Peirce against the racetrack, measured

Ask how closely a cloth can be set and the answer is a geometry problem: threads of some thickness passing over threads of some thickness, and the question of when they touch. Ask it of a real cloth and the geometry problem needs a shape for the thread, and there is no correct answer to what that shape is.

This essay runs two of the standard choices side by side on the same yarn. The point is not to decide between them. It is to find out how much the choice costs, which turns out to depend entirely on what is being asked.

Peirce’s model, stated in full #

Peirce’s 1937 geometry is the oldest and still the reference. It makes three assumptions and they are all worth naming.

The thread has a circular cross-section of diameter $d$. It does not flatten, whatever it is pressed against.

The thread’s path is made of circular arcs joined by straight lines. Where it passes a crossing thread it wraps around it on an arc of radius $D/2$, where $D = d_1 + d_2$ is the sum of the two diameters; between crossings it runs straight.

The two systems together fill the thickness of the cloth: the crimp heights satisfy $h_1 + h_2 = D$.

From those, writing $p$ for a thread spacing, $l$ for the length of thread per repeat, $h$ for the crimp height and $\theta$ for the weave angle, Peirce’s relations are

$$p_2 = (l_1 - D\theta_1)\cos\theta_1 + D\sin\theta_1$$

$$h_1 = (l_1 - D\theta_1)\sin\theta_1 + D(1 - \cos\theta_1)$$

with the same pair for the weft and the closure condition tying them together.

Eliminate $l$ and the crimp height becomes a function of the weave angle alone,

$$h_1(\theta_1) = (p_2 - D\sin\theta_1)\tan\theta_1 + D(1 - \cos\theta_1)$$

whose derivative works out as $(p_2 - D\sin\theta_1)\sec^2\theta_1$. That is non-negative exactly on the range where the straight portion has not gone negative — so $h$ is monotone there, and solving for $\theta$ by bisection is exact rather than hopeful.

The equations are one short #

There is a gap in the model that gets glossed over, and it is not an artefact of this implementation.

Two unknowns — the two weave angles — and one closure condition. That leaves the system underdetermined. Peirce’s equations do not decide how the crimp divides between warp and weft. They say the two crimp heights must add to the thickness; they do not say which system takes more of it.

That is not a defect. It is a correct statement about the geometry: a cloth in which the warp is held tight and the weft slack, and a cloth with the reverse, are geometrically distinct and equally admissible. What decides the division is tension — the loom’s, and the finishing’s — and tension is not in the model.

Something further has therefore to be supplied, and different treatments supply different things. Some assume equal crimps. Some assume the two thread tensions are equal, which gives a different answer. The choice taken here is to specify the crimp ratio, because that is what a fabric analysis actually measures: unravel a piece, straighten the threads, and the two crimps come out directly.

The solutions above take the ratio as one, a balanced cloth. Setting it to three gives a cloth with a warp crimp of fourteen per cent against a weft crimp of five, which is roughly what a warp-tight shirting looks like — and the sett, the thickness and the cover all move with it.

The racetrack, and what flattening does #

Kemp’s modification, from 1958, replaces the circle with a rectangle capped by semicircles — a racetrack. The yarn is $a$ wide in the plane of the cloth and $b$ thick through it, with the same cross-sectional area as the circle it replaces.

The reason for the change is not aesthetic. A yarn in a real cloth is squashed. The threads crossing it press it flat, and they press it flatter the more often they cross it — which ties the section shape to the interlacing count, so it spreads in the plane and thins through the thickness, and a model that keeps it circular is describing a yarn that has not been woven yet.

Peirce’s relations survive the change with two substitutions. $D$ becomes $b_1 + b_2$, the sum of the thicknesses rather than of the diameters, because that is what fills the cloth. And the thread crossing a flattened neighbour runs flat across it for a distance $a - b$ before the arc begins, so that length is added to the spacing.

The flattening ratio $a/b$ is the model’s new parameter, and setting it to one gives the circle back exactly — which is the only real test an implementation of this kind gets.

The weave angle at the jam is sixty degrees, in both #

At jamming the straight portions vanish: the threads touch everywhere and $l = D\theta$. The equations collapse to $p = D\sin\theta$ and $h = D(1-\cos\theta)$.

For a balanced cloth the closure condition gives $h_1 = h_2 = D/2$, so $\cos\theta = \tfrac12$ and

$$\theta = 60°$$

That is not an empirical number. It is forced by the closure condition and holds in the circular model, in the racetrack model, and in any model that keeps arcs of radius $D/2$ and asks the two crimp heights to be equal. The weave angle of a jammed square plain weave is sixty degrees, and no property of the yarn enters it.

Which is the first thing worth taking away: some of what looks like a modelling choice is not one.

What the models disagree about #

Now the comparison. Same yarn, same closure condition, same jam, different section — and three quantities to ask about.

The sett barely moves. From a circle to a racetrack twice as wide as thick, the jammed spacing changes by under five per cent, and across the whole family drawn here the spread is about an eighth.

It also does not move the way the phrase “flattening lets more threads in” would suggest. A flattened yarn is thinner through the cloth, which shortens the arc it has to travel and would let the threads closer — but it is also wider in the plane, and the extra width is added straight onto the spacing. The two effects nearly cancel. Running the arithmetic finds a shallow minimum at a flattening ratio of about 1.15, where the jammed spacing is 1.725 diameters against the circle’s 1.732: four parts in a thousand denser than a round yarn, and looser than a round yarn at every flattening above about 1.3.

Nobody would have guessed the sign of that, which is the argument for computing it.

The minimum has a closed form, and it is a good check on both models #

The shallow minimum is worth solving rather than sweeping, because it comes out in three symbols and reproduces every number quoted above.

At the jam the spacing is the arc term plus the flat run: twice the thickness times the sine of sixty degrees, plus the width less the thickness. Holding the cross-sectional area equal to the circle’s fixes the thickness once the width-to-thickness ratio is chosen, and substituting leaves the spacing as one function of that ratio alone. Setting its derivative to zero gives

a ÷ b = 1 + √3 − π/2 ≈ 1.161,

and the spacing there is the square root of π times (4√3 − π), all over two — 1.7246 diameters against the circle’s √3 = 1.7321. Four parts in a thousand, which is the figure the sweep reports.

The same expression says where the advantage runs out. Setting the spacing equal to the circle’s gives a second root at

a ÷ b = 1 + (12 − 2√3·π) ÷ π ≈ 1.356,

so a racetrack is denser than a circle only between one and about 1.36 and looser above it — which is the boundary the sweep locates near 1.3.

Two things follow that are worth more than the numbers.

The minimum exists for a reason that has nothing to do with yarn. The spacing is a linear function of the width-to-thickness ratio divided by the square root of a linear function of it, and any expression of that shape has exactly one turning point. The cancellation the previous section describes qualitatively is forced by the area constraint, not by anything about how threads behave.

And it is a test the implementation can fail. A sweep that reported its minimum anywhere but 1.161, or its crossing anywhere but 1.356, would be reporting an error in the substitution rather than a property of the model — which is the only kind of check available when two models are being compared and neither is known to be right.

The cover moves enormously. The circular model gives 58 per cent; the racetrack at three-to-one gives 80. That is not a refinement, it is a different answer to a question people actually ask, because cover is what decides whether a fabric is windproof, opaque or filtering.

The thickness moves as far, downwards. A third less at a flattening of two.

So the answer to “does the model matter” is: not for the sett, decisively for everything else. And the shape of that answer is the general lesson. The model matters most for the quantity nobody thought to ask about. Anyone comparing the two models on the number they were introduced to argue about — the maximum sett — would conclude the choice was academic, and would then quote a thickness that was out by a third.

The same models in an unbalanced cloth #

Everything above is a square plain weave, which is the case where the closure condition does most of the work. Loosening the balance is where the two models start to be asked different questions, and it is worth one section because it is the commercial case.

Two things move. The thickness is now set mostly by the warp’s crimp height, so a cloth woven under warp tension is thinner than a balanced one in the same yarn — which is the ordinary experience of a shirting against a sheeting. And the jamming spacings become different in the two directions, so the cloth jams warpwise before it jams weftwise, or the other way round, and the sett that can be achieved is limited by whichever comes first.

That second point is where the racetrack’s extra parameter earns its keep. In an unbalanced cloth the two systems flatten by different amounts — the tighter system is pressed harder — and a model with one section shape for both is describing a cloth that does not exist. Neither model here allows separate flattening ratios for the two systems, and that was written down as unfinished rather than glossed. It is finished: the compression energy minimises over the two aspect ratios independently, so the warp and the weft flatten by different amounts and the amounts are computed rather than chosen. What that turns up is that the two models are not rivals at all — the circle is the relaxed cloth and the racetrack is the pressed one, and the aspect ratio is a record of a force rather than a modelling choice.

Against the trade rule #

There is a third model in play, and it is the one in daily use.

Ashenhurst’s rule, which the earlier essay on setting uses, says the threads in a repeat occupy their own diameters plus one diameter for every interlacing, because a crossing thread has to pass through. It is not a geometry at all; it is a bookkeeping argument about how much room the bends take.

For a jammed square plain weave it gives a spacing of two diameters. Peirce’s circular geometry gives 1.73. The trade rule is therefore about thirteen per cent more conservative — it says a cloth cannot be set as closely as the geometry allows.

That difference is in the right direction and probably for the right reason. Real cloth cannot be woven at the geometric jam: at the jam the threads have zero straight portion and the loom cannot beat another pick in. The rule of thumb has an allowance in it, and Peirce’s model does not.

Which is the honest way to compare them. Ashenhurst’s rule and Peirce’s geometry are not competing answers to one question; one is a weaving limit and the other a geometric one, and the ratio between them is a measure of how much of the fabric’s construction is machine rather than shape.

The elliptical model, and why it is not here #

A third section shape is in the literature and is not computed on this site. Olofsson and others use an ellipse of the same area, which is a more natural squashing than a racetrack and is harder to work with: an ellipse rolling against an ellipse does not have a closed-form contact geometry the way a circle against a circle does, and the jamming condition has to be solved numerically in a way the racetrack’s does not.

It is left out deliberately and it is worth saying why rather than quietly omitting it. Adding a model whose implementation would need more care than the two here, in order to produce a third number lying between them, would make the spread look better characterised than it is. The two computed models bracket the effect; a third would not narrow the conclusion, which is that the section shape is worth about a third of the thickness and almost none of the sett.

Where the models came from #

Both models come out of industrial research laboratories rather than universities, and knowing that explains a good deal about their shape.

Frederick Peirce worked at the Shirley Institute in Manchester, the British cotton industry’s research body, and his 1937 paper The Geometry of Cloth Structure is the founding document of the subject. It reads like an engineer’s paper: the geometry is developed as far as closed form will take it, tables of approximations are given for hand calculation, and the assumptions are stated and then used. What it does not contain is any apology for the circular section, because in 1937 there was no realistic alternative that could be computed by hand.

Kemp’s racetrack, twenty-one years later, is precisely a response to that constraint being lifted a little. The racetrack is the simplest shape that admits flattening and still has closed-form contact geometry — the flat part contributes a straight length and the ends contribute the same arcs the circle did. It is a modification chosen for tractability, and it is worth saying so: nobody believes yarn sections are racetracks.

The elliptical treatments came later still, with numerical methods, and by then a third thing had happened: photomicrographs of cloth sections in resin had made it possible to look at what yarn sections actually are. The answer is that they are irregular, vary along the thread, and depend on where in the weave they are measured — a thread at a crossing is flatter than the same thread between crossings.

So the honest description of the whole family is that it is a sequence of approximations to a shape that has no single value, chosen at each stage for what could be computed. That is not a criticism. It is the reason naming the model is a rule on this site rather than a nicety: there is no model here that a number could be said to come from unconditionally.

What none of these models has #

The list is longer than the list of what they do have, and it is the reason a fabric analysis is still done with a microscope.

No friction. Nothing in any of them says how hard it is to push the threads to the jam, or whether they will stay there. A fabric relaxes after weaving, and where it relaxes to is a question about friction and about yarn bending stiffness.

No bending stiffness. The thread path is a geometric construction, not a solution of an elastica. A real yarn bent round a crossing thread resists, and the resistance sets the crimp as much as the geometry does.

No balance. Every calculation above is a plain weave. A warp-faced cloth jams differently, because the system on the face bends less often and needs less room to do it in.

No variation. Every thread is the same diameter and every spacing identical. Real cloth is not, and the jamming condition in a real fabric is met somewhere before it is met everywhere.

And no answer at all to the crimp division, as noted above. That is worth repeating because it is the model’s own admission rather than an outside criticism: the equations are one short, and the missing statement is about tension.

Where the ladder goes next #

The rung below is how close threads can be set, which asks the same question with the trade’s own rule and gets an answer thirteen per cent more cautious. The rung beside it is the yarn count systems, which supply the diameter every model here consumes.

The natural companion is crimp and interchange, where the same geometry is put to work on what happens when the cloth is pulled — and where the crimp division the equations decline to fix becomes the whole subject.

And for what the models are ultimately for, thread count is not quality: a cover factor is the output of a calculation like this one, and the number the trade quotes instead is the input with the model thrown away.