Textile Structure
how cloth is put together
Concept

Period — where it appears

A translation that leaves a pattern exactly as it was, and therefore a statement about the pattern rather than about the paper it is drawn on. The set of them forms a lattice, and the smallest rectangle that lattice admits is the repeat a draft could have been written on.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

A beat at 19.1 mm from grids at 0.5 mm. Two grids at 0.5 and 0.5 mm pitch, the second turned by 1.5°, over a 30 mm window. The dashed rules are one predicted beat period apart. The pattern between them is 38 times the pitch of either grid and neither grid has anything at that scale.

Watered silk is a beat

Fold a ribbed cloth on itself and press it, and a figure appears at a scale neither ply has — fifty times the rib pitch, wandering across the piece. It is the difference of two wavevectors, it is enormous because the angle is tiny, and no two pieces match because no two are folded at the same angle.

pattern · Moire Which dentings leave a mark. Every combination of ends per dent and weave repeat, with how many ends the grouping takes to come back into step. A small number means the reed treats every repeat the same way and the grouping shows as a stripe at the dent pitch; a large one means the grouping walks across the weave and there is nothing periodic for the eye to find. The rule is one word: dent so the two share no factor.

The reed leaves its own mark

A reed does not space a warp evenly. It groups it, several ends to a dent, and the grouping beats against the weave repeat — so a denting that shares a factor with the repeat treats every repeat identically and shows as a stripe, and one that does not is invisible.

pattern · Moire The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 6% of a dent, which is 25 µm. The upper band's errors are independent; the lower band's repeat every 8 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.7 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up.

A random error hides and a periodic one shows

A warp whose threads vary by fifteen per cent looks perfectly even. One dent of the reed a tenth of a millimetre wide makes a streak that gets the piece rejected. The same amount of error, arranged two ways — and the ratio between them is √(2n/π), with nothing fitted in it.

pattern · Variation One weft fault at three periods, a fraction of a millimetre apart. A 260 mm slice of a 1500 mm cloth at 22 picks per centimetre, 220 picks deep, with a thick place recurring along the weft. Each pick takes a whole width of yarn, so the marks land 35 to a pick across the full width and step sideways by the remainder of the width divided by the fault's period. On the left that remainder is zero and every thick place in the piece falls in the same columns — a warp-way stripe made entirely by a weft fault. In the middle the period is five hundredths of a millimetre longer and the same fault draws steep diagonals. On the right it is a third of a millimetre longer again, the marks land nowhere near each other, and the fault reads as texture. Nothing about the yarn distinguishes the three; the cloth's width does. The slice is drawn rather than the whole width because thirty-five marks a pick fill a panel solid at any step but zero, which is a true picture of a dense pattern and a useless one of its structure.

A slub finds the width of the cloth

A thick place recurring along a weft yarn does not make a bar. It makes diagonals — and when the cloth's width happens to be a whole number of fault periods, it makes stripes down the piece instead, from a fault that is entirely in the weft.

pattern · Moire The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 5% of a dent, which is 21 µm. The upper band's errors are independent; the lower band's repeat every 4 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.8 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up.

A course is one thread and a warp is many

A woven fabric draws its warp from two thousand packages side by side, so a yarn's drift averages out across the width. A weft knit takes whole courses from one package, so the same drift becomes a band — and the standard remedy for that turns an invisible error into a visible one.

knits · Knit Three places a mistake can be made, and three shapes it leaves. A loom holds a design in three separate objects, and a single mistake in each of them produces a fault of a completely different size — not because the mistakes differ, but because of how many intersections each object controls. On a 50 m piece of muslin 1500 mm wide, holding 396,000,000 intersections: one end drawn on the wrong shaft is wrong at every pick for the whole length, 110,000 of them; one pick made in the wrong shed is wrong across the whole width once, 3,600; and one shaft tied wrongly to one treadle is wrong wherever that shaft's ends meet that treadle's picks, which is everywhere — 6,187,500, or 1.6% of the cloth. The ratio between the extremes is 1719 to one, and the largest of the three is the one nobody sees happen, because every thread is exactly where it should be.

Three mistakes and the shape each one leaves

A loom holds a design in three separate objects, and one error in each of them is the same size of error. What they cost differs by a factor of seventeen hundred — and the largest of the three is the one nobody can see happening, because every thread is exactly where it should be.

fancy · Shafts A colour order against a 2/2 twill. The visible face of a 2/2 twill under 2 colour orders, drawn at the repeat the divisor arithmetic allows and outlined at the repeat the surface has. A filled cell is a dark thread on the face, which is the warp's colour where the warp is up and the weft's where it is not — so none of these patterns is in the draft, and the draft is the same in all of them. The colour period and the weave repeat beat exactly as a reed's grouping beats against a weave: the surface repeats on the least common multiple of the two, which here is 8×8 and 4×4. What the panels cannot show is colour: the two threads are drawn as filled and empty, and two colours of similar value make a pattern far weaker than this.

A colour order beats the weave it is threaded on

The reed's grouping beats against the weave repeat and the arithmetic is a least common multiple. A colour order is a second grouping of the same warp and the arithmetic is identical — but where the reed's beat is a fault to be dented out of a cloth, the colour order's beat is the pattern the cloth is sold for. Across 472 colour orders on four weaves the divisor bound is the surface's exact repeat in 470 or more, and the handful that beat it have no pattern left at all.

pattern · Moire The magnification a cloth will carry. The largest magnification a moiré can be read at, against the irregularity of the cloth making it. The points are measured: a grating whose spacings are drawn from a seeded lognormal is laid against a perfect one, the fringes are found from the phase difference, and the gain is recorded at which their count first departs from what the ideal beat predicts. The product of the irregularity and that gain comes out at 0.82 to 0.84 across every irregularity tried, so the ceiling is 0.84 divided by the coefficient of variation — the solid curve. The dashed curve is the accumulated-wander model, in which position errors random-walk and the ceiling goes as the inverse square; it is wrong by a factor of 42 at a two per cent irregularity. What the plot cannot show is what a cloth's spacing irregularity actually is: it has not been measured, and the curve is therefore a prediction with an unmeasured input.

A moiré is a vernier, and it magnifies the error too

Two gratings a per cent apart in pitch beat at a hundred pitches, so a moiré reads a pitch difference at a hundred times — which is what a vernier is. The magnification is free and its ceiling is not: the fringes split when one thread's own spacing error reaches 0.84 of the pitch difference the beat is built on, so the usable gain is 0.84 divided by the cloth's coefficient of variation, inversely and not inverse-squarely. At an ordinary yarn's spacing irregularity, a moiré carries a magnification of five.

pattern · Moire Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

pattern · Moire

Named alongside it

The objects these essays reach for when they reach for this one.

BeatMoireCoefficient of variationPopulationAppearanceDentingMagnificationMeasurementReedSurface patternCoprimeGreatest common divisor

All concepts