The most a cloth can give back

A cloth’s interchange budget is crimp it can give up, and crimp is made by setting threads close enough that they have to bend around each other. So the budget should rise without limit as a cloth is set closer, and the question “how much extension can a woven cloth have with no thread stretching?” should have the answer “as much as anybody is willing to weave”.

It does not. The budget rises, reaches a maximum and falls away, and both the maximum and the place it happens are single numbers with no yarn count in them.

The claim #

The largest interchange budget a square plain cotton cloth can have is 7.2857 per cent, at a warp cover factor of 0.407782, and there is no yarn count in either number.

Solved to six figures at five, ten, twenty, forty, sixty and a hundred tex, the cover at the maximum is the same to eight decimal places and the budget at it is the same to six. The sett it occurs at moves by a factor of 4.47 across that range — from 48.81 ends per centimetre at five tex to 10.91 at a hundred — and the answer does not move at all.

Why there is a maximum at all #

Interchange is a trade. The warp gives up crimp and the weft has to take it on, and both halves have to work.

At an open sett there is nothing to trade. The threads barely deflect around one another, the warp’s crimp is a fraction of a per cent, and there is almost nothing to straighten out. The cheesecloth in this site’s table is set at ten ends per centimetre and has 1.91 per cent of extension in it, which is the entire crimp its warp carries.

At a close sett there is nowhere to trade to. Taking the warp’s crimp out means pushing it into the weft, and a weft in a dense cloth is already deflecting most of the way it can — its straight run between crossings is nearly gone, and once it has vanished the cloth stops whatever the warp still had in hand. The sheeting has 14.61 per cent of warp crimp and reaches 4.03 per cent, because its weft has had enough at that point.

Between those two failures there is a best cloth, and the curve is smooth enough that “best” is a single point rather than a plateau. The machinery checks that: the sweep that brackets the feasible setts also counts turning points in the budget across them, and refuses to run a golden-section search on anything with more than one.

Why it is a cover and not a sett #

Every length in Peirce’s geometry is a multiple of the yarn diameter. A thread’s path is an arc of a circle whose radius is the sum of the two diameters, plus a straight run; the closure condition says the two systems supply that sum between them; the diameter is the only place a yarn count enters at all.

So a budget cannot depend on a spacing except through the ratio of that spacing to a diameter — which is the reciprocal of a cover factor. The count divides out as a matter of algebra, and the numerical agreement over a twentyfold range is a check on the arithmetic rather than a discovery.

What is worth noticing is how thoroughly the sett fails to be the interesting quantity. A weaver setting a fine yarn at 48 ends per centimetre and a coarse one at 11 has made two cloths with nothing obvious in common, and they have the same interchange budget to the sixth figure.

The obvious guess about where the maximum is, and why it is wrong #

A cloth’s budget is stopped by one of two events — the warp going straight, or the weft jamming — and which of the two binds changes over at a definite cover. This collection computed that changeover for a different reason: it is the point where a weft thread carries exactly a quarter of the circumference of the circle it bends round, and it comes out at a cover of 0.337408, again with no count in it.

It would be tidy if the maximum were at the changeover. A quantity limited by two competing bounds often is.

It is not. The changeover is at 0.3374 and the maximum is at 0.4078, and between them the budget is still rising — the weft has become the binding bound and the cloth goes on gaining for another fifth of its cover before the weft’s limit starts to bite harder than the warp’s crimp is growing. The two are different questions and the coincidence would have been a coincidence. It is worth saying plainly because the tidy answer is available, is easy to state, and is false by a fifth.

What balance costs #

The number above is for a square cloth. Unbalancing one takes budget away fast.

The mechanism is the same one: a cloth with few picks has few places to put the crimp its warp gives up, so the weft jams sooner. What is unexpected is how little the cover moves while the budget halves. Across balances from square to two-to-one the optimum cover shifts by five per cent of itself.

That makes the practical statement a one-number statement rather than a two-number one. A designer who wants a cloth to recover well should set the warp near a cover of 0.41 and should balance the cloth, and the second instruction matters more than the first.

What the maximum is worth knowing #

Two things, and they pull in different directions.

The first is that it puts a ceiling on the whole business. Whatever a fabric is made of and however it is set, a plain woven cloth has at most seven per cent of extension available for free, and a garment that is stretched further than that at a knee or an elbow is going to keep something. Seven per cent is not much. A knitted fabric has an order of magnitude more, for reasons that have nothing to do with crimp and everything to do with the loop being a mechanism of a different kind, and the gap between the two numbers is most of what separates a woven garment from a knitted one in use.

The second is that the ceiling is nearly reached by ordinary cloth, which is the more interesting half. Four of the eight constructions in this site’s table sit above ninety per cent of their own maximum. Weaving is a craft that has been optimising something for a very long time without necessarily knowing which quantity it was optimising, and the sett a weaver arrives at by feel is set by hand, by cover, by how the cloth beats up and by what the loom will take. That those constraints should land near the interchange optimum is not obvious and is not claimed to be causal.

What the optimum is not sensitive to, and what it is #

A maximum is only useful if a designer can miss it and still get most of the benefit, so it is worth reading the curve’s shape rather than only its peak.

The peak is broad. The budget is within five per cent of its maximum over a cover range from about 0.35 to about 0.47, which is a third of the way across the whole weavable span. So a cloth set anywhere in that band has essentially the best interchange available to it, and the instruction “set near 0.41” is a much weaker requirement than a six-figure optimum suggests.

The fall-off is asymmetric. Below the peak the budget falls away steeply, because an open cloth has almost no crimp to trade; above it the fall is gentler, because a close cloth still has crimp and is merely running out of room to put it. So a designer in doubt should set closer rather than opener — which is convenient, since every other reason to choose a sett points the same way.

That asymmetry also explains the table’s own distribution. The four cloths above ninety per cent of their maxima are all on the closed side of their own optima, and the two furthest below — the cheesecloth at thirty per cent and the sheeting at sixty — are at opposite ends. The cheesecloth is far open of its peak and is paying the steep penalty; the sheeting is far closed of it and is paying the shallow one, which is why a cloth as different from the optimum as a sheeting still keeps three fifths of the budget.

And nothing about the peak’s position is available by feel. A cover of 0.41 is not a handle, a firmness or an appearance; it is a spacing measured in yarn diameters, and it is invisible to every sense a weaver uses at the loom. That is what makes the six-figure agreement across counts worth stating rather than merely correct — the quantity that decides the answer is one nobody would have arrived at by trying constructions.

What was counted, and how #

The feasible span of setts is found by scanning, and taking the longest contiguous run rather than the outermost feasible points, which is not fussiness. At very open setts Peirce’s geometry flickers in and out of solvability, and a search that took the first and last feasible sett as its bracket would search across a region containing no cloth.

Inside the span the budget is unimodal, which is asserted rather than assumed, and the maximum is found by golden section to two hundred iterations. The optimum is then recomputed at each of six counts and the agreement is asserted to a tolerance of a millionth.

The muslin is a curiosity and is reported as one. Twenty-four ends and twenty-two picks per centimetre in 20 tex is within a rounding error of the construction that maximises interchange for that yarn, and it is the row of the table labelled “the ordinary cotton cloth”. Whether that is a fact about weaving or an accident of a table with eight rows in it, this collection cannot say, and the honest position is that eight rows is not a sample.

A defect the sweep found, and what it was #

The first version of this arithmetic reported negative budgets at open setts — a cloth whose extension bound was minus seventy-three per cent, which is a cloth shorter than itself.

The cause was in the site’s own tensile geometry and had been sitting there since the ladder was built. The crimp height of a thread at weave angle θ is (l − Dθ) sin θ + D(1 − cos θ), and its derivative is (l − Dθ) cos θ. The library’s own comment said the bracket is non-negative on the interval from zero to the jam angle and concluded that the height rises across the whole of it. The bracket has two factors, and only the first is non-negative there: cos θ turns negative at a quarter turn.

So a thread whose jam angle is past a quarter turn reaches its greatest crimp height before the jam rather than at it, and the bisection that inverts the height function was inverting something that goes up and comes back down. It returned answers. They were plausible. The extension bounds computed from them were not — which is the third solver on this site to fail by treating two ways of being unreachable as one.

Three of the eight cloths in the table wrap past a quarter turn — the cheesecloth at 158 degrees, the voile at 105, the batiste at 87 just under — and for every one of them the height at the true maximum exceeds the cloth’s whole thickness, so the locus clamped it and no figure on this site ever saw the difference. It took a sweep over open constructions, where the jam angle runs past π and the cosine has changed sign twice, to make it visible. The correction is to take the maximum at the smaller of the jam angle and a quarter turn, which is what the compression ladder’s own version of the same function had been doing all along.

Every quantity this collection has published is unchanged, which was checked against a full rebuild rather than argued.

Where the model stops #

The maximum is a maximum of the geometry and not of anything a wearer notices. A cloth at the optimum cover recovers best from a strain past its budget, and says nothing about how much force that strain took, how the cloth drapes, whether it covers, or what it weighs. Those are other ladders and they point in other directions — a cover of 0.41 is an open cloth by shirting standards.

It is a plain weave throughout. A twill or a satin has a different arc-and-straight construction, so the numbers would move; whether the maximum survives at all is not settled here, because a longer float changes the relationship between crimp and cover in a way this geometry does not model.

And the balance is a sett ratio and not a count ratio. Every cloth in the sweep has the same yarn in both directions. A cloth with a fine warp and a coarse weft is a two-parameter family this rung does not explore, and the poplin — the only such row in the table — is also the row furthest from its own optimum in a way that has two causes rather than one.

The generalisation #

A quantity that is bought from two sources at once has a maximum, and the maximum is rarely where either source runs out. Interchange needs crimp to give and room to receive; setting a cloth closer buys the first and spends the second. The same structure governs a gearbox’s usable ratio range, an amplifier’s bias point, a wing’s aspect ratio and the doping of a semiconductor: two monotone effects with opposite signs, one product, one interior optimum.

The second lesson is narrower and is about dimensionless groups. When a maximum turns out to be at a fixed value of a ratio rather than of either quantity in it, that is a statement that the ratio is the real variable — and it is worth looking back to see what else in the subject was quoted in the wrong units. This collection has quoted setts for fourteen fields. A great many of those statements are really statements about cover, and the ones that are not are the interesting ones.

Who found it, and when #

The cover factor is nineteenth-century weaving practice, formalised by Peirce in 1937 alongside the geometry, and its independence of count is the whole reason it was invented. That the quarter-circle changeover has no count in it is this collection’s own, from the ladder that built the extension bounds.

The maximum itself does not appear in the standard accounts, which is unsurprising. Textile mechanics quotes the extension available from crimp as a property of a fabric in front of it, not as a function to be maximised over constructions, and the question “which cloth has the most” is not one a mill has any reason to ask.

Where the ladder goes next #

The next rung on this anchor leaves extension for curvature, and finds that the same split decides what happens at a fold, where the cloth-level route runs out at a radius of millimetres and hands the problem to the fibres.

Sideways, the budget being one quantity shared between two directions is the subject of what a pre-tension actually costs, and the sett shifts that the finishing imposes mean that the construction a loom is set to is not the one the budget belongs to.