A figured warp needs a beam for every share of its figure

A damask is the only figure that costs its beam nothing, and the reason is a crimp. A warp end that bends at every crossing carries more length than one that floats, so a figure of one weave on a ground of another consumes warp at two rates, and on one beam the difference accumulates down the length of the figure. An eight-end satin figure on a plain ground puts the two regions 14.15 per cent apart, and a loom that absorbs 1.2 millimetres of slack bounds such a figure at 8.5 millimetres along the warp. A damask’s two regions are the same satin used two ways and take up identically, so nothing bounds it.

That account ended on the obvious escape. Put the figure’s ends on one beam and the ground’s on another, let each run at its own rate, and the accumulation has nowhere to go. It noted, without pursuing it, that the escape has a hole in it: 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.

That hole is most ends of most figures. The question is how many beams a design needs when its ends cross the figure, and the answer turns out to be a count with a clean description — and a count that jumps from one to its maximum at the first crimp difference anybody could measure.

An end’s consumption is a ramp that rises only in the figure #

Take a block design: a grid of blocks, each a repeat of the ground weave, some of them figure. Down one column of blocks a warp end passes figure blocks and ground blocks in some order, and at every millimetre of figure it consumes Δ more warp than it would in the ground, where Δ is the crimp difference.

So measured from an end that never enters the figure, an end’s extra consumption through a repeat is a ramp that climbs at slope Δ through each figure block and runs level through each ground block. At the end of the repeat it stands at Δ times the figure length it crossed, and the next repeat starts it climbing again from there.

Two ends on one beam are let off at one rate. Whatever one of them consumes beyond the other has to come out of slack the loom can absorb, and the account below took that slack as 1.2 millimetres. So two columns can share a beam only if their ramps never stand more than a slack apart.

Two conditions, and the second is the first run for a whole piece #

That condition has two parts, and they behave very differently.

Inside one repeat, two columns’ ramps may part and come back together. A check’s two columns take turns in the figure: one climbs while the other waits, then the other catches up. They part by at most the two blocks each spends in the figure, and at 4-millimetre blocks that is 1.13 millimetres, which fits.

Across repeats, two columns that end a repeat at different heights part by the same amount again in every repeat, without limit. A column in the figure for three blocks of twelve and a column in it for four part by a block’s worth every repeat, and a hundred-metre piece holds two thousand repeats of a 48-millimetre design.

So over a piece, two columns can share a beam only if they spend exactly the same share of the repeat in the figure, and among those only if their ramps stay within the slack inside a repeat. The first condition is the second one run for the whole length of the piece.

The two are not independent, and the pair of check figures shows the dependence exactly. A check’s two columns share a beam while twice a block’s drift fits the slack: at 14.15 per cent that is blocks up to 4.24 millimetres. At 4 millimetres the check needs one beam; at 8 it needs two. The design has not changed, only its scale.

The beam count, and the two counts it sits between #

Compatibility in that sense is not transitive — column A may stay within a slack of B and B of C while A and C part by more — so the fewest beams is a minimum clique cover of the columns: the fewest groups in which every pair is compatible. It is found exactly by search, and on every design here the search is small.

Two easier counts bracket it. The distinct shares — how many different fractions of the repeat the columns spend in the figure — are a lower bound, because columns of different shares can never share a beam over a piece. The distinct columns — how many different patterns of figure and ground the columns carry — are an upper bound, because identical columns can always share.

The distinct columns are also a number with a meaning on another loom. What a figure costs the loom found that a block design’s shaft count on a dobby is a repeat of the ground weave for every distinct block-column, whatever the figure’s size. So the beam count is bounded above by the dobby’s cost and below by a coarser count of the same columns — the dobby asks which columns differ, and the let-off asks only by how much of the repeat.

A stripe needs its columns and a band needs one #

The two stripes in the census are the limiting cases, and they come out as they must.

A warp-way stripe puts whole columns of blocks in the figure and whole columns in the ground. Every end lives wholly in one region, which is the case the second beam was invented for: two distinct columns, two shares — all and none — and two beams, one per region, at any block length.

A weft-way band puts whole rows in the figure. Every end crosses every band, in step with every other end, so every column is the same column: one beam, whatever the crimp difference, because nothing ever drifts. It is the woven cloth’s version of a stripe that costs its beam nothing — not by having identical crimps, as a damask does, but by making every end pay the same.

Between those, the check and the chequer are the designs where the two counts part: two distinct columns, one share between them, and a beam count of one or two depending on whether a block’s drift fits the slack. A chequer’s columns part by one block where a check’s part by two, so a chequer shares a beam up to 8.48-millimetre blocks and a check only up to 4.24.

A round figure needs a beam for every height #

Figures that are not stripes or checks have columns of different heights, and height is share.

A disc twelve blocks across has columns of four heights, mirrored across its centre, and needs four beams. Its mirrored columns are identical, so its distinct columns and its distinct shares are the same four, and the bracket closes: there is no phase freedom to exploit, because two columns of equal height in a disc are the same column.

As the disc grows the heights multiply. Twenty-four blocks across needs eight beams, forty-eight needs fifteen, ninety-six needs twenty-nine — close to a quarter of the width, because half the columns mirror the other half and a pixelated circle repeats some heights near its middle.

A jacquard is where figures that large live. The jacquard is every end its own shaft, and its budget is its hooks: a 1,200-hook harness on an eight-end ground holds a design 150 blocks across. A round medallion filling it, woven with figure and ground of different crimps, would want some forty-five beams. No loom carries forty-five warp beams, and the figured silks that such harnesses were built for are not woven that way.

The beams are not the same size #

A beam count says nothing about how the ends divide among the beams, and for a round figure the division is lopsided in a way a loom builder would notice first.

In the disc twelve blocks across, the two outermost columns on each side of the centre are the shortest, the four central columns the tallest, and heights between fill the rest. So the beam feeding the centre carries four columns of blocks, the beams for the two intermediate heights carry four and two, and the outermost beam carries two. On an eight-end ground a column of blocks is eight ends, so the four beams hold thirty-two, thirty-two, sixteen and sixteen ends of a ninety-six-end repeat, and in a cloth of many repeats each beam holds that share of the warp across the whole width.

That is not a problem of count but of let-off. Each beam turns at its own rate — the centre’s fastest, because its ends spend the most of the repeat in the figure, and the outermost’s slowest — and a loom with four let-off motions keeping four sheets of warp at four rates is a loom with four tensions to control. The beam count is the number of let-off motions a design needs, and each of them is set by one number: the share of the repeat its columns spend in the figure, times the crimp difference.

It also says which beams a designer can merge by redrawing. Two heights a single block apart differ in share by one block in twelve, and a figure redrawn so that those columns match — a disc squared off at one step of its outline — loses a beam for every pair of heights it equalises. A round figure is the most expensive outline there is, and a figure drawn in steps of a few heights is the cheapest one that still reads as a shape.

Any crimp difference at all costs every beam #

The most striking thing about the count is how it depends on the crimp difference, which is hardly at all.

A damask needs one beam for every design, because its two regions take up identically and no column ever drifts. An eight-end satin figure on a five-end satin ground — two satins, crimps 0.31 per cent apart, a difference no finisher would see — needs four beams for the disc, exactly as many as a satin on a plain ground forty-five times further apart.

That is the first condition doing its work. A share difference drifts every repeat, and two thousand repeats multiply any nonzero drift past any slack. The count does not fall as the crimps close; it falls only when they are equal. Over a piece, the beam count of a figured cloth is a step function of its crimp difference with one step, at nought.

A sample can share a beam that a piece cannot #

The piece length is what the step hides, and bringing it back shows where a small difference stops being free.

With the eight-end satin on the five-end, a sample under a metre long weaves on one beam, and the full four arrive by three metres. With a satin on a twill three per cent apart, one beam lasts less than fifty millimetres. With a satin on plain, four are needed from the first repeat.

So a sample can prove a design that a piece cannot weave. A pattern woven as a trial length on one beam with near-matched weaves shows no fault at all, and the same design put into production on the same beam drifts its columns apart a few metres in. The trial’s success is a statement about its length, not about the design.

What the trade did instead of beams #

The arithmetic leaves exactly three ways to weave a large figure from a small number of beams, and the history of figured cloth used all three.

Make the crimps identical. That is the damask, and a damask is its own complement: figure and ground are one satin used two ways, so every column consumes alike whatever its share, and one beam serves any design.

Keep the warp out of the figure. A lampas carries its figure in extra wefts bound by a second warp, and a brocade lays a pattern weft that floats as far as the next figure. In both, every warp end weaves the same ground structure everywhere, so every column’s share of anything is the same, and the second beam holds a second system of ends rather than a second class — each beam’s ends uniform, which is the weft-way band’s trick applied to a whole design.

Put every end wholly in one region. That is the warp-way stripe, and it is the construction a pile warp and a leno’s doup ends use for their own reasons: a separate beam for ends whose consumption differs, each beam feeding ends that never change class.

What the arithmetic rules out is the fourth thing a designer would first try: a figure of one weave on a ground of another, drawn freely across a warp and fed from a beam or two. Beams are counted by share, and a free drawing has as many shares as it has heights.

One count, three constructions #

This closes a small circle across the fancy field. Two layers need two beams because a double cloth’s two layers consume at two rates; a leno needs a second beam because its doup ends consume more than their partners; a terry needs one because its pile ends are fed several times faster. Each of those is a warp-way stripe in the sense above: ends that belong permanently to one class, one beam a class.

A figured cloth is the case where the classes are not permanent. An end changes class every time its column enters or leaves the figure, and the beam count is then not the number of regions but the number of distinct ways a column spends its repeat — up to the slack a loom can absorb, and exactly, over a piece.

What was counted, and how #

A design is a grid of blocks, each block a length along the warp. The crimp difference between figure and ground is the account below’s: Peirce’s geometry at each region’s own bending pitch, for 0.25-millimetre threads at 0.5-millimetre spacings. For each distinct column the extra consumption over an all-ground end is accumulated block by block; two columns are compatible when their difference, taken as its drift per repeat times the repeats the piece holds plus its span inside one repeat, does not exceed the slack. The beams are a minimum clique cover of the distinct columns under that relation, found by exhaustive backtracking.

The count was confirmed to be one for a damask on every design; to lie between the distinct shares and the distinct columns for eight designs at block lengths from one to sixteen millimetres; never to fall as blocks lengthen; to equal the distinct columns for a warp-way stripe and to be one for a weft-way band at any block length; to switch a check from one beam to two exactly at the block length where twice a block’s drift equals the slack; and to equal a disc’s distinct heights.

Where the count is too strict #

A warp stretches. The slack here is a fixed length the loom absorbs, which is the account below’s model. A real end under tension takes a difference as extra strain, and a tighter end straightens a little and consumes less, so part of any drift is corrected by the cloth rather than absorbed by the loom — at the price of columns woven at slightly different tensions, which is a fault of its own.

Blocks are uniform. A real figure’s outline is not a staircase of whole blocks, and its columns change region part-way through a block.

The crimps are a model’s. Peirce’s geometry at a region’s bending pitch gives a crimp difference for any pair of floats, and a finished cloth relaxes both regions by amounts the geometry does not supply. None of that changes the step at nought: two regions that differ at all drift over enough length.

Still open: how much a warp’s own stretch absorbs #

The count treats slack as a length and a warp’s tension as something that does not move. A real let-off controls tension, and a column that runs ahead pulls harder, straightens and takes up less — a feedback that shrinks any drift until the tension difference it costs balances it. The question the count leaves is how large that tension difference is for a disc woven satin-on-twill on one beam, and whether it is small enough to weave through or large enough to show as streaks where the columns change height. That needs the cloth’s crimp-interchange response to a tension difference between neighbouring ends, which the budget a cloth has for two directions begins to describe and nothing here has applied column by column.

Who described it, and what is new here #

Separate beams for pile, for doup ends and for the layers of a double cloth are ordinary weaving practice, and the damask, lampas and brocade are the classical figured constructions. Counting the beams a figured design needs as a clique cover under drift, bracketing it between distinct shares and distinct columns, and finding that a crimp difference of any size costs every share over a piece while a short sample can hide it, was done here.