A group is one thread for cover and two for bending

The rung below this one established what a rib and a hopsack are — plain weave with its threads doubled — and ended on a limitation it could not resolve:

The group is not a thick thread. Two ends with nothing between them lie touching in a finished cloth, and the model says only that no weft passes between them. Whether they behave as one thread of twice the diameter depends on twist, on hairiness and on how much the finisher has done to the cloth.

Half of that is right and half of it is an evasion. Three of the four quantities a group would be compared on have exact answers, with no friction in them at all, and the reason the question felt unanswerable is that the three answers disagree with each other.

Cover adds, exactly #

Cover is a width. Two threads of a quarter millimetre lying touching occupy half a millimetre of the cloth’s width, and one thread of half a millimetre occupies half a millimetre.

The two are identical and there is nothing approximate about it. Cover at a given sett is a sum of widths and a group’s width is the sum of its members’, so a k-fold group covers exactly what a thread of kd covers.

That is the reading the eye takes and it is why a hopsack looks coarse. The cloth’s width is divided into half as many objects, each twice as wide, and the visual scale of the surface has doubled.

Yarn adds too, and that is the trap #

Mass is an area. Two threads of d carry 2 × πd²/4 of yarn; one thread of 2d carries πd², which is twice as much.

So a pair covers exactly what a double-diameter thread covers using half the yarn, and every comparison between the two is a comparison between two different amounts of material.

That is the trap the limitation was really about. Asking “does a pair behave as one thread of twice the diameter” is asking whether k threads behave as k² threads’ worth of yarn, and of course they do not — the question has an amount of yarn hidden in it.

The comparison worth making is at equal yarn, against a single thread of d√k: one thread carrying exactly what the group carries. And there the pair covers 1.414 times as much, which is where the coarseness actually comes from.

Bending does neither, and the range is the answer #

Bending rigidity is a second moment, and second moments are the reason none of this is intuitive.

A round thread’s rigidity goes as the fourth power of its diameter, so one thread of 2d is sixteen times as stiff as one of d. Two threads of d free to slide over each other contribute their own rigidity each: two. The same two fused into a single body contribute their own rigidity plus the parallel-axis term for each one’s offset from the pair’s centre line: ten.

So a pair is between an eighth and five eighths of the thick thread it looks like, and where in that range is exactly the friction question. That range is the honest answer to the limitation: not “it depends”, but it lies between 0.125 and 0.625, and the friction decides where.

A factor of five is a large uncertainty and a bounded one. The criterion this collection is built on cannot see friction either, and the same response applies: name the two limits, say which quantity chooses between them, and stop.

Which makes the doubling a trade rather than a simplification #

Put the two comparisons at equal yarn side by side and the construction stops looking like a simplification of plain weave and starts looking like a choice.

A k-fold group, against one thread carrying the same yarn:

• covers √k times as much;
• bends, free, at 1/k of the rigidity.

At k = 2 that is 1.41 times the cover and half the stiffness; at k = 4, twice the cover and a quarter the stiffness.

So grouping buys cover and spends stiffness, at fixed material. That is a real design trade with an exact exchange rate, and it is the reason a hopsack is a soft open cloth and not merely a coarse one — the softness is not a side effect of the coarseness, the two are the same purchase.

And it says what a rib does, which is the same trade in one direction only. A warp rib groups the picks and not the ends, so it buys cover along the length and spends stiffness there, and leaves the other direction where plain weave left it. A rib is an anisotropic version of the same purchase, which is why it drapes softly across the cord and stiffly along it.

The brackets ruled under those four drafts are the whole subject and they are ruled rather than drawn, because point paper cannot show a group at all: every square in a draft is the same size, so two ends with nothing between them are two columns of the drawing and one thread’s width apart in the cloth. That is the reason the arithmetic above has to be done rather than looked at, and it is why the rung below could reach the grouping and not its consequences — the grouping is in the matrix and the diameter is not.

Taking the doubling further makes the point harder to miss. At four threads to a group the drawing is thirty-two squares wide, the cloth’s unit is thirty-two intersections, and the harness is still two shafts — and every one of those three numbers is available from the matrix while the thing a reader would actually notice, that the cloth has become coarse, is not.

Which of the two limits a cloth is near, and how to tell #

The factor of five is bounded and it is not comfortable, so it is worth asking whether a weaver could place a cloth inside it without a testing machine.

The two limits differ in one observable: whether the pair bends as a unit. At the free limit the two threads slide over one another where the cloth is folded, so the outer one runs further and the pair takes a kink at the fold; at the fused limit they bend together and the fold is a smooth arc.

That is visible. Fold a hopsack sharply and look along the crease: a loose one shows the two members of each group at different heights on the fold and a milled one shows a single ridge. It is a qualitative test and it separates the two limits, which is what a bounded uncertainty needs.

And the direction of the finish is not in doubt. Everything a finisher does to a woollen — milling, pressing, resin, decating — increases the contact between neighbouring threads, so it moves the cloth towards the fused limit. Nothing routine moves it the other way. So a cloth’s rigidity from this source is a ratchet: it rises through finishing and stays risen, which is why an unfinished hopsack handles so differently from the same cloth off the perch.

The one thing that does move it back is wear. Abrasion and flexing break down the surface contact between grouped threads, so an old hopsack is softer than a new one by more than the fibre damage accounts for. That is a claim about a mechanism rather than a measurement, and it is the sort of claim this collection tries to mark as such.

The rib is the anisotropic case and the numbers say by how much #

A hopsack groups both systems and a rib groups one, and the arithmetic above gives the rib’s anisotropy exactly.

A k-fold warp rib has its picks in groups of k and its ends single. So in the weft direction the cloth’s threads are grouped and in the warp direction they are not, and the two directions have different cover and different stiffness from the same yarn.

At k = 2 and equal yarn per unit area, the grouped direction covers 1.41 times what the ungrouped one does and bends, free, at half its rigidity. So a warp rib is 41 per cent better covered across the cloth than along it and half as stiff there, which is a very large anisotropy from a construction that changes no thread and adds no material.

That is the quantitative form of something the trade states as a handling note — a repp drapes across the cord and stands up along it — and it is worth having as a number because the number is available and the note is not actionable. The unbalanced cloth is the collection’s own account of asymmetry from the sett; this is asymmetry from the grouping, at the same sett.

What the matrix could and could not have told anybody #

The rung below found that six of nine matrix measures cannot tell a 2/2 hopsack from a 2/2 twill, and that the three that can are about the notation or the loom rather than about the cloth. This rung says what the missing measures were.

They were not matrix measures at all. Cover, yarn and bending rigidity all need a diameter, and a diameter is exactly what a draft does not have. The matrix knows that two adjacent ends have identical columns and therefore that nothing passes between them; everything after that is geometry.

So the honest division is a clean one. The matrix supplies the grouping and the geometry supplies its consequences, and the three consequences are a width that adds, an area that adds, and a second moment that does neither.

That is a better place to leave it than “it depends on the finish”, and it also says what a finish could possibly do: a finish moves the bending rigidity within its factor-of-five range and cannot move the cover or the yarn at all. Milling, calendering and resin all work by increasing the coupling between adjacent threads, which is a move towards the fused limit — so a heavily finished hopsack is stiffer than a loom-state one by up to five times, with no change in its cover.

The same arithmetic explains a cabled yarn #

There is a construction that makes the identical purchase deliberately and inside the yarn rather than inside the cloth, and setting the two side by side is the cleanest test of whether the arithmetic is about grouping or about weaving.

A cabled yarn is a fold of folds: singles twisted into a fold, folds twisted into a cable. At every level the same thing happens — several threads lie together, nothing separates them, and the assembly covers as their widths add and bends as their second moments do not.

So a two-fold yarn is a thread group with twist on it, and the twist is precisely the mechanism that moves it towards the fused limit. A zero-twist pair is at the free limit and a hard-twisted fold is well towards the fused one, which is why a folded yarn is stiffer than two singles of the same total count and why the stiffness rises with the folding twist.

That is a prediction rather than a restatement, and it is checkable against something the collection already has: the folding twist costs almost nothing in strength across its whole usable range. Strength barely notices the folding twist and stiffness should notice it a great deal, because strength is about the fibres’ grip and stiffness is about whether the two members bend as one body.

The two quantities are therefore decoupled in a folded yarn for the same reason they are decoupled in a thread group, and a spinner choosing a folding twist for torque balance is choosing a stiffness without being told.

Which is the general form of the whole essay. Wherever threads lie together with nothing between them — in a fold, in a cable, in a rib’s group, in a hopsack — cover adds, yarn adds, and rigidity depends on a coupling nobody specifies. The construction is the same one at four scales.

What was counted, and how #

The three quantities are elementary and are computed rather than quoted. Cover is a sum of diameters, mass is a sum of areas, and the second moments are πd⁴/64 for a circle with the parallel-axis term added for each thread’s offset in the fused case.

Both comparisons are built, and the reason is the trap above: comparing a group against a thread of the same width compares two different amounts of yarn, and comparing it against a thread of the same yarn compares two different widths. Neither comparison is the right one on its own and the pair of them is.

The assertions are two-sided. The group must cover exactly what the thick thread covers and carry exactly a kth of its yarn; it must bend at less than the fused limit and more than the free one, both below the thick thread’s; and at equal yarn it must cover more and bend less. A version asserting only that the group is less stiff would have been satisfied by its covering less as well, which is the opposite of the finding.

And the trend is asserted over three group sizes, because a two-thread claim demonstrated at two threads is a claim about a drawing.

What a mill could do with the exchange rate #

The trade already uses the purchase and does not price it, so the last thing worth doing is to write the price down in the form a mill would use.

To cover a given width with a given weight of yarn, group the threads. Grouping k to a bundle covers √k times as much as the same yarn as single threads, so a cloth wanting cover at low weight is a hopsack rather than a plain weave — which is what a scrim, a cheesecloth backing and a light casement all are.

To stiffen a cloth without adding yarn, ungroup it. The same yarn as single threads is k times as stiff, free, as it is in groups of k. So a cloth wanting body at low weight is set with single threads and a firm weave rather than with grouped ones.

And to change one without the other, finish it. Cover is fixed by the geometry and cannot be moved by anything short of shrinking the cloth; rigidity moves through the factor of five between the two limits. So a finisher has one lever, it works in one direction, and it works on the quantity the designer cannot control at the drawing board.

That is a clean division of labour and it is not how the trade describes itself. A designer chooses cover and a finisher chooses handle, and the arithmetic says those are exactly the two quantities the two of them can respectively move.

The one place it goes wrong is a cloth designed for cover and finished for handle: milling a hopsack towards the fused limit stiffens it fivefold and the designer chose the construction for softness. The two decisions are made in different rooms about the same number, which is the sort of thing worth a sentence in a specification and rarely gets one.

Where the model stops #

The threads are circles and they are not. A yarn in a cloth flattens where it is gripped, and a pair of yarns lying together flattens against each other as well — which raises the fused limit and lowers the free one, both in the direction of making the range wider rather than narrower. Peirce’s circular thread against the flattened one is the site’s own version of that correction and it is not applied here.

The free and fused limits are limits and not a model. A real pair is coupled by friction at a contact that runs the whole length of the group, and how much shear that contact carries before slipping is a capstan question with a normal force this collection does not have. What can be said is which limit a cloth is nearer: a loose, unfinished, low-twist cloth is nearer the free limit and a milled one is nearer the fused.

Bending rigidity is a thread’s, and a cloth’s is not the sum of its threads’. The cloth’s own bending length involves the interlacings, the crimp and the friction at every crossing, and this essay computes only what the group contributes at one place. The bending bracket is the collection’s own account of how far that gets.

And nothing here is about twist. A group of two two-fold yarns and a group of two singles have different internal coupling and the same geometry, so the twist enters the friction question and not the three exact answers.

Who found it, and when #

That a group of threads is not a thick thread is known to anybody who has handled a hopsack, and the reason usually given is friction. That much is trade knowledge and is correct as far as it goes.

The fourth-power scaling of bending rigidity is elementary mechanics and is a century and a half old. Applying it to a thread group appears not to be done, and the reason is probably that the two communities that would do it — textile mechanics and weave design — divide at exactly this line: bending is studied on whole fabrics and grouping is studied on drafts.

What this collection adds is the pair of comparisons and the observation that they answer differently. The question “is a group one thread or two” has three answers and they are one, two and somewhere between two and ten — and until the comparison is fixed at equal yarn, none of them means anything.

Where the ladder goes next #

The group has been taken apart as a bundle of threads. It is also a shape: a pair of picks lying together stands proud of the cloth around it, and the cord a reader sees running across a warp rib has a height — which is not the yarn’s diameter, is not the weave’s, and comes from a place neither the matrix nor this rung has looked.