrix alone does not. So a check can walk the generators, sort them by whether a spacing or a diameter reaches them, and require every caption on a figure from the first group to name a state.

That is a caption gate of exactly the kind this collection already runs — the one that forbids a typed count has the same shape, and so does the one that requires every figure to carry a description. It cannot check that the state named is the right state, any more than the count gate can check that a collection is the right collection; what it can do is make the omission impossible, which is what the sixty-one essays needed.

The class boundary is the same boundary the site is built on, which is why the rule is so simple. This collection’s own division is between what a draft decides and what a yarn decides, and it turns out that the first is state-free and the second is not — so does this quantity need a state and does this quantity need a yarn are the same question.

And it says which essays are safe #

Applying the rule retrospectively sorts the collection without reading it.

Every census is safe. The four-by-four enumerations, the separation rate, the plane groups, the colour collisions, the shaft counts, the twill classes: all matrix quantities, all state-free, none of them touched by anything a finishing works does.

Every setting, cover and thickness essay is not. Those are the ones the omission reaches, and it reaches them uniformly rather than selectively — which is the good case, because a uniform omission moves every number in one direction by one factor and leaves every comparison intact.

That is the honest summary of the damage. The orderings survive and the levels do not, and the collection’s own habit of preferring ratios to values turns out to have been protecting it from a mistake it had not yet noticed.

The three states a standard names #

Since the state has to be named, it helps that the trade has already named them, and the names are more careful than they look.

Dry-relaxed is the fabric left alone: taken off the machine, laid flat, and given time. It is the easiest state to reach and the hardest to reach reproducibly, because it is defined by what has not been done to the cloth rather than by anything positive, and a piece that has been folded, rolled or stacked is a piece that has had something done to it.

Wet-relaxed is the fabric soaked, drained and dried flat with no mechanical action. Wetting is what lets the yarn move: water swells the fibre, lubricates the contacts, and releases whatever the drying tension of the previous operation had set.

Fully relaxed — the term is used loosely and the procedures behind it are not one procedure — adds mechanical action, usually tumbling, until further treatment changes nothing measurable. It is the only one of the three defined by a stopping condition rather than by a recipe, which is what makes it the right reference and the most expensive to establish.

The distance between the first and the last is not small, and the knit essay in this field computes it exactly for a plain knit: a fifteen per cent change in area between dry-relaxed and fully relaxed, identical for every fabric regardless of what it is made of.

Where the model stops #

Four limits, and the first is the largest.

The loom state is idealised as a crimp reduction and nothing else. A real cloth on a real loom is also being beaten up by the reed, abraded at the drop wires, and held at a width that varies across the piece. Reducing all of that to two fractions is a model, and its justification is that the crimp is what the dimensions depend on rather than that the rest does not happen.

Peirce’s geometry is one model of several, as this site has said at length. Kemp’s racetrack gives different thicknesses for the same cloth and therefore different crimps. What survives the choice is the shape of the argument — thread length conserved, crimp as the only variable, a ceiling at c/(1 + c) — and not the third decimal place.

Nothing here is time-dependent. Real relaxation is not one event: a cloth loses some of it in the first wash, more in the fifth, and settles asymptotically at a state that takes tumbling and repeated wetting to reach. The states in this field are equilibria, and the path between them is not modelled at all.

And the fibre is treated as inextensible and unchanging except where swelling is explicitly turned on. A wool fibre under load in water does things this arithmetic knows nothing about.

Who found it, and when #

The bookkeeping is old. Peirce’s 1937 paper set out the geometry and, in the same work, the observation that the two crimps trade against one another. What the trade calls crimp interchange was understood as a mechanism well before it was computed.

Compressive shrinkage — the operation that pre-empts all of this by doing it in advance — is Sanford Cluett’s, patented in 1930 and sold under his name ever since. That the machine exists at all is the clearest evidence that the effect was recognised early and taken seriously: nobody builds a machine to remove four per cent of the length of everything they sell unless the four per cent is real and expensive.

What is comparatively recent is the insistence on naming the state. The standards that define residual shrinkage, and the ones that specify the relaxation procedure a specimen must undergo before it is measured, are post-war and were written because dimensions quoted without a state had become a commercial problem rather than a scientific one.

Where the ladder goes next #

Three ladders leave this essay and they leave in different directions.

The first is the arithmetic of shrinkage itself, which begins with relaxation and runs through why the two directions differ, what bounds them, and what a pre-shrunk label is actually promising.

The second is the surface: the yarn flattened, the fibre swollen, and the floats raised — three operations that leave every dimension where it was and change what the cloth is.

The third is felting, which is the one operation in this field that is not reversible in any sense and needs a mechanism the site did not have.

They meet again at the end, in the essay about what a dimension means when nobody has said which state it was measured in.