Textile Structure
how cloth is put together
The collection

Every essay — page 3

Page 3 of 19, 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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Cloth doing a job

What happens when a fabric has to meet a specification rather than merely be described. A reinforcement that must hold fibre and still admit resin, a filter that must retain the soil and still pass water, a seam that must grip harder than the load without perforating the cloth, a pattern cut for a shape it does not yet have. Every requirement here is a pair of inequalities on quantities the earlier fields compute — and the useful answer is sometimes that no fabric satisfies both.

What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 900-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 12 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.

A grade charges by the length and a cutter pays by the panel

The four-point system scores a fault by how far it runs — one point to three inches, four beyond nine — and caps a linear metre at four points however many faults it holds. A cutting room pays by how many panels the fault lands in. Below a panel's length every fault costs exactly one panel while its score runs from one to four; above it the panels grow without bound and the score does not move at all. Two fifty-metre pieces built to the same 267 points a hundred square metres lose 33 per cent of their panels and 92.

6 figures · Faults The first 12 metres of a bolt, cut three ways. A bolt of 50 metres carrying 40 faults, with its first 12 metres drawn: the fault positions above, and below them the 900-millimetre panels a cutter gets cut blind, cut with the same rigid tiling slid to its best offset, and cut around the faults with the map in hand. Over the whole bolt the three yield 28, 30 and 40 sound panels of 55. Sliding the tiling buys 2; breaking it buys 12, which is 43 per cent more cloth from the same roll. What the strip cannot show is the width, across which the same argument runs again with a different panel dimension.

A fault map is worth most where the grade is worst

A cutter who knows where the faults are can slide the marker or break it, and only one of those is worth anything: sliding a rigid tiling to its best offset recovers two panels of fifty-five, and letting the tiling break recovers twelve — two panels in five more cloth from the same roll. The gain has a maximum in the middle of the range, because there is nothing to recover on a clean bolt and nothing to be done on a ruined one. And the prediction the grading essay made, that a map is worth most on a bolt whose faults are bunched, is false: bunching leaves clear runs for the blind cutter too.

6 figures · Faults

What cloth is

A fabric is a structure before it is a material. Interlacement, the draft as a matrix, and the question of whether a cloth holds together at all.

The plain. The plain 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.

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 · What cloth is The 2/2 twill. The 2/2 twill 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.

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.

6 figures · What cloth is Whether the cloth is one cloth. Two drafts. Both interlace everywhere, both have short floats, and both look like perfectly ordinary weaves. One is a single fabric and the other is two fabrics lying on each other, and the bars beside each strand say which layer it belongs to.

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.

7 figures · Integrity The plait braid. A braid written as a word of crossings and drawn from it. Each strand is coloured by the component of the above-and-below relation it belongs to, so a word describing two independent braids rather than one shows as two colours. This one is 1 braid.

Braids and the third thread system

A braid is one set of strands interlacing with itself at an angle, so the question of whether it holds together is the same question a weave answers — and most crossing sequences answer it badly.

7 figures · Integrity 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.

What the matrix cannot say

A weave is a binary matrix, and the whole of this collection rests on it. So it is worth setting down, in one place and precisely, what that encoding decides exactly, what it decides while appearing to decide something else, and the four quite different reasons a real fabric can fall outside it altogether.

6 figures · What cloth is A web at 6 fibre lengths squared per unit area. Straight fibres dropped at random positions and random angles. The largest connected group is drawn solid; everything not joined to it is drawn faintly. Whether that group reaches both edges is what decides whether this is a sheet or a heap.

Nonwovens, and what holds them together instead

A web of fibres laid down at random has no repeat, so the exact test for whether a fabric holds together has nothing to work on. What replaces it is a threshold, and the threshold is sharp.

8 figures · Integrity How often a draft falls apart. Every four-by-four draft in which each end and each pick interlaces at least once, sorted by its longest float, with the fraction that describe more than one cloth. The counts are produced by running the enumeration rather than by recalling it.

Every cloth there is, at four by four

Sixty-five thousand matrices, twenty-two thousand weaves, and about a dozen with names. The complete census of the smallest interesting repeat is a map of a whole small world, and almost none of it has ever been woven.

6 figures · What cloth is 2/2 twill: written at 4×4, repeating on 4. On the left the draft as written, with the smallest rectangle that tiles it outlined. On the right the same draft with every translation that leaves it unchanged marked from the top left corner: there are 4 of them, so a fundamental domain holds 4 intersections rather than the 16 the rectangle claims and the 16 the point paper carries. Nothing in the drawing on the left says so.

What a repeat repeats

Point paper is ruled in squares, and the rectangle a draft is written on is a decision by whoever drew it. The unit a cloth actually has is smaller — two intersections for a plain weave, four for a 2/2 twill — and it is not a rectangle at all.

6 figures · Repeat 3 cloths on 6 ends. A repeat of 6 ends and 6 picks holding 3 complete cloths, each with 2 ends and 2 picks of its own. The bars beside the strands say which cloth each belongs to; the ceiling at this size is 3, and the longest float is 5 because the face warp passes over every pick below it.

How many layers a draft can have

Of the 22,874 four-by-four drafts this site sweeps, 22,730 are one cloth and 144 are two. None is three, and none can be — a repeat of n ends holds at most n halved cloths, because every cloth needs two ends and two picks of its own before it interlaces at all.

6 figures · Integrity 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.

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.

6 figures · Notation The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica this site does not have.

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

6 figures · Contact Where a thread stops sliding and starts breaking. A pick of sheeting gripped over a length of cloth, drawn one crossing at a time. The resistance is 0.30 times the 0.599 N each crossing presses with, so it rises with the length held; the breaking load of 3.74 N does not. The two are equal at 7.4 mm. What the drawing cannot show is that μ is a range rather than a constant, so the mark is a band and its position is exactly inversely proportional to the friction.

A thread is held one crossing at a time

Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.

6 figures · Contact A thread withdrawn from a plain weave and from a satin. One pick of a sheeting being pulled from 1.60 mm of cloth, in a plain weave and in a five-end satin, at a friction coefficient of 0.30. The drawn width of each thread is the tension in it at that point, to a common scale; the faint rules are where the thread actually turns, which is at its interlacings and nowhere else. A plain weave turns 2.80 times per millimetre and a satin 1.12, so the plain weave's tension compounds 2.78-fold over this length against the satin's 1.46. What the drawing cannot show is that the wrap angle is taken from a plain-weave geometry in both panels: a satin's crimp is genuinely smaller, so its real turns are gentler than these and its grip weaker still.

A thread is gripped where it turns

The arithmetic this site has used for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.

6 figures · Contact Stick, slip, and the ratio between the two coefficients. The force in a thread as a cloth is agitated, with a static coefficient of 0.300 and a kinetic one of 0.225 — a ratio of 0.75, which is what fibre on fibre measures. The force climbs until it reaches the static limit, the contact breaks away, and while it is sliding it resists only at the kinetic limit. So a cloth that is being shaken can be left anywhere in the narrower band, and a cloth at rest anywhere in the wider one. What the trace cannot show is how much this buys: the band does not narrow in the ratio of the coefficients, because the restoring force stiffens away from the minimum, and the real narrowing is nearer 0.87 than 0.75.

Two coefficients, not one

Every friction on this site is a single number, and the account of why a cloth relaxes better when it is agitated depends entirely on there being two. Separating static from kinetic changes what a resting state is: a cloth at rest is held by one coefficient and a cloth being shaken by the other, and the band it can be left in narrows — by less than the ratio, because a cloth's restoring force is not linear in its extension.

6 figures · Friction A cotton fibre dry and wet. One cotton fibre in its dry state and saturated with water, both drawn at the same scale in both directions. It is 20% wider and 1.2% longer, so its cross-sectional area rises by 44% if the section stays similar to itself. The length difference is drawn and is nearly invisible, which is the point: a swelling that were the same in both directions would make a cloth bigger and change nothing about its structure, and this one changes every ratio of a diameter to a spacing in the cloth. What the drawing cannot show is the section: a cotton fibre is not a cylinder, and the directly measured area swelling of 40% to 42% does not agree with the square of the width change, which is a fact about the fibre.

What water does to a thread

A cotton fibre in water is a fifth wider and a hundredth longer. If it grew equally in both directions a wet cloth would simply be a bigger cloth and nothing structural would follow; because it does not, every ratio of a diameter to a spacing in a cloth moves, and they all move the same way.

7 figures · Water A yarn's voids against its fibres' swelling. One cotton yarn's cross-section at a packing factor of 0.60, drawn dry and with every fibre swollen by 20% while the yarn's own outline is held. The fibres now occupy 0.864 of the section, which is above the 0.75 a heavily compacted assembly reaches and above the 0.65 of a spun yarn, so the yarn cannot stay this size: it must grow by at least 7.3%. What the drawing cannot show is disorder — the fibres are laid on a lattice here to make the areas exact, and a real yarn's fibres are neither round nor evenly spaced, which is why the bound is quoted over three packing limits rather than at one.

A yarn's voids are not enough

A ring-spun cotton yarn is sixty per cent fibre and forty per cent air, and its fibres gain forty-four per cent of area in water. The obvious thought is that the air takes it. The arithmetic says the air cannot, and gives a floor on how far the yarn itself must grow with no measurement of a yarn in it anywhere.

7 figures · Water The two routes a poplin has to a strain. A poplin drawn in section at three places: as woven, at the end of what its crimp can supply, and past that. Between the first two the warp's crimp falls from 8.97% to 4.85% and the weft takes on what it gave up, and the thread length is 0.4953 mm in both — nothing has stretched, and the cloth is 3.93% longer. Between the second and the third the geometry cannot move because the weft's straight run has vanished, so the cloth's extra 2.0% is the thread's extra 2.0%. What the drawing cannot show is which of the two a piece of cloth has had: the first two states look different and the last two look the same, and it is the last two that differ in whether the cloth comes back.

A cloth gives back less than it took

Everything this collection computes about a deforming fabric is reversible, and no fabric is. The repair is not a new material property: a woven cloth has two routes to a strain, one of them costs its threads nothing and comes back in full, and where the first route runs out is a number about the sett with no fibre in it at all.

7 figures · Memory The interchange budget against cover, at three yarn counts. The extension a plain cloth can reach with no thread stretching, plotted against its warp cover factor, for 10, 20, 40 tex yarn. The three curves coincide, because every length in Peirce's geometry is a multiple of the yarn diameter and a spacing measured in diameters is a cover factor — so the count divides out exactly and the maximum is at 0.407782 for all of them, at a budget of 7.2857%. That the curve has a maximum at all is the finding: crimp is what a cloth spends, so more of it should be better, and past this cover the weft has nowhere to put what the warp gives up. What the plot cannot show is the balance, which moves the height of the maximum a long way and its position hardly at all.

The most a cloth can give back

The extension a cloth can find without stretching a thread comes from crimp, so setting a cloth closer ought to give it more. It does, up to a point, and then takes it away again — and the point is a cover factor of 0.4078 at a budget of 7.2857 per cent, identical to six figures for every yarn count from five tex to a hundred.

6 figures · Memory The channel through a muslin in plain. A cut across a muslin woven plain, at 373 µm per hundred pixels, showing one hole between two warp ends. The ends sit at the same level here, so the clear width between them is 250 µm at the waist against 250 µm straight through. The profile on the right is that width at every height in the 334 µm the threads occupy: the passage is an hourglass, its narrowest section is 250 µm across, and the band that every level contains — what a straight line of sight can use — is 250 µm. The two differ by 0.0 per cent, and the difference is the weave and nothing else.

A hole is a channel, not an opening

Every hole drawn so far has been drawn from above, and the sentence underneath every one of them says the same thing: the hole between four threads is the spacing less the diameter, and every hole is the same size. That is a picture of a cloth's shadow. A cloth has a thickness, so its hole has a length, and the narrowest place along it is not the place a person looking through can see.

6 figures · Openings A muslin's warp, drawn at the diameters it actually has. 18 ends of a muslin's warp yarn, drawn at a seeded sample of their own diameters — mean 167 µm, coefficient of variation 15% — and spaced exactly, because the reed does not vary. The threads look even enough. The gaps do not: the widest here is 280 µm and the narrowest 229 µm, against a mean of 250 µm, and their coefficient of variation is 7% — larger than the yarn's, by the ratio of the diameter to the gap. That amplification is the whole reason a close cloth's holes are so much less uniform than its threads, and it gets worse as the cloth is set closer, because the gap in the denominator is the thing being shrunk.

A cloth is a population, not a thread

Every number in this collection was computed from a diameter, and no yarn has one. Putting the distribution back changes some answers by nothing at all, some by a few per cent, and some by a factor — and which of the three happens is decided by one derivative.

6 figures · Variation The evenness a cotton yarn cannot be better than. Lay staple fibres down at random and count how many cross a plane: the count is Poisson, its variance is its mean, and the coefficient of variation of the mass per unit length is therefore 1/√n with no material and no machine in it. The lower curve is that floor. At 20 tex there are 118 fibres in the section and the floor is 9.93 per cent; at 5 tex there are 29 and it is 19.86. The upper curve is what a yarn spun at an index of irregularity of 1.35 actually measures, which is the floor times a constant — so the whole shape belongs to the counting and none of it to the spinning. The exponent is exactly −½ and is asserted as such rather than fitted to the curve.

The spread was never free

This collection has taken a yarn's irregularity off a delivery note and used it as an input. It is not an input. Counting the fibres in a cross-section puts a floor under it that no spinner can beat, and the floor is one over the square root of the count.

6 figures · Variation The two diameters of a 20 tex yarn. The pressure inside a twisted yarn is zero at its surface, so the outermost fibres are held by nothing but their own buried ends and some of them stand off as loops and ends. A yarn therefore has two diameters: a mass diameter of 167.1 µm, which is a volume divided by a length and is the one every other calculation on this site uses, and a contact diameter of 217.1 µm, which is what a neighbouring thread, a finger or an air stream meets. The gap is a hair layer of 25.0 µm on each side and it is measured, not computed — nothing here predicts hairiness. What is computed is the consequence, and it is 29.9% of the diameter every cover factor on this site was built from.

The hairs are what touch

The pressure inside a twisted yarn falls to exactly zero at its surface, so the outermost fibres are held by nothing and some of them stand off. A yarn therefore has two diameters — the one its mass gives and the one a neighbour meets — and only the first is in the arithmetic.

6 figures · Hairiness