Textile Structure
how cloth is put together
Concept

Integrity — where it appears

Whether a draft describes one fabric rather than several lying on one another, decided exactly from its above-and-below relation. It is the property this collection is built around, because a cloth that fails it looks entirely reasonable on point paper.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

A preform through its thickness, and whether it is one piece. An orthogonal three-dimensional preform in section, with the binder's path, beside the digraph the integrity criterion consumes: one node per level, an arrow from each level to the one above it, and the binder's own contacts. The warp and weft here are straight and do not interlace at all, so the whole of the connectivity is the binder — and the count of separable pieces is what the criterion returns, unchanged from the weave it was written for.

The third index is not a repeat

The fancy weaves left three-dimensional weaving open as a possible fifth escape from the binary matrix. It is not one. The criterion that decides whether a plain weave is one cloth decides a five-layer preform unchanged — and what a third dimension actually takes away is periodicity, because a thickness has a top and a bottom and a repeat does not.

applied · What cloth is half-cardigan as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here.

Knit, tuck and miss

A weave is a matrix over two symbols and a weft knit is a matrix over three. The site's central question survives the translation intact: a weave falls apart when its above-and-below relation is disconnected, and a knit falls apart when a needle never knits.

knits · Stitch notation Milling, on the nonwoven's own scale. Migrated fibre accumulating with agitation, measured against the stick-percolation threshold this site uses for nonwovens. Above the line the fibre network holds on its own, which is why a milled cloth can be cut without fraying — the weave is no longer the only thing keeping it together.

Milling holds the cloth a second time

This site's integrity criterion returns exactly the same answer for a melton as for the loose twill it was woven as, and it is right both times. What has changed is that a second network now holds the fabric together, made of migrated fibre rather than of thread crossings — and it is the one that decides whether a cut edge frays.

finishing · Felting How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it.

A dimension without a state

Every essay in this field turned on the same omission, and collecting them produces a result none of them had on its own — the quantities a cloth's geometry determines divide cleanly into those a finishing works can move and those it cannot, and the division is not the one the vocabulary suggests.

finishing · State 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 How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

fancy · Integrity The pressure a wetting generates in a close cloth. The pressure a sheeting's yarn is compacted at, against how far its fibres have swollen. Below 9.29% the cloth accommodates the swelling as a shape change and the pressure is nothing; above it there is no state at constant thread length, so the yarn must be compacted back to a diameter the geometry can hold and van Wyk's cube law prices it. At cotton's 20% it is 7.80 MPa. The other route out — stretching the threads until they are long enough to wrap the swollen partner — needs 9.1% of strain against a breaking strain of 6.6% computed from the site's own tenacity and modulus, so the thread would break first and there is one route rather than two. What the curve cannot show is its own uncertainty: van Wyk's constant runs from 0.003 to 0.011, so the height of this curve is known to a factor of nearly four and its shape is not.

A wetting supplies the force the criterion needs

This collection's criterion decides whether a cloth is one cloth, exactly, and cannot see friction — so pricing what it misses needed a contact force, and the only one available had a measured fabric thickness inside it. A wetted close cloth generates one from geometry alone, and it lands within seven per cent of the measured route's answer.

fancy · Integrity How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

mechanics · Compression 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 run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

applied · Damage 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

Named alongside it

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

Contact forceEnumerationFloatFrictionShadingSpecificationToneCensusLatin squareCapstanCloth thicknessCriterion

All concepts