Textile Structure
how cloth is put together
Theme

The thread: 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.
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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

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