The diameter was quoted at one twist

There is one number in this collection that has never been measured, never been derived, and never been checked against anything. It is 0.6, it is the packing factor, and every diameter here has been computed through it.

Where it came from is stated honestly enough in the essay that introduced the two thread models: Peirce published a rule giving a cotton yarn’s diameter as one over twenty-eight times the square root of the count, this collection inverted it, and the packing factor that comes out is 0.601 for cotton.

That is a respectable provenance and it hides two problems. Peirce’s constant was fitted, to real yarns, by measurement — so it carries whatever those yarns happened to be. And packing depends on twist, so it carries whatever twist those yarns happened to have.

The claim #

A packing factor decides four of this collection’s quantities and does not touch the fifth, and the exponents are exact.

The one that does not move is the useful part of the answer. An areal weight is a sett times a count and never passed through a diameter, so it is the one number in a cloth’s specification that this uncertainty cannot touch — which makes it the right thing to check a construction against when anything else disagrees.

What a packing factor is, and why it is not a constant #

A yarn is fibres and air. The packing factor is the fraction that is fibre: divide the mass per unit length by the density and by the yarn’s cross-sectional area, and what is left over is the air.

It is not a material property. A yarn spun harder is packed tighter, because the twist presses the fibres together — the radial pressure inside a twisted yarn goes roughly as the square of the twist angle, and a body under pressure compacts. A yarn spun softly is loose. A yarn that has been wet and dried is different again. A yarn in a cloth, flattened at its crossings, is different from the same yarn on a bobbin.

So the honest statement is that φ is a state rather than a constant, and this collection has treated it as a constant from its first essay.

The sensitivity, quantity by quantity #

Everything below is exact, because every relation involved is a power law and the exponents compose.

The diameter goes as φ^(−½), straight from the volume arithmetic: the same mass in a looser packing occupies a wider circle, and area goes as the square of width.

A cover factor goes the same way, being a sett times a diameter. So a cloth designed to a cover factor of 0.7 at φ = 0.6 is at 0.81 if the yarn is really packed at 0.45 and 0.63 if it is at 0.75 — and those are different cloths. A cover of 0.81 is nearly closed and 0.63 is comfortably open.

A jammed sett goes as φ^(+½), being a reciprocal of a diameter. How closely threads may be set is therefore uncertain by the same 29 per cent, in the opposite direction — a tighter yarn can be set closer, which is right and is a real effect a weaver relies on.

A bending rigidity goes as φ^(−2), because the coherent bound goes as the diameter to the fourth. That is a factor of 2.78 across the range, which is larger than the uncertainty in the fibre modulus and comparable with the whole bracket that essay could not close — except that the free bound is the one that governs in cloth, and the free bound has no packing factor in it at all. That is a second reason to be glad the bracket resolved the way it did.

And an areal weight does not move. Nor does a fibre count, nor an evenness floor, nor anything else that is a count rather than a size.

Which twist was it measured at? #

The question the title asks has an answer that can be narrowed and not settled.

Peirce’s rule was published for cotton yarns in the 1930s, in a context of ordinary weaving constructions, so the yarns behind it were at ordinary weaving twist factors — a surface angle somewhere around 20° to 25°. Inverting his constant gives φ = 0.601 for cotton, 0.697 for wool and 0.662 for polyester, the differences being the fibre densities rather than anything about the yarns.

The extrapolation has a direction and it is worth stating even though the size is unknown. A softly twisted yarn is packed more loosely than 0.6, so its real diameter is larger, its cover higher, its jam sooner. A hard-twisted yarn is packed tighter and everything goes the other way. So:

• This collection systematically understates the cover of soft-twisted cloths — flannels, hosiery, weft-faced goods — and overstates their jammed setts.
• And it does the reverse for hard-twisted ones — voiles, organdies, crepes — which is one more reason the crepe arithmetic in this collection is at its least reliable.

That is a qualitative correction with a known sign, which is worth having, and it is the most this can honestly offer.

The chain it sits in #

It is worth drawing the whole chain, because the packing factor is not at the start of it and not at the end, and where it sits decides how much damage it can do.

A fibre has a fineness and a density, both measured. Divide a yarn’s count by the fibre’s and the result is the fibre count, which is exact and carries no packing. Multiply the fibre count by one fibre’s rigidity and the result is a thread’s bending rigidity in cloth, which is also exact and also carries no packing.

Take the same count and turn it into a diameter and the packing enters. From there it reaches the cover, the jam, the hole between four threads, the crimp geometry, the cloth’s thickness, and everything computed from any of those.

So the chain has a clean division in it: the counting half is exact and the geometric half is not. That is a useful thing to know about a body of arithmetic, and it is more useful than a better estimate of the constant would be, because it says which results are safe rather than making all of them slightly less unsafe.

What a 29 per cent band does to a conclusion #

A sensitivity is only useful if it says which conclusions survive it, and the survival rate here is higher than a 29 per cent band on every diameter suggests.

Comparisons survive and values do not. Two cloths of the same yarn compared at two setts share the packing factor exactly, so it cancels out of every ratio between them: the ordering of the eight cloths by cover, by jam margin, by hole size and by everything downstream is unaffected. Nearly every finding in this collection is a comparison of that kind, which is why the constant has been able to sit unexamined for so long without doing damage.

A threshold is where it bites. A statement that a construction jams, or that a specification is unreachable, or that a hole is over a rating, is a comparison against an absolute — and there the 29 per cent goes straight through. The jammed sett is uncertain by that factor, so a cloth computed at 95 per cent of its jam might be at 74 or past it.

And a dimensionless group is safest of all. The cover factor at which the interchange budget is maximised is a cover, so it moves with the packing; but the statement that the optimum is at a fixed cover whatever the count does not, because it is a claim about which variable the answer is a function of. Every claim of that shape on this site is untouched.

So the band divides the collection’s results into three, and the division is worth knowing: orderings are safe, thresholds are not, and claims about which variable matters are safest of all. That is a more useful map than a better constant would be, and it is available for the cost of one exponent per quantity.

It also says where a measurement would be worth making. A packing factor measured on the yarns a particular mill actually spins would sharpen every threshold in this collection and no ordering at all — so the value of the measurement depends entirely on whether anybody is using the site’s arithmetic to decide a threshold, and for the comparative work that is most of it, the constant could be wrong by its whole range without anything needing to change.

Which is the reason to publish the sensitivity rather than to go and measure. A body of work that is mostly comparisons does not need a better constant; it needs a clear statement of which of its results are not comparisons, and that statement is what this rung is, written once and applied everywhere rather than repeated as a caveat.

Why it is not fixed here #

Two things would be needed and neither is available.

A model of packing against twist. The relation is measurable and is measured: packing rises with twist factor towards an asymptote somewhere between 0.6 and 0.7, and the shape of the rise depends on the fibre, the spinning system and the fineness. Fitting a curve to it would replace one unmeasured constant with a fitted function, and this collection’s rule for such a case is to compute the exact half and model the other half explicitly — which here would mean adding a fitted function to an already fitted constant.

A measurement of packing that is not circular. The difficulty is that a packing factor is defined through a diameter, and a yarn’s diameter is exactly the thing that cannot be measured cleanly, because a yarn has two of them. An optical measurement includes some hair layer and gives too low a packing; a compressed measurement gives too high a one. Every published packing factor carries an unstated choice between them.

So the position taken here is the one the collection takes elsewhere in the same situation: name the constant, price the sensitivity, state the direction of the extrapolation, and refuse the fit.

The two ways a constant can be inherited #

There is a general point here about numbers taken from elsewhere and it is worth separating from the particular case.

A constant can be inherited as a measurement — somebody measured a quantity, reported it with an uncertainty, and the uncertainty travels with it. A fibre density is like that: it is 1.52 for cotton, it is known to three figures, and anyone using it knows what they are using.

Or it can be inherited as a fit — somebody found that a formula with a particular constant in it described their data, and the constant now carries every unstated feature of that data. Peirce’s 28 is like that, and so, therefore, is the 0.6 this collection derived from it.

The difference is that a measurement’s uncertainty is stated and a fit’s is not. Nothing in “one over twenty-eight root the count” says which yarns, at what twist, measured how, or over what range of counts. The formula is perfectly serviceable inside the region it was fitted to and says nothing at all about its own edges.

So the risk with an inherited fit is not that it is imprecise. It is that it is precise about the wrong thing. This collection uses the constant for hosiery yarns and crepe yarns and carpet yarns, none of which Peirce was looking at, and the arithmetic gives three figures in every case.

That is the general form of what this essay is about, and it applies to two other numbers here as well: the friction coefficients, which are a range because they are measurements, and the contact efficiency in the fibre-grip argument, which is a fit and is labelled as one everywhere it is used. Labelling it everywhere is the discipline this essay exists to extend to the packing factor.

What to do with a cloth that disagrees #

The practical value of a sensitivity analysis is that it says which measurement to trust when two disagree, and here the answer is unusually clean.

If a computed cover and a measured cloth disagree, suspect the packing. A 29 per cent range in the diameter is a large discrepancy, it is the largest single unknown in the geometric chain, and it has the right shape: it moves cover and sett together and in opposite directions.

If a computed weight and a measured cloth disagree, do not suspect the packing. It cannot be the cause. Look at the crimp, the sett or the count.

And if a computed rigidity disagrees, the packing is not the first suspect either, because a thread in cloth bends at the free bound, where the packing does not appear. Suspect the fibre modulus, which is a range rather than a value.

That is a small diagnostic and it is the kind of thing an explicit sensitivity buys: three quantities, three different first suspects, decided by which exponent each one carries.

What was counted, and how #

The exponents are asserted, not fitted. The measured log-slope of the diameter against the packing must equal exactly −½ across the range, to twelve decimal places, and the areal-weight column must be exactly one at every point — because the claim is about the form of each dependence and a curve that merely went the right way would support none of it.

The three implied packing factors are computed by inverting Peirce’s constant for each fibre’s own density, so the differences between them are the densities and nothing else. That is worth checking rather than assuming, since a fibre-dependent packing would mean the constant carried something about the fibres as well as about the yarns.

Where the model stops #

Nothing here models packing against twist. That is the whole of the gap and it is deliberate.

The packing is treated as uniform across the section, and it is not: a ring-spun yarn is denser at its core than at its surface, so a single number is an average whose weighting depends on what is being computed. A cover factor wants the outermost radius; a mass wants the average; a bending rigidity wants the fourth moment.

A yarn in a cloth is flattened, so its packing there is not its packing on a bobbin. This collection carries a flattening ratio as a separate argument and treats the two as independent, which they are not — a soft-twisted weft flattens readily and a hard-twisted warp does not, so the flattening ratio is itself a twist consequence and is being passed as a free parameter.

And the range 0.45 to 0.75 is a reported range rather than a derived one. Nothing here establishes that real yarns lie inside it; it is what the literature reports and it is used as the width of the sensitivity rather than as a claim.

Where the ladder goes next #

Nowhere new, which is the point of a sensitivity: it is a piece of bookkeeping about arguments already made rather than a new argument. What it changes is how the existing ones should be read — with a 29 per cent uncertainty on every diameter, a 2.78-fold one on every rigidity computed from a coherent bound, and none at all on a weight.

The one place it points forward is the packing-against-twist relation, which is the missing piece and would close a chain that currently has a fitted constant in the middle of it: a count and a twist factor would then give a diameter, and a diameter is what everything else in this collection is computed from.