Textile Structure
how cloth is put together
The collection

Every essay — page 19

Page 19 of 23, continuing through the fields in the same order.

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Weaves

Plain, twill and satin, and what actually separates them — float length, interlacing count, and a satin that cannot exist on six ends.

What combination adds to the manuals' reach. The three counts. The manuals' own operations on their own seeds reach 9 of the 426 four-by-four cloths. Admitting stripes, checks and figures of any two seeds — 17,787 constructions, of which 177 repeat inside four ends and four picks — takes it to 28, a gain of 19. That is a threefold rise and it leaves 398 cloths unreached, which is 93.4 per cent of the catalogue. What the chart cannot show is combinations of combinations, which are excluded on purpose: the closure's seeds have to be what a chapter actually teaches or the count measures something else.

What combining two weaves reaches

The account before it found that the manuals' own operations on their own basic weaves reach nine of the 426 four-by-four cloths, and recorded one exclusion honestly: combination — striping, checking and figuring — was left out, because a combination of two four-end weaves is eight ends wide and so is not a four-by-four cloth at all. Admitting it triples the reach and leaves ninety-three per cent of the catalogue outside.

6 figures · Derivation The crossover length in four weaves. The gripped length at which a pick of a sheeting breaks rather than slides, in four weaves at a friction coefficient of 0.30. It is a millimetre or two for a plain weave and 8.0 mm for an eight-end satin, which is what a cut edge of each does. The interlacing rate appears in the answer only as a factor outside the logarithm, so the ratios between the four are exactly the ratios of their interlacing rates — in this model and in the sum of independent contacts alike. The rung below expected the float correction to change that ordering; it does not. It changes the size, by a factor of 3.7. What the rows cannot show is that all four use one cloth's crimp, so a satin's genuinely gentler turns are not in them.

A float presses on nothing

The rung below expected the float correction to change how a satin's hold compares with a plain weave's, by something like the ratio of their interlacing rates. It does not change the comparison at all — the interlacing rate leaves the answer outside the logarithm and divides straight out of any ratio. What it changes is the absolute answer, by a factor of four, for every weave alike.

6 figures · Float The 6-end satin. The 6-end satin on point paper, drawn over 2 repeats with its marks joined in reading order inside the first. The joining segments are not parallel and not equal, because there is no number of picks the mark advances by at every end. That is what irregular means, and it is not visible in the squares alone. At this order there is one distinct satin and none is regular. The longest float is 1 in the warp and 5 in the weft, and the cloth is one cloth. What the drawing cannot show is that this is the only one: that is a statement about 36 arrangements and is made by the enumeration.

The six-end satin that does exist

There is no six-end satin, and this collection proved it in its founding essays. The proof is about satins with a move number. Drop that word — keep one mark per end and no two marks touching — and six ends has exactly one satin, unique up to where the repeat is started, and four ends still has none at all.

6 figures · Satin A seersucker in section. A seersucker in section across four stripes, at a feed ratio of 1.30 — the slack warp let off 30 per cent faster than the tight one — over a 6.0 mm stripe. The surplus has nowhere to go in the plane, so it buckles, and the standard small-amplitude result gives 2.09 mm of rise, which is 4.2 times the cloth's own thickness of 0.500 mm. Nothing has to relax for this to appear: unlike a honeycomb, a seersucker comes off the loom already puckered, and washing deepens it rather than creating it. What the drawing cannot show is that the buckle's shape is an assumption — a sinusoid pinned at the stripe's edges — while its amplitude follows from the surplus and the half-wavelength alone.

A seersucker is made at the loom

Every other relief weave in this collection gets its shape after the loom, from a difference of crimp between two regions of a few per cent. A seersucker's surplus is thirty per cent and is put in as the cloth is woven — an order of magnitude more, which the square root turns into a factor of four in depth and no more than that.

6 figures · Texture weaves A 3-pick fold across a 8-end satin, move 3. A 8-end satin, move 3 on point paper with a fold 3 picks wide drawn across it at pick 3. A warp end's crimp is made where it turns from over to under, and the dots mark the turns that fall inside the fold. Averaged over the ends there are 0.75 of them, and 4 of the 8 ends in the repeat have none at all — those ends are marked with a line, and each of them crosses the fold dead straight with no crimp to give up. Over every fold position, 50.0% of end-and-place pairs are like that. What the drawing cannot show is what happens to those ends instead, which is that the whole of the fold's length difference goes into their fibres.

How sharply a weave lets a cloth fold

A fold's length difference is paid for out of crimp, and crimp is not spread evenly along a thread — it is made at the interlacings and nowhere else. So what a fold has to spend is not the weave's average crimp but whatever is inside the few picks the fold crosses, and in an eight-end satin half the warp ends have nothing there at all.

6 figures · Float The 14 drafts whose holes are all one size. Every four-by-four draft in which each end and each pick interlaces — 22874 of them — built and asked whether all sixteen of its holes pass the same thing. At a muslin's construction 14 of them do. Set the same yarn square and 170 do; turn the cloth over and it is 14 again. Only 2 drafts are in all three lists, and they are the plain weave and its complement, marked. The other 12 are uniform because this cloth's warp is set closer than its weft, so the gap across the ends is the smaller of the two and binds every hole whatever the picks are doing — a fact about the sett wearing a fact about the weave's clothes. Two of them carry floats of three, which is as long as this repeat allows.

Only a plain weave has one size of hole

A weave's holes come in kinds, and the kinds are read off the matrix. Asking which weaves have only one kind looks like a question with an obvious answer and a one-line proof. Every draft at four by four was built and asked instead, and the count came back fourteen — of which twelve turn out to be telling the truth about the sett rather than about the weave.

7 figures · Openings A slot and a square of the same area do not pass the same air. A hole of 62500 square micrometres, drawn out from a square to a slot twenty times longer than it is wide, at constant area throughout. The open area is unchanged by construction and the flow is not: it falls to 39 per cent of the square's. Two things move the same way and neither is a correction to the other — the hydraulic diameter falls as the rectangle is drawn out, and the shape factor rises from 14.23 for a square towards 24 for an infinitely thin slit, which is Shah and London's result quoted rather than derived. This is why a weave's float matters to what it passes even where its cover does not: a float lays parallel threads side by side and the hole beside it is a slot.

A satin's hole is a slot

Two cloths at the same cover have the same open area, exactly, and do not pass the same air. A float lays parallel threads side by side and the hole beside them is long rather than square, and drawing a hole out at constant area cuts what it passes to two fifths — because the hydraulic diameter falls and the shape factor climbs, and both of them move the same way.

6 figures · Float A plain weave with one end missing. A plain weave on the left and the same cloth with one end broken and not pieced up on the right, drawn over 2 repeats so that the fault can be seen as the cloth has it: absent from every repeat, for the whole length of the piece. The picks that were held by the missing end are now held by whatever is on either side of it, so the longest float across the ends goes from 1 to 3 end widths — measured in the width the cloth had rather than in the narrower repeat, because the place the end used to occupy is still there. Every pick still changes side somewhere, so the cloth holds together — which is what happens in 55% of all the ways a four-by-four draft can lose an end.

What a missing end does to the weave

Every four-by-four draft there is, with each of its four ends taken away in turn: ninety-one thousand cloths, and not one of them falls into layers. What happens instead is worse, and the criterion has never had to report it before.

7 figures · Integrity One pick of a 2/2 twill made in the wrong shed. Pick 2 of a 2/2 twill laid in the shed belonging to another pick. The thread is there, it is beaten up in its place, and it is simply not the pick the design asked for — so the fault is a bar the whole width of the cloth and one pick deep. This one leaves the cloth sound, with its longest float at 3. Across every four-by-four draft and every possible wrong shed — 1,372,440 substitutions — 63.2% leave a cloth that still hangs together, 36.6% leave a thread loose, and 0.25% split the cloth. The same wrong shed is harmless in one draft and destroys another, so nothing about the size of the mistake predicts the size of the fault.

A mispick is one row in the wrong place

Every four-by-four draft, with each of its picks replaced by every shed the loom could have made instead: a million and a third substitutions. Two thirds leave a cloth that still hangs together, a third leave a thread held by nothing, and the same wrong shed is harmless in one draft and fatal in another.

7 figures · Derivation The float a point fault gives a weave, over every draft there is. Every four-by-four draft that describes one cloth — 22,730 of them — laid out as 8 ends by 8 picks, with each intersection of its repeat reversed in turn and the longest float that produces kept. There are two answers and no others. 90 drafts hold the fault to a float of 3; 22,640 — 99.6% of every weave there is — hand it a float of 7. The mechanism behind the second number is that a reversal at a binding point joins the two floats on either side of it, so a weave with runs separated by single binding intersections gives a single mistake the sum of two of its own floats. The 90 that escape are the drafts whose floats are short and evenly spaced — the plain weave and the ribs — which is not the class of weave the trade recommends for hiding a fault.

Which weave hides a fault

The trade says a busy weave hides a mistake. Every four-by-four draft there is was laid out as cloth and given every wrong lift it could have, and ninety-nine in a hundred hand that single mistake a float of seven — because a reversal does not lengthen a float, it joins two.

6 figures · Float Z twist at 1600 turns per metre. A 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 40.0°, and it is the only quantity in this family: the yarn is 13.8% shorter than the fibre in it and carries 59% of the strength the same fibre would give lying straight.

The other crepe is in the yarn

A crepe weave puts the texture in the matrix. A crepe yarn puts it nowhere the matrix can see: the cloth is a plain weave, and the surface comes from a thread twisted so hard that it shortens by a seventh and spends the rest of its life trying to untwist.

6 figures · Texture weaves A round thread reflects a beam into a fan. A round thread seen end-on, with a beam arriving from straight above. Every point across the thread has its own normal, tilted by the angle it sits at, and mirrors the beam through twice that angle — so a single direction in becomes a whole fan out, spread across the thread and not at all along it. Seen from the front that fan is a highlight lying along the thread, exactly as long as the length of thread that is straight, which is the float. Nothing about this depends on the fibre. A cylinder of any material returns a fan, and the only way to narrow it is to stop the section being a cylinder — which is what a calender does.

A float reflects into a line

A cylinder cannot return a beam to a point. Its normals sweep the whole half-turn across it and nothing at all along it, so a straight thread throws light into a fan — seen as a highlight lying along the thread, exactly as long as the length of thread that is straight. The lobe of a satin is twice as narrow along the thread as across it; the lobe of a plain weave is exactly round.

6 figures · Shine How much every four-by-four draft can shine. All 22,874 four-by-four drafts in which every end and every pick interlaces, at sheeting's construction and a tolerance of 2°, counted by specular area. The range runs from 0.02% to 0.93%, a factor of 60.0, and the distribution is not smooth — it clusters, because the quantity behind it is a count of whole crossings and takes only certain values. The dullest drafts in the catalogue are the plain weaves, which have no plateau at all and shine only from the crowns of their turns; the brightest carry the most float on the face, with the fewest turns interrupting it. Lustre over the catalogue is a length census, and nothing about the yarn enters it.

Lustre is a length times a width

The specular area of a cloth factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply. Neither factor knows anything about the other, and over the four-by-four catalogue the first alone spans a factor of sixty.

6 figures · Shine A calender multiplies the highlight by 32, and all of it is width. A 2/2 twill in sheeting pressed at increasing force, with the specular area recomputed at each state from the standard compression model. It rises from 0.91% of the plan to 28.9%, a factor of 32, while the cloth thins from 381.6 µm to 186.1 µm. The gain is not in the length of the crowns: that moves by 5 per cent. It is in their width, which moves by 31.8 times, because pressing puts a flat top on the section and a flat top has one normal rather than a fan of them. The finish does not polish the thread. It changes the dimension of the highlight, from a line to a band, and the arithmetic says so by refusing to put any of the gain in the other factor.

A calender buys the width

Press a cloth and its lustre multiplies by twenty-four. None of that comes from the length of its crowns, which moves by six per cent; all of it comes from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refuses to put any of the gain in the other factor.

7 figures · Shine Turn the cloth and the highlight changes hands. A 2/2 twill in sheeting turned under a light, with the specular area of each system counted separately at each angle. The warp peaks at 0° and the weft at 83°, a quarter turn apart, and neither returns anything worth seeing where the other peaks. The reason needs no dye and no interference: a warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. A cloth woven with one colour in the warp and another in the weft therefore shows one colour at one angle and the other a quarter turn away, which is the whole of shot silk — a geometric effect that has been sold as a mysterious one for three hundred years.

Turn the cloth and the shine changes hands

A warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. Turn the cloth a quarter turn and the weft takes over. That is the whole of shot silk — a geometric effect with no dye that changes and no interference in it.

6 figures · Shine A crepe's search has 4,416 winners and the surface separates them. All 5,040 rearrangements of the base this collection's crepe is built on, scored by how unevenly their crown line is spread over the repeat. 4,416 of them reach the correlation floor, which is the criterion the crepe was chosen by — so that criterion is not choosing, it is tying, and the search takes the first of a very large set. 28 of the rearrangements have a perfectly even surface, the bar at zero, and 16 of those are also at the correlation floor. The crepe actually drawn, marked, sits at 0.236 — the thirty-eighth percentile, better than most and not at the floor. The improvement is available, it costs nothing, and no criterion this collection had could see it.

A crepe is flat in its draft and not in its surface

A crepe weave is chosen by pushing the draft's correlations as flat as they will go. That criterion turns out to tie: on the base this collection uses, 4,416 of the 5,040 rearrangements reach the floor. Sixteen of them additionally spread their crown line perfectly evenly — and the crepe actually drawn is not one of the sixteen.

6 figures · Texture weaves A shot effect needs a fibre with no ends. The peak-to-trough contrast of an eight-end satin in sheeting as the cloth is turned in the light, against how much hair stands on it. Bare, the contrast is 37 to one, because a straight thread's normals lie in the plane across it and the warp and the weft therefore reflect a quarter turn apart. A hair layer does two things and only one of them matters: it blocks, which takes the same factor off the peak and the trough and changes no contrast at all, and it returns light of its own, which is added to both. A hair population points every way at once, so its return has no azimuth in it — and adding a constant to both ends of a ratio of 37 destroys the ratio. On an ordinary spun cotton the contrast is already down to 5.1 to one; singeing recovers it to 29; raising kills it outright at 1.00. The one fibre with no staple length is the one fibre with no fibre ends, and every shot fabric ever woven is made of one.

A hair layer veils a highlight

An eight-end satin's shine swings by a large factor as the cloth is turned, because a straight thread's normals lie in the plane across it. Put fibre ends on it and the swing disappears — not because the hairs block the light, which changes no contrast at all, but because they return light of their own that has no direction in it.

6 figures · Shine A woven crossing, and the number that never changes. A warp end and a weft pick at 6% crimp, drawn with the thickness expanded three times so the interlacing can be seen. Each goes over its neighbour and comes back; neither passes through the other. The Gauss linking integral over the pair, closed far outside the crossing, returns 0.0000. It returns that at every crimp and for every weave, because crimp moves a thread up and down across its neighbour and a curve that goes over and comes back has done nothing a linking number can see.

A woven cloth is not linked at all

Every thread in every woven cloth passes over its neighbours and comes back. None of them passes through. So the linking number of any two threads in any weave is zero, at any crimp, permanently — and almost everything a cloth does that a knitted fabric does not follows from that one number being nought.

7 figures · Topology A woven cloth asked the same question, and the answer is nearly one. The closest approach two crossing threads make, for four cloths from an open voile to a dense duck, in units of the separation they have where they touch. A value of one means the closest approach is exactly at the crossing and the cloth fits together; anything below one is an overlap. The values run from 1.000 to 0.955, so the worst overlap in the table is 4.5% of a contact separation — against 22% for a knitted fabric. The overlap rises with the crimp, which is what identifies the mechanism: the vertical gain from moving away from a crossing is the crimp, and a cloth that barely crimps has nothing to gain by moving.

A woven cloth asked the same question

A knitted fabric's two adjacent courses occupy the same space by a fifth of a diameter. A woven cloth's two systems overlap by nothing at all in an open cloth and by four and a half per cent in a dense one — and the difference is that they cross rather than run alongside.

7 figures · Contact What a group of 2 threads behaves as. A group of 2 threads of 250 µm with nothing separating them, beside the single thread of the same width and the single thread of the same yarn, all drawn to one scale. The bars are the group's four readings as ratios. Cover is a width and adds, so the group covers exactly what a 500 µm thread would — with 50 per cent of its yarn. Bending rigidity is a second moment and does not add: 2 threads free to slide give 12.5 per cent of the thick thread's and the same 2 fused into one body give 62.5, so a real group is somewhere between and where depends on friction. Against the thread of the same yarn the two readings point opposite ways: the group covers 1.41 times as much and bends 0.50 times as stiffly if free. What the drawing cannot show is the friction that decides where between the two limits a finished cloth sits.

A group is one thread for cover and two for bending

The rung below leaves a limitation standing: two ends with nothing between them lie touching, and whether they behave as one thread of twice the diameter was said to depend on twist, hairiness and finish. Three of the four measures have exact answers with no friction in them and no two agree. A pair covers exactly what a double-diameter thread covers, with half its yarn, and bends at between an eighth and five eighths of its rigidity — and at equal yarn it covers forty per cent more and bends half as stiffly.

5 figures · Rib weaves A warp rib's cord, at doublings of 2 and 4. One warp end drawn in section along the cloth, over the pick groups of a warp rib, at doublings of 2 and 4 and at one scale. The cord's crests are the pick groups and its two dimensions come from different places. The height is the weft's own crimp amplitude and Peirce's closure condition caps it at the two yarn diameters together — 500 µm here — so it runs 304 µm at a doubling of 2 and 361 at 4, which is 61 and 72 per cent of the ceiling. The pitch is the doubling times the pick spacing and has no ceiling at all. So a larger doubling gives a taller cord and a wider one, and wider faster: the aspect falls from 0.40 to 0.29. What the section cannot show is what a finish does, which flattens the cord without changing either the pitch or the ceiling.

A cord's height has a ceiling and its width has none

A warp rib's cord is a wave, and its two dimensions come from two different places. The height is the weft's own crimp amplitude, and Peirce's closure condition caps it at the two yarn diameters together — 500 µm for a quarter-millimetre yarn, of which a 2/2 rib reaches 304 and a 6/6 rib 391. The pitch is the doubling times the pick spacing and has no cap at all. So a bolder rib is taller and wider, and wider faster: the aspect falls from 0.41 to 0.22.

5 figures · Rib weaves Grouping the two systems by different amounts. Plain weave with its ends grouped by one number and its picks by another, over a grid of both. The four weaves the trade names are the corners of this space — plain at one and one, a warp rib down the first column, a weft rib along the first row, a hopsack on the diagonal — and the interior is the oblong matt, which has a name and no literature. Every cell weaves on two shafts, so the harness cannot tell any of them apart; the fundamental domain is exactly 2ab, so the notation's cost is the product; and the longest float is the larger of the two groupings. The two densest setts move with the two groupings separately, so the sett ratio is one exactly on the diagonal and nowhere else — a 3×1 matt sets at 1.50 and its transpose at the reciprocal. What the grid cannot show is the cord: the diagonal has no directional relief at all and everything off it does, in the direction of the larger grouping.

The four named weaves are corners of a family

Plain, warp rib, weft rib and hopsack are one construction with two knobs, and the trade turns both together or neither. Group the ends by two and the picks by three and the result is an ordinary cloth with a name, no literature and a fundamental domain of twelve intersections on two shafts — and the family's three quantities all have closed forms: two shafts everywhere, a unit of exactly 2ab, and a longest float of the larger grouping.

5 figures · Rib weaves How hard each grouping holds its own threads. The grip a cloth has on one of its own threads inside a 10 mm seam allowance, for six members of the doubled family at 24 threads per centimetre and a friction coefficient of 0.3. Grip accumulates multiplicatively at every crossing — the capstan equation on Peirce's own weave angle — so it is exponential in the crossings, and the crossings are the interlacing rate times the intersections in the allowance. The family's interlacing rate has a closed form, (a + b)/2ab, which is half the sum of the two reciprocals — so the two groupings enter symmetrically and each one saturates. The dashed line is the thread's own strength: a cloth whose grip falls short of it lets the thread slide out rather than break, which is seam slippage. A 2/2 hopsack is below it at this allowance and a 1/4 warp rib is above, on cloths whose firmness differs by an eighth. What the bars cannot show is the friction coefficient, which is measured and is not a constant of cloth; the ordering holds at every value anybody reports and the sizes do not.

What nothing separates comes out together

The doubled family's interlacing rate has a closed form — half the sum of the two groupings' reciprocals — so a seam's grip on its own threads is the exponential of a harmonic mean, and it saturates in each grouping separately. At a ten-millimetre allowance a 2/2 hopsack holds a thread at seventy-four times the applied tension and a 1/4 warp rib at two hundred and eighteen, on cloths whose firmness differs by an eighth.

5 figures · Rib weaves The lattice under 8/3 and 10/3. Satin marks drawn as points over two repeats, the 8-end satin on a move of 3, whose closest marks are √8 apart and next √10, so its marks line up 45° off the weft; and the 10-end satin on a move of 3, whose closest marks are √10 apart and next √10, two equal directions at right angles and so no single diagonal. For a regular satin the blue arrow is the shortest lattice vector and the red the next, and the faint lines run along the shortest through every mark — the diagonal the marks make. What the drawing cannot show is whether an eye finds that diagonal in woven cloth, where the marks are not points but short interruptions of a float, and where the yarn's own twist lies across them at an angle of its own.

Most satins still have a diagonal

A satin is chosen so that no diagonal forms, and its move is ranked by how far apart its interlacings sit. But the interlacings of a regular satin lie on a lattice, every lattice has a shortest step, and the marks line up along it. Only when two shortest steps tie is there no row to follow — and between five and forty ends the best move manages that at twelve of the thirty-five orders.

8 figures · Satin