The crimp ratio is not a measurement

There is a line in this site’s own source that has been true and unexamined since the setting field was built. It sits in the docstring of the function that solves Peirce’s geometry, and it says: two unknowns and one closure condition leaves the system one short, which is a property of the model rather than of this implementation.

The consequence is stated in the next sentence. Peirce’s equations do not decide how the crimp divides between warp and weft. Some further statement is needed, and the one taken is the crimp ratio — which, the docstring goes on, is what a fabric analysis actually measures.

That is an honest way to run out of theory: take the missing quantity from the bench, say so, and carry on. It is also the only place here where a number is quoted rather than derived and nothing above it notices. Every figure that solves a Peirce state is drawn at a crimp ratio of one, in essays that argue about crimp to two decimal places.

The claim #

A cloth sits at the state of least bending energy on its own locus, and that fixes the crimp ratio exactly. The ratio is not a measurement, it is a consequence, and for a cloth whose two yarns are the same count it can be computed without knowing anything about the yarn at all.

Two halves follow from it and they are worth separating before either is argued.

The first is a theorem. A square cloth divides its crimp equally. Same count in both directions, same sett in both directions, and the answer is one — at every stiffness, for every fibre, whatever the packing factor.

The second is a measurement of something else. No cloth in this site’s table is square, so none of them is covered by the theorem, and each has a relaxed state that can be computed. What that state is not is the state a figure of its quoted construction draws — which took a further stretch of work to establish and is the subject of the last section here.

The equation Peirce is missing, and what kind of thing it is #

Peirce’s model gives each thread a path: a straight run between crossings and a circular arc of radius D/2 around the thread it crosses, with D the sum of the two diameters. Writing p for a spacing, l for the thread length in one modular length, h for the crimp height and θ for the weave angle,

p₂ = (l₁ − Dθ₁) cos θ₁ + D sin θ₁ h₁ = (l₁ − Dθ₁) sin θ₁ + D (1 − cos θ₁)

with the same pair for the weft, and the closure condition h₁ + h₂ = D, which says the two threads between them fill the thickness of the cloth.

Count them. Four unknowns — two thread lengths and two crimp heights — and three equations once the two spacings are given. What is missing is not a fourth geometric relation. There is no fourth geometric relation; the geometry is satisfied by a whole one-parameter family of states, and the tensile ladder has been drawing that family under a different name. It is the locus of states a cloth can reach without any yarn changing length.

The missing statement is therefore not about shape. It is about which of the available shapes the cloth prefers, and a preference is an energy.

The energy, which is one multiplication #

The energy of a bent rod is ½·B·κ² integrated along it, with B the bending rigidity and κ the curvature. Peirce’s path makes that integral trivial, because the path has only two kinds of piece: the straight runs, whose curvature is zero and which contribute nothing, and the arcs, whose curvature is exactly 2/D and whose length is exactly Dθ. So one modular length of crimped thread stores

u = ½ · B · (2/D)² · Dθ = 2Bθ / D

There is no approximation in that, no numerical integration and no fitted constant. It is the closed form of the model the site has been solving since the setting field was built, and it says something almost too simple to notice: the energy in a crimped thread is proportional to its weave angle. Not to the crimp, not to the crimp height, not to the amount of extra thread — to the angle through which it turns.

The cloth’s energy is the sum over both systems, because the closure condition ties them: thickness taken by the warp is thickness the weft cannot have. Minimise

u = 2(B₁θ₁ + B₂θ₂) / D

along the locus and the state that comes back is where the cloth sits.

Why a square cloth divides its crimp equally #

The theorem takes three lines and does not need the energy to be evaluated at all.

A cloth whose two yarns are the same count in the same fibre has d₁ = d₂ and B₁ = B₂. If it is also set at the same spacing in both directions, then p₁ = p₂ and the whole construction is symmetric under exchanging warp and weft. That symmetry carries the locus to itself, reversing its direction, and it carries the energy function to itself. A symmetric function on a symmetric interval has its stationary point at the fixed point of the symmetry, which is the state where θ₁ = θ₂ — where the crimps are equal and the ratio is one.

Nothing in that argument mentions the value of B. It survives multiplying both rigidities by any number at all, which is exactly what moving from one end of a yarn’s stiffness bracket to the other does. So the prediction holds at the free bound, at the coherent bound, and everywhere between.

The machinery asserts it rather than reporting it. Six square cloths spanning a factor of five in count and a factor of three in sett, each run at three positions in the bracket: eighteen predictions, every one of them 1.000, with a spread of six parts in a quadrillion across the whole set.

And why no real cloth is square #

Every cloth in the table is set with a denser warp than weft. A sheeting is 28 ends against 26 picks per centimetre, a muslin 24 against 22, a poplin 32 against 22. That is not a quirk of the table; it is what cloth is like, because the warp is under tension on the loom and the weft is not, and because the reed sets the warp once while the take-up sets the weft continuously.

The consequence for the crimp is short. A densely set system leaves its crossing partner short spans to bend across, and a short span is an expensive one: the same crimp height over a shorter run needs a larger angle, and the angle is what the energy counts. So the more closely set system takes the larger share of the crimp — and since the warp is always the more closely set one, the warp always crimps more.

The numbers, at the free bound. Every row is the state that cloth relaxes to, at that state’s own sett — not the crimp division at the construction in the second column, which is a different question with a different answer and is taken apart below:

The departures are small for the closely set cloths and enormous for the open ones, and the ordering is not simply by sett ratio — a voile and a muslin are set at the same 24 × 22 and give 1.93 and 1.48, because the cover is different and the cover decides how much room the geometry leaves.

The two cloths whose prediction is an interval, and the six whose is not #

The poplin is the only cloth in the table with unequal counts: 15 tex warp against 20 tex weft. For it, B₁ ≠ B₂, and how far apart they are depends on where in the yarn’s stiffness bracket the yarn sits. At the free bound a yarn’s rigidity is proportional to its count and the ratio is 0.75; at the coherent bound it is proportional to the square and the ratio is 0.56. So the poplin’s predicted crimp ratio is not a number but an interval, 12.97 to 36.18, and the honest form for it is the interval.

For the other six the weights are equal, and a weighted sum with equal weights is the same function whatever the weight. The prediction is identical at both ends of a bracket a factor of four hundred wide — checked to twelve figures, because a bug that made the energy depend on the two rigidities separately would be invisible in any single run.

That split is the useful part of the whole result. A correction that needs a material constant is a correction nobody can apply, because the constant is a range and the range is wide. A correction that is pure geometry is one anybody can apply to their own cloth with a ruler and a yarn count. Six of the eight are the second kind.

How firmly the ratio is settled, which is a second quantity #

A minimum is a point. How much it matters is the depth of the well around it, and the depths in the table run over a factor of fifty.

A batiste’s energy at the ends of its locus is 87 per cent above its minimum. A cheesecloth’s, at one end, is 1.8 per cent above. The batiste is held firmly at its predicted ratio and the cheesecloth is barely held at all — and that is the quantitative form of something every fabric analyst knows and nobody writes down, which is that the crimp figures for an open scrim scatter and the figures for a close cloth do not.

It also connects two ladders that were built for different reasons. The depth of the well against friction is exactly what decides the band a relaxed cloth comes to rest in: a cloth slides down its own energy hill until the slope is shallower than friction, and a shallow hill means it stops early and anywhere. So the same number that says how firmly the crimp ratio is settled says how repeatable the cloth’s finished dimensions are, and the two were arrived at from opposite directions.

What was counted, and how #

Everything above is computed rather than sampled, and the arithmetic has three stages.

The reference state. Each cloth’s quoted construction — two counts and two setts — becomes two diameters by the site’s standing volume arithmetic at a stated packing factor, and two spacings by division. Peirce’s equations are then solved by bisection on how the thickness divides, and the solution is fed back through the equations it was solved from: the worst residual is at the level of machine precision, which is the check that the solver solved what it claimed to.

The locus. From that reference state, the site’s existing tensile machinery generates every state reachable at constant thread length — 641 of them for the tables here — and reconstructs both thread lengths at every sampled point, not merely at the ends. A deformation path that conserves length at its endpoints and drifts in the middle is exactly the failure a two-state figure cannot show.

The energy. Each state’s two weave angles go into 2Bθ/D and the two are added. The minimum is found by scanning, which quantises the answer at the locus’s own sample spacing — so the tolerance on the square-cloth theorem is that spacing rather than machine epsilon, and asking for more would be asking a scan to be something it is not.

Three assertions guard the result rather than decorating it. The square-cloth prediction must come out at one at every position in the bracket, or the symmetry argument is wrong. The equal-count cloths must give identical predictions at both ends of the bracket, or the energy has picked up a dependence it should not have. And every cloth’s minimum must be interior to its own locus — a minimum at an endpoint would mean the model was saying “as much crimp in the warp as the geometry allows”, which is a degenerate answer wearing a computed one’s clothes. All eight are interior.

Where the model stops #

Bending is not the only energy in a cloth, and the second one is available now. The threads are also flattened where they cross, and that flattening has an energy — which turns out to prefer a round section in a relaxed cloth, so the omission here is smaller than it looked.

Nothing here is an elastica. Peirce’s path joins arcs to straights, so its curvature jumps at every join, and a real bent rod’s does not. The bending energy of such a path is perfectly well defined and is what has been used; the forces along it are not recoverable by differentiating it, which is why the contact force at a crossing is computed by a different route entirely and the relaxed case is recorded as missing.

The model has no history. It says where the least-energy state is. It does not say a cloth is there, and the finishing field’s whole argument is that a cloth’s state depends on what has been done to it — what came off the loom is not the relaxed state and neither is what came out of the wash.

The correction that was recorded as owed, and why it is not owed #

This essay ended, when it was written, by recording that the ratio of one every Peirce solution here is drawn at was wrong for all eight cloths, and that correcting them would move a hundred figures and forty essays’ numbers — a change to make deliberately and check cloth by cloth.

Checking cloth by cloth refuted it. The number above cannot go on those figures.

The locus is a set of states at constant thread length, and it asserts at every sample that one spacing rises exactly as the other falls. So its least-energy point is at a different sett — one per cent away for six cloths, and 32 × 22 against 29.8 × 24.3 for the poplin. Every row above is a cloth at its own relaxed construction, and writing it onto a drawing of the quoted one puts the ratio of one state on a picture of another.

Ask instead at the quoted construction — spacings held, thread lengths free, as every Peirce figure here draws — and for six of the eight cloths the energy falls to the end of the range, where the weft has gone straight and the warp carries everything. A ratio of infinity, and a fabric nobody weaves. The two with an interior minimum land at 2.92 and 3.95 against the 1.22 and 1.17 above — a different question, and a much larger answer.

And the check that would have caught it soonest: the reference state is built by calling for a crimp ratio, which fixes the thread lengths the locus is made of — so feeding the answer back in moves it. Six cloths drift upward by a tenth; the cheesecloth and the poplin run away.

So the figures are unchanged, and not because changing them is expensive. The crimp ratio at a cloth’s own construction is not decided by bending energy, the default of one is an assumption labelled as one wherever it is used, and this route does not improve it. What decides it is the tension the cloth was woven under — which is what Peirce said.

The theorem survives untouched, and so does the relaxed state — which is simply not what a figure of a quoted construction shows.

The generalisation #

Strip the cloth out and the shape of the argument is this: a kinematic model with a spare degree of freedom is not incomplete, it is a model of a set rather than of a state, and the way to close it is to ask what the degree of freedom costs.

That is not a textile fact. It is what happens whenever a geometric description admits a family — a linkage with a redundant joint, a packing with a free rotation, a network with a slack member — and no amount of further geometry supplies the missing statement, because it is of a different kind.

With a caution this essay had to learn. An energy closes such a model only over the set it is minimised on, and naming that set is the whole of the work. Minimise over the wrong one and the answer is real, well behaved, and about a different object.

The second half generalises too. When the free parameter is fixed by a symmetry, the material property cancels — a symmetric problem’s stationary point is at its fixed point whatever the coefficients — so the interesting cases are the ones where the symmetry is broken. That half needed no correction, and the reason is worth noticing: it is a statement about a fixed point, and a fixed point does not care which set it was reached on.

Who found it, and when #

Peirce’s geometry is from 1937, and the incompleteness is his own: his paper solves for the shape of the crossing and takes the division of crimp as given, because a fabric analysis supplies it and there was no reason to want it otherwise. Everything the model was built for — cover, jamming, thickness, the maximum sett — is available with the ratio measured.

The idea of closing a fabric-geometry model by minimising an energy is Olofsson’s, from 1964, and belongs to a line of work that treats a cloth as an elastic structure rather than a shape. That literature computes load–extension curves and bending behaviour, and it is where the fabric-mechanics tradition of the 1960s and 1970s went. What it did not much do — because it was after harder quantities — was turn round and ask the elementary question this rung asks: what does the energy argument say about the parameter the geometry was already short of?

The answer here is this site’s, and its shape is characteristic. The interesting half is not the number for any one cloth. It is that the correction splits into a part that needs a stiffness and a part that does not, and that the second part covers three quarters of the table.

Where the ladder goes next #

The immediate next rung is the one this makes possible: with an energy along the locus, the slope of that energy is a force, and the locus acquires a load–extension curve — where the flat start every cloth’s curve has turns out to be the same symmetry argument as this one, read at a different point.

Sideways, the same energy against friction gives the band a relaxed cloth rests in, and the same bending rigidity puts a number on what the shed costs the warp and on what the reed is fighting at the fell.

Further out, the missing compression term is the obvious next model. The site has a racetrack section and no stiffness for it; adding one would make the flattening a second energy, and the two together would predict not just the crimp ratio but the thickness — which is currently an output of the closure condition rather than a result.