Textile Structure
how cloth is put together

A weave is a matrix.

Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not a convenient way of writing a weave down — it is the weave, and it means almost every question about a fabric has an exact answer: how long the floats are, how close the threads can be set, and whether the thing holds together at all. These are essays about cloth as a structure, with the arithmetic done rather than asserted.

The weaves everything else is built fromPlain, twill and satin on point paper. Reading across, the floats lengthen and the interlacings fall — and every property a weaver cares about follows from that one trade.plainfloat 1 · 8 interlacingsfirmness 1.002/2 twillfloat 2 · 16 interlacingsfirmness 0.505-end satinfloat 4 · 20 interlacingsfirmness 0.40a filled square means the warp is on the facecounted from the matrices, not from the pictures
Fig. 1 The three weaves everything else is built from, drawn on the point paper a weaver would use — a filled square means the warp is on the face. None of these was drawn by hand. Each was generated from its own rule and then examined: the floats counted, the interlacings counted, and the cloth itself checked for the failure that a draft cannot show, which is whether the threads are tied into one fabric or several.

Start anywhere

19 essays

plainfloat 1 · 8 interlacingsfirmness 1.002/2 twillfloat 2 · 16 interlacingsfirmness 0.505-end satinfloat 4 · 20 interlacingsfirmness 0.40a filled square means the warp is on the facecounted from the matrices, not from the pictures What cloth is

A fabric is a structure, not a material

Cotton is a material. Cloth is an arrangement, and almost everything a fabric does follows from how the threads are put together rather than from what they are made of.

6 figures plainfloat 1 · 8 interlacingsfirmness 1.002/2 twillfloat 2 · 16 interlacingsfirmness 0.505-end satinfloat 4 · 20 interlacingsfirmness 0.40a filled square means the warp is on the facecounted from the matrices, not from the pictures Weaves

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.

10 figures 100 ends and 100 picks per inch in bothfine yarncover 78%coarser yarncover 97%cover from diameter and spacing, circular sections Setting and geometry

Thread count is not quality

It counts threads. It says nothing about how much of the cloth they cover, it can be inflated by counting plies, and it ranks fabrics in nearly the wrong order. The quantity it is mistaken for is cover factor, and that one is computable.

6 figures shear 30°bias +22.5%across -29.3%area 87%locks at 60°every segment checked against its own lengthno thread stretches Mechanics and drape

The bias is a mechanism

Cloth cut at forty-five degrees stretches by a third and springs back, while the threads in it stretch by nothing at all. Almost every explanation given for this is wrong, and the right one is not about elasticity.

6 figures weaveleave the faces aloneexchange themplain1616basket16162/2 twill16163/1 twill16none5-end satin20noneevery candidate operation tried against the repeat Pattern and colour

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

7 figures a course runs acrossa wale runs down5 wales × 4 coursesevery loop passes through the one below itweft knit Knits and other structures

The loop

A knit is one thread bent into loops, each drawn through the loop below. Nothing in it is straight, which is why it stretches in every direction while a woven cloth stretches only at an angle.

6 figures 2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4 What cloth is

The draft is a matrix

Point paper is not a diagram of cloth. It is a grid of yes-or-no decisions about which threads to lift, which makes a weave a binary matrix and almost every question about it arithmetic.

9 figures 5-end satinwarp floats of 4weft floats of 120 markedruns measured round the repeatfloat 4 Weaves

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.

7 figures as wovenwarp crimp 8.0% · weft 4.0%pulled 3% along the warpwarp crimp 4.9% · weft 7.1%warp thread 108.0 and weft 104.0 — unchanged in bothboth thread lengths checked, not assumed3% pull Setting and geometry

Crimp, and why cloth narrows when it is pulled

A thread in cloth is longer than the cloth it crosses. Pull the fabric one way and that extra length is taken out of one direction and put into the other, so the cloth gets narrower without a single fibre stretching.

7 figures the dots mark which way each course facesstockinette — every course the same0 face changes · curlsgarter — alternate courses reversed5 face changes · lies flatface changes counted, not the curl itself Knits and other structures

Why stockinette curls

A knitted loop is not symmetric front to back. Work every course the same way and the asymmetries add up along the edges; alternate them and they cancel. That is the whole difference between a fabric that rolls and one that lies flat.

6 figures 0%10%20%30%40%0°15°30°45°60°cover 40%locks at 66°, +38%cover 60%locks at 53°, +34%cover 80%locks at 37°, +26%extension along the biasshear anglethe yarns never stretch; the net changes shape Mechanics and drape

The locking angle

The bias runs out at an angle that yarn diameter and thread spacing decide between them. It is the number behind whether a cloth will go round a curve, and it has nothing to do with how strong the fabric is.

7 figures only the order the coloured threads are put in changesthe weave2/2 twillfour dark, four light▪▪▪▪▫▫▫▫two dark, two light▪▪▫▫the face colour computed, not designed2/2 twill Pattern and colour

Colour and weave

Thread the same weave with a different order of coloured ends and a pattern appears that is nowhere in the draft. Houndstooth is an ordinary two-and-two twill, and nothing about its interlacement is unusual at all.

7 figures the bars mark which separable cloth each thread belongs toan ordinary 2/2 twillone clothevery end and pick interlacestwo separable clothsconnectivity of the above-and-below relation, computed What cloth is

Does it hang together

A draft can interlace everywhere, have short floats, and describe two fabrics lying on each other rather than one. Nothing about the drawing says so, and the test that does is exact.

6 figures ordermoves that give a satinhow many3 endsnone — every step shares a factor with the order04 endsnone — every step shares a factor with the order05 ends2 326 endsnone — every step shares a factor with the order07 ends2 3 4 548 ends3 529 ends2 4 5 7410 ends3 7211 ends2 3 4 5 6 7 8 9812 ends5 7213 ends2 3 4 5 6 7 8 9 10 111014 ends3 5 9 114enumerated from the coprimality condition, not recalled3–14 Weaves

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 weaveinterlacingsdensest settingplain1.0020 per inch2/2 basket0.5027 per inch2/1 twill0.6724 per inch2/2 twill0.5027 per inch3/1 twill0.5027 per inch5-end satin0.4029 per inch8-end satin0.2532 per incha geometric jamming model, circular sections, yarn diameter 1/40 inch Setting and geometry

How close can threads be set

There is a maximum. Push more threads into a cloth than the geometry allows and they simply will not go, and the limit depends on the weave as much as on the yarn.

7 figures a knit — the break runs up the wale3 loops freeda weave — the break stays put1 thread freedcounted from the structure, not from experience Knits and other structures

Ravel, fray and run

Cut a woven cloth and one thread comes loose. Break one loop in a knit and every loop above it follows. The two structures fail in opposite ways, and the reason is topology rather than strength.

7 figures a sphereworst shear 22.2°cells closelocks at 57°nothing jamsthe fishnet construction, every segment checkeddoubly curved Mechanics and drape

Why clothes need darts

A flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. The amount of shear is decided by the curvature, and when it exceeds what the threads allow, something has to be cut out.

8 figures 2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4 Pattern and colour

The diagonal is not a thread

A twill's diagonal is the most looked-at feature in weaving and the most misdescribed. No thread in the cloth runs diagonally. What runs diagonally is a pattern of which thread is on top, which is a different sort of object.

6 figures weaveinterlacingsdensest settingplain1.0020 per inch2/2 basket0.5027 per inch2/1 twill0.6724 per inch2/2 twill0.5027 per inch3/1 twill0.5027 per inch5-end satin0.4029 per inch8-end satin0.2532 per incha geometric jamming model, circular sections, yarn diameter 1/40 inch Weaves

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

Threads running through

themes, not chapters

A weave is a matrix

Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not an encoding of a weave — it is the weave, and it makes almost every question about it decidable.

8 essays

Does it hang together

A draft can look entirely reasonable on point paper and describe cloth that falls into two layers, or has threads lying loose on the surface. The failure is invisible and the test is exact.

2 essays

The float decides

How far a thread travels on the face before it goes under is the quantity behind lustre, drape, snagging, abrasion and how close the cloth can be set. Almost nothing else has that reach.

4 essays

Exactly this many

There is no regular satin on six ends, and the reason is a one-line argument about common factors rather than a rule of thumb. Counts in this subject are more often theorems than tallies.

2 essays

A mechanism, not a material

Cloth on the bias stretches half as long again while no thread in it stretches at all. The deformation is kinematic — a trellis closing — and almost every explanation given for it is wrong.

6 essays

Measured, not claimed

Thread count, breathability, the strength of satin: the trade is full of numbers that do not mean what they are taken to mean. Where a claim is refuted here, it is refuted by a computation.

10 essays

Structure before fibre

Most of what a fabric does follows from how it is put together rather than from what it is made of. The same yarn gives a shirting or a satin, and they behave nothing alike.

9 essays