A finish spends a spread before it spends a mean

A yarn arrives with a variation on it, and every property of the finished cloth inherits some of that variation. Not all of it, and not the same amount: a cloth’s mass per unit area inherits almost none, its bending stiffness inherits four times as much as the yarn had, and its thickness inherits half.

The multiplier is not a matter of taste or of measurement. It is one number, computable from the function connecting the two quantities, and it has a name that says exactly what it is: the log-slope.

The claim #

If y goes as the kth power of x, then a small spread in x becomes k times that spread in y. Exactly, in the limit of small spreads, and near enough at the ones a yarn has.

So an operation is variance-reducing or variance-amplifying according to whether its exponent is below or above one, and this is a property of the operation rather than a design choice. Nobody selects a finish for its log-slope. Every finish has one anyway.

The interesting case is the one where the answer runs against the purpose of the operation, and this collection has one: calendering. A calender is run to make a cloth smooth, level and uniform in appearance. Measured on this collection’s own compression solver, its log-slope on thickness is 1.4 — so it takes the yarn’s variation and hands the cloth forty per cent more of it than it started with.

The argument #

Differentiate the logarithm. If y = f(x) then

$$\frac{\mathrm{d}\ln y}{\mathrm{d}\ln x} = \frac{x f'(x)}{f(x)} = k,$$

and a small proportional change in x becomes k times as large a proportional change in y. Since a coefficient of variation is a proportional spread, it is multiplied by the same k.

That is the whole derivation, and its plainness is what makes it useful: the number that decides how much variation an operation passes on is a number that can be read off a graph of that operation. No distribution has to be assumed, and nothing has to be integrated. It stops being exact only when the spread is large enough for the curvature to matter, which for these functions is well beyond where a yarn sits.

The distinction that decides whether any of it matters #

A cloth’s mass per unit area does not vary by thirty per cent. Anyone who has weighed one knows that, and the arithmetic above appears to say otherwise.

It does not, and the missing step is worth stating carefully because it is what separates the properties that inherit a spread from the ones that do not.

A property averaged over an area loses the spread. Weighing a square metre of cloth averages over tens of thousands of crossings, and the spread of that average is the crossing’s spread divided by the square root of the number averaged — which is to say, nothing. The thirty per cent is real and it is the variation between individual crossings.

A property evaluated at a point keeps it. The thickness under a small presser foot, the width of one hole, the stiffness of the cloth at the particular line a fold happens to fall on: each of those is a local quantity, and each carries the yarn’s spread multiplied by its own exponent.

And a property that is an extreme keeps more than it. A thickness gauge finds the highest crossing under its foot, which is a tail quantity rather than a typical one.

So the exponent table is not a list of how much fabrics vary. It is a list of how much the same place in the fabric varies, and whether that shows depends entirely on whether the measurement averages.

What a calender actually does to the spread #

Here the measurement disagrees with the expectation, and the expectation was reasonable.

A calender presses the cloth between rolls at a heavy line load. Thick places have more to give, so pressing them ought to bring them down towards the thin ones; the operation ought to have an exponent below one and to leave the cloth more even than it found it. That is essentially why a calender is run.

Running the site’s own compression solver at three constructions and a range of loads gives the opposite.

The mechanism is lateral and this collection has met it before. A thread flattens by spreading sideways, and what it can spread into is the room its neighbours leave. A cloth of thicker yarn at the same sett has less of that room: it is nearer its own jam across the width, so it resists flattening more, not less. The thick cloth is stiffer in compression precisely because it is thick, and the press therefore takes proportionally less out of it.

Which is not to say a calender fails at its job, because the job is not what the arithmetic above measures. What a calender is bought for is lustre and hand: it flattens the threads, moves the cloth onto a different section model, and reflects light along the flattened faces. That is a change in the mean, and it is large and real. What the exponent says is only that the operation does not also level the population, and that the reflex assumption that a heavy press evens a cloth out is wrong at the sign.

What was counted, and how #

The elasticity rule is checked rather than assumed: for a family of power maps, the spread of the output is computed by integrating over the population and compared with the exponent times the input spread. The two agree closely at small exponents and depart at large ones, which is the second-order term arriving, and the departure is drawn rather than mentioned.

The calendering result comes from the compression solver already in this collection, run at three constructions — the cloth’s own, and the same cloth with every thread fifteen per cent thicker and fifteen per cent thinner — at each of a range of loads. Two assertions run with it and each catches a different failure: a firmer press must make a thinner cloth at every step, which would fail if the solver were being handed the wrong construction; and the log-slope must rise with the load, which is the finding, asserted as a monotone relation rather than as a value.

A defect in the machinery was found by asking this question, and it is worth recording because of its shape. The solver memoised its results under the name of the cloth, and every construction invented by scaling one keeps that name — so the first answer was returned for all three constructions, to the last decimal place, and the log-slope came out as exactly zero at every load. A flat result is the most easily believed of all wrong results. The same defect had been found and fixed in one other file months earlier, and left standing in two, because nothing had ever exercised the path.

Why the calender’s exponent moves with the load #

The log-slope rises from 0.91 unloaded to 1.40 pressed, and the movement is as informative as either endpoint, because it says the amplification is a consequence of the press rather than a property of the cloth.

Unloaded, a cloth’s thickness is a sum of two diameters, so it inherits the diameter’s spread at an exponent of about a half — and the measured 0.91 is above that because the two systems’ crimps share the thickness and a thicker yarn takes more of the share. Either way it is below one, and the cloth’s own geometry is damping what the yarn handed it.

Loaded, the thickness is what the compression solver leaves, and the solver’s answer depends on how much lateral room the threads have. That room falls as the yarn gets thicker at a fixed sett, so a thick cloth resists proportionally harder — and the harder the press, the more of the answer is resistance rather than geometry. The exponent therefore climbs with the load and crosses one somewhere in the middle of the range.

So there is a press at which a calender neither narrows nor widens, and it is a light one. Below it the operation is levelling, above it amplifying, and the setting a mill uses for lustre is well above it. That is a genuinely actionable statement: a cloth calendered lightly for hand is being levelled, and the same cloth calendered heavily for lustre is not.

It also says the effect is not universal to pressing. Any operation that presses a cloth against a bed rather than between two rolls — a decatising, a pressing between plates — loads the cloth without the same lateral confinement, and the mechanism that produces the amplification is the confinement. Whether those operations sit above or below one has not been computed here and the mechanism predicts they sit lower, because a thread with room to spread is a thread the press can reach.

And it explains why the finding was invisible. An operation whose exponent crosses one inside its own working range shows no consistent effect at all in a mill’s records: some settings level a cloth and some do not, the difference is a second moment nobody measures, and the aggregate reads as noise. A quantity that changes sign inside the range it is used over is exactly the quantity a practice will never notice.

Which operations narrow, which widen, and which do neither #

The rule sorts a finishing route into three kinds, and the sorting is more useful than any individual number because it says where to look.

Operations that are a power with an exponent below one narrow. Anything whose output goes as a root of its input passes on less variation than it received. A thickness built as a sum of two diameters is the clearest case in this collection: its exponent is a half, so the cloth’s unloaded thickness is already half as variable as its yarn.

Operations that are a power with an exponent above one widen. A calender under load, at 1.4. Anything that acts on a bending stiffness, at four. In general, an operation that acts on a quantity already carrying a high exponent inherits that exponent as well, so a route that measures a stiffness, acts on it, and measures again is compounding rather than adding.

Operations that truncate do neither, and are the ones most often mistaken for levelling. Singeing burns off protruding fibres; cropping shears them; a shearing machine takes the surface down to a set height. None of those is a power law at all: each removes everything above a threshold and leaves everything below it exactly as it was.

That third class deserves its own sentence, because it is what most of the finishing route consists of. Truncation changes the tail and barely moves the mean, which is the reverse of the operations above, and it is why singeing and cropping have such a large effect on appearance for such a small effect on any measurable property. What the eye reads on a cloth’s surface is very largely a tail quantity — the fibres that stand furthest out, the crossings that sit highest, the places that catch light — so a treatment that removes a tail transforms the appearance while leaving weight, thickness and cover almost untouched.

And it explains a mismatch that recurs across finishing: an operation that visibly improves a cloth often cannot be detected in its specification. A cropped cloth and an uncropped one from the same loom differ in nothing that a test method reports, and differ obviously to a buyer. The specification measures means; the buyer is reading the tail.

The practical form of this is a question to ask about any step in a finishing route: is this operation a shift, a stretch, or a cut? Only the third leaves the population’s centre where it was, and only the first two can be undone by adjusting a setting.

Where the model stops #

A whole cloth is scaled, not one thread in it. The two are different questions with different answers: a single thick end among ordinary ones can push its neighbours aside and flatten more than this says, while a cloth wholly of thick yarn cannot. The solver reaches the second, which is the conservative limit for the claim being made, and there is no instrument here for the first.

The exponent is measured at one construction. A more openly set cloth has more lateral room and should show a smaller amplification, possibly none — the mechanism predicts that the effect vanishes as the cloth opens — and that has not been swept.

The rule itself is a linearisation. It is exact in the limit of small spreads and good to a few per cent at fifteen; at the spreads a close cloth’s holes have, which run past forty, it is out by enough to matter and the full integral has to be done instead.

And nothing here touches the fibre. A calender’s effect on a real cloth is partly plastic and partly elastic, depends on temperature and moisture, and is partly recovered afterwards. The exponent measured here is the geometric part of an operation that has a material part as well.

The generalisation #

Every transformation has a variance multiplier and it is not a design decision. Reading it off is one derivative, and it decides whether a process step makes a population more uniform or less — a question that is usually answered by intention rather than by arithmetic.

The habit worth taking away is to ask two separate questions of any operation and never to let one answer both:

• What does it do to the mean? That is what it is for, and it is what the operator watches.
• What does it do to the spread? That is its log-slope, and nobody watches it.

They can point in opposite directions, as they do here, and the second is the one that decides how the process behaves at its own limits — where the tail of the population meets the tolerance.

And the second lesson is that a flat result deserves suspicion in proportion to how convenient it is. An exponent of exactly zero at every load, from a solver that had never been asked this question before, was reported by a working piece of machinery and was entirely spurious. What caught it was that the answer was too clean, which is the only warning such a failure gives.

Who found it, and when #

The elasticity rule is elementary calculus and is applied throughout measurement and error analysis; there is nothing new in it. What is new here is the list of exponents for a cloth’s own quantities, which could only be made once the functions existed, and the measurement of a calender’s own exponent, which came out against the expectation that prompted it.

The mechanism behind that surprise is older than the measurement: that a densely set cloth resists flattening because its threads have nowhere to spread is implicit in every account of cover and jamming since Peirce. Putting the two together is what turns a plausible expectation about a finishing process into a number with a sign.

Where the ladder goes next #

The spread that a finish passes on is the same spread that decides what a thickness gauge reads and how close a warp can be set, so this rung is the operation and those are its consequences.

Sideways, the whole population argument turns from how much a quantity varies to how the variation is arranged, where the same amount of error behaves entirely differently depending on whether it repeats — which is the difference between an even-looking cloth and a streaky one.