A damask is the only figure that costs its beam nothing

A damask is its own complement: its figure and its ground are the same satin used two ways, one warp-face and one weft-face, with the same longest float, the same shaft count and the same threading. That account counted what the pattern costs in structure and found it costs nothing — the cloth’s most famous effect is carried by direction alone.

There is a second budget a figured cloth spends, and it is spent one thread at a time for as long as the loom runs. Figure and ground consume warp at different rates, because a warp end’s take-up is its crimp and a region’s crimp depends on how often its threads bend. Over a long figure the difference accumulates, the ends in one region go slack while the ends in the other go tight, and the cloth cannot be woven from one beam.

The damask escapes that too, and it escapes it exactly rather than nearly. The same complementation that leaves the shaft count alone leaves both float lengths alone, so the two regions’ crimps are not similar; they are the same number.

A region’s crimp is its float #

Peirce’s geometry asks what shape a thread takes when it passes over and under threads at a stated spacing, and this collection’s relief weaves already run it at a modified spacing: a warp end floating over four picks passes three of them without turning, so the distance between its bends is four weft spacings. Handing Peirce the longer spacing answers the question about a floated region, and the crimp it returns is the region’s own.

The number falls away fast. At a half-millimetre spacing with quarter-millimetre threads, a plain weave’s warp crimp is 14.35 per cent, a 2/2 twill’s is 3.21, a five-end satin’s is 0.50 and an eight-end satin’s is 0.196. Nearly all of the fall happens in the first three crossings, which is the shape of the curve and is the reason the results below are so lopsided.

The difference accumulates, so it bounds a length #

A crimp is extra warp length per unit of cloth length. Two regions of one warp on one beam with crimps $c_1$ and $c_2$ consume warp at rates differing by $c_1 - c_2$, and the difference is proportional to how far down the piece the figure runs.

Over a figure ℓ millimetres long, an end passing through the figure and an end passing through the ground differ by ℓ times the crimp difference. A loom can absorb some of that — the warp between the fell and the back rest is elastic, the let-off gives a little, and a millimetre of slack on a free length of twelve hundred is under a tenth of a per cent of strain. Take the allowance as a millimetre and a fifth and the bound is one division.

An eight-end satin figure on a plain ground may be eight and a half millimetres long and no longer. That is not a figure; it is a speck. A 2/2 twill figure on a plain ground is bounded at eleven millimetres, which is not much better.

So the arithmetic settles a question about which figured cloths exist. A cloth whose figure and ground are different weaves cannot be woven from one beam, and the two beams such cloths are woven with — one for the figure warp, one for the ground — are not a convenience but a requirement, computed from the crimps. That is the same conclusion two layers need two beams reaches for a double cloth and terry needs two beams for a pile, arriving here for a figure that is only one layer thick.

And a damask is exactly nought #

The damask’s escape is an identity and the identity is worth writing out, because “the two crimps are close” and “the two crimps are the same number” are different claims and only the second gives an unbounded figure.

In an n-end satin every end is down at exactly one pick in n, so every end turns once every n picks: the warp’s bending pitch is n weft spacings. And every pick is up at exactly one end in n, so every pick turns once every n ends: the weft’s bending pitch is n warp spacings.

Complement the draft — which is what turning the cloth over does, and what a damask does between its figure and its ground — and every mark becomes a space. The end that was down at one pick in n is now up at one pick in n, so it is still down at n − 1 and still turns once every n picks. Both bending pitches survive the complementation unchanged.

So the region-crimp calculation is handed the same two numbers for the figure and for the ground, and returns the same crimp: 0.196 per cent on an eight-end satin at these spacings, on both sides of every boundary in the cloth. Not 0.196 and 0.197. The same double.

That is why a damask tablecloth can carry a figure a foot across and a brocade cannot. It is not that the damask’s figure is simpler or its loom better; it is that the two halves of a damask are the one pair of weaves in the whole catalogue whose warp consumption is identical by construction.

What the near pairings buy #

Between the damask’s nothing and the plain ground’s eight millimetres there is a usable range, and it is worth reading off because it says what a designer restricted to one beam can actually do.

An eight-end satin figure on a five-end satin ground — two satins of different orders, which is an ordinary way of building a two-tone figured cloth — differs by 0.31 per cent and allows a figure 392 millimetres long. That is a figure most of a piece’s width and as deep as anybody would design, so the constraint never binds.

A five-end satin figure on a 3/1 twill ground differs by 0.90 per cent and allows 133 millimetres, which is a real constraint on a large motif and no constraint at all on a small one.

An eight-end satin figure on a 2/2 twill ground differs by 3.02 per cent and allows 40 millimetres. A motif four centimetres deep is a spot rather than a figure, so this pairing is effectively single-beam only for small work.

The pattern in those three is the curve’s shape read backwards. Two weaves that are both floated are close in crimp however different they look; a floated weave and a plain one are far apart whatever else they have in common. So the practical rule is not about how different the two weaves are as patterns — it is about whether either of them interlaces at every crossing.

Across the width the same mismatch is a pucker #

The figure runs in two directions and only one of them has a beam in it. The other has the weft, and the weft is one thread crossing both regions, so it cannot consume different lengths in different places — it is the same thread.

What gives instead is the cloth’s own width. Two regions side by side across the warp want different weft crimps, so they want different widths, and a cloth held at one width by its reed has surplus material in the region that wanted to be narrower. Surplus material in a plane leaves the plane, which is a seersucker made at the loom and is the mechanism behind every relief weave in this collection.

So the same crimp difference has two completely different consequences depending on which way the boundary runs:

Along the warp it is a rate of consumption, it accumulates with length, and it shows as a slack or tight end.

Across the weft it is a width, it does not accumulate, and it shows as a pucker whose height is a square root of the surplus rather than a multiple of it.

Only the first bounds anything. A pucker is a finished appearance and often a wanted one; a slack end is a fault that gets worse the longer the loom runs. That asymmetry is the reason figured cloths are designed with their weave boundaries running across the piece more freely than down it, and it follows from the warp having a beam and the weft not having one.

What the step in the surface has to do with it #

An earlier essay already found the two regions of a figured cloth differing in one respect, and it is worth separating the two findings because they come from the same place and are not the same thing.

A figured cloth has a step in its surface: a region that interlaces less often is pressed less often, flattens less and stands thicker, so a satin figure on a plain ground stands about fifty micrometres proud. That is a thickness difference, it is a property of each region taken on its own, and it does not accumulate.

The take-up difference is a length difference, it is a property of the warp passing through both regions, and accumulating is the whole of what makes it a constraint. A cloth can have the first without the second — which is what a damask is: figure and ground at the same crimp, and therefore at the same thickness, so a damask has no step in its surface either, and the earlier account’s finding that its effect is carried by direction alone is the same identity read a third way.

The three readings are one fact about complementation. It preserves the shaft count, it preserves the float lengths, and it therefore preserves the interlacing rate — and the interlacing rate is what decides the pressing, the thickness and the take-up. A cloth built as an exact complement is a cloth with one value of every one of those, everywhere.

An open cloth forgives a mismatch and a close one does not #

Every bound above is at one construction — half-millimetre spacings with quarter-millimetre threads, which is a closely set cloth. The crimps are properties of the geometry rather than of the weave alone, so opening the sett moves all of them, and it moves them in one direction.

Across a realistic range of spacings — 0.45 to 0.7 millimetres, which is twenty-two threads a centimetre down to fourteen — the satin-on-plain bound runs from six millimetres to eighteen and the satin-on-satin bound from 317 to 770. So a design that is single-beam at one sett is single-beam at every sett in the range, and one that is not is not: the sett changes every answer by a factor of three and changes no decision.

That is worth stating because it is the opposite of the way most of this collection’s constructions behave. A jamming sett, a cover factor, a bearing curve, a permeability — all of them have thresholds that a change of sett walks a cloth across. This one does not, because the quantity that matters is a ratio of two crimps computed at the same spacing, and opening the cloth lifts both of them together.

The one thing an open sett does change is the weft side. A pucker’s height goes as the square root of the surplus, and the surplus is a difference of weft crimps, so an open cloth’s relief is larger as well as its tolerance — which means opening a figured cloth’s sett buys warp-direction freedom and costs weft-direction flatness, in the same operation.

The generalisation, which is about rates rather than about cloth #

The shape of this result is worth separating from the weaving, because it recurs wherever one supply feeds two processes.

A difference in a level is a defect and a difference in a rate is a deadline. The step in a figured cloth’s surface is a level: fifty micrometres, everywhere the figure is, however long the figure runs. The difference in take-up is a rate: a fraction of a per cent per unit of length, which is nothing at all for one millimetre and is a broken warp at a metre. The first is a property of the design and can be accepted or not; the second is a property of the design multiplied by how long anybody runs it, and cannot.

The practical form is a question worth asking of any shared supply: is the mismatch in what the two consumers hold, or in what they draw? If it is in what they hold, the answer is a tolerance. If it is in what they draw, the answer is a length — and the only two ways out are to equalise the rates exactly or to give each consumer its own supply. Weaving takes the second route routinely and calls it a second beam; the damask is the one place it takes the first, and it does so not by adjusting anything but by being built out of a symmetry that makes the two rates the same object.

That is the useful half of the finding. An exact equality obtained from a symmetry is unconditional — it does not need the sett, the yarn, the tension or the finish to cooperate, because none of them enters the argument. Every other pairing in the table is a tolerance that a change of any of those could move, and the damask is the one whose freedom nothing can take away. A weave is a matrix, complementation is an operation on it, and what an operation preserves is worth more than what a measurement happens to agree about — which is this collection’s standing preference, arriving here as a statement about how big a tablecloth’s flowers may be.

What was computed, and how #

Each region’s crimp is Peirce’s geometry handed the region’s own bending pitch — the thread spacing times the float, in each direction — which is the relief weaves’ own modelling step and is stated there as a step. The difference of the two warp crimps is the differential consumption per unit of cloth length, and the length a figure may run is the loom’s slack allowance divided by it. The identity for a damask is the observation that complementing a satin leaves every thread turning at the same interval; it is checked by computing both regions rather than by declaring it.

Four things are tested. A damask’s figure and ground take up identically, with the difference exactly nought and the bound infinite, which is the essay’s claim and would fail on any rounding. Every other pairing in the table differs and is bounded, so the identity is a property of the damask rather than of the arithmetic. A longer float carries less crimp at every step from one to sixteen, which is the check on the modified spacing being handed in the right direction. And a figure of a far weave is bounded at millimetres where a near one is bounded at centimetres, by a factor of at least ten, which is the result read as an inequality so that a collapse of the two cases would fail it.

The thread diameters, the spacings and the loom’s slack allowance are inputs. Every length in the essay is proportional to the last of them.

Where the model stops #

The slack allowance is a guess. A millimetre and a fifth over a free warp length of twelve hundred is under a tenth of a per cent of strain, which is a defensible order and is not a measurement of any loom’s let-off. Every bound here scales with it exactly, so a reader with a better number can read off the consequence.

The take-up is computed at the loom and the cloth is finished afterwards. Both regions relax when the cloth comes off, and they relax by different amounts, which is what the relief-weave arithmetic computes and what this account does not carry through. The accumulation happens on the loom, so the bound is a loom quantity; the finished cloth’s appearance is a separate calculation.

A boundary is a line and a region is not uniform near it. The threads within a float’s length of the boundary belong to neither region cleanly, and a damask’s edge floats further than its figure at the one ground position that breaks the nesting. Nothing here computes a boundary’s own crimp.

And the ends are taken as independent. A slack end in a warp of several thousand is held by its neighbours through the friction of the reed and the cloth, so the first millimetre of slack is shared rather than borne alone, which makes the real bound more forgiving than this one by an amount nothing here computes.

Still open: whether a two-beam cloth is bounded by anything #

The whole argument is about one beam. A cloth with two beams lets its figure warp and its ground warp run at their own rates, and the bound vanishes — which is why such cloths exist.

What replaces it is not obvious. Two beams solve the accumulation and leave the boundary: an end that runs in the figure for part of the piece and in the ground for the rest is on one beam throughout, and no arrangement of beams helps it. So a two-beam figured cloth is a cloth in which every end is permanently a figure end or a ground end, which is a constraint on the design rather than on its size — and it is the constraint that makes a lampas a lampas rather than a large brocade.

Whether that constraint has an arithmetic of its own — how many distinct end-classes a design needs, and therefore how many beams — is the same question what a figure costs the loom answers for shafts, asked of beams instead. It has not been asked here.

Who worked it out #

That figure and ground take up differently, and that figured cloths of mixed weaves need two beams, is weaving practice of long standing and is in every account of compound cloths. Peirce’s geometry is from 1937 and its application to a floated region is new here, arriving from the relief weaves. The bound on a figure’s length, and the identity that puts a damask’s two regions at the same crimp exactly, were computed directly.