A woven outline is a staircase

A jacquard machine is sold by its hooks. Six hundred, twelve hundred, twenty-four hundred: one hook per end of the repeat, each independently controlled, and the number is quoted the way a screen’s pixel count is quoted, as the fineness of what the machine can draw.

Take a 1,200-hook machine weaving 140 cm wide. That is a hook every 1.17 millimetres, which sounds coarse until it is compared with what the cloth can actually hold — and then it turns out to be far finer than anything the fabric will accept, because a figure’s smallest feature is not one end. It is one block, a block is at least one repeat of the ground weave, and on an eight-end satin at 24 ends and 22 picks to the centimetre that is 3.33 mm across and 3.64 mm up.

The machine resolves 2.86 times finer than the cloth can use. Buying a finer machine changes nothing.

Why the block is the floor #

The previous rung of this ladder established the condition and this one uses it. A figure whose blocks are at least a whole repeat of both weaves cannot separate the cloth; below that, it can, and the commonest failure leaves a thread lying on the face for the length of the repeat.

That makes the repeat a floor on the block, and the floor is hard rather than conventional. A designer who steps an outline by half a repeat is not being bold about detail. They are drawing a design that a census of 65,536 profiles says has a one-in-sixty chance of describing more than one cloth, and a much better chance than that of carrying a float long enough to snag — which is exactly what a jacquard designer’s rule about nothing longer than four is written to prevent.

So the smallest feature a figured cloth can have is a block, the block is at least a repeat, and the repeat in millimetres is the repeat in threads divided by the sett. Every term is a decision somebody has already made for other reasons — the sett to get the cover and the handle wanted, the weave to get the face wanted — and the resolution falls out of them.

The directions an outline can take, counted #

A straight edge on a block grid is a repeated step: j blocks across for every k blocks up. Two steps whose fractions reduce to the same thing are the same direction, and a design N blocks each way can only hold steps with j and k at most N. So the distinct directions available are the reduced fractions with both parts bounded by N — a Farey structure, and countable exactly.

At six blocks there are 23 directions. At eight, 43. At twelve, 91. The count is twice the sum of Euler’s totient over the whole numbers up to N, less one, which is computed here directly from the definition and then checked against the closed form, because a count that agrees with its own closed form is a count and one that does not is a bug.

The gaps are where the interesting part is. The directions are dense near the warp and near the weft, and sparse in between — and the sparse region is exactly where the steps are longest, because a step of one block across in one block up is a single large tread while a step of one in twelve is nearly a straight line. So the outline is roughest, and the choice of angle coarsest, in the same place.

That is why a damask’s diagonals read as staircases and its near-vertical edges do not. It is not that a designer took less trouble over the diagonals — and it is the same asymmetry a fashioned knitted edge has, for the same reason.

The step in millimetres #

The resolution claim is worth doing carefully, because it involves two quantities in different units and the comparison is the whole point.

A jacquard’s hook pitch is the fabric width divided by the hook count: 140 cm over 1,200 hooks is 1.17 mm. That is the finest thing the machine can distinguish across the width.

The block’s pitch is the ground weave’s repeat divided by the sett. Eight ends at 24 per centimetre is 3.33 mm; eight picks at 22 per centimetre is 3.64 mm. That is the finest thing the cloth can hold.

Their ratio is 2.86, and the sign of the comparison is what matters: the cloth is the coarser of the two, so the machine’s resolution is not the binding constraint and improving it improves nothing. On a plain-weave ground the numbers invert — a repeat of two at 24 ends per centimetre is 0.83 mm, finer than the hook pitch, and there the machine does bind. Which is to say the answer is not a property of jacquard weaving; it is a property of a particular cloth, and both cases are ordinary.

What was counted, and how #

The direction count is an enumeration over pairs, not a formula evaluated. For each j and k up to N the fraction is reduced, duplicates are discarded by their reduced form, and what is left is sorted by angle — the angle being computed on the cloth, with the across-step divided by the warp sett and the up-step by the weft sett, so that the ratio of the two setts enters where it belongs. The largest gap is then read off the sorted list.

The closed form is computed separately, as twice the sum of the totient function over 1 … N, less one, and the two are required to be equal. They are, at every N the essays place.

The step in millimetres comes from the weave’s own minimal repeat rather than from the rectangle it happens to be written on, which matters: a plain weave written on a four-by-four grid would otherwise claim a step twice its true one. The minimal repeat is computed and then checked by tiling it back out and comparing against the original writing square by square, because a factorisation is worth what its check is worth.

And the hook pitch is arithmetic on two numbers a machine is sold by, stated rather than derived.

The two resolutions a designer works with #

It is worth separating two things a designer thinks of as one, because the essay’s whole comparison depends on them being different.

The first is the grid the design is drawn on. A jacquard design is prepared on squared paper or its screen equivalent, one square per end and per pick, and at 1,200 hooks that grid is as fine as the machine. A designer draws a curve on it and it looks like a curve.

The second is the grid the cloth can hold, which is the block. Everything between two block boundaries is one weave; there is no such thing as half a block of satin. So the design grid is a fiction at the scale a designer draws on — a perfectly useful fiction, because the design has to be recorded at thread resolution to be woven at all, but a fiction about what the fabric can express.

The step from one to the other is where a figure’s outline is decided, and it is usually taken by software or by a technician rather than by the person who drew the curve. That is the practical reason the resolution claim is worth making: a designer working on the fine grid has no way to see the coarse one, and the person who applies it is not the person who chose the shape.

A hook count is a block count, once the ground is chosen #

The right way to quote a jacquard to a designer follows directly, and it is not the number on the machine.

A 1,200-hook machine grounded on an eight-end satin has 1,200 ÷ 8 = 150 blocks across its repeat. The same machine grounded on a plain weave has 600. That is the number a design is actually drawn in, it is the N the direction count is a function of, and it changes by a factor of four with a decision made about the face.

At 150 blocks the Farey count runs to something over thirteen thousand distinct directions — the count grows as the square of N, at three over π² times it, so it passes ten thousand well before a design has used a quarter of the machine. Direction is not the scarce resource at that size. What is scarce is the step: the finest available departure from vertical is one block in 150, and one block is 3.33 mm, so the shallowest edge a full-repeat design can draw still has treads a third of a centimetre long. The angles are dense and the treads are not, and a reader sees treads.

That is the honest form of the resolution claim. Fineness of angle improves as the square of the block count and is comfortably sufficient at any real repeat; fineness of feature does not improve at all, because it is one block whatever the design’s size.

Why the block cannot be bought down #

A designer wanting a finer step has exactly two levers and both of them are somebody else’s decision.

Shorten the repeat. A plain-weave ground steps four times more finely than an eight-end satin at the same sett, and it is a completely different cloth: four times the interlacings, no float to speak of, no lustre, and a firmness that is the opposite of what a damask is for. The resolution is a consequence of choosing a face and cannot be separated from it.

Set the cloth closer. The step is the repeat divided by the sett, so raising the sett shortens it in proportion — and this is where the lever runs out. Bringing an eight-end satin’s 3.33 mm step down to a plain weave’s 0.83 mm needs 96 ends per centimetre against 24, and a cloth of that yarn jams long before it gets there. The satin’s resolution cannot be reached by setting closer, at any sett the threads admit.

So the two grounds are not two points on a continuum that a designer moves along by tightening the cloth. They are separated by a factor the sett cannot make up, and the only way across is to change the weave — which is to say, to make a different fabric.

There is a third lever that is not a lever, and it is worth naming because it is the one usually reached for. Weaving the same design larger does not make it finer. Doubling every block count doubles the design’s size in centimetres and leaves every tread exactly where it was, so the staircase is unchanged and the figure is simply bigger — which improves the appearance for the same reason that standing further back does, and improves nothing about the cloth. The step is an absolute length and the only quantities in it are the repeat and the sett.

Where the model stops #

Resolution here is a statement about structure, not about appearance. A staircase whose treads are 3.3 mm is not automatically visible; whether a reader sees steps depends on contrast between figure and ground, on viewing distance, on the yarn’s hairiness and on the finish. None of that is here. What is here is that the steps exist at that size and cannot be made smaller without changing the weave.

The block is a floor, not the block. A designer may perfectly well use blocks of two or four repeats, and many do, for reasons of appearance and of shaft economy. The claim is only that they may not go below one.

The Farey count is a count of straight edges. A real outline is a curve, and a curve’s local direction changes along it, so the relevant question for a curve is not which directions exist but how finely the tangent can be tracked — which is a different quantity and is not computed here.

And nothing here says what a figure boundary does physically. Two weaves with different interlacing rates take up thread differently, so a boundary is a place where the cloth is not flat, and a stepped boundary is a stepped ridge. That needs two reed pitches and a model of take-up on the loom, which this site does not have — the same gap a striped cloth’s join has.

The generalisation #

The shape of this is a rule about which layer of a system sets its resolution, and it is worth stating without the cloth in it.

A pipeline in which a fine mechanism feeds a coarse medium has the medium’s resolution and not the mechanism’s, and buying a finer mechanism is buying nothing. That much is obvious when it is said. What is not obvious is that the coarse layer here is the one nobody thinks of as a resolution at all: the weave is chosen for its face, its firmness and its float, and its repeat is a consequence rather than a specification. Nobody selects an eight-end satin in order to get a 3.33 mm step, and nobody is told that they have.

The second half generalises differently. A staircase on a lattice can take only the directions the lattice’s Farey set contains, and those directions are unevenly spaced with the widest gaps in the middle of the range. That is true of every rasterised line ever drawn, and the textile case is unusual only in that the raster is a physical object with a size in millimetres rather than a convention.

Who found it, and when #

The rule that a figure boundary steps to the ground weave’s repeat is in every jacquard designer’s practice and in the manuals, usually as a statement about floats: step too finely and the figure’s edge carries long floats that snag. That is correct and it is the commonest failure by a factor of thirty. The structural half — that a fine step can also split the cloth — is the first rung of this ladder.

Quoting a jacquard by its hooks is at least as old as the machine, and it is the right way to quote a machine: hooks are what limits the repeat width, which is a real and binding constraint on what patterns exist. Reading it as the resolution of the cloth is the confusion this essay is about, and it is an easy one to fall into because the two are measured in the same units.

The Farey structure of the achievable directions is an observation about a lattice rather than about textiles, and it is old. What is this site’s is running it at the block scale of a real cloth, with the two setts in it, so that the answer comes out in degrees on the fabric rather than in fractions on paper.

Where the ladder goes next #

The blocks ladder has three rungs and this is the third. What it leaves open is the physical boundary — a step is a ridge, and how large a ridge needs the take-up arithmetic that is the setting field’s rather than the pattern field’s.

Sideways, the same lattice quantisation with a completely different mechanism is a fashioned knitted edge, where the steps are whole wales at whole courses and the achievable angles turn out to be the same for every plain knit ever made. The machine whose hooks this essay is about is in its own essay, and what it actually abolishes is a budget rather than a resolution. And the diagonal a reader thinks they see running through a twill is not a thread either, which is the same kind of misreading one level down.