The energy has no crimp ratio to give

The crimp ratio is not a measurement put the account’s missing equation in: Peirce’s geometry is two thread systems, four unknowns and three equations, so it cannot say how the crimp divides — and giving the threads a stiffness supplies the fourth, because the cloth should sit where its bending energy is least.

It left one thing open, in a sentence at the end: “a construction whose preferred ratio is outside its admissible interval sits at a boundary, and boundary states have not been studied here at all.”

They have now, and there are more of them than the sentence suggests. Six of the eight cloths in this account’s own table are boundary states, and the two that are not have energy wells shallower than the model’s own precision.

The minimum is a number and the profile is the argument #

The energy at a fixed sett is a function of one variable: how the crossing height D divides between the two systems. Write the warp’s share as a fraction of D, and the bending energy per unit area is

$$u = \frac{2B_1\theta_1 + 2B_2\theta_2}{D},$$

with each weave angle following from that system’s share of the height and its own pitch. Sweeping the share and taking the least value is what the account’s fourth essay did.

A least value does not say whether it is a minimum. It says which of the states tried was lowest, and the states tried run from one end of the feasible interval to the other. So there are two quite different things a “least-energy state” can be:

• a stationary point, where the energy has a genuine well and the cloth is held in it;
• an endpoint, where the energy simply falls all the way and stops because the geometry runs out.

The first is a prediction. The second is a statement that the model wants something the geometry forbids, and the number it returns is a property of the forbidding rather than of the cloth.

Six of eight, and the state is not a cloth #

Running the profile over the table of cloths here:

Six of the eight fall to the same end, and that end is the warp taking the entire crossing height. Its weft crimp is not small; it is nought to six decimal places, and its warp crimp is between 6.9 and 40.9 per cent.

No woven cloth has a dead straight weft. Every crimp this account has measured, computed or quoted has both systems bent, and crimp interchange — the mechanism the whole setting field runs on — requires both to have some to trade. So the boundary state is not a cloth the model is describing; it is the direction the model wants to go, stopped by the geometry rather than by anything physical.

And the two interior wells are below the model’s own precision #

The sheeting and the filter cloth do have stationary points, and the depth column is what to read.

The sheeting’s well is 3.27 per cent deep and the filter cloth’s is 0.52. Those are the differences in bending energy between the least state and simply going to the end.

Now set them against what the energy is known to. A yarn’s stiffness is a bracket, not a number: the free bound, with the fibres sliding, and the coherent bound, with the yarn a solid rod, differ by the fibre count over the square of the packing factor — a factor of 408 for these yarns. Every energy above is computed at the free bound, and the real one is somewhere in that bracket.

A well half a per cent deep, in a quantity known to within a factor of four hundred, is not a state. It is a number the arithmetic produces and the model cannot see the bottom of. The sheeting’s 3.27 per cent is better and it is not better by enough.

So the honest summary of the fixed-sett energy over this table is: six cloths have no interior state and two have one the model cannot resolve. The fourth essay’s fourth equation arrives, and for the constructions this account actually carries it arrives with nothing in it.

Two regions, and the frontier between them #

Sweeping the two ratios that a construction has — the counts and the setts, each against a balanced reference — shows that the boundary is not an accident of this table.

Of the 28 solvable constructions in the grid, 25 fall to an end, and they fall to both ends: some to the weft dead straight and some to the warp. The few between them are the frontier where the two regions meet.

That is a phase diagram rather than a scatter. A cloth is in one regime or the other, and the ratio the energy gives is 1 or 0 — all the crimp in one system — with a thin band between them where a stationary point exists at all. The physical reading is straightforward: the energy always wants to flatten the stiffer, more tightly pitched thread completely, and which one that is flips across the frontier.

It also explains why the ratio the account’s fourth essay reported for the two interior cloths — 2.92 for the sheeting and 3.95 for the filter — is so far from the 1.22 and 1.17 the constant-thread-length locus gives. They are not two estimates of one quantity. They are answers to two questions, and the fixed-sett one is being asked near a frontier where the answer swings between 0 and ∞.

How flat is flat, in the units the cloth is described in #

A depth in energy is hard to weigh, so it is worth converting it into the quantity the answer is quoted as.

Take the band within half a per cent of the least energy and ask what crimp ratios it contains.

The filter cloth’s half-per-cent band runs from 0.552 to 0.852 of the crossing height, and the crimp ratio across it runs from 1.36 to 31.8 — a factor of twenty-three in the answer, for half a per cent in the thing being minimised. Widen the band to one per cent and the ratio’s upper end goes to a million, because the band reaches the degenerate endpoint.

The sheeting’s half-per-cent band runs from 0.573 to 0.728, and its ratio from 1.54 to 6.48 — a factor of four.

So the model’s precision in the ratio is a factor of four on the better of the two cloths and twenty-three on the worse. Quoting a crimp ratio to three figures from this argument is quoting the resolution of the sweep rather than the resolution of the physics — a ratio of 2.92 reads as a measurement and a bracket of 1.5 to 6.5 reads as what it is.

That is the same shape as the bending bracket itself. A yarn’s stiffness is a bracket because the fibres may slide or may not, and the honest form of the answer is an interval with its ends computed. A crimp ratio from this energy is an interval too, and this essay’s contribution is to say how wide it is rather than to pick a point in it.

What was missing, and it is not a better solver #

The temptation is to read all this as a numerical problem — a shallow well needs a finer sweep, a boundary needs a better search. It is not.

The energy is bending only, and bending alone has no interior preference. Flattening a thread costs its partner nothing in this model; the two systems interact through the closure condition and through nothing else. So the energy is a sum of two terms each of which is minimised by giving the whole height to the other system, and a sum of two such terms lands at an end unless the geometry happens to balance them.

What a real cloth has that this does not is tension. A weft is beaten in against a warp held at tension, and the warp’s tension is what stops it taking the whole of the crossing height — it is what the shed is already spending and what the beat-up works against. Peirce said exactly this: the crimp ratio is decided by the tensions the cloth was woven under, and it is not in the geometry.

So the boundary states are the model saying so. An energy with no tension term in it has no interior minimum to give, and the six cloths that fall to an end are six cloths telling the model what it is missing rather than six cloths with a straight weft.

That is a better result than a fourth equation would have been, because it names the missing term. A tension term would enter as work done against the warp’s own tension when its thread length changes, which is a quantity this account computes on two other accounts — and putting it into this one is the obvious next piece of work.

What every number computed from a ratio inherits #

The practical consequence runs through the whole site, and it is worth stating precisely rather than alarmingly.

Every figure drawn at a crimp ratio of one is drawn at a chosen ratio, and this essay says the energy cannot improve on the choice for six of the eight cloths. That is not a defect in those figures; it is the reason this account has always stated which ratio produced a number.

The quantities that do not depend on the ratio are unaffected, and they are most of them. The thickness, the cover, the jamming sett and the crimp sum are all fixed by the closure condition, which is one equation short of a ratio and complete without one.

The quantities that do depend on it are the ones about the two systems separately: the warp’s crimp against the weft’s, the take-up against the width contraction, and anything about how a cloth narrows when it is pulled. Those are computed at a stated ratio and are a family rather than a value.

The cloth that was called impossible is the cautionary case and it is the right note to end the accounting on. That essay found a poplin recorded as having no solution, and the fault was a search that treated a state it could not reach as evidence of having gone too far. The boundary states here are the same shape of thing seen correctly: a search reaching the edge of its interval is reporting the edge, and whether that is an answer depends entirely on why the edge is there.

The one place the boundary is the right answer #

There is a construction for which the degenerate state is not a failure of the model, and naming it keeps the result honest.

A warp rib and a weft rib are cloths in which one system really is nearly straight. The four named weaves are corners of a family: grouping the ends by two and leaving the picks single gives a cord in which the grouped threads lie side by side with nothing between them, and the thread that wraps them takes almost the whole of the crimp. A cord’s height is the weft’s own crimp amplitude and the warp within a group is straight to within its own diameter.

So the boundary state is a real state for a real construction, and the model reaching it is the model being right about that construction and wrong about the eight plain-woven cloths in the table.

Which is the diagnosis stated the other way round. What holds a plain weave’s two systems both bent is that each is beaten and tensioned against the other, and a rib’s grouped threads are not — they are held by their neighbours in the group rather than by their crossings. A model with no tension in it is a model of a rib, and the six cloths falling to the boundary are six plain weaves being described as ribs.

What was counted, and how #

The profile is the fourth essay’s own energy, unchanged: $2B_1\theta_1 + 2B_2\theta_2$ over $D$, with the weave angles solved from the height split by bisection against Peirce’s own height relation. What is new is returning the whole curve rather than its least point.

The feasible interval is found rather than assumed. A height split is feasible when both systems have a weave angle that reaches it, and for the sheeting that rules out everything below 0.294 — so its left-hand endpoint is a geometric refusal and not a choice of range.

The depth is measured against the better endpoint, which is the honest comparison: the question is whether the interior state is meaningfully below what the cloth would do if it simply went to the end.

The map is required to have two regions. The census requires that most constructions land at an end and that both ends are reached; a grid in which every cloth fell the same way would be a grid with one region and a quite different result, and the check is what separates the two.

And the bracket is quoted from this account’s own bending work rather than estimated, so the comparison between a well’s depth and the model’s precision is between two numbers this account computes.

What this cannot say #

The energy has no tension in it, which is the whole of the diagnosis and is also the limit on the diagnosis. A model with a tension term might have interior minima everywhere, at a frontier that has nothing to do with this one, and nothing here predicts where.

Nor is there any friction. Threads at a crossing press on one another and resist sliding, so a cloth may be held at a ratio that is not a minimum of anything — held by hysteresis, which is how a cloth gives back less than it took. A shallow well and a rough energy landscape are indistinguishable in their consequences: both say the cloth’s ratio is set by its history.

And the grid is a grid. The frontier between the two regions is located to within a cell, which is enough to say there are two regions and not enough to write its equation. That equation is the condition $B_1$ and the two pitches satisfy when the energy’s two endpoint values are equal, and it is one line of algebra nobody has written down here.

Who found it, and when #

Peirce’s 1937 geometry is the source of the whole difficulty and he stated it himself: the equations do not determine the crimp ratio, and what does is the tensions under which the cloth was woven. Every account since has either assumed a ratio or measured one.

The energy argument is this account’s and so is its refutation. The fourth essay proposed the bending energy as the missing equation and reported where it minimises; this essay profiles the same energy and finds that on most constructions it does not minimise anywhere inside.

The part worth carrying is the shape of the failure. A model that returns an endpoint has not failed numerically and it has not failed silently — it has said, in the only way a variational model can, that the term deciding the answer is not in it. Six cloths saying the same thing in the same direction is a much clearer message than a scatter of plausible ratios would have been, and it points at the one term Peirce named eighty years ago.

Still open: what a tension term does to the frontier #

The obvious next piece of work is to put the warp’s tension in, and the shape of it is already determined by what is here.

A tension $T_1$ in the warp does work when the warp’s thread length changes, and the warp’s thread length is a known function of the height split — so the term is $T_1\,\partial l_1/\partial f$, added to the energy’s derivative. It opposes exactly the direction the bending energy wants to go, which is what makes it the missing term rather than a refinement.

Two things would follow immediately. The minimum would move inward from the endpoint by an amount increasing with the tension, so the ratio would become a function of a loom setting rather than a property of the cloth — which is what Peirce said and what every weaver knows. And the frontier in the map above would move, because the two regions are decided by which endpoint is lower and a tension term lowers only one of them.

The tension is available. What the shed costs in newtons computes a warp tension from the loom’s geometry, and a pick density is a force budget computes what is left of it at the fell. Both are in this account and neither has ever been used in a crimp calculation, which is the gap this essay exists to name.