Textile Structure
how cloth is put together
Generator

What each way of getting at the catalogue reaches

What each way of getting at the catalogue reaches
What each way of getting at the catalogue reaches. The 426 four-by-four cloths as an area, with three subsets drawn inside it to scale: the 110 a loom of 3 shafts and 4 treadles can weave, the 9 the manuals' derivations reach, and the 5 that have a fraction name. Areas are proportional to counts.

The 426 four-by-four cloths as an area, with three subsets drawn inside it to scale: the 110 a loom of 3 shafts and 4 treadles can weave, the 9 the manuals' derivations reach, and the 5 that have a fraction name. Areas are proportional to counts.

9 essays call derive-count. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Every figure listed here is drawn from this one rule.

What four notations can say. Four ways of writing a weave down, with how many of the 426 four-by-four cloths each can express and whether expressing one identifies it: point paper, 426, identifies; a four-part draft on 3 shafts and 4 treadles, 110, identifies; a fraction name, 5, does not identify; a longest-float specification, 426, does not identify. A longest-float specification names every cloth and separates them into only 3 classes. What cloth is

Four ways to write a weave down

A weave's name in the trade is a pair of numbers — 2/2, 3/1, five-end satin — and the notation is so universal that it is easy to forget it is a notation. It has an image and the image is computable: of the 426 cloths at four by four, five have a name of that kind, and there are four names for the five.

 The basic weaves at four by four. Plain weave and the three twills a repeat of four admits, each drawn with its longest float and its layer count computed from the matrix. The fifth frame is empty: a satin needs a move coprime with its order and neither one nor one less than it, and four ends admits 0 such moves. So the smallest interesting repeat contains two of the three weaves every manual begins with. Weaves

The three basic weaves do not generate the rest

Every weaving manual opens with the same sentence: there are three basic weaves, and everything else is derived from them. The complete catalogue of the smallest interesting repeat is in hand, so the claim can be checked instead of repeated. Starting from plain weave, every twill and every satin, and applying every derivation the manuals name, reaches nine of the 426 cloths that exist there.

 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. Weaves

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.

 Which specifications a 30 tex yarn can be woven into. Warp sett across against weft sett down, in threads a centimetre, for a 30 tex cotton yarn. Each cell is one specification. 220 are left empty because no cloth in the catalogue can be woven at those two setts; 16 are filled and outlined in the warp's colour because every one of the 426 can; and 20 are filled and outlined in the float's colour because some can and some cannot. The admissible region is a rectangle because the warp sett and the weft sett are bounded by two different counts. What cloth is

A specification can name a cloth that is not there

The three notations measured so far write drafts, and every draft is a cloth somebody could weave. A mill works from none of them: it works from a count, two setts and a weave quoted together, and that is the first notation whose image has holes in it. Of 2,560 specifications across the trade's own working range, 2,097 name no cloth at all — and in the 240 where the weave field decides anything, both numbers the trade quotes to decide it rank the catalogue wrongly.

 What one wrong symbol reaches. Five kinds of symbol, with how many of them a document holds and how many intersections of a 1800-by-1800 piece one of them decides. A threading digit reaches 810,000; a jacquard card's hole reaches one. The product of the two columns is 3,240,000 in every row, because each set of symbols decides every intersection of the piece exactly once. What cloth is

The shortest notation has the longest mistakes

Four essays of this account have asked what each notation can express. None has asked what happens when one is written down wrong, and the answer is an identity rather than a tendency: a notation's symbols partition the cloth, so the count of symbols times the reach of one is the piece, in every notation, with nothing to trade. One threading digit decides 810,000 intersections of a square metre and one of a jacquard's holes decides one — and only the threading can be wrong while the cloth is right.

 One wrong threading digit against two. What becomes of a threading when one digit is wrong and when two are, over every draft in the four-by-four sweep: one digit wrong, 252,968 cases, 45.0 per cent refused by the drawdown, 54.8 per cent a different cloth, 0.228 per cent the same cloth; two digits wrong, 1,074,972 cases, 59.1 per cent refused by the drawdown, 39.6 per cent a different cloth, 1.285 per cent the same cloth. What cloth is

A second wrong digit is silent only by cancelling the first

One wrong threading digit leaves the cloth exactly as it was 576 times in 252,968, all on three-shaft drafts. The obvious guess about two was that silence would become the rule, since two changed digits can swap two shafts and swapping shafts is the harness's own freedom. Counted over all 1,074,972 pairs, two wrong digits are silent 1.29 per cent of the time — five and a half times as often, on fifteen times as many cloths — and never by adding one silent slip to another. Every silent pair is two audible errors cancelling, and at two errors point paper and lifting plans, which can never be wrong silently by one symbol, are silent almost as often.

 Silent threading errors under three lifting plans. For every four-shaft threading of eight ends, under three lifting plans: 2/2 twill, 36,320 of 40,824 threadings can be mis-threaded silently by two digits, 0.97% of double errors and 0.20% of single errors are silent; 1/3 twill, 29,728 of 40,824 threadings can be mis-threaded silently by two digits, 0.55% of double errors and 0.20% of single errors are silent; unrelated rows, 864 of 40,824 threadings can be mis-threaded silently by two digits, 0.0084% of double errors and 0.0000% of single errors are silent. A twill's plan slides one row a pick at a time, so its shafts are interchangeable by a slide along the picks, which is a writing of the same cloth. What the chart cannot show is a plan with more shafts. What cloth is

Silence lives in the lifting plan

Two wrong threading digits leave a four-end cloth exactly as it was 1.3 times in a hundred, and the question left was whether a longer repeat makes that rarer or commoner. At eight ends the answer depends on something the question did not name. With a lifting plan of unrelated rows, silence all but vanishes; with a twill's plan — one row slid a pick at a time — nine four-shaft threadings in ten can be silently mis-threaded, and even single errors are silent one time in five hundred. The repeat hardly matters. The plan's symmetry does.

 Which lifting plan exposes a threading error another plan hides. For every four-shaft threading of eight ends and every double threading error silent under the row's lifting plan, the share that the column's plan exposes — weaving a cloth that is caught or visibly different — when the same threading is re-pegged. Silent under the 2/2 twill (99,872 pairs): 2/2 other way 0%, 1/3 twill 44%, 3/1 twill 44%, broken 70%, point 34%, plain 0%, hopsack 34%, unrelated 99%. Silent under the 1/3 twill (56,352 pairs): 2/2 twill 0%, 2/2 other way 0%, 3/1 twill 0%, broken 76%, point 50%, plain 0%, hopsack 47%, unrelated 98%. Silent under the broken twill (29,888 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 55%, 3/1 twill 55%, point 0%, plain 0%, hopsack 53%, unrelated 97%. Silent under the point twill (66,208 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 57%, 3/1 twill 57%, broken 55%, plain 0%, hopsack 47%, unrelated 99%. A zero means the column's plan hides every error the row's does; the plan of unrelated rows exposes all but the 864 pairs that are the threading itself started elsewhere. What the table cannot show is which plans a given mill pegs one warp with. What cloth is

A proof plan has no slide in it

A threading error that a 2/2 twill hides is invisible in the cloth and still wrong on the loom, and it shows only when the loom is re-pegged. Re-weaving every silent error under eight other plans says which do the showing. The twill run the other way and plain weave never expose one. A 1/3 twill exposes 44 per cent, a broken twill 70, and a plan of eight unrelated rows all but 864 — and those 864 are not errors at all, because each is the right threading started at another end. So the plans nest: whatever a broken or a point twill hides, the 2/2 hides too, and a threading proved under a 2/2 twill has passed the weakest test there is.

 Twelve eight-shaft lifting plans' silent errors, nested. The double threading errors silent under twelve eight-shaft lifting plans, over every threading that puts one end on each of eight shafts: 1/7 twill, 2/6 twill, 3/5 twill, 1/2/3/2 twill, satin, move 3, satin, move 5: 1,536; 4/4 twill: 1,888; 3/1/1/3 twill: 448; 2/1/1/4 twill: 256; broken 4/4 twill: 384; point 4/4 twill: 576; unrelated rows: 0. Plans in one box hide exactly the same errors. An arrow means every error silent under the first is silent under the second: 1/7 twill into 4/4 twill, 3/1/1/3 twill into 4/4 twill, 2/1/1/4 twill into 1/7 twill, 2/1/1/4 twill into 3/1/1/3 twill, broken 4/4 twill into point 4/4 twill, point 4/4 twill into 4/4 twill. The order is not total — the 3/1/1/3 twill and the satins each hide errors the other does not — but it has a top, the 4/4 twill, which hides every error any of the others hides. The plan of unrelated rows hides none. What cloth is

An eight-shaft error hides in the row's mirror

At four shafts the lifting plans nest, with the 2/2 twill hiding every threading error the others hide. An eight-shaft loom has satins as well as twills, and their slides are different slides, so the order was expected to break. It does not. Over every threading that puts one end on each of eight shafts, the 4/4 twill hides every double error any of eleven other plans hides, and a satin hides exactly what a 1/7 twill hides, error for error. What decides it is not the slide. Every silent error is an exchange that turns the threading into a copy of itself, the slide accounts for one in six of them, and the rest hide in the row's mirror.

The whole library · All essays