A figure woven on one beam shows its columns off the mirror

A figured warp pays in tension before it needs a beam found that a figure whose regions take up warp at different rates can often share one beam. The ends that want more yarn pull harder, give up crimp and consume less, until every end consumes alike. It priced the tension that costs. For an eight-end satin figure on a five-end satin ground the price was a few grams an end; for a five-end satin on a 3/1 twill ground it was 0.021 to 0.069 newtons, between four and thirteen per cent of the loom’s working tension. That was the borderline case, and whether to accept it turned on a question the essay could not answer: whether a tension difference between neighbouring ends shows in the cloth.

Two things change once the question is followed through. The first is a correction. That essay let only the ground give crimp, and named the figure’s side as the thing it left out; a satin figure’s crimp is the easiest crimp there is to pull out, and letting it give lowers every tension and turns the borderline pairing into a safe one. The second is the answer. At the mirror a tension difference is invisible, however large. A few degrees off the mirror, along the warp, it is not: the tight columns stop reflecting before the slack ones do, and the figure’s columns appear as bands.

The figure gives crimp too #

A figured end spends part of the repeat in figure and part in ground. Pulled harder, both parts straighten, each along its own locus of constant thread length — the curve the locus gets a force turned into a load–extension curve. Each region is taken, as the tension essay took it, as Peirce’s plain geometry at its own float length.

The two regions are not equally stiff. A 3/1 twill ground carries 1.4 per cent of warp crimp and a five-end satin figure 0.5 per cent. The satin’s crimp is small, but it is held by very little: at 0.02 newtons, with the yarn’s rigidity at the lower end of its placed band, the satin region has given up nine tenths of its crimp and the twill under two thirds of its own. So when a column that crosses both is pulled, most of the first length it gives comes out of its figure.

The tension essay counted only the ground’s contribution. With both regions giving, a column reaches the common consumption at a lower tension, and the correction is not small. The borderline pairing’s tightest column needs 0.013 newtons at the lower placement of the yarn’s rigidity and 0.042 at the upper, against 0.021 to 0.069 when only the ground gave — two and a half to eight per cent of the loom’s 0.52-newton working tension, inside a tenth at both ends of the band. By the tension essay’s own criterion, a five-end satin on a 3/1 twill shares a beam. The eight-end satin on a five-end satin ground falls from 0.006–0.020 to 0.004–0.012 newtons. The eight-end satin on a 2/2 twill falls from 0.30–1.0 to 0.22–0.72, still far past a tenth.

Each column settles at its own tension #

The disc is the design the beam essays have used throughout: a round figure twelve blocks across on a square of twelve by twelve. Its columns differ in how much of the repeat they spend in figure. The four columns through its middle are figure from top to bottom; the outermost are two thirds ground.

On one beam every column must take the same length of warp per length of cloth, and the column that wants least sets the rate: here, the all-figure columns, since the satin carries less crimp than the twill. They run slack. Every other column is pulled until its two regions together have given up the difference, so the tension rises with the column’s share of ground: 0.002 newtons for columns five-sixths figure, 0.005 for two-thirds, 0.013 for the outermost at the lower placement, and three and a quarter times those at the upper.

That is the gradient the tension essay described without computing, and it is a gradient only because the figure gives. Had the figure been stiff, every column with any ground at all would have needed the same full tension and the map would have been two flat levels, slack in the middle and tight everywhere else.

A tight column’s warp lies lower, by an invisible amount #

An end pulled straighter lies lower. Its crowns stand at half its crimp height plus half its diameter above the cloth’s mid-plane, and taking crimp out takes height out with it.

In the ground the crowns run from 242 micrometres in the columns nearest the figure’s middle down to 214 in the outermost — a relief of about 30 micrometres — and in the figure from 250 at the slack centre to 181 at the edges, about 70. Those are the sizes a figured cloth has a step in its surface found between a figure and its ground, fifty micrometres, and a figure shows by its shine, not its step found that such a step returns almost no light: it is a gentle slope spread over a block several millimetres wide, and it tilts nothing enough to send a lamp anywhere new. The tension’s relief is gentler still, spread over a whole column. The relief does not show, and the question the tension essay posed about a ridge has that much of an answer.

At the mirror, a tension changes nothing #

If the streak shows, it shows as shine, as the damask’s figure does. The first result there is a null, and it is exact rather than approximate.

At the mirror — lamp and eye at equal angles either side of the vertical — a thread reflects from two places. It reflects from its plateaux, the flat tops of its floats, and from a strip of arc at each crossing, where it turns from one face to the other. The crossing’s strip has a length set by the thread’s diameter and the angular tolerance of the lamp and eye, about $D\varepsilon$, and the weave angle does not enter it: a crossing that turns through a steep angle and one that turns through a shallow one both sweep through the flat direction once, and reflect over the same arc while they do. A column pulled straighter has shallower crossings and exactly the same reflecting area.

The computed difference between the tightest and slackest ground columns at the mirror is under half a per cent, all of it from the slightly changed spacings. Looked at straight on, under a lamp placed for the mirror, a figured cloth woven on one beam shows nothing of its tension.

Off the mirror, the tight column goes dark first #

Move the lamp, or the eye, a few degrees so the half-vector between them tilts along the warp, and the plateaux stop reflecting at once: a flat float top sends its light to the mirror direction only. What still reflects is the crossings, each over the part of its arc whose slope matches the tilt. A crossing’s slope runs from nought to its weave angle $\theta$, so a crossing reflects only while the tilt is within $\theta$ plus the tolerance.

That is the mechanism a lamp off the mirror lights a satin only across its floats found for a satin: along its floats a satin goes dark as soon as a lamp leaves the tolerance, because its crossings are shallow. Here it separates columns rather than weaves.

The tight outer column’s ground crossings turn through 6.8 degrees and the slackest ground column’s through 9.1. Between 8.8 and 11.1 degrees of tilt, the tight column’s warp reflects nothing and the slack column’s still does. Inside that window the contrast in the warp’s reflection is complete: one column is lit and its neighbour two blocks away is dark. The figure has its own window, lower, because a satin’s crossings are shallower: its tight edge columns turn through 2.6 degrees and its slack centre through 5.8, so between 4.6 and 7.8 degrees the disc’s centre still shines and its sides do not.

The lustre map at the head of this essay is these two windows drawn on the design. At 6.2 degrees the disc appears narrower than it is, its outer columns dark. Past 9 degrees the figure is dark throughout, and the ground shows bands: lit above and below the figure’s middle columns, dark beyond them, a shape that follows the figure’s profile down the whole length of the piece.

Every pairing but a damask has a window #

The same account runs for the other pairings the tension essay priced.

The eight-end satin on a five-end satin ground, the pairing the tension essay called free, needs at most 0.004 to 0.012 newtons — a gram or so an end — and it still has windows: 6.2 to 7.4 degrees in the ground and 3.8 to 5.6 in the figure. The eight-end satin on a 2/2 twill, which needs its beams on any account, has the widest: 6.4 to 10.9 degrees in its ground.

The narrowness matters as much as the existence. A window a degree wide is crossed by a fold in a millimetre or two of cloth, so on a drape the eight-on-five pairing’s bands would be thin lines along the crests rather than broad areas, and a thin line of changed lustre on a satin is less conspicuous than the same contrast over a block. A damask has no window at all. Its figure and its ground are one satin and its complement with identical crimps, so every column runs slack and every crossing turns through the same angle. A damask is the only figure that costs its beam nothing found that it needs no beams; it is also the only figure that needs no tension, and so the only one that draws no ghost of its own columns.

That is a second reason for the damask’s place in the trade, and a stronger one than the first. A mill can always buy a second beam. It cannot buy a figured cloth that looks the same from every angle unless its figure and ground take up alike.

The yarn’s stiffness cancels from the windows #

The tension essay’s forces came as a band, a factor of three wide, because a yarn’s rigidity is known only to where a yarn’s stiffness is a bracket, not a number places it. The borderline pairing’s verdict turned on where in that band the yarn sat.

The windows do not depend on it at all. The rigidity multiplies the force along both regions’ loci alike, so a stiffer yarn needs proportionally more tension to reach every state, and every column reaches exactly the same state as before. The column’s share of ground decides how far along its locus it goes, and the locus decides the weave angle. So the relief, the weave angles and the tilt windows computed at the lower placement come out identical at the upper one, and only the newtons differ. Whether a figure’s columns show is a question about the weaves and the design, and not about the yarn.

The weft fills in some of the dark #

Everything above counts the warp’s crossings. The weft’s are there too, and they reflect differently. A tilt along the warp is a tilt across the weft, and a round weft crown offers every angle across itself, so the weft keeps reflecting at every tilt in both columns alike.

In the regions as modelled, where each is plain geometry with warp and weft equally crimped, the weft returns about as much as the warp does at the mirror. So at 9 degrees the tight column’s total specular reflection is a third less than the slack column’s, not nothing. A twill or satin ground is warp-faced, its face mostly warp floats and its weft buried under them, and the weft’s share of what reaches the eye is smaller than the model’s even split; the contrast lies between that third and the whole. Either way it is far above the one or two per cent at which a sharp boundary between two areas of different brightness becomes visible under a steady light.

A drape sweeps through every window #

A figured cloth is seldom looked at under one lamp at one angle. A curtain, a tablecloth over an edge or a garment has surfaces tilted by every angle over a few centimetres, and a draped cloth sweeps its surface through every window at once. Somewhere on every fold the tight columns are dark and the slack ones lit, and as the cloth moves the bands move with the fold.

That is why a streak of this kind is characteristically a fault seen in use rather than at inspection. A cloth examined flat on a table under overhead lights is examined near the mirror, where the tension is invisible. The same cloth hung in folds shows ghost bands down the ground in line with the figure, and the bands vanish again when it is laid flat. Turn the cloth and the shine changes hands described the same angular behaviour for a figure’s own contrast; here it belongs to a defect.

What a seersucker does with the same mechanism #

A seersucker is made at the loom runs this mechanism on purpose and far past any window: two beams, two tensions, and one stripe’s warp so much straighter than the other’s that the difference comes out as puckers a millimetre high. A tension streak is a seersucker too small to pucker. Its warps differ in straightness by tens of micrometres of crown height, which a finger cannot feel and a flat view cannot see, and which a raking view picks out because a crossing’s angle, not its height, is what decides where it sends light.

How the numbers were produced #

The regions are the tension essay’s. Each is a plain cloth of 0.25-millimetre yarn at a sett of twenty over its float length, the figure at five and the ground at three for the borderline pairing, with its load–extension curve along the constant-thread-length locus from the least-energy state. The rigidity is placed at both ends of the band where a cantilever and a washing measurement agree.

The columns are the disc’s. For each, the tension was found by bisection at which its ground share of the ground’s extension and its figure share of the figure’s together equal what the slackest column gives up, which is nothing. Each region’s state at that tension supplied its crimp height, its weave angle and its spacings, and a surface built from that state was asked for its specular area with the half-vector tilted along the warp, the lamp and eye’s tolerance taken as two degrees, the crossings reflecting over the part of their arc whose slope the tilt matches.

What the calculation is required to do. A damask’s columns must all run slack. In every other pairing the column with most figure must run slack and tension must rise with the ground’s share. At the mirror the tightest and slackest columns’ ground must shine alike to within a per cent. And one thing that could have failed: tilted into the middle of the ground’s window, the tight column’s warp must reflect nothing while the slack column’s still reflects, for the eight-on-five pairing as well as the borderline one.

What the account assumes #

The regions are stand-ins. A 3/1 twill is not plain weave at three spacings; its floats have plateaux the stand-in lacks, which add to the mirror’s reflection and nothing to the tilted one, so they dilute the contrast near the mirror and leave the windows alone.

Friction is left out. The most a cloth can give back found a band of resting states rather than one, and a crimp held by friction at its crossings gives up less than the energy alone says; the tensions would be higher and the regulation less complete. Since the windows come from states rather than forces, friction moves them only to the extent it stops a column reaching its state.

The loom state is taken as the cloth’s. The regulation happens on the loom under tension. Off it, the tension goes and the cloth relaxes, but each column has only the warp it was given: its crimp cannot come back without yarn it does not have, and the straighter warps stay straighter. How the finishing relaxation shares that fixed length between a column’s figure and its ground has not been computed.

Who found which part #

Figured weaving on one beam, and the streaks it risks, are old practice, and so is the rule that some figures need a separate beam. The regulation of a figured warp by tension, and its price, are the tension essay’s; letting the figure give crimp as well as the ground is the correction that essay named.

What is added here is the corrected tensions and the gradient they make across a design; the relief they leave, and why it cannot be seen; the exact null at the mirror, because a crossing’s reflecting arc does not depend on its angle; and the tilt window along the warp, which the yarn’s stiffness does not move, in which a regulated figure’s tight columns go dark and its slack ones do not.

Still open: how bright a window has to be to be seen #

The windows are a few degrees wide, and the contrast inside them is between a third and the whole of the warp’s reflection. Whether a person notices depends on something no geometry supplies: how much of the room’s light arrives within those few degrees of the right direction. A single bright lamp makes a sharp window; a room lights a satin at the harmonic mean of its lamps found that a room of many sources averages its specular contrasts down, and an overcast window averages them further.

The streak’s visibility under a room’s light, rather than a lamp’s, is that calculation run with the columns’ two weave angles in place of two weaves, and it would say whether a regulated figure’s bands are a fault seen only under a spotlight or one seen in every ordinary room.