Peirce and Kemp are one cloth at two moments

Peirce against the racetrack set the two section models beside each other and drew the lesson that has stood on this site ever since: they agree on the sett to about a tenth and disagree on the thickness by nearly half, so the model matters most for the quantity nobody thought to ask about. That essay was careful never to say which one was right, and it was careful for a good reason. Nothing here could compute the aspect ratio of a flattened thread, so choosing the racetrack meant choosing a number, and choosing a number is not choosing a model.

That is now resolvable, and the resolution is that the question was badly posed.

The two models are the same model #

Write down the flattened geometry properly and this becomes arithmetic rather than opinion. A thread crossing a flattened thread runs horizontally along its flat top for a distance F before curving, so Peirce’s two equations gain one term each:

$$p = F + (l - D\theta - F)\cos\theta + D\sin\theta$$

$$h = \phantom{F + {}} (l - D\theta - F)\sin\theta + D(1 - \cos\theta)$$

with D now the sum of the two flattened thicknesses. Set the aspect ratio to one and F goes to zero, b goes back to d, and every term returns to Peirce’s. This is checked against this site’s own circular solver at all eight cloths in the table and agrees to two parts in ten thousand million million — which is not a claim about Kemp’s model but a claim about the implementation, and is the reason it is worth running.

So there is one geometry with a parameter in it, and the parameter is the aspect ratio. Peirce is that geometry at f = 1. Kemp is the same geometry at f > 1. They cannot disagree; they are the same function evaluated at two points.

What decides which point #

Giving the section an energy answers it, and the answer is short.

At no load a cloth of two equal counts sits at exactly f = 1 — at every transverse modulus tried, at both ends of the bending bracket, for seven of the eight cloths in this table. Flattening does not pay. It buys a little crimp through the flat run and loses more through the arc, because Peirce’s arc has radius D/2 and shrinking D sharpens the bend faster than it shortens it.

Under load a cloth flattens, monotonically, without a threshold worth speaking of — a few thousandths of a newton per crossing, against the 1.35 N a warp end at ordinary weaving tension applies.

So the circle is the relaxed cloth and the racetrack is the pressed cloth, and there is nothing left to choose between.

Which of the two a cloth in the world is #

Almost always the pressed one, and the amount is now readable off a measurement.

A cloth’s thickness is a routine specification line, a circular section over-predicts every one of them by between a third and four fifths, and reading the gap backwards through the energy gives the force each cloth is carrying — 0.19 to 0.85 newtons per crossing — and the aspect ratios that go with it: between 1.56 and 2.33.

That range is what published cross-sections of ordinary cotton fabrics have shown for as long as people have been embedding cloth in resin and photographing it. It was not arranged. The model was fitted to one constant against one measurement and returns the flattening that microscopy sees.

What that does to the original comparison #

The previous rung’s finding survives and changes meaning.

It found that the two models agree on the jammed sett to 12.6 per cent and disagree on the thickness by 47. The agreement was read as reassurance and the disagreement as a warning, and both readings were right about the arithmetic and wrong about what the two models are.

They are one cloth at two loads. The quantity the two “models” agree on is the quantity that survives being pressed, and the quantity they disagree about is the quantity that records the pressing. A sett is set by the reed and the take-up and cannot be much changed by squashing what is already in place; a thickness is nothing but how much room the threads take up through the cloth, which is exactly what squashing changes. The agreement and the disagreement were never about model choice at all.

There is a second consequence and it is about how figures on this site should be drawn. Every Peirce solution here has used a circular section, and that is now known to be right for a relaxed cloth of equal counts and wrong for any cloth that has been through a finishing works — which is most of the cloth anybody handles. The figures are not being changed, and the reason is stated rather than left implicit: moving them would move about a hundred drawings and the numbers quoted in some forty essays, and it is a change to make deliberately and check cloth by cloth. What has changed is that the choice is now a choice, with a name and a magnitude attached, instead of a default nobody could interrogate.

And what it does to the cover #

The cover factor is where this bites hardest, because cover is the one setting-field quantity that is made of thread width.

A flattened thread is wider by exactly f times κ — 1.67 times its round diameter at an aspect ratio of two — so a cloth’s directional cover rises in the same proportion while nothing about its construction changes. A sheeting at a round-section cover of 0.52 goes to 0.59 at the light flattening a modest calender gives it, and to 0.72 at a heavy one. No thread has been added, no sett has changed, and the cloth is a third of the way from open to covered.

That is the arithmetic behind a fact every finisher knows and no specification records: a cover factor is a property of a cloth’s history as well as of its construction, and two fabrics with identical setts and counts can have covers a third apart because one has been through a nip.

It is also why the jamming sett is a moving target. Threads jam when they touch, a flattened thread touches sooner across the cloth and later through it, and the sett at which a given construction jams is therefore a function of how hard it has been pressed. The previous rung’s comparison already showed flattening raising the cover and lowering the thickness; what is added here is that both movements have a cause, and it is not the modeller’s.

The jam moves, and less than expected #

A cloth cannot be set closer than its threads can pass one another, and that limit is the setting field’s hardest constraint. Since flattening changes a thread’s width and its thickness in opposite directions, it might be expected to move the jam a long way. It does not.

For a sheeting’s 25 tex warp, the jammed sett is 30.9 ends per centimetre with a round section, 30.7 at an aspect ratio of 1.5, 30.0 at 1.8 and 28.5 at 2.4. A factor of nearly two and a half in aspect ratio buys an eight per cent change in the jam, and it goes the wrong way from the intuition: flattening a thread makes a cloth jam sooner, not later, because at the jam the straight portions have vanished and what is left is the flat run plus two quarter-circles, and the flat run is wider than what it replaced.

Over the same range the cover at jam goes from 0.577 to 0.766 and the thickness from 0.374 mm to 0.224 mm — a third off each. So the three quantities separate cleanly: the sett barely moves, the cover moves a third, and the thickness moves a third the other way. Which is the same lesson the previous rung reached by comparing two models, arrived at by moving one parameter in one model, and it explains why: the sett is fixed by how far apart the thread axes must sit, and squashing a thread barely changes that, while cover and thickness are made of the thread’s outline and are nothing else.

The eight per cent is worth one more sentence, because it is the reason a finisher can calender a cloth without destroying it. If flattening moved the jam substantially, a heavily calendered fabric would be a jammed fabric and would have no drape left at all.

Calendering moves a cloth towards its own jam #

The eight per cent is treated above as a reassurance — the jam barely moves, so a calender does not destroy a fabric. It is worth following the sign, because the direction is the uncomfortable one and it explains a handle everybody has felt.

Flattening makes a cloth jam sooner. The sheeting’s jammed sett falls from 30.9 ends per centimetre round to 28.5 at an aspect ratio of 2.4, so the ceiling comes down while the cloth’s actual sett — which is fixed by the reed and the take-up — does not move at all.

A cloth is normally woven at some fraction of its jam, sixty to eighty per cent depending on what it has to do, and that fraction is what a designer is really choosing. Pressing the cloth raises it without anybody touching the construction:

a cloth at 70 per cent of its greige jam is at 76 per cent of its calendered one.

Six points of headroom, gone, from an operation whose stated purpose is lustre. And headroom is exactly what a cloth’s drape is made of — a fabric near its jam has threads with nowhere to move, so the shear that gives it drape is locked and the cloth handles as a sheet rather than as a fabric.

That is a mechanism for a thing usually attributed to the wrong cause. A heavily calendered cotton feels board-like, and the explanation offered is that the threads have been flattened and the surface glazed. Flatness is what it looks like; what it feels like is a cloth closer to jammed, and the two are separate consequences of the same nip. A cloth that was flattened without being brought nearer its jam would be shiny and would still drape.

Three things follow that a finisher could use.

The effect is worst on the cloths that were already close. A sheeting at 80 per cent goes to 87; a scrim at 40 goes to 43. So the fabrics least able to afford the loss of headroom are the ones that lose most of it, which is the ordinary shape of a proportional cost.

And it is recoverable, unlike the flattening. The jam moves back when the section rounds up, so a washed cotton recovers its headroom along with its thickness — which is why a calendered cotton softens in the first wash by more than the loss of shine alone would explain.

And it says which cloths cannot be calendered hard. A construction already at 85 per cent of its jam has nowhere to go: a heavy nip takes it past its own ceiling, and a cloth at its jam is one whose threads are in contact along their arcs, which is a fabric that creases rather than drapes. The trade’s rule that a densely set cloth takes a lighter nip is usually given as a matter of not crushing it. The arithmetic says it is a matter of headroom, and headroom is computable from the construction before the cloth reaches the machine.

What was counted, and how #

The reduction to Peirce is a check across two files rather than inside one: the flattened geometry is solved here and the circular geometry in the yarn library, and the two are required to agree term by term at f = 1 at every cloth in the table. That catches the whole class of error in which a transcribed geometry looks right and is off by a term.

The relaxed state is checked across two files too. This ladder finds the least-energy state with the thickness free to change and the load set to zero; the bending-only argument of the rung below finds it along a locus at fixed thickness. Where the section comes out round the two are solving the same problem by different code, and they agree to a quarter of a per cent on six of the eight cloths. The two exceptions are named rather than allowed for: the poplin genuinely does not relax round, and the cheesecloth’s energy well is so shallow that its minimum sits at the end of the locus and the two searches quantise it differently.

The section arithmetic is checked to conserve area at every aspect ratio, to one part in a million million. A racetrack that quietly changed the yarn’s packing factor would be modelling compaction rather than shape change and would still look like a racetrack.

Where the model stops #

The energy has one fitted constant and it absorbs the modelling error above it. The transverse modulus is read out of a fabric thickness, so anything systematically wrong in the geometry — the uniform section along the thread, the small-strain energy at strains near a half — is swallowed by it. The shape of the conclusion does not depend on the constant; the numbers do.

The relaxed section is exactly round only where the two counts are equal. The poplin, whose warp is finer than its weft and set half as far apart again, settles at an aspect ratio of 1.02 at a stiff yarn and 1.45 at a limp one — so for an unbalanced cloth the relaxed section is not decided by geometry at all. Seven of the eight cloths here are balanced and the eighth is stated.

Nothing here is plastic. This model says a cloth taken out of a calender springs back to round, and real cloth does not, because a real calender is hot and sets the flattening rather than merely producing it. That is a rung in the finishing field and it is not in this arithmetic.

And the racetrack is still a stylised section. A real yarn section is lenticular and slightly irregular, not a rectangle with semicircular ends, and the difference matters most for the very quantity this rung is about — how wide the flat is. Kemp’s shape is chosen because it conserves area exactly and has one parameter, not because anybody thinks it is the shape.

The generalisation #

When two models of one thing disagree, check whether one of them is the other with a parameter moved.

That is a cheap test and it settles more disagreements than it has any right to. Here it turns out that the circle is the racetrack at f = 1, so the models were never in competition and the whole long hedge was about a value, not about a model. The hedge was not wrong — it was correct about a genuine uncertainty — and it was pointed at the wrong object, which made it look permanent when it was not.

The general form is worth carrying: a disagreement about which model to use, where one model is a special case of the other, is really a disagreement about a number. Numbers can be measured. Models argued about as though they were incompatible cannot, and the argument runs indefinitely.

The second lesson is about what a free parameter is doing in a function signature. The racetrack’s aspect ratio sat in one here for as long as this collection has run looking exactly like a material property — a thing about yarn, to be looked up. It is a thing about what happened to the cloth. Nothing in the signature said so, and nothing was going to.

Who found it, and when #

Peirce’s 1937 geometry is explicit about its circular section and about the fact that real yarns are not circular; he says so and proceeds anyway, which is the right decision for a paper that is establishing a geometry.

Kemp’s 1958 racetrack was proposed precisely to fix that, and the flattened-thread equations that follow from it are Hearle’s and others’. Every account of them takes the aspect ratio as an input to be measured off a section.

What is this site’s is the observation that the two are one geometry, that the parameter separating them has an energy, and that minimising the energy puts the relaxed cloth at the circle — which turns a nine-year-old choice between models into a reading of a thickness gauge.

Where the ladder goes next #

Sideways, the same solver that made this comparison possible refused something on its way past: a cloth this site’s own table has called impossible for a long time, which turns out to be perfectly weavable and to have been lost to a bisection that walked the wrong way.

The cover arithmetic here is what the calendering rung spends, and the thickness inversion is what gives a relaxed cloth its contact force.