Textile Structure
how cloth is put together
The collection

Every essay — page 18

Page 18 of 23, continuing through the fields in the same order.

What cloth is Weaves Setting and geometry Knits and other structures Mechanics and drape Pattern and colour Compound and figured cloths After the loom Cloth doing a job

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Setting and geometry

Yarn diameter, crimp and cover. How close threads can be set, why thread count is not what it is taken for, and which model produced any number.

A jersey's bending length against its tightness. The wale-wise bending length of plain jersey in the fully relaxed state at 10, 20, 40 tex, across the whole tightness band from 1.3 to 1.6, at the two ends of the yarn's stiffness bracket. At the free bound every count and every tightness gives 10.13 mm. At the locked bound the lines separate by count — 55 mm at 10 tex, 70 mm at 20 tex, 88 mm at 40 tex — and are still flat. The weights along the lines run from 97 to 239 g/m². What the plot cannot show is friction between the loops, which is where a tight jersey's firmness has to come from.

A jersey's drape does not know its loop

A woven cloth's bending length at the free bound turned out to hold neither its count nor its weight — only the fibre. A plain jersey goes further. Its stiffness per width and its weight per area are both a number of loops per millimetre times something about one loop, so the loop cancels at every stiffness the yarn could have; at the free bound the count cancels too, and what is left is the fibre and which relaxed state the fabric is in.

6 figures · Weight How likely one staple length of yarn is to be weaker than a tension, under three laws. The probability that one independent try — a staple length of a 20 tex cotton yarn — is weaker than a tension, on a logarithmic scale, with the tension as a fraction of the yarn's 500 mm breaking load. Three laws are drawn, a normal, a lognormal and a Weibull, each fitted to reproduce the same two tensile tests at 100 and 500 mm exactly. Near the breaking load, where a tensile test samples at about one chance in 18, they lie together. At one chance in 1.4×10⁸, where a warp of 4,000 ends by 1,000 metres samples, they are far apart: at the back shaft's 52 per cent the three give normal 1.3×10⁻⁷, lognormal 1.1×10⁻¹¹, Weibull 4.0×10⁻⁶ per try. What the chart cannot show is which of the three a real yarn follows, which no tensile test can decide.

A warp breaks in a tail no tensile test reaches

A 500 mm tensile test is the weakest of about eighteen staple lengths of yarn. A warp of four thousand ends by a thousand metres is the weakest of a hundred and forty million, and an end breaks wherever one of them is weaker than the shed's tension. Three strength laws fitted to the same two tensile tests agree within five per cent at every gauge a tester can clamp, and disagree by a factor of 375,000 on how many ends a warp will break at the back shaft.

6 figures · Assembly What a 2-fold yarn weighs against the sum of its singles. The count of a 2-fold cotton yarn against the sum of its singles' counts, as the folding twist runs from three tenths to nine tenths of the singles' twist, at twist factors of 3000, 3800, 4500 (turns a metre times the root of the tex). Below a ratio near two thirds the fold is lighter than the sum, above it heavier; the curves cross the sum at 0.661 at 3000, 0.658 at 3800, 0.654 at 4500, and in the small-angle limit at exactly 0.6667. The trade's folding band, 0.6 to 0.75, is shaded, and across it the fold is within −0.6% and +1.3% of its notation. What the curves cannot show is a real fibre's migration between the core and the surface of each single, which the ideal helix leaves out and which shortens the path both lengths are computed from.

Two thirds is where a fold weighs its singles

A folded yarn's count is written as its singles' count divided by the number folded, as though folding changed no length. It changes two. The fold's helix wraps each single round the yarn's axis, which puts more single into every metre, and the untwisting straightens each single's fibres, which puts less fibre into every metre of single. Each is three to seven per cent at a weaving twist. They cancel at a folding ratio that is exactly two thirds in the small-angle limit — the folder's own number — and six elevenths for three singles and two fifths for four.

7 figures · Count What a 2×2 cable weighs as its cable twist rises. A cotton cable of 2 folds, each of 2 singles, with the folds set at 0.707 of the singles' twist and the cable twisted the singles' way at ratios from nought to 0.60, at twist factors of 3000, 3800, 4500. A little cable twist makes the cable lighter, because it takes twist out of folds set above their own crossing; then the cable's own helix wins and the count climbs. It crosses the singles' sum at 0.260 at 3000, 0.250 at 3800, 0.239 at 4500. At the rule as quoted, a cable twist of 0.500, the cable is +2.21%, +3.54%, +4.95% heavy; at the rule read in the finished thread, 0.293, it is +0.16%, +0.32%, +0.54%. What the curves cannot show is a real cable's folds pressed out of round, which shortens the cable's helix and moves every crossing right.

A cable weighs its singles at the twists the finished thread keeps

Cabling untwists every fold and hands that twist back to the singles inside it, so beneath its own helix a cable is a fold folded at the difference of two twists — and the cable's helix is a length nothing below it can pay for. Twisted where the cabling rule is quoted, a two-by-two weighs three and a half per cent over its notation. Read in the finished thread, with every level's surface at one angle, the same rule sets the folding frame exactly where a plain fold's is and the cable at 0.293, and the cable comes back to within half a per cent.

7 figures · Count What a thread of twos weighs as its levels of twist are added. A two-fold, a cable of two two-folds, a cord of two such cables and a cord of two cords, all of 20 tex cotton singles, each twisted so that every level's surface sits at one angle in the finished thread, at twist factors of 3000, 3800, 4500. At 3,800 they weigh fold −1.35%, cable +0.32%, cord +2.02%, cord of cords +3.75% against the sum of their singles: a fold light, a cable on its notation, and every level after it one helix heavier than the last. Dashed, the same four twisted to the rule as quoted on the frames, +0.46%, +3.54%, +4.54%, +6.98%. What the chart cannot show is the folds' and cables' real shape, which is taken round.

The singles pay once and every level adds a helix

A cable twisted so that every level's surface sits at one angle weighs its notation, and it is tempting to read that as a law: equal angles, honest count. It is not one. With every surface at one angle the singles keep the same twist however many levels sit above them, so their saving is paid once, while every level adds a helix of the same length. A two-fold comes out light, a cable on its notation, and a cord of two such cables two per cent heavy — the cable's agreement is the walk passing through zero. For a cord to weigh its notation, everything above its singles has to sit at about half their angle.

7 figures · Count

Weaves

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

The 5-end satin. The 5-end satin on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.

Plain, twill and satin

Three rules, and everything else in weaving is a variation on them. What separates the three is not appearance but one trade — how often a thread changes face against how far it runs when it does not.

6 figures · Foundation weaves Where the floats are in the 5-end satin. Every longest run of warp on the face, marked. A float is smooth because nothing interrupts it, which is the same reason it snags: there is a length of thread lying on the surface with nothing holding it down.

The float decides

How far a thread runs on the face before it goes under is one number, and it sits behind lustre, drape, snagging, abrasion, tear strength and how densely the cloth can be set. Almost nothing else in the subject has that reach.

6 figures · Float Which satins exist. For each order, the moves that give a satin — a step coprime with the order, and not the steps of one and one less which give a twill instead. Four and six admit none, so no regular satin exists on four ends or on six.

There is no satin on six ends

Weavers have known it as a rule for centuries. It is a theorem with a one-line proof about common factors, and it rules out four ends as well.

6 figures · Satin What an interlacing costs. Each weave's interlacing count beside the closest it can be set in the same yarn. The two run opposite ways, because a thread that changes face often has to bend often and a bend takes room.

Interlacings and firmness

Every time a thread changes face it has to bend, and a bend takes room. That one sentence decides how densely a cloth can be set, how firm it feels, and why the two run in opposite directions.

6 figures · Firmness The 2/2 twill both ways round. The same twill stepped one end to the right and one end to the left. Every quantity the matrix can produce is identical for the two, so the direction is real in the cloth and invisible in the arithmetic.

Twill direction, and how it is named

The two twills are mirror images, and every quantity the matrix can produce is identical for both. What separates them is the yarn, which has a handedness of its own — and the angle, which is forty-five degrees only in a square-set cloth.

6 figures · Foundation weaves A 2/2 twill reversed — the clean way. A twill reversed at intervals, generated from the rule rather than drawn. The count beside it is the number of adjacent ends that come out identical, which is the defect a weaver sees as a thick line at the seam and which no amount of looking at the grid announces.

Broken and herringbone twills

Reversing a twill is the cheapest way to turn a rule into a figure, and there are two ways to do it that look identical on paper. One of them leaves two adjacent ends doing exactly the same thing.

6 figures · Foundation weaves A double cloth. Two complete fabrics woven in one repeat, and what happens when a single intersection is turned over. The layer count beside the draft is computed from the matrix drawn, and the census below it is every single-square change tried in turn.

Backed and stitched constructions

A double cloth is two fabrics woven at once, and the check that verifies it says exactly two. Turn one intersection over and it says one — which means one mistake is enough to destroy the construction, and the enumeration says half of them would do it.

7 figures · Integrity The highlights of a 8-end satin, move 3. Every unbroken run of warp on the face, drawn as the band of light it returns. A yarn is a cylinder and a cylinder reflects a line rather than a point, so a float's highlight is as long as the float — and how long that is, in inches, is the float divided by the picks per inch the weave can be set to.

Why satin shines

Lustre is usually filed under fibre, and silk gets the credit. It is a property of the weave: a cylinder reflects a line rather than a point, so the highlight a cloth returns is exactly as long as its longest float.

6 figures · Float What one cut costs. The length of thread set loose by a single abrasion cut through a warp float, with each weave measured at the densest setting that weave allows in the same yarn. The bars are inches of freed thread; the note beside each is the float in picks and the setting that float permits.

Floats and abrasion

A satin is said to wear badly. It does not wear quickly — its flat face spreads the rubbing over more thread than a plain weave's crowns do. What it does is fail badly, and those are different quantities moving in opposite directions.

6 figures · Float What the geometry says about a tear. A slit in a woven cloth, with the ends ahead of the tip gathered into the group that will break together. How many can gather is the opening divided by the slack in each gap, and the slack is the thread spacing less the thread diameter — an expression with no weave in it at all.

Does a loose weave tear better

The trade says a twill tears stronger than a plain weave of the same yarn, because fewer interlacings let the threads group. The geometry of that grouping has no weave in it at all — and where the geometry does speak, it predicts the opposite.

6 figures · Float What a float limit of 4 leaves. For each repeat, the fraction of the distinct twills on it whose longest float is within the limit. The constraint is usually stated as a rule about a drawing; it is really a statement about how much of the catalogue exists, and the catalogue shrinks as the repeat grows.

Designing to a float limit

Every jacquard designer works to a rule of the form nothing longer than four. It reads as a constraint on a drawing. It is really a statement about how much of the catalogue exists, and the catalogue shrinks as the repeat grows.

6 figures · Float Which move to use on 16 ends. For each satin move the order admits, the distance from an interlacing to its nearest neighbour, measured on the torus the repeat lives on. A move whose interlacings crowd gives the eye something to find; the one that scatters furthest is the one to weave.

Which satins are worth weaving

Manuals give the moves a satin admits as a list, as though the survivors were interchangeable. They are not. Measure how far apart the interlacings sit and the traditional counter turns out to be the best move at the orders a weaver mostly uses — and to fail first at thirteen, where the square root names four and five scatters further.

6 figures · Satin 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.

How many shafts a draft needs

Every quantity counted so far is a property of the cloth. This one is a property of the loom — the number of distinct columns in the matrix — and it is very probably the strongest single predictor of which of the twenty-two thousand four-by-four drafts anybody ever wove.

8 figures · Shafts How many twills there are. Every way of writing a twill on each repeat, reduced by the two operations that leave the cloth unchanged: starting on a different pick, and turning it over. What is left is the number a designer actually chooses between.

How many twills a repeat admits

Sixty-four ways of writing a twill on eight ends, and twenty-one twills. The difference is two operations that leave the cloth unchanged, and a catalogue that does not quotient by them is counting notations rather than fabrics.

6 figures · Foundation weaves Float lengths in a honeycomb. The same draft twice. On the left, filled where the warp is on the face — which is all point paper says. On the right, every intersection shaded by the length of the float it belongs to, from one at the palest to 6 at the strongest. The gradient on the right is the whole mechanism of a relief weave and it is invisible on the left.

A honeycomb gets its cells in the wash

The obvious mechanism is take-up on the loom, and the arithmetic says it is wrong: every end of a diamond passes through the long floats and the tight ones alike. What is left is finishing, and a cell is a region that wanted to shrink less than the cloth around it.

8 figures · Texture weaves How much a draft agrees with itself. On the left the draft; on the right its correlation at every offset, one cell per offset, with the offset of nothing at the top left. Warp-up counts as plus one and weft-up as minus one, so the number in each cell is agreements minus disagreements out of 64. The correlations away from the origin sum to exactly -64, whatever the draft — structure can be moved about and not removed.

A crepe cannot be structureless

A crepe weave is designed to have no line in it anywhere. The correlations of a draft with itself sum to a number fixed by the repeat alone, so structure can be spread and never removed — and on eight ends the floor turns out to be half the repeat, set by a fact about binary words with nothing textile in it.

6 figures · Texture weaves Mock leno, 3 threads to a bundle. Threads that interlace identically have no weft passing between them, so nothing holds them apart and they lie touching. The reed still sets the average spacing, so the space they leave collects at the bundle's edge — a hole 0.60 by 0.60 mm, made without one thread crossing another. Drawn to scale on a fixed 9 mm square of cloth at 20 threads per centimetre and a 0.3 mm yarn.

A hole with nothing crossing

A real leno holds its holes open by crossing one thread over another, which is a topological arrangement and cannot come undone. A mock leno makes the same holes by grouping threads that nothing separates, and everything about it is friction.

6 figures · Texture weaves Plain weave, doubled. Plain weave, then the same weave with its picks grouped in 2, its ends grouped in 2, and both — which are a warp rib, a weft rib and a hopsack — and a 2/2 twill beside them for comparison. The rules under the drafts bracket the threads that lift together on every pick and so lie touching. Each doubling multiplies the cloth's own unit by 2: 2, then 4, then 8 intersections. The hopsack and the twill interlace equally often and are drawn from different rules. The setts under each draft are the densest that weave may be set at with a 0.25 mm yarn, and a rib's two are 0.750 apart where every other weave here is square.

A cord is a stripe with no colour in it

Warp rib, weft rib and hopsack are plain weave with its threads doubled, in the warp, the weft, or both. Six of the nine measures this site takes off a matrix cannot tell a 2/2 hopsack from a 2/2 twill, and the three that can are not the ones a weaver quotes.

8 figures · Rib weaves 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.

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.

6 figures · Derivation