A dimension without a state

The compound-cloths field ended with an essay that collected its field’s boundary, and the collecting produced a result the individual essays did not have: the four constructions that escape the matrix are the four commitments the matrix makes, each declined in turn.

This field ends the same way and produces the same kind of result. Fourteen essays have each turned on a quantity being quoted without the state it was measured in. Put them together and the quantities sort themselves into two classes, cleanly, and the sorting rule is not the one the vocabulary implies.

The two classes #

Invariant under every operation in this field:

• The draft — which thread is over which, at every intersection. No finishing operation exchanges an over for an under.
• Everything the draft decides: float lengths, interlacing counts, the plane group, the shaft requirement, and whether the cloth is one cloth.
• The yarn count, in the sense of mass per unit length, up to what desizing and scouring remove.
• The number of ends and picks in the piece — a count, not a density.
• The loop length in a knit.

State-dependent, and moved by ordinary operations by ten per cent or more:

• Every dimension: length, width, area.
• Mass per unit area, which is a mass over a state-dependent area.
• Threads per unit length, in both systems, and therefore the cover factor.
• Crimp, which is the mechanism of all of the above.
• Thickness, moved by calendering directly and by relaxation indirectly.
• The yarn diameter, moved by swelling and by flattening.

The rule, and why it is not obvious #

The sorting rule is this: a quantity is invariant if it is a count or a topological fact, and state-dependent if it has a length in it.

Stated that baldly it sounds trivial. It is not, because the vocabulary of the trade cuts across it in exactly the wrong place.

Consider what a fabric is normally described by. Thread count sounds like a count and is a density — threads per inch — so it moves. Weight sounds like a mass and is a mass per unit area, so it moves. Cover factor sounds like a structural property and is a sett times a diameter, so it moves twice over. Meanwhile weave sounds like a stylistic choice and is the most invariant thing about the fabric.

The words that sound like properties of the cloth are mostly ratios with a length underneath, and the word that sounds like a preference is the invariant. That is why the state gets left off: nothing in the vocabulary signals which quantities need it.

The knitted side already solved this #

There is one part of this subject that does not have the problem, and comparing them is instructive.

Munden’s result put a knit’s dimensions on one invariant — the loop length — and gave the constants of proportionality a state label. A knitter specifies loop length, quotes the state, and the dimensions follow. The state is not optional in that vocabulary; the constants literally differ between states and a number without one is unusable.

The woven side has the same invariant available and does not use it. Crimp is the exchange rate between a thread’s length and the cloth’s, exactly as the loop length is for a knit, and a specification carrying the loom-state construction plus the crimp would be state-independent and convertible to any state.

Nobody writes one. The reason is practical rather than conceptual: measuring crimp means destroying a sample and straightening threads out of it, while measuring a length is free. So the woven trade specifies the easy state-dependent quantities and manages the consequences with allowances, and the knitted trade specifies the harder invariant one and does not need them.

That is a genuine finding of this collection and it was not visible from inside either field.

What this means for everything above this field #

Sixty-one essays preceded this one and they quote quantities from both classes. The honest accounting is:

Every count on this site is unaffected. The 22,874 four-by-four drafts, the 426 distinct cloths, the twelve plane groups, the 144 separable drafts, the satin orders, the shaft censuses, the braid words — none of these has a length in it and none of them cares what state a fabric is in.

Every claim about which weave does what is unaffected, because those are claims about the draft.

The geometric quantities are loom-state quantities and are now labelled. The setts, cover factors, crimps, thicknesses and jamming calculations in the setting and mechanics fields describe an unfinished cloth. Their relations — crimp rises with sett, cover rises with diameter, jamming occurs at a weave angle of 60° — hold in any state, because they are consequences of the geometry rather than of the numbers.

That is the reassuring part and it is not a coincidence. This site has consistently computed relations and enumerated counts rather than tabulating measurements, and both of those survive a change of state. A collection built on measured values would have needed a much larger correction.

The instrument that stays silent #

The integrity criterion has been this site’s sharpest tool for a long time, and it is worth stating plainly what it did in this field: nothing.

It returns the same answer before and after every operation here, correctly. A melton and the loose twill it was woven as are both one cloth. A mercerised poplin and the greige poplin are both one cloth. A raised flannel is one cloth, and so is the fabric it was raised from, and so is the fabric after the nap has worn off.

This is the third time the criterion’s silence has been the finding rather than a disappointment. The the compound-cloths field found it blind to friction — a V tuft and a W tuft, a leno and an open plain weave, indistinguishable and different fabrics. This field finds it blind to fibre entanglement, so a milled cloth’s second network is invisible to it.

Both blindnesses are the same blindness and it is the criterion’s virtue. It consumes contacts between threads and a direction at each, which is less than any other model here assumes, and that is why one implementation decides a plain weave, a satin, a double cloth, a braid word, a warp-knit lapping, a pile fabric, a double plush, a leno gauze and now a melton. The price of assuming little is seeing only what follows from little.

The method that follows has been the same each time and is worth stating as a method: do not extend the criterion. Compute the missing quantity separately and print the two side by side. The compound-cloths field printed the criterion’s verdict beside the capstan’s number. This field prints it beside a percolation reading. Neither blends them into a score, because a blended score would have no model behind it and would hide exactly the disagreements worth seeing.

What was counted, and how #

This essay computes nothing new. It re-runs figures from the field’s other essays with the arguments they were given there, which is deliberate: a collecting essay that introduced a new calculation would be a fifteenth essay rather than a collection.

What it does add is a check that could fail. assertFinishRefuses feeds the field’s machinery six inputs it must refuse — a loom tension taking out more crimp than exists, a swelling factor below one, friction coefficients with the scales pointing the wrong way, an unknown relaxation state, a racetrack thinner than it is wide, and a loop length of zero — and asserts that all six throw.

That is this site’s founding habit, applied to a new library: an assertion that has never rejected anything proves nothing, so every family ends by feeding its own machinery input it must refuse.

What is left undone, and named #

Two things this field would want and does not have, recorded here rather than left implicit.

A crimp model for weaves other than plain. Peirce’s geometry assumes every crossing is a crossing, so the relaxed crimp of a twill or a satin is argued from interlacing counts rather than solved. That is enough to order the weaves and not enough to put a ceiling on any particular one.

A fabric-level flattening model. A calender nip closes on crimp crowns rather than on a free yarn, so the cover and thickness figures in this field are computed from a yarn model where a fabric model is wanted. Both directions of the error are known and neither is corrected.

Naming them is this site’s practice at the end of a stretch of work, and the reason is recorded in earlier ones: a shortfall written down is a shortfall a later essay can close, and a shortfall left implicit is one that gets rediscovered.

Where the model stops #

Everything in this field rests on the crimp bookkeeping, and the crimp bookkeeping rests on thread length being conserved. Where it is not — a yarn consolidating under repeated wetting, a fibre shrinking chemically, a resin cross-linking the cellulose in place — the arithmetic does not apply and the essays say so individually.

The state labels are three names on a continuum. Dry-relaxed, wet-relaxed and fully relaxed are the vocabulary the knitted trade settled on, and the procedures behind them vary between laboratories more than the names suggest.

And the invariant list has an asterisk. Desizing removes several per cent of a sized warp’s mass, and a resin finish adds mass, so even the yarn count is invariant only up to the chemistry. What is genuinely untouchable is the draft and everything topological that follows from it.

What a specification would have to carry #

If the argument in this essay is right, then a fabric specification that could not be misread would carry four things, and only one of them is normally present.

The construction in the loom state — ends and picks per unit length as woven, and the yarn counts. Invariant up to desizing, and known to the weaver exactly.

The weave. Invariant absolutely, and normally present.

The crimp in both directions, or equivalently the finished construction and the state it was measured in. This is the piece that is missing and it is the piece that makes the rest convertible.

The finishing route, because the operations in this field are not commutative and their order changes the result.

That is more than anybody wants to write on a swatch card, which is a fair objection and is why it is not done. The point is not that the trade should adopt it. The point is that a fabric described by a mass per unit area and a thread count has been described by two numbers that a finishing works can move by ten per cent each, and that anybody comparing two such fabrics is comparing their finishing routes as much as their construction.

The four-part specification collapses to three and a half #

The specification proposed above has four parts and is rejected as more than anybody will write on a swatch card. It is shorter than that, because two of the four are not independent.

The crimps are not free. At the least-energy state the crimp division is fixed by the two bending rigidities and the two spacings — a ratio of (B₂/B₁)(p₂/p₁)³, which the force balance supplies — and the closure condition then fixes both crimp heights. So the counts and the setts determine the crimps, and quoting the crimps separately is quoting a derived quantity.

So the specification is the loom construction, the two counts, and the weave, all three of which are already on a mill’s own cloth particulars, plus one number saying how far along the relaxation the finishing left the cloth.

And that number has a name the trade already uses: the residual shrinkage. A cloth quoted with two per cent residual is a cloth stopped two per cent short of its own minimum, and from that plus the construction every state-dependent quantity follows in any state.

Three numbers a weaver already records and one a finisher already measures. That is a considerably less unreasonable request than four new columns, and the reason it looks like four is that the crimps have been treated as an input where they are an output.

And the pair already quoted can be checked #

There is a cheaper thing a buyer can do immediately, without asking for anything new, and it follows from the same fact.

A specification normally carries a mass per unit area and a thread count, and those two are not independent: the areal mass is the setts times the crimps times the counts, so the pair must lie on one curve. A quoted pair that does not lie on it is a pair from two different states, or from two different cloths.

The check is one multiplication. Compute the areal mass the quoted construction implies and compare it with the quoted areal mass; the residual is the state discrepancy, and its sign says which way.

Too light and the cloth has lost mass — desized, scoured, not resined — or was measured before it finished relaxing. Too heavy and mass has been added, or the count was measured relaxed and the mass was measured on the roll.

That is a consistency check on data every specification already carries, it needs no new measurement, and as far as this collection can tell nobody performs it. The two numbers are habitually read as two independent descriptions and they are one description written twice.

Which is the field’s rule in its most useful form #

The rule this essay states — quote the state with any quantity that has a length in it — is a discipline for writing. The check above is the same rule as a discipline for reading, and it is the half a reader can act on.

Two state-dependent quantities on one specification are a redundancy, and a redundancy is a test. Where a document quotes a sett and a mass, or a cover and a thickness, or a crimp and a construction, the pair is over-determined by exactly one number — and that number is the state.

So a reader who cannot get the state told can compute it, from the disagreement between two quantities that were supposed to describe one cloth. The omission the whole field is about is recoverable from the documents that omit it, which is a better outcome than the field’s own conclusion suggests and needs nothing from anybody but the arithmetic.

What this field added to the site #

Each field has added a capability, and it is worth saying what this one added, because it is not a new object.

The foundation added the matrix and the integrity criterion. Expansion added the yarn, the section models and the nonwoven. Scale added the link graph and the census machinery. Fancy weaves added the constructions the matrix cannot hold, and the capstan to measure what the criterion cannot see.

This field added a second index on every quantity. Nothing new is drawn and no new object is counted; what changed is that a number now has a state as well as a value, and the essays here are the account of which numbers need one.

That is a smaller-sounding contribution and it reaches further than the others, because it applies retrospectively to all sixty-one essays that came before. Work that adds a qualification to what exists is doing something different from work that adds new material, and this site’s ledger records it the same way, which is a limitation of the ledger rather than of the work.

The rule, stated once #

Since the point of a collecting essay is to leave something usable behind, here is the rule this field produced, in the form the rest of the site should follow.

Any quantity with a length in its definition is quoted with the state it was measured in. Setts, covers, crimps, thicknesses, areal densities, diameters. If the state is the loom state, say so; if it is a relaxed state, say which.

Any quantity that is a count or a topological fact needs no state and should not be given a spurious one. Float lengths, interlacing counts, shaft requirements, plane groups, layer counts, census totals.

And where a figure draws two states, draw them at one scale, so a reader can see the difference before reading the number. That is what relaxation does and it is why it is the field’s hero figure: a pair of panels scaled to fill equal boxes would have hidden the entire finding.

The rule costs a clause per figure and it is the cheapest thing in this field. Its value is that it makes the omission visible — a figure that cannot say which state its numbers are in is a figure whose author has not decided, and that is worth discovering while the figure is being written rather than much later.

Who found it, and when #

No one found this, in the sense that there is nothing here to attribute. The state-dependence of fabric dimensions has been known for as long as fabrics have been washed, and the standards that require a specified relaxation before measurement are the institutional response to it.

What is unusual is the sorting — putting the invariant quantities and the state-dependent ones in two columns and noticing that the vocabulary cuts across the boundary. That is available to a collection that computes both kinds and unavailable to a practice that measures one kind and manages the other, which is probably why it is not standard.

It is also the sort of result that could only appear at the end of a field. Each of the fourteen essays before this one noticed the omission in its own quantity. None of them could see that the omissions had a shape, and the shape is the finding.

Where the ladder goes next #

This field is done and the next one is the point at which stopping would leave something complete: cloth in use — composites, technical textiles, garment cutting — where the fabric stops being an object to be measured and becomes a thing that has to do a job.

The quantities this field has just finished labelling are the ones that field will be quoting, so it inherits the labelling rule rather than the correction. That is the right order, and it is the reason this essay is at the end of a field rather than appended to the beginning of the next one.