The closest approach is not the crossing

Every cloth geometry ever written places two threads a stated distance apart at a crossing. Peirce’s does; the racetrack does; the elastica that replaced them does; and this collection’s knitted geometry does.

Not one of them checks whether the crossing is where the two threads come closest.

The question #

Two curves are given. They touch, or nearly touch, at a designated point. Is that point where they are closest?

For two straight lines, yes trivially. For two curves it depends, and what it depends on is worth setting out because it decides the answer for every fabric this collection has.

Move a small distance s along one curve from the designated point. Two things change.

The along-the-curve separation grows. How fast depends on the angle between the two curves: if they run parallel, moving along one moves parallel to the other and the separation grows slowly or not at all. If they cross at right angles, moving along one moves directly away from the other and the separation grows as s.

The across-the-curve separation changes too, by however much the two curves have moved relative to one another in the transverse direction. That is set by their curvature, and to leading order it goes as s².

So the trade-off is between a term that is linear in s and one that is quadratic, and which wins depends on the coefficient of the linear one — which is the angle between the curves.

The two cases #

Perpendicular curves. The linear term has its full coefficient. Moving away from the crossing costs distance immediately, and the quadratic gain cannot catch up until s is large — by which time the curves have separated for other reasons. The minimum stays at the crossing.

Parallel curves. The linear term vanishes. Moving along costs nothing in that direction, so any transverse gain at all moves the minimum, however small. The minimum leaves the designated point immediately.

Between them, the coefficient falls with the cosine of the angle, so the transition is smooth and the parallel case is the limit rather than a special one.

Which fabric is which #

The two ways of making cloth fall on opposite sides of this.

A woven cloth’s two systems are perpendicular. Warp and weft cross at right angles by construction, so the linear term has its full coefficient and the closest approach is at or very near the crossing. Computed for four ordinary cloths, the answer is exactly at the crossing for the two open ones and a per cent or a few off for the two dense ones.

A knitted fabric’s adjacent courses are parallel. They run in the same direction, one course spacing apart, and the interlacing is a point where they pass. So the linear term vanishes and the minimum moves — and it moves by a fifth of a diameter.

That is the whole of why one geometry is self-consistent and the other is not, and it has nothing to do with how carefully either was built.

Why the transverse gain exists at all #

The quadratic term needs a source, and in both fabrics it is the same one: the two curves are going in opposite directions through the thickness.

In a woven cloth at a crossing, the warp is at its highest and the weft at its lowest. Move away and the warp descends while the weft rises, so the gap closes. The rate is the crimp.

In a knitted fabric at an interlacing, one course is arriving at a crest and the other at a trough. Move away and the same thing happens.

So both fabrics have a transverse gain and only one of them can collect it, because only one of them has the linear cost to pay first.

Where the woven case does slide #

The perpendicular case is not immune, and where it fails is instructive.

The linear cost is fixed by the geometry and the quadratic gain rises with the crimp. So a cloth that crimps hard enough eventually overcomes the cost, and the four cloths computed here show exactly that: no overlap below about ten per cent mean crimp, and four and a half per cent overlap at eighteen.

That is a threshold in the crimp rather than in the sett or the count, which is what the arithmetic predicts: the gain is quadratic in the crimp amplitude and the cost is not.

So the rule is not “perpendicular systems are safe” but “perpendicular systems are safe until the crimp gets large”, and ordinary cloth is below the threshold.

What to check before building a geometry #

The transferable output is a rule for anybody laying out a fabric model, and it is short.

If the structure’s two systems cross, place them at the crossing and check the crimp. The construction is nearly right and the check is cheap.

If they run alongside, the designated point is not the answer and the model has to be checked over its whole length.

This collection did the first without knowing it was a check and skipped the second entirely, which is why its woven geometry has been sound for eighteen phases and its knitted one has been quietly impossible since the day the third dimension arrived.

The third structure #

Two other fabric types are worth locating on this axis, because they are the intermediate cases.

A warp-knitted fabric — tricot, raschel — threads its loops sideways as well as vertically, so its neighbouring loop structures run neither parallel nor perpendicular but at an angle set by the lapping movement. The linear coefficient is the cosine of that angle, so a warp knit should overlap by an amount between the two, and by less for a longer lap — which is a prediction about a fabric this collection has barely modelled.

A braid’s strands cross at an angle set by the braid angle — typically forty-five degrees — so its linear coefficient is about seven tenths of the perpendicular one, and it should sit nearer the woven end.

Neither has been computed. The warp knit is the more interesting because it is the case the explanation makes a prediction about rather than one it was derived from, and the collection has the machinery to try it.

Why this is worse than it sounds for a knitted model #

The parallel case has a property the perpendicular one does not, and it makes the defect harder to design around.

In the perpendicular case the overlap is bounded by the crimp: however hard a cloth crimps, the two threads can only close the gap by the amount they move through the thickness, and that is a fraction of a diameter.

In the parallel case there is no such bound. Two curves running alongside one another can approach arbitrarily closely, because the only thing keeping them apart is the transverse offset and nothing stops that offset going to nought.

The knitted case is saved from being worse than it is only by the fact that the two courses diverge quickly once they are past the interlacing. If the loop’s shape were slightly different — flatter crests, a longer free run — the overlap could be much larger.

That is why the flattening the fabric demands moves so much with the tightness factor: a tighter fabric has flatter crests and the parallel run is longer, so the overlap grows quickly rather than slowly.

An unbounded failure mode with a mild coefficient is a worse thing to have in a model than a bounded one with a large coefficient, because it will get worse in cases nobody has computed.

What was counted, and how #

The knitted case is a point-to-point minimum over two sampled solved courses, at a hundred and twenty samples per half period over two wales.

The woven case is a grid search over both threads’ whole periods, at two hundred and sixty samples each, with the paths laid out as sinusoids of the crimp heights Peirce’s geometry gives.

The two use different path models deliberately. The knitted paths are the collection’s own solved elastica, because it has one; the woven paths are sinusoids, because a Peirce path has a curvature discontinuity at every join and a discontinuity is exactly the kind of thing that produces a spurious minimum in a distance calculation.

The comparison between them is therefore a comparison of arrangements rather than of solvers, which is what the argument needs.

A test the argument makes on this collection’s own machinery #

The angle explanation predicts something that can be checked without any new measurement, and it is worth stating because it is the difference between an explanation and a story.

Rotate one of the two knitted courses — lay the second course at a small angle to the first rather than parallel to it — and the overlap should fall, by an amount set by the cosine of the angle entering the linear coefficient.

That is not a fabric anybody makes and it is a perfectly good numerical experiment: the machinery lays out two courses from one solve and displaces the second, and displacing it with a rotation instead is a line of code.

If the overlap falls smoothly to nothing as the angle rises towards a right angle, and matches the woven value there, the angle explanation is right and the two fabrics are two points on one curve.

If it does not, something else is different between them — the crimp amplitude, the wavelength, the shape of the curve near the contact — and the explanation is incomplete.

Nobody has run it and it is twenty minutes.

Where the model stops #

The analysis is local. The linear-versus-quadratic argument is about small displacements from the designated point, and the minima found numerically are not always small displacements away. It says which case will slide and it does not predict how far.

Only two curves at a time. A real fabric has many, and a thread is near several neighbours at once. The nearest of several is not the nearest of two.

And the angle is the angle between the curves’ tangents at the designated point, which for a knitted fabric is not exactly zero: the two courses’ tangents at an interlacing differ slightly because one is arriving at a crest and the other at a trough. The linear coefficient is therefore small rather than nought, and the analysis treats it as nought.

Why the two fabrics were built the same way and came out differently #

There is a piece of intellectual history in this that is worth drawing out, because it explains why the defect was not obvious.

Both geometries were built by the same reasoning. Decide where the threads touch; place them there; let the paths between be whatever the mechanics says. That reasoning is correct and it is what every fabric geometry in the subject does.

What differs is only the arrangement it is applied to, and the arrangement is a fact about the craft rather than about the modelling. Weaving crosses its threads and knitting runs its courses alongside, and no amount of care about the geometry changes that.

So the knitted geometry is not worse work than the woven one. It is the same work applied to a harder case, and the case is harder for a reason nobody had articulated.

That is a more comfortable conclusion than “somebody was careless” and it is also the more useful one, because it says what to do differently: not be more careful, but check the case.

What the same argument says about a rib #

The obvious extension within knitting is to a two-bed fabric, and the argument makes a prediction there too.

A rib’s adjacent courses are still parallel — they run in the same direction, one course spacing apart — so the linear coefficient still vanishes and the minimum still slides.

What changes is the transverse gain. A rib’s wales alternate between two beds, so its courses travel much further through the thickness than a jersey’s do: the climb is a bed gap rather than a yarn diameter. That is a larger quadratic term, so a rib should overlap more than a jersey at the same tightness.

Whether it does has not been computed, and it is the same machinery with a different structure passed in. If it overlaps more, the flattening a rib demands is more severe than a jersey’s — which would be a testable difference between two fabrics anybody can section.

The generalisation #

The rung’s value is a rule that could have been applied before any of this ladder’s measurements were made, and was not because nobody had stated it.

A model that specifies a distance at a point should be checked at every other point.

That sounds like a counsel of perfection and it is a cheap one. The check is a minimum over two sampled curves, it takes a moment, and it would have caught this collection’s knitted defect on the day the geometry was written rather than four ladders later.

The reason it went unmade is that the specified distance was the hard part. Getting the interlacing right — the climb, the half diameter each way, the thickness that follows — was the work, and once it was right the rest of the curve looked like consequence rather than like something to check.

That is a general hazard with constructions: the part that was difficult gets checked and the part that was automatic does not, and defects live in the second.

What a geometry should carry to make the check possible #

The rule “check every point” is only actionable if a geometry produces something checkable, and not all of them do.

A geometry that produces a path — a function from arc length to a position — can be checked, because two paths can be compared point by point. This collection’s elastica and its knitted solve both produce paths.

A geometry that produces only dimensions — a crimp height, a thickness, a spacing — cannot be. Peirce’s original is stated as a set of relations between measurements rather than as a path, and its path has to be reconstructed before the check can be made at all.

That is a real difference and it argues for a habit: a fabric geometry should return a curve, not a table of numbers. A curve can be drawn, differenced, integrated, checked for self-intersection and compared with a neighbour’s. A table of numbers can only be compared with another table.

This collection went to paths several ladders ago for a different reason — because it wanted forces, and a force is a derivative of an energy over a path. The self-consistency check is a second dividend of the same decision, and it was not anticipated.

Who found it, and when #

The geometric observation — that a minimum distance between two curves is not generally at a designated contact point — is elementary and is not anybody’s discovery.

Its application to fabric geometry does not appear in the literature this collection has, which is unsurprising: fabric geometries are written to reproduce measured dimensions, and a self-consistency check on the thread paths is not something a dimension-fitting exercise calls for.

What is this collection’s own is asking the question of both its fabrics with one instrument, finding the two answers on opposite sides of a threshold, and identifying the angle between the systems as what decides it.

One more structure the rule places #

For completeness, the rule places a third fabric type and the placement is a little surprising.

A double cloth — two woven cloths made at once and stitched together — has, within each of its two layers, the ordinary perpendicular arrangement, and its stitching threads run at whatever angle the design puts them.

But the two layers are parallel to one another, and where they are stitched the two cloths’ threads run alongside. So a double cloth should have the parallel case at its stitching points and the perpendicular case everywhere else.

That predicts something checkable: the yarn in a double cloth should be more flattened at its stitching points than elsewhere, by more than the local pressure alone accounts for, because the arrangement there demands it.

It also says that a densely stitched double cloth is a harder geometry than a plain weave of the same yarn, which is consistent with the practical difficulty of setting one — and this collection has written about that difficulty without a mechanism for it.

Where the ladder goes next #

The knitted overlap is a fifth of a diameter and the flattening it demands is this collection’s most usable new number. What that flattening does to a woven cloth’s sett is a different question with an answer this collection can compute, because a jamming condition is a statement about how much room threads take.

What a sett is when the yarn is not round puts a predicted flattening into a condition that has always assumed one.