A seersucker is made at the loom

A honeycomb comes off the loom nearly flat and gets its cells in the wash. This collection made that claim in the texture-weaves ladder and gave the reason: every end of a honeycomb passes through the long floats and the tight interlacings alike over one repeat, so every end takes up almost exactly the same length. That is why a honeycomb weaves from one beam, and it is why its relief is a finishing effect rather than a weaving one.

A seersucker is the other case entirely, and the two together are the whole of how relief gets into a cloth.

Two beams and a surplus #

A seersucker has two warps. They are threaded in stripes across the cloth, they come from two beams, and the beams are let off at different rates: the slack warp is fed thirty to fifty per cent faster than the tight one into the same length of cloth.

The extra thread has nowhere to go. It cannot lie in the plane — the cloth’s length is fixed by the tight warp and by the picks — so it leaves the plane, and the slack stripes rise into puckers between the tight stripes that hold the cloth down.

Nothing has to relax for this to happen. The surplus is in the cloth as it is woven, and it is visible on the loom. Washing deepens the pucker, because the finishing shrinkage adds to the surplus, but it does not create it.

That is the difference from a honeycomb, and it is a difference in where in the process the surplus is put rather than in the mechanism by which a surplus becomes a shape.

The arithmetic is the same as the honeycomb’s #

A region of cloth with more material than the plane can hold buckles, and the standard small-amplitude result for a sinusoid pinned at both ends of a half-wavelength L with an excess length fraction ε is

$$A = \frac{2L}{\pi}\sqrt{\varepsilon}$$

That is the same relation this collection uses for a honeycomb’s cells, and it is used here unchanged. Only the surplus differs, and the half-wavelength — a seersucker’s stripe is millimetres and a honeycomb’s cell is a fraction of one.

Holding the half-wavelength fixed so that the comparison is of surpluses and of nothing else:

Fourteen and a half times the surplus and 3.8 times the depth, and the discrepancy between those two factors is the square root doing its work.

And it saturates #

The square root is not a detail. It is why a seersucker looks the way it does and why the trade’s feed ratios sit where they do.

Doubling the surplus multiplies the amplitude by √2, which is 41 per cent. So going from a feed ratio of 1.3 to 1.6 — twice the surplus, and a considerably harder cloth to weave, because a very slack warp is difficult to control at the reed — buys 41 per cent more depth and no more.

At a 6 millimetre stripe the amplitudes run 1.21 millimetres at a ratio of 1.1, 2.09 at 1.3, 2.70 at 1.5, and 3.82 at 2.0. The curve is steep where the surplus is small and flat where it is large.

A cloth-maker’s whole usable range is on the flat part, and the useful lever is therefore not the feed ratio at all. It is the stripe width, which enters linearly: doubling the stripe doubles the amplitude exactly, with no square root in the way.

That is a real design consequence. A deeper seersucker is made by widening the stripes rather than by slackening the beam, and a narrow-striped seersucker is necessarily a shallow one however the beams are run.

What the pucker is worth against the cloth’s own thickness #

An amplitude in millimetres does not say whether a relief is a relief. This collection’s standing measure for that is the ratio to the cloth’s own thickness, because a cell a tenth of a thickness deep is not a honeycomb.

At a feed ratio of 1.3 and a 6 millimetre stripe the amplitude is 4.2 times the cloth’s thickness. At 1.1 it is 2.4 times and at 2.0 it is 7.6.

By comparison a honeycomb’s own peak relief, at its own cell width rather than at a seersucker’s stripe, comes out at a fraction of a thickness — which is why a honeycomb is a texture under the hand and a seersucker is a stripe visible across a room.

Both are reliefs and they are not in the same class of object. Calling them both “relief weaves” is a classification by outcome that hides an order of magnitude.

The steepness, and where the model gives out #

The buckling formula is a small-amplitude result and it is worth checking how far into it these answers stand.

The steepness — the amplitude over the half-wavelength — runs from 0.20 at a feed ratio of 1.1 to 0.64 at 2.0. A sinusoid of steepness 0.2 is a gentle wave and the small-amplitude approximation is fine; one of 0.64 is a fold, and the approximation is being asked to do work it was not derived for.

So the numbers at the low end of the range are trustworthy and the ones at the top are indicative. What survives everywhere is the square root, because that comes from the geometry of arc length against chord and not from the small-amplitude expansion — the relation between excess length and amplitude is a square root at any amplitude, with the constant changing.

That distinction is asserted rather than described: two feed ratios whose surpluses are in the ratio four must give amplitudes in the ratio two, to machine precision, and a later change that made the amplitude depend on the sett or the yarn would break it and should.

The other seersucker, which is chemistry #

There is a second way to make the same cloth and it is worth knowing about, because it produces something that looks identical and behaves quite differently.

A chemical crêpon is made by printing caustic soda onto a plain cotton cloth in stripes. The treated stripes mercerise: the fibres swell permanently, the yarn’s packing falls, and the treated cloth contracts. The untreated stripes do not, so they have surplus material relative to their neighbours and they pucker.

The mechanism is the same buckling and the surplus arrives from the shrinkage difference between treated and untreated cloth, which for a full caustic treatment is several per cent — between a honeycomb’s two per cent and a seersucker’s thirty.

But the two cloths are not equivalent. A woven seersucker’s surplus is thread, permanently: the slack stripe genuinely has more yarn per unit length of cloth, and no amount of washing, pressing or stretching removes it. A chemical crêpon’s surplus is a shrinkage difference, and pressing a crêpon flat under heat and moisture flattens it — sometimes for good.

That is the practical rule of thumb the trade states as “a real seersucker never needs ironing”, and it is a statement about where the surplus is stored rather than about the fabric’s finish.

Why a honeycomb cannot be made this way #

It is fair to ask why, if the loom route is fifteen times better, anybody uses the finishing route.

Because a honeycomb’s relief is not a stripe. Its regions are a lattice of cells, each a few millimetres across, alternating in both directions. Making that at the loom would need not two beams but one beam per column of cells, all let off at different rates, and the rates would have to alternate along the cloth as well — which is a machine nobody builds.

The finishing route needs none of that. It needs one beam and a draft whose float lengths vary, and the cloth sorts itself out when it contracts. That is an enormous simplification and it is bought at the cost of a surplus limited to the difference between two regions’ crimps, which is a few per cent because both regions are made of the same threads at the same sett.

A terry is the third case and it splits the difference: two beams, as a seersucker has, but with the slack warp’s surplus taken up as loops standing off the surface rather than as a buckle of the whole cloth. Its surplus is far larger again — a terry’s pile warp is fed several times the ground warp’s length — and the loops are not buckling, they are simply long.

So the three constructions are three answers to one question about where a surplus goes, and this collection now has all three with the same arithmetic behind them.

What the stripe partition costs the loom #

A seersucker is a striped warp, and this collection has an arithmetic for what a striped warp costs — because two weaves in one cloth need shafts for both, and the cost is not always the sum.

Here it is cheaper than that. The two stripes of a seersucker are usually the same weave, most often plain, and differ only in which beam they come from. So the harness cost is a plain weave’s and the stripe is free in shafts.

What it costs instead is at the back of the loom: a second beam, a second let-off, and a threading plan that alternates between them. That is a machine cost rather than a harness cost, and it is why a seersucker is a straightforward cloth for a mill and an awkward one for a hand weaver — the reverse of most stripe effects, where the harness is the constraint.

There is also a third cost that is neither. The two warps have different tensions at the reed, so the reed’s beat lands differently on the two stripes, and the pick spacing in a slack stripe is not quite the pick spacing in a tight one. The mechanism is the beat-up’s own and this collection has not put numbers to the difference.

What was counted, and how #

The pucker’s depth is the square root of its surplus, asserted as an identity across three pairs of feed ratios to 1e-12, because it is a closed form and the seersucker adds nothing to it but a surplus.

A loom-made relief has more surplus than a finishing-made one, asserted as an ordering rather than a size, with the finishing surplus read off a solved honeycomb rather than assumed. What would break it is a change that made a honeycomb’s regions differ by tens of per cent, which would mean the finishing model had come loose.

And the amplitude ratio exceeds two, which is a floor set well below the computed 3.8 because what is being asserted is that the routes are in different classes rather than that they are in a particular ratio.

The honeycomb’s own surplus is the largest a region has to lose, read off the cells the relief solver returns, and the two routes are put through the same buckling function at the same half-wavelength so that nothing but the surplus differs.

Where the model stops #

The buckle’s shape is assumed. A sinusoid pinned at the stripe edges is a reasonable picture and it is not solved for; a real pucker’s profile depends on the cloth’s bending stiffness and on how firmly the tight stripe holds it, and neither is in the arithmetic.

The half-wavelength is taken as the stripe width. That is right if the pucker rises once between its edges and wrong if it buckles into two or more waves, which a wide stripe in a limp cloth would. Where that transition happens is a bending problem this collection has the pieces for and has not assembled.

There is no weaving in it. A slack warp at a high feed ratio is genuinely difficult: it wanders at the reed, it is prone to broken ends, and it needs a heavier reed and a slower loom. Nothing here prices any of that, and it is very likely the real limit on feed ratios rather than the square root.

And the two warps are treated as identical apart from their feed. A real seersucker often uses a different count or a different twist in the slack stripe as well, which changes its stiffness and therefore its buckling, and this collection’s arithmetic has no yarn property in it at all.

Where the stripe-width lever runs out #

The design conclusion above is that the useful lever is the stripe width, because it enters linearly while the feed ratio enters under a square root. That is right, and it has a limit, and the limit is the thing the model stops short of: the point at which a stripe buckles into two waves rather than one.

The pieces are available. A stripe of width L holding a surplus ε and buckling into m half-waves has amplitude 2L√ε ÷ mπ — falling as one over m — and a bending energy that works out proportional to m². So bending alone always prefers a single wave, and something has to push the other way.

What pushes is the pucker’s own weight. A taller pucker holds more cloth further from the plane, and that cost is proportional to the amplitude, which falls as 1/m. Adding the two and minimising gives m³ proportional to the weight term over the bending term, and after the substitutions — using B = W·c³, which is the definition of the bending length —

m ≈ L ÷ (c · ε^(1/6)), to a constant of order one.

The number of waves across a stripe is its width divided by its cloth’s bending length, with the surplus entering only through a sixth root, which over any range that matters is very nearly not at all.

Put ordinary numbers in. A cotton shirting has a bending length of about seventeen millimetres; a surplus of thirty per cent gives ε^(1/6) = 0.82, so the single-wave regime lasts to a stripe of roughly fourteen millimetres. Real seersucker stripes are three to ten, comfortably inside it, which is why the essay’s linear lever works over the whole range the trade uses.

And why widening past it makes things worse #

The consequence is not that the lever flattens. It is that it reverses, once, and then resumes.

Widening a stripe raises the amplitude in proportion until the second wave appears, at which point the amplitude halves. So a seersucker whose stripe is pushed past the threshold is shallower than one at the threshold, and it stays shallower until the width has doubled again.

That gives a design rule with a number in it, which the square-root argument on its own cannot produce:

the deepest seersucker of a given cloth has a stripe about as wide as that cloth’s bending length, and both slackening the beam further and widening the stripe further are wasted.

It also predicts what a too-wide stripe looks like, and the prediction is testable by eye. A stripe past the threshold does not read as one bold pucker; it reads as a shallow, slightly irregular double swell, because m = 1 and m = 2 are close in energy near the transition and the cloth will take either depending on how it was handled. Irregularity is the signature of being near the threshold, not of poor weaving — which is a useful thing to be able to say about a fault that is otherwise blamed on the beam.

And it connects the two halves of the design problem, which have until now been separate. The feed ratio is limited by weaving, the stripe width by bending, and the bending limit involves the finished cloth’s stiffness — so a cloth that is going to be softened in finishing has a lower threshold than the one that came off the loom, and a seersucker that looked right greige can lose its definition in a softener. That is a real and reported behaviour, and it now has an arithmetic reason rather than an attribution to the finish.

The generalisation #

When two processes produce the same outcome by the same mechanism, the interesting comparison is of their inputs and not of their outputs.

Both routes here are a surplus becoming a buckle, and both go through the same function. Comparing the finished cloths would have given a factor of four and an impression that a seersucker is a somewhat deeper honeycomb. Comparing the surpluses gives a factor of fifteen and the reason: one process has access to a difference between two beams and the other only to a difference between two crimps.

The square root is what makes those two comparisons different, and it is also the practical moral. A mechanism with a square root in it converts a large advantage in the input into a modest one in the output, which cuts both ways — it makes the finishing route more competitive than its surplus suggests, and it makes further slackening of the loom route almost worthless.

Who found it, and when #

Seersucker is very old — the word is Persian by way of Hindi and the cloth predates any of the mechanics here — and the two-beam construction is standard in every weaving manual. That the pucker is a differential take-up is not in dispute and is how the cloth is specified.

The buckling relation is the standard small-amplitude result for an inextensible strip with excess length, which is elementary and is used in this collection for the honeycomb as well.

What is this collection’s is putting the two routes through the same arithmetic at the same half-wavelength, the resulting factor of fifteen in surplus against four in depth, and the observation that a designer’s usable lever is the stripe width rather than the beam ratio because one enters linearly and the other under a square root.

Where the ladder goes next #

Into the pattern field, where a shading changes two things at once: the tone of a cloth in exactly even steps, and its lustre in steps that are not even at all.