What a sett is when the yarn is not round

A jamming condition is one of this collection’s founding results. Threads cannot be set closer than they can be packed, the packing is decided by their diameters, and the condition follows from geometry with no material constant in it.

Everything about that is right and it contains a word that does not mean one thing. A thread in a cloth does not have a diameter. It has a wide axis and a narrow one, and the two differ by a third at ordinary flattenings.

Which diameter a jam is about #

Threads in a woven cloth jam in two different ways and the two use different diameters.

Side by side. Two adjacent warp ends touching one another along the cloth’s width. That is a jam across the yarn’s wide axis, because the yarn is flattened in the cloth’s plane and its wide axis lies in that plane.

Over and under. A warp end and a weft pick pressed together at a crossing, which is what sets the cloth’s thickness. That is a jam across the narrow axis.

So the two jams see two different numbers, and a condition written with one diameter is right about one of them and wrong about the other by a third.

Which way it goes #

The direction matters and it is not the obvious one.

A cloth jams side by side sooner than a round-yarn condition says, because the flattened yarn is wider than round. At a flattening of 0.78 the yarn is 1.28 times as wide, so the maximum sett is 1.28 times lower.

And it jams over and under later, because the flattened yarn is thinner through the cloth. The thickness a jammed cloth reaches is 0.78 of what a round-yarn geometry gives.

So flattening makes a cloth less densely settable and thinner, and both are consequences of one section change.

That is not what a reader would guess. The instinct is that a flattened yarn packs better — it does, in the sense that a jammed cloth of flattened yarn has more fibre in it per unit thickness — and it packs worse in the direction that decides the sett.

What the collection has been doing #

Both conditions are in this collection already and each uses a diameter without saying which.

The jamming condition for how close threads can be set uses a diameter for the side-by-side jam and takes it from the count and the packing factor — which is the round diameter, so the condition over-estimates the sett a cloth can reach.

The jammed thickness calculation uses a flattening parameter explicitly, swept over a range, because everybody knows a thread in a cloth is flattened. That one is right and its input was a free parameter.

So the collection has been inconsistent between two calculations about the same object: one carries a flattening and one does not, and both were written in the same phase.

That is worth recording as a defect found rather than as a subtlety, because it is one.

What this work supplies #

The contact ladder predicts a flattening for a knitted fabric from its own geometry, and finds that a woven cloth’s geometry demands almost none — from nothing in an open cloth to four and a half per cent in a dense one.

So a woven cloth’s flattening is not predicted by its arrangement; it is a memory of the loom, and it has to be measured or assumed.

That is a negative result and it is useful here: it says the jamming condition cannot be closed by geometry, and that the flattening it needs is a process input.

Which is a more honest position than either of the two calculations currently occupies. One assumes round and one sweeps; the right answer is to take a measured flattening and use its two axes in the two conditions.

The condition, rewritten #

Putting it together gives a form the collection could use.

Let a thread’s round diameter be d and its flattening f, so its wide axis is d/√f and its narrow one d·√f at constant area.

The side-by-side jam is at a spacing equal to the wide axis, so the maximum sett is √f times the round-yarn value.

The over-and-under jam sets a thickness of two narrow axes plus a crimp height, so the jammed thickness is √f times the round-yarn value too — the same factor, arrived at from the other axis, because holding the area makes the two moves reciprocal in the axes and identical in the square root.

That is a tidy result and it is worth noticing: at constant area, a flattening multiplies both jamming quantities by the same factor, which is the square root of the flattening.

So the two conditions move together and their ratio — which is what a cover factor is — does not move at all.

Which explains why nobody noticed #

That last is the reason a condition written for a round yarn has survived a century of use on flattened yarn.

The quantity the trade actually uses is the cover factor: a sett times a diameter, which is a dimensionless measure of how full a cloth is. If both the maximum sett and the diameter move by the same factor, the cover factor at jamming does not move at all.

So a cloth’s jammed cover factor is the same whether its yarn is round or flat, and every empirical rule about cover factors is unaffected.

What is affected is anything that uses an absolute sett or an absolute thickness, and those are exactly the quantities that get quoted with a diameter and a construction rather than as a dimensionless index.

That is a familiar shape in this collection: a dimensionless quantity survives a modelling error that an absolute one does not, and the trade’s habit of working in dimensionless indices has protected it from a mistake nobody had identified.

What a jammed cloth’s thickness really is #

Following the thickness half through gives a number the trade measures and this collection has always predicted too high.

A jammed cloth’s thickness is two thread thicknesses plus whatever crimp height the geometry gives. With round threads, that comes out well above what a gauge reads on a real cloth — the collection has recorded the gap and closed it by assuming a flattening.

The arithmetic here says the assumption is doing the right thing and says by how much: √f, which at a flattening of 0.78 is 0.88, so a twelve per cent reduction on the round-yarn thickness.

Measured thicknesses sit about twenty per cent below the round prediction, so the flattening accounts for most but not all of the gap.

What is left is the crimp height, which is computed from a geometry that assumes round threads at the crossing — so it is over-estimated too, and correcting it would close some of the remainder.

That is a chain of two corrections in the same direction and it is exactly the kind that gets attributed to a single fudge factor when neither is computed.

What was counted, and how #

The section arithmetic is elementary: an ellipse of the same area as a circle, with its minor axis f times the circle’s diameter, has a major axis of the diameter over the square root of f — so the two axes are d/√f and d·√f.

The jamming conditions are this collection’s own and are restated rather than recomputed.

The woven approach measurement is this collection’s, from four cloths with Peirce’s geometry solved for each and the thread paths laid out as sinusoids of the crimp heights it gives.

The conclusion that both quantities move by √f is a substitution rather than a result, and it is checked by observing that the cover factor — their ratio — is unmoved.

Why the two jams do not happen together #

A structural point worth drawing out, because it explains why a cloth has two conditions rather than one.

A cloth reaches its side-by-side jam when its threads in one system touch one another along the cloth. That is a condition on that system’s sett alone.

It reaches its over-and-under jam when the two systems cannot be pressed closer through the thickness. That is a condition on both systems together.

A cloth can be at either without the other. A very fine, densely set cloth of thin threads may be side-by-side jammed in the warp and nowhere near the thickness jam; a heavy canvas may be the reverse.

So the flattening’s two effects — a lower maximum sett and a lower jammed thickness — apply to different cloths and do not usually bite at once.

That is worth knowing before applying the correction: which of the two matters depends on which jam the cloth in question is near, and a cloth near neither is unaffected by either.

Where the model stops #

The section is an ellipse and a real one is not. A racetrack, which is what this collection usually draws, has different axes for the same area, and the factor is not exactly the square root.

The area is held. A yarn jammed hard enough loses air rather than changing shape, and at that point the whole framing is wrong.

And the flattening is an input. For a woven cloth this collection’s own measurement says the geometry does not supply one, so the number has to come from a section or from a process argument.

Nor is the crimp’s own contribution to the thickness re-examined. A jammed cloth’s thickness is two narrow axes plus a crimp height, and the crimp height is computed from a geometry that assumes round threads.

That last is a real inconsistency and it is not closed here.

The knitted version of the same question #

A knitted fabric has a jamming condition too and it is worth asking whether the same correction applies.

A knit jams when its courses cannot be brought closer, which this collection takes as a course spacing of one yarn diameter — a condition through the fabric’s thickness, so it is the narrow axis that applies.

At a flattening of 0.78 the narrow axis is 0.88 of the round diameter, so a knitted fabric can be pulled to a course spacing twelve per cent smaller than the round condition allows, and its extension ceiling rises correspondingly.

That is a correction in the opposite direction from what the extension ceiling needs — the ceiling is already three times too high — so it makes the collection’s worst quantitative discrepancy slightly worse.

It is small against a factor of three and it is worth recording rather than hiding, because it is the honest consequence of applying this rung’s arithmetic to the fabric this work has spent most of its effort on.

And it is one more piece of evidence that the extension ceiling’s problem is not a section problem at all, which is what the ladder concluded from the other direction.

What the loom decides and the geometry does not #

The rung’s uncomfortable half is that a woven cloth’s flattening is a process input, so the corrected condition needs a number the geometry cannot supply.

That is worth taking seriously rather than treating as an inconvenience, because it says something about which quantities in this subject are structural and which are historical.

A knitted fabric’s flattening is structural. Its own arrangement demands it, it follows the tightness factor, and any fabric of that construction has it.

A woven cloth’s flattening is historical. Its arrangement demands almost nothing, and the flattening a section shows was put there by the beat-up, the warp tension and the finishing.

So two fabrics that look alike in a micrograph have their sections for entirely different reasons, and only one of the two can be predicted from a construction.

That is a real asymmetry between the two crafts and this work found it by asking one question of both. A woven cloth’s designer inherits a section from the mill; a knitted fabric’s inherits one from the geometry.

The generalisation #

The rung is an instance of a habit worth having whenever a quantity has a direction.

A word that names a length must say which length.

A yarn’s diameter is one number for a round yarn and two for a flattened one, and every condition that uses it has to say which. Once that is said, the conditions are easy; before it is said, they look like one condition with a well-defined input.

This collection has met the same shape twice already. A fabric’s thickness is a structural one and a measured one and they differ by a hair layer. A fabric’s dimension needs a state. And now a thread’s diameter needs an axis.

In every case the fix is the same and costs a word: name the quantity rather than the object.

What a measurement would need to report #

The rung’s practical output is a specification for a section measurement, and it is short.

Both axes, not one. A section that reports “the diameter” reports whichever axis the observer measured, and the two differ by a third.

Which axis is which relative to the cloth. The wide one should lie in the cloth’s plane and the narrow one through it, and a section that found them the other way round would be reporting something odd about the cloth or about the mounting.

And the area, or equivalently the product of the two axes, because a yarn that has lost air is not a yarn that has changed shape and the two need different treatment.

Three numbers where the literature usually reports one. All three come off the same micrograph and the extra cost is one measurement per section.

That is a small ask and it would let every jamming calculation in this collection be closed against a measurement rather than a sweep.

What this does to a cloth’s design arithmetic #

A designer choosing a construction works with four quantities and the correction touches three of them, so it is worth saying which way each moves.

The maximum sett falls by the square root of the flattening — twelve per cent at 0.78 — so a cloth cannot be set as densely as a round-yarn calculation says.

The jammed thickness falls by the same factor, so a heavy cloth comes out thinner than predicted.

The areal weight does not move at all, because it is a count times a sett times a crimp and none of those is a diameter.

And the cover factor does not move at jamming, because both its terms moved together.

So two of four move and two do not, and the two that move are the ones a designer is most likely to be pushing against — because a designer is usually asking for the densest or the heaviest cloth a construction will give.

That means the correction bites hardest exactly where a design is most constrained, which is the least convenient place for it and the most useful place to know about it.

Who found it, and when #

Peirce’s jamming conditions are from 1937 and are the foundation of every account of woven cloth setting.

That a thread in a cloth is flattened, and that a flattened thread has two diameters, is universal knowledge and is why racetrack sections exist.

What is this collection’s own is noticing that its own two conditions treat the same object inconsistently — one round, one swept — and that the resolution is a factor of the square root of the flattening applied to both, which leaves the cover factor unmoved and every absolute quantity changed.

Why the flattening cannot simply be assumed away #

A last defence of taking the trouble, because a reader may reasonably ask why not just use an effective diameter and be done.

An effective diameter works if only one condition is being applied. Take the wide axis, call it the diameter, and the side-by-side jam is right.

It fails the moment two conditions are applied to the same cloth, which is exactly what a cloth-design calculation does: it asks for a sett, a thickness, a cover and a weight, and those use the two axes differently.

A single effective diameter tuned to get one of them right gets the others wrong, and the errors have opposite signs — so a calculation that fits its diameter to a measured sett will over-predict the thickness, and one fitted to a measured thickness will under-predict the sett.

That is a familiar failure and this collection has met it before: a single number standing in for a distribution is right for one question and wrong for the rest, and which question it is right for is decided by whoever fitted it.

Two axes cost nothing and remove the whole difficulty.

Where the ladder goes next #

The cover factor is unmoved at jamming and it is not unmoved everywhere. A cloth that is not jammed has a cover that depends on how wide its threads are, and a flattened thread is wider than a round one of the same area.

What a flattened yarn does to its cover is the same section arithmetic applied to a quantity the trade uses constantly.