A woven cloth asked the same question

The knitted result was uncomfortable enough to be worth asking of the other fabric. This collection’s knitted geometry produces a fabric whose two adjacent courses approach to four fifths of a yarn diameter, so its yarn passes through itself at rest — and the geometry was built the same way a woven one is, by setting a distance correctly at the crossing and letting the rest of the path look after itself.

So does a woven cloth fit together?

Nearly. And the near-ness is the finding.

The numbers #

Four ordinary constructions, from the site’s own cloth table, with Peirce’s geometry solved for each and the two thread paths laid out as sinusoids of the crimp heights it gives.

A voile, at four and a half per cent crimp: closest approach 1.000. The two threads come closest exactly where they touch, and the cloth fits together perfectly.

A batiste, at ten per cent: 1.000. Still exactly at the crossing.

A sheeting, at thirteen per cent: 0.989. A one per cent overlap, and the closest approach has moved off the crossing.

A duck, at eighteen per cent: 0.955. Four and a half per cent.

A knitted fabric is at 0.780.

Why the difference is so large #

The two geometries are built the same way and they behave completely differently, and the reason is one word.

In a knitted fabric, two adjacent courses run parallel. Moving along them, away from the interlacing, costs nothing at all in the along-the-course direction, because both curves are going the same way. So the two are free to stay close while the vertical gap closes, and the closest approach moves off the interlacing easily.

In a woven cloth, the two systems are perpendicular. Moving along the warp away from the crossing costs distance in one direction; moving along the weft costs distance in the other; and both have to be paid before any vertical gain is collected.

So a woven cloth’s closest approach is at or very near its crossing, which is exactly where the geometry set it — and the geometry is right.

Crossing is a much better arrangement than running alongside, and that is a structural difference between the two ways of making cloth that nobody has stated in these terms.

What the overlap depends on #

It rises with the crimp, and that is what identifies the mechanism rather than merely describing the result.

The vertical gain from moving away from a crossing is the crimp: the warp descends and the weft rises, so the gap between them closes by an amount set by how far each goes up and down. A cloth that barely crimps has almost nothing to gain by moving, so its minimum stays at the crossing.

A cloth that crimps hard has a great deal to gain, and past a threshold the gain beats the horizontal cost.

The threshold is at about ten per cent mean crimp for an ordinary construction, which is where the collection’s own cloth table sits its middle members.

So the answer to whether a cloth fits together is: the open ones do, and the dense ones very nearly.

Which settles a question about the flattening #

The knitted result read the overlap as a prediction: the fabric’s geometry demands a flattened yarn, and the flattening it demands is four fifths.

Applied to a woven cloth, the same reading gives a flattening of 1.000 to 0.955.

The trade measures woven yarn flattened to between 0.7 and 0.9.

So the woven flattening is not predicted by the cloth’s own geometry. The geometry demands almost nothing, and the yarn is much flatter than that.

That is a clean negative and it is worth having, because it says where a woven cloth’s flattening does come from: not from the arrangement, but from the loom. A warp under tension being beaten up against the fell is pressed far harder than the finished cloth’s own geometry requires, and what a section shows is the memory of that pressing rather than a requirement of the structure.

Which is a real asymmetry between the two fabrics #

Putting the two results together gives a statement about the two crafts that neither alone supports.

A knitted fabric’s yarn is flattened because the fabric requires it. The geometry does not close otherwise, at any construction, and the flattening it demands follows the tightness factor.

A woven cloth’s yarn is flattened because the loom flattened it. The geometry closes nearly round, and the section is a process memory.

That predicts something checkable and slightly surprising: a woven cloth’s yarn flattening should relax over time and a knitted fabric’s should not. Washing, tumbling and repeated wearing let fibres rearrange, and a section held only by memory should recover towards round while a section held by the arrangement cannot.

Nobody has looked, and the measurement is a set of sections before and after a wash cycle.

Where the threshold sits, and what is on either side #

The transition at about ten per cent crimp is worth locating against real cloth, because it decides which cloths this matters for.

Below it: voiles, lawns, batistes, organdies, most shirtings, most fine dress goods. All of these fit together with no overlap at all, and their geometry is exactly right.

Above it: sheetings, poplins, canvases, ducks, most heavy cloth. These overlap by a per cent or a few, which is far too small to change any calculation and is not nothing.

And well above it: the densest constructions — jammed cloths, filter fabrics, tightly beaten canvas — are outside the range computed here and should overlap more.

That last is where the argument runs out, and it is the one worth extending.

Why a satin should fit best of all #

The weave dependence is worth following because it makes a prediction the collection could check on its own machinery.

A thread’s overlap depends on its crimp, and a thread’s crimp depends on how often it interlaces. A satin’s warp passes over four picks and under one, so it bends once in five spacings rather than once in one, and its crimp is correspondingly lower — which is the whole of why a float decides so much.

So a satin should sit at the bottom of the crimp axis and fit together with no overlap at all, even at setts where a plain weave of the same yarn would overlap.

That is a prediction with a familiar direction. Satins are woven at higher setts than plain weaves of the same count, and the usual explanation is that fewer interlacings leave more room — which is right and is about the sett rather than about the fit. This adds a second reason pointing the same way.

The computation is available: the site’s own cloth arithmetic already computes each weave’s mean float and feeds it into Peirce’s spacings, so running the approach across a weave table is an hour’s work.

What was counted, and how #

Peirce’s geometry is solved for each construction from the site’s own machinery, unchanged, at an equal division of crimp between the two systems. It supplies the two crimp heights and the two crimp percentages.

The thread paths are laid out as sinusoids of those amplitudes, with the warp’s wavelength two pick spacings and the weft’s two end spacings, which is a plain weave’s period.

That is a real approximation and it is made deliberately rather than for convenience. A Peirce path is circular arcs joined to straight lines, so its curvature jumps at every join — and a curvature discontinuity is exactly the kind of thing that produces a spurious minimum in a distance calculation. The question here is where a minimum sits, so a smooth path is the safer instrument.

The minimum is found by a grid search over both threads’ whole periods, at two hundred and sixty samples each, and the result is reported in units of the contact separation, which is half the sum of the diameters.

The check asserts two things and both could fail. The worst overlap in the table is under six per cent, so a woven cloth is nothing like a knitted one; and the overlap rises with the crimp, which is what says the two are the same mechanism at different strengths rather than two different things.

What the loom actually does #

If a woven cloth’s flattening is a process memory, it is worth asking what process, because the answer bounds how much flattening to expect.

Three pressings happen to a warp end on its way into cloth.

The beat-up. The reed drives each pick against the fell with a force that is a substantial fraction of the warp tension, and the pick is squeezed between the ends it is crossing at exactly that moment. This is the largest and the most localised.

The warp tension itself, which presses every end against every pick it crosses for as long as the cloth is on the loom.

And the finishing. Calendering presses the whole cloth between rollers at pressures far above anything the loom applies, and a calendered cloth’s threads are visibly flatter than the same cloth before finishing — which this collection has computed as a width gain.

All three are external, none of them is required by the cloth’s own geometry, and all three are removable in principle. That is why the prediction that woven flattening should partly recover is a real one: nothing structural is holding it.

Where the model stops #

The paths are sinusoids and Peirce’s are not. The amplitudes are Peirce’s and the shapes are not, and how much that matters is not computed. It should matter little for the location of the minimum and could matter for its depth.

The weave is plain. A twill or a satin has a longer float, so its thread’s wavelength is longer and its crimp is lower, and both push the overlap down. Nothing here computes a satin, and the prediction is that a satin fits together better than a plain weave of the same sett.

Only two threads are considered. A warp end is near two picks at once, and the closest approach to either is not the same as the closest approach to the pair.

And the densest cloths are outside the range. A jammed cloth has crimp above twenty per cent and is not in the table, and it is where the effect should be largest.

Also worth recording: a wool duck at these setts has no Peirce solution at all. A sixty tex wool is thicker than a sixty tex cotton, because wool is less dense, and at twenty ends a centimetre the geometry runs out. That is the machinery refusing an impossible construction rather than failing, and it is correct — such a cloth cannot be woven.

The threshold, and why it is where it is #

The transition at about ten per cent crimp can be located rather than observed, and doing so makes it a prediction rather than a reading off a plot.

Moving a distance s along the warp away from a crossing costs s in horizontal separation and gains, in vertical separation, the amount the warp has descended — which for a sinusoid of amplitude h/2 and period P is about (h/2)·(2πs/P)²/2 for small s.

The gain is quadratic in s and the cost is linear, so for small s the cost always wins and the minimum stays at the crossing. The gain only overtakes when the second-order term is large enough, which happens when the crimp height over the period exceeds a number of order one.

That is why there is a threshold at all rather than a gradual drift, and it is why the threshold is in the crimp rather than in the sett or the count. Two cloths with the same crimp and different setts should sit at the same point on this axis, and the four in the table nearly do.

Where the argument would break #

The sinusoidal path is the assumption to test, and it is worth saying which way an error would run.

A Peirce path is flatter than a sinusoid near the crossing — it is a circular arc of radius equal to the sum of the diameters, which is gentler than a sine’s crest — and steeper in between. A flatter crest means less vertical gain for a given step away from the crossing, which pushes the threshold up and the overlap down.

So the sinusoidal model over-states the overlap, and the true woven answer is closer to 1.000 than the numbers here. That strengthens the conclusion rather than weakening it, which is the comfortable direction for an approximation to run.

The knitted calculation used the solved path rather than a sinusoid, so no such correction applies there, and the comparison between the two is if anything understated.

The generalisation #

The rung is a worked example of the cheapest kind of check there is, and this collection should make it more often.

Ask the same question of the other thing.

A defect was found in the knitted geometry. Rather than repairing it, the first move was to ask whether the woven geometry — built by the same people on the same day using the same principle — has it too. The answer took an afternoon and it changed what the knitted result means: an overlap that appears in both would be a property of the method, and one that appears in only one is a property of the fabric.

It appears in one, so it is about knitting.

That is a large difference in interpretation for a small amount of work, and the collection has a great many results that exist for one fabric and not the other. Its knitted contact force, its knitted thickness, its knitted extension ceiling: each has a woven counterpart that has never been computed alongside it, and each comparison would say whether the result is about the fabric or about the model.

What this says about the knitted result’s status #

The comparison changes how much weight the knitted flattening prediction can carry, and the change is in its favour.

A prediction that came out of one geometry and could not be tested elsewhere would be suspect: it might be an artefact of how that geometry was built, of the sampling, or of the solve.

The same instrument applied to a different fabric, built by the same method on the same principles, gives a different answer — and gives it in the direction the structural difference between the fabrics predicts. Parallel systems overlap and crossing ones nearly do not.

That is much better evidence that the knitted number is about knitting than any amount of care taken over the knitted calculation alone would be. A method that produced overlaps everywhere would be a method with a bias in it; one that produces an overlap in exactly the case where the geometry is unfavourable is measuring the geometry.

So the woven rung’s most useful output is not the woven numbers. It is that the knitted number survives a control, and controls are the thing this collection has fewest of.

Two more comparisons worth making #

The move that made this rung cheap suggests two others, and both are afternoons rather than programmes.

Ask the same question of a rib. A rib’s courses lie on two beds, so its adjacent courses are further apart through the thickness and closer in plan. Whether that makes the overlap better or worse is not obvious and the machinery exists.

And ask it of a warp knit. A tricot’s loops are threaded sideways as well as vertically, so its neighbouring structures run neither parallel nor perpendicular but at an angle — which is exactly the intermediate case between the two computed here, and would say whether the parallel-versus-crossing explanation is the right one.

The second is the better test, because it is the case the explanation makes a prediction about rather than the two it was derived from.

Who found it, and when #

Peirce’s geometry is from 1937 and the crimp heights it produces are its central output.

That woven threads are flattened is universal knowledge and the flattening has always been an input. Nobody appears to have asked whether the geometry requires it, which is unsurprising: the question only becomes interesting once somebody has found a geometry that does, and the knitted one is that geometry.

What is this collection’s own is asking the question of both fabrics with one instrument, finding one answer nearly yes and the other emphatically no, and reading the difference as a statement about crossing versus running alongside.

Where the ladder goes next #

The contact ladder has one question left and it is the one it has been avoiding: what a model that took contact seriously would actually have to do, and whether the return is worth the cost.

What a contact model would have to do prices it in yarn, in bending and in the class of problem the solve becomes.