What nothing separates comes out together

Four rungs of this ladder have been about what a thread group is: a partition, a bundle with its own cover and stiffness, a cord with a ceiling on its height, and a corner of a two-parameter family. None of them has asked what a group does in use.

It does exactly what the definition says. A group is a set of threads nothing separates, so at a cut edge nothing separates them there either — and the cloth’s hold on any one of them is weaker for the same reason the group exists.

The firmness has a closed form and it is a harmonic mean #

The two-parameter family makes the arithmetic available in a way the named weaves do not.

An end of an a×b matt carries its marks in one run of b picks out of 2b, so it changes face twice per repeat whatever b is; a pick does the same with a. Counting both directions and dividing by the intersections:

interlacings per intersection = (a + b) / 2ab = ½(1/a + 1/b).

That is half the sum of the two reciprocals — a harmonic form — and the two groupings therefore enter symmetrically and reciprocally. Checking the corners: plain gives 1, a 2/2 hopsack 0.5, a 1×2 warp rib 0.75, all of which the rung this ladder starts from reports from the count.

The reciprocal form is the interesting half. Grouping a system heavily buys very little more than grouping it moderately: as b grows without bound the firmness falls only to 1/2a and stops there.

Which makes the seam grip saturate #

A seam holds a cloth’s own threads by friction at their crossings, and the capstan equation says the grip accumulates multiplicatively — so it is exponential in the number of crossings, and the crossings are the firmness times the intersections in the allowance.

So the family’s whole seam behaviour is the exponential of a harmonic mean, and it saturates in each grouping separately:

The first doubling of the picks costs a factor of nine in grip; the second costs three; the third costs less than two. Each further doubling loosens the seam less than the one before, and the whole first column converges towards a floor.

Set against that, doubling the other system takes a fresh bite: 1×4 grips at 219 and 2×4 at 25, which is a factor of nine again. The saturation is per knob and not overall.

And it says which members slip #

The number that matters is not the grip but whether it exceeds the thread’s own strength. Below that the thread slides out and the seam opens; above it the thread breaks first and the seam fails some other way.

At a ten-millimetre allowance, a quarter-millimetre yarn at 24 threads per centimetre and a friction coefficient of 0.3:

A 2/2 hopsack is below the line and a 1/4 warp rib is above it.

Those two cloths differ in firmness by an eighth — 0.500 against 0.625 — and they differ in behaviour by a category. That is what an exponential does to a small difference, and it is the reason a seam slips before it breaks is a fault of some cloths and not of others.

Which means the allowance is the lever and it is a blunt one #

The trade’s response to seam slippage is a wider allowance, and the arithmetic prices it.

Grip is exponential in the allowance, so a wider seam buys grip fast — and it stops buying anything the moment the grip reaches the thread’s own strength, because past that point the thread breaks instead of sliding and further cloth is holding nothing.

Plain weave reaches that point at 5.3 mm of a ten-millimetre allowance, so 4.7 mm of the seam is doing nothing. A 1×2 warp rib reaches it at 7.1 mm and a 1×8 at 9.5 mm; a hopsack does not reach it at ten millimetres at all.

So the family divides into three regimes rather than two:

• plain and the light ribs, where the allowance is more than enough and the extra is waste;
• the heavier ribs, where the allowance is about right;
• the hopsacks, where a ten-millimetre allowance does not reach the break point and the seam will slip.

That is a specification a garment engineer could use, and it says the useful thing: the allowance a cloth needs is set by its firmness and the excess is measurable.

The other half, which the grip cannot see #

Everything above is about one thread, and it treats a hopsack’s end exactly as it treats a twill’s — which is right, because a group’s members are held by their own crossings and not by each other.

What differs is what happens when one goes.

In a twill the ends are on four distinct columns, so a slipped end leaves a gap one diameter wide with two threads either side that are not going anywhere. In a hopsack an end’s group-mate has the identical column and nothing between them, so it slides sideways into the gap — and the gap reappears at the group boundary, where a weft does pass, at the group’s own pitch.

So a grouped weave frays at the group pitch and in group-sized steps, and a plain or twill weave frays thread by thread. The rate is the same and the appearance is not, which is why a hopsack’s cut edge looks so much worse than its firmness predicts.

That is a mechanism and not a measurement: the criterion this collection is built on is blind to friction, and so is everything downstream of it, including the lateral contact that lets a group-mate slide. The claim is stated as a mechanism and marked as unmeasured, which is where this collection leaves things it cannot count.

The mock leno is the same computation with the sign reversed #

There is a construction on this site that uses the identical machinery to the opposite end, and setting the two beside each other is what makes the grouping’s role clear rather than incidental.

A mock leno groups its threads so that nothing passes between them and then relies on that to open a hole: the group closes up, the space it vacates opens, and the cloth acquires an ordered array of gaps. The grouping is computed exactly as it is here — runs of identical columns — and the rib family and the mock leno share the code that finds them.

The difference is entirely in what the cloth is for. A rib uses the grouping to make a cord and a mock leno uses it to make a hole, and both are the same fact about the matrix: adjacent threads with identical columns have nothing holding them apart.

That symmetry has a consequence for this rung. A mock leno’s threads are held by fewer crossings for exactly the reason a hopsack’s are, so a mock leno slips at least as badly, and its holes are held open by the same absence that lets its threads slide.

Which is why mock leno is a decorative construction rather than a structural one, and why every account of it warns about seam performance without giving a number. The number is the one above, at a firmness the grouping decides.

What a finish does to all of it #

The grip is exponential in μ, and μ is the one quantity here that a finisher moves.

Everything that roughens or swells the yarn raises it. Milling, raising, resin finishing and heat setting all increase the friction at a crossing, and the grip goes up as the exponential of that increase — so a finish that raises μ from 0.2 to 0.3 multiplies the hopsack’s grip by a factor of three and takes it across the line.

And everything that smooths or lubricates lowers it. Mercerising, calendering, softening and silicone finishing all reduce it, in the direction that makes a marginal cloth slip.

So a cloth’s seam performance is decided in the finishing room by a quantity nobody in that room is measuring, and the effect is exponential rather than proportional. A softener applied for handle can move a cloth from the safe side of the line to the wrong one, on a construction whose firmness has not changed at all.

That is the practical warning the family’s arithmetic supports, and it is worth stating with its own limitation: the sizes here are at stated values of μ and the direction is what survives. What can be said with confidence is that the sensitivity is exponential, which means a finish that changes the handle noticeably has changed the grip by a large factor.

What was counted, and how #

The closed form is asserted against the count at every member built. The expression (a + b)/2ab is derived from the runs and then checked against interlacings, which counts face changes in the matrix — because a formula that agrees with an enumeration is a formula and one that does not is a bug.

The grip is applied.js’s seamGrip, unchanged: the capstan equation on Peirce’s own weave angle, with the crossings taken from the interlacing rate. Nothing about the seam model is re-derived here; what is supplied is the family.

The saturation is asserted as a difference of differences, not as a limit: the fall in firmness from 1 to 2 must exceed the fall from 2 to 4, which must exceed the fall from 4 to 8. That is what “each doubling buys less” means and it is checkable at finite b, where the limit is not.

And the floor is asserted too — the firmness cannot fall below half the reciprocal of the other grouping, which is what makes the saturation a bound rather than a trend.

How this fits the ladder’s own opening finding #

The rung this ladder starts from ended on something uncomfortable: six of the nine measures this site takes off a matrix cannot tell a 2/2 hopsack from a 2/2 twill, and the three that can are about the notation or the loom rather than about the cloth.

This rung says what the missing cloth measure was and where it came from.

It is the firmness, which is one of the six blind ones — and it is blind for a good reason: a hopsack and a 2/2 twill really do interlace at the same rate, 0.500, and really do hold a thread with the same force. The grip is not the thing that separates them.

What separates them is a quantity the matrix does supply and nobody was taking off it: the thread grouping, which the rung below called “the one that is actually about the cloth’s behaviour, and the one the site added last and for another purpose entirely.”

So the family’s arithmetic divides cleanly. The firmness decides whether a thread comes out and the grouping decides what it takes with it, and the two are independent — which is why a hopsack and a twill of identical firmness fray so differently and why no single number was ever going to separate them.

That is a better resolution than “the measures are inadequate”. The measures are exactly adequate; they were being asked one question and the cloth answers two.

What the allowance would have to be #

The grip is exponential in the allowance, so the width a seam needs to reach the break point is a logarithm — and that makes it a short and useful expression.

The grip is exp(μ · 2θ · crossings) and the crossings are the interlacing rate times the intersections in the allowance, so the allowance at which the grip reaches the thread’s own strength is

allowance = ln(strength ratio) ÷ (μ · 2θ · firmness ÷ spacing).

Everything in that is either the cloth’s or the yarn’s, and the only one of the family’s parameters in it is the firmness — which is ½(1/a + 1/b).

So the allowance a member of the family needs is inversely proportional to its firmness, which is to say directly proportional to the harmonic mean of its two groupings. At a plain weave that is 5.3 mm and at a 2/2 hopsack it is over ten; the family’s whole spread in allowance is the reciprocal of its spread in firmness, exactly.

That is a rule a garment engineer could put on a specification and it is a simple one: double the doubling, double the allowance. It is not what the trade says — the trade says wider allowances for loose weaves and leaves the amount to judgement — and it says a number rather than a direction.

The limitation is the one that runs through the whole rung: μ is not measured for a particular cloth, so the constant in that expression is unknown and the scaling is exact. A mill that measures one member of the family has the constant and can then compute every other member without measuring again, which is the useful shape for a result of this kind to have.

What it means for a cloth that stripes two members #

A stripe is a partition of the ends, and the family’s flat harness cost makes striping two members free — which raises a question this rung can answer and the harness cannot.

A seam crossing a striped cloth crosses two firmnesses. A plain band grips at 5,537 and a hopsack band beside it at 74, on the same seam, at the same allowance, held by the same stitches.

So the seam does not have a grip; it has a grip per band, and it fails where the weakest band is. The cloth’s seam performance is the worst band’s and not the average, which is a weakest-link statement of exactly the kind a bundle makes about threads.

That is worth having because a striped cloth’s specification is an average — a mean firmness, a mean cover, a mean weight — and its seam is not. A designer striping a firm ground with an open cord has made a cloth whose seams behave like the cord, and nothing in the cloth’s own numbers says so.

Where the model stops #

μ is measured and is not a constant of cloth. Every number above is at 0.3, which is a plausible cotton-on-cotton value and is not a property anybody can look up for a particular cloth. The ordering survives every value reported and the sizes do not, which is the same division the seam ladder makes throughout.

The thread’s own strength enters as a ratio. “A hundred times the applied tension” is a stand-in for a breaking load and is what makes the slide-or-break question decidable at all; a real comparison needs the thread’s tenacity and the tension the seam sees, neither of which is here.

The group’s lateral contact is not modelled. The fraying mechanism in the section above needs the friction between two threads lying side by side with no wrap between them, which the capstan equation cannot supply — it is a contact and not a wrap. That is the same boundary the pile constructions run into and it is where this collection stops.

And the seam’s stitches are not in it. Everything above is about the cloth’s hold on its own threads; a seam also has a sewing thread, a stitch density and a stitch type, and the stitch itself weakens the cloth it passes through.

Who found it, and when #

Seam slippage is a well-studied failure with its own test methods, and the rule of thumb — loose weaves and long floats slip, plain weaves do not — is universal and correct.

The capstan framing is this collection’s and belongs to the seam ladder rather than to this rung. What is added here is the family: the doubled weaves’ firmness has a closed form, the form is a harmonic mean, and a harmonic mean saturates.

That last step is what makes it a result rather than a table. A designer told that grouping loosens a seam will assume the loosening is proportional to the grouping and will worry about a 4/4 hopsack far more than a 2/2 — and the arithmetic says the 2/2 has already spent most of the loss, and that grouping the other system is what costs the next factor of nine.

Where the ladder goes next #

Five rungs have taken the doubled family from a matrix to a seam, and every one has treated the group as a set of threads that nothing separates. The construction that does the opposite — the same grouping used to open a hole rather than to close one — is the mock leno, and it is the same computation with the sign reversed.