Textile Structure
how cloth is put together
Concept

Enumeration — where it appears

Running a count rather than quoting one, which on this site means every draft of a size rather than a sample. The distinction matters because a quoted count cannot disagree with a change to the code that produced it and a run one can.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

A spread shading on 8 ends. The 7 tone steps of a spread shading on an 8-end repeat, each built by adding one more coset of the 8-end satin with move 3. Every coset has exactly one mark in every end and every pick, so the fraction of warp on the face is k over 8 exactly at every step, with no averaging in it. The numbers beneath are the longest float, and they run 7, 3, 3, 1, 3, 3, 7 — so the lustre scale is not the tone scale. The midtone is a plain weave and the extremes are satins, which is why a spread shading is at its most matt exactly in the middle. What the drafts cannot show is the surface: a long float stands proud of a short one, so an evenly toned shading is not an even surface either.

A shading changes two things at once

The tone steps of a shaded damask are exactly even — each adds one satin coset, so the fraction of warp on the face is k over n with no averaging in it. The lustre steps are not even at all: on eight ends the longest float runs 7, 3, 3, 1, 3, 3, 7, so a series that grades smoothly in tone is at its most matt exactly in the middle.

pattern · Shading 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.

weaves · Satin 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.

weaves · Openings 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.

weaves · 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.

weaves · 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.

weaves · Float The cyclic decomposition of a 4-end repeat. A 4-end repeat split into 4 parts, each with exactly one warp mark in every end and every pick, drawn above with each intersection numbered by the part it belongs to. That object is a Latin square, and it is what a shading actually requires: a tone step is a union of parts, so it has exactly k marks in every end and pick and its tone is k over 4 exactly. This is the cyclic square, which is what a satin's cosets write — and at four and six ends there is no satin, so the same square has to be reached through a twill instead. The drafts below are the tone steps in the best order this square admits, whose longest floats run 3, 1, 3. What the drawing cannot show is that the numbering is arbitrary: relabelling the parts gives the same square and a different chain, which is exactly the freedom the order is chosen out of.

A tone step does not need a satin

Every account of shading builds its tone steps out of satin cosets, and at four ends and at six there is no satin to build them from. The construction was never about satins: what a tone step actually needs is that every end and every pick carry the same number of marks, which makes a chain of them a Latin square. A four-end repeat has twenty-four of those and a six-end repeat 1,128,960.

pattern · Shading One tone of an 8-end cell, arranged three ways. The same 32 marks in the same 8 × 8 cell, placed three ways, with the number of marks in each end printed beneath it. On the left the clustered dot a halftone screen makes: 4 of its threads carry every mark or none, so they never leave a face, and the criterion reports 16 separable layers rather than one cloth. In the middle a cloth built from the tone step by moving 2 marks sideways within their own picks — every pick still carries 4, the ends run 3 to 5, and that difference is a warp stripe of 25.0% contrast the design did not draw. On the right the tone step: every end and every pick at exactly 4. What the drawing cannot show is how visible the middle one's stripe is, which depends on the sett and on the viewing distance and is not computed here.

A weave is a halftone screen with n greys

An eight-by-eight cell of dots gives a printer sixty-five levels of grey. The same cell in cloth gives seven. Sixteen of the missing fifty-eight go to the requirement that every thread reach both faces and forty-two go to the requirement that every thread carry the same number of marks — so evenness, not interlacing, is what a weave pays for its tone scale.

pattern · Shading What a profile draft can reach at 4 by 4. The share of the 22,874 interlacing 4-by-4 drafts that a profile draft can express, at two block sizes. A profile is a grid of blocks each carrying a figure weave or a ground weave, so its image is every draft reachable by any choice of the two weaves and any assignment — which is enumerated here rather than argued: 4,096 combinations at the larger block, and the distinct results counted. With two-by-two blocks it reaches 306 drafts, which is 1.34 per cent. With one-by-one blocks the profile is the draft and it reaches all of them, which is the control. What the bars cannot show is that the reachable drafts are the useful ones: every figured cloth ever woven is in the small set, and the notation is narrow because designs are.

A profile draft is a notation whose alphabet is weaves

The rung below measured four notations for a single weave and left open the notations for something larger. A profile draft is the first of them: a grid of blocks, each carrying a figure weave or a ground weave. Its image is enumerable and it is tiny — every pair of two-by-two weaves against every assignment of two-by-two blocks reaches 306 of the 22,874 interlacing four-by-four drafts, which is 1.34 per cent. And the 306 are the ones anybody weaves.

cloth · Notation Every 6-end decomposition, by the float its best chain holds the middle tones to. All 1,128,960 Latin squares of order 6 with their first row in order — every way of splitting a 6-end repeat into 6 parts with one mark in every end and every pick — each asked for its chains of tone steps. 2,816 can hold every tone between the extremes to a float of 2, and they fall into 64 classes once the repeat's starting corner and reading direction are set aside, the cyclic square a twill writes among them; 800,658 cannot do better than 4. 576 can put a plain weave at the midtone, and every chain that does floats three on either side of it. Every one of the 434,540 distinct tone steps met is one cloth. What the rows cannot show is which of the classes a designer would choose, since the float profile is one criterion among several.

A six-end shading can be even or have a plain centre, not both

A six-end repeat can be split into the parts a shading is built from in 1,128,960 ways, and every one of them has now been walked. Only 2,816 — sixty-four distinct shadings — hold every tone between the extremes to a float of two, and seven in ten cannot do better than four. Five hundred and seventy-six can put a plain weave at the midtone, and not one of those can keep twos beside it: taking a part out of a plain weave, or adding one, always leaves a float of three.

pattern · Shading The draft for 2/2 twill, as a loom holds it. The 2/2 twill written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

A lifting plan says nothing without a threading

The second of the notations for something larger than a weave is a pair, not a notation: a threading and a lifting plan, and neither alone expresses anything. The pair's image is exactly the drafts with no more distinct columns than there are shafts — 98 at two shafts, 5,282 at three, all 22,874 at four — and it is not nested with the profile draft's in either direction. The profile reaches 192 drafts that need all four shafts, and misses 64 of the 98 a two-shaft loom weaves.

cloth · Notation How many cloths any one cloth derives into. The 426 four-by-four cloths sorted into the orbits the manuals' derivations cut them into. 12 orbits hold 1 cloth; 83 orbits hold 2 cloths; 62 orbits hold 4 cloths. The largest orbit in the whole catalogue holds 4, so no cloth derives into more than 3 others by any sequence of the named operations, however long. The derivations generate a group of 256 elements and it cuts the catalogue into 157 pieces. What the bars cannot show is which cloths are in which orbit, which is the next figure.

No cloth derives into more than three others

Every weaving manual opens by saying the three basic weaves generate the rest. This collection counted the reach and found nine of 426, and left the nine as a count. It is not a count: every derivation the manuals name is a relabelling of the grid or a complementation of it, both invertible, so they generate a group — and that group cuts the 426 cloths into 157 closed pieces of which the largest holds four. The claim is not merely wrong about how much derivation reaches; derivation cannot reach more than four cloths from anywhere, by any sequence of operations, however long.

weaves · Derivation Which satin orders are row-free, from 5 ends to 40. Every satin order from 5 to 40, marked where the best move's lattice has two shortest steps of equal length rather than one — which is the condition under which the interlacings do not line up into a row. The row-free orders are 5, 10, 13, 15, 17, 24, 25, 26, 29, 34, 35, 37: twelve of the 35 orders that admit a regular satin at all. An n-end satin floats over n − 1, so a float limit is a ceiling on the order, and the ceilings for limits of 8, 12, 16 are drawn. What the strip cannot show is the spread, by which the orders are ranked and which decides which move is best within each.

A float limit leaves one row-free satin

A satin is chosen so that no diagonal forms, and an earlier essay found that most of them fail: the interlacings lie on a lattice, every lattice has a shortest step, and the marks line up along it unless two steps tie — which happens at twelve of the thirty-five orders from five to forty. The other constraint was named and not applied. An n-end satin floats over n − 1, so a yarn that will not carry a float longer than eight admits four orders in all, and exactly one of them is row-free: the five-end satin, which is the one everybody already weaves.

weaves · Satin What an irregular satin buys, order by order. For each order, the best regular satin's and the best irregular satin's scatter at the order's own best spread — the largest share of the closest pairs that point in one direction, where one is a line and less is a scatter. 5 ends: regular 0.50, irregular none at the best spread; 6 ends: regular none exists, irregular 0.25; 7 ends: regular 1.00, irregular 0.33; 8 ends: regular 1.00, irregular none at the best spread; 9 ends: regular 1.00, irregular 0.25; 10 ends: regular 0.50, irregular none at the best spread; 11 ends: regular 1.00, irregular none at the best spread. Irregularity buys something at 6, 7, 9 and nothing at the rest, and where it buys it scatters over four directions with no more than a third in any one. What the bars cannot show is whether a reader sees the difference, which is a question about a visual system.

An irregular satin scatters where a regular one lines up

A regular satin's marks lie on a lattice, so its closest pairs all run along one vector and make a row. An irregular satin has no lattice at all, so its closest pairs may point several ways at once — and at seven and nine ends, where every regular satin at the best spread has a row, an irregular one reaches the same spread with its closest pairs scattered over four directions and no more than a third in any one. At eight and eleven ends there is no such satin: the best spread is reached by regular satins alone, and irregularity has nothing to offer.

weaves · Satin

Named alongside it

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

CensusFloat lengthSatinWeave matrixFloatIntegrityMove numberShadingToneFaultFirmnessLatin square

All concepts