Textile Structure
how cloth is put together
Generator

What a 15% spread does to each of this site's own quantities

What a 15% spread does to each of this site's own quantities
What a 15% spread does to each of this site's own quantities. Seven quantities this collection computes from a thread's diameter, each pushed through a population of threads whose mean is a muslin's and whose coefficient of variation is 15%, and reported as the excess of the population's average over the value the average thread gives. The order is decided by one thing: the curvature. A cover factor is linear in the diameter and is unaffected at any spread whatever — the zero in this table is exact and is the control on the method. A mass goes as the square and is 2.25% high; a bending rigidity goes as the fourth power and is 14.3% high. Every convex quantity is under-reported by the mean thread and no concave one is over-reported by less, which is Jensen's inequality doing the only thing it does.

Seven quantities this collection computes from a thread's diameter, each pushed through a population of threads whose mean is a muslin's and whose coefficient of variation is 15%, and reported as the excess of the population's average over the value the average thread gives. The order is decided by one thing: the curvature. A cover factor is linear in the diameter and is unaffected at any spread whatever — the zero in this table is exact and is the control on the method. A mass goes as the square and is 2.25% high; a bending rigidity goes as the fourth power and is 14.3% high. Every convex quantity is under-reported by the mean thread and no concave one is over-reported by less, which is Jensen's inequality doing the only thing it does.

14 essays call spread-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. What cloth is

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.

 The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 6% of a dent, which is 25 µm. The upper band's errors are independent; the lower band's repeat every 8 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.7 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up. Pattern and colour

A random error hides and a periodic one shows

A warp whose threads vary by fifteen per cent looks perfectly even. One dent of the reed a tenth of a millimetre wide makes a streak that gets the piece rejected. The same amount of error, arranged two ways — and the ratio between them is √(2n/π), with nothing fitted in it.

 One weft fault at three periods, a fraction of a millimetre apart. A 260 mm slice of a 1500 mm cloth at 22 picks per centimetre, 220 picks deep, with a thick place recurring along the weft. Each pick takes a whole width of yarn, so the marks land 35 to a pick across the full width and step sideways by the remainder of the width divided by the fault's period. On the left that remainder is zero and every thick place in the piece falls in the same columns — a warp-way stripe made entirely by a weft fault. In the middle the period is five hundredths of a millimetre longer and the same fault draws steep diagonals. On the right it is a third of a millimetre longer again, the marks land nowhere near each other, and the fault reads as texture. Nothing about the yarn distinguishes the three; the cloth's width does. The slice is drawn rather than the whole width because thirty-five marks a pick fill a panel solid at any step but zero, which is a true picture of a dense pattern and a useless one of its structure. Pattern and colour

A slub finds the width of the cloth

A thick place recurring along a weft yarn does not make a bar. It makes diagonals — and when the cloth's width happens to be a whole number of fault periods, it makes stripes down the piece instead, from a fault that is entirely in the weft.

 The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 5% of a dent, which is 21 µm. The upper band's errors are independent; the lower band's repeat every 4 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.8 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up. Knits and other structures

A course is one thread and a warp is many

A woven fabric draws its warp from two thousand packages side by side, so a yarn's drift averages out across the width. A weft knit takes whole courses from one package, so the same drift becomes a band — and the standard remedy for that turns an invisible error into a visible one.

 How much of a 20 tex yarn's unevenness survives being woven. Weaving averages, and the averaging is a square root. A patch of cloth 3 mm across contains 7.2 warp threads and as many picks, each contributing its own mass independently, so the patch's coefficient of variation is 3.53% against the yarn's 13.4% — a reduction of 3.8-fold. The rule at the top is the yarn's own figure. The curve steepens past the staple length, where a patch starts to contain independent samples along each thread as well as across them, and the second regime is the one a large area of cloth is judged in. Pattern and colour

A cloth cannot be more even than its yarn

It can be very much more even than its yarn, and by exactly the square root of the threads in view. Which raises a question the fineness argument left open — and the answer is that at a fixed cloth weight the count cancels out entirely.

 The stripe a knitting machine chooses. A circular machine takes its yarn from a fixed number of feeders arranged round the cylinder, and feeder k lays every F-th course. So any difference between packages — a shade, a count, an evenness — is reproduced in the fabric with a period of exactly F courses, and at 20 courses per centimetre that is a band every 48.0 mm on a 96-feeder machine. The machine chooses the period, not the yarn. The spacing is proportional to the feeder count and inversely proportional to the course density, as the turn of the cylinder requires whatever the bars show, and a one-feeder machine produces no band at all from the same packages. Knits and other structures

Why a knit shows a thick place

A woven cloth has hundreds of separate warp ends and averages a yarn's faults among them. A knit has one thread and a machine that repeats — so a difference between two packages becomes a stripe, and the machine chooses its period.

 An operation multiplies a spread by its own log-slope. A transformation does not leave a population's spread alone: if y goes as the kth power of x then a small spread in x becomes k times that spread in y, exactly in the limit and nearly so at the CVs a yarn has. So an operation with an exponent below one narrows the population it acts on — a thickness that goes as the square root of a load comes out at half the spread it went in with — and one with an exponent of four widens it fourfold. This is the same derivative that decided every bias in this ladder, read for its magnitude rather than for its curvature, and it is why a finish can be a variance-reducing operation without anyone having chosen it for that. The straight line through the origin is the whole of the rule; the departure from it at the right-hand end is the second-order term arriving, which is where the linearisation stops being one. After the loom

A finish spends a spread before it spends a mean

Every operation on a cloth multiplies the variation it inherits by its own log-slope, so an operation with an exponent below one makes the cloth more even and one above it makes the cloth less even. Calendering, which is bought for evenness, has an exponent of 1.4.

 What a tear asks for is the weakest of a few. A tensile test pulls every thread in the width and averages them. A tear pulls the handful in the triangle at the tip of the cut, and what lets it move is whichever of those is weakest — so the quantity that governs is the minimum of a small sample, and two things follow that no mean can show. Its expectation is below the mean: at CV 15% the weakest of 4 is 0.853 of the mean thread, and the weakest of 40 is 0.718. And its scatter is enormous compared with a tensile test's: 10.9% against the 0.75% a mean of four hundred threads would show. A tear strength that varies by a tenth between specimens of the same cloth is not a badly run test. It is the only answer an extreme of four can give. Cloth doing a job

A tear asks fewer threads than a pull

A strip test averages a hundred threads and a tear interrogates four. That difference alone accounts for the two things everybody knows about tear testing — that it reads low, and that it scatters — without anything about the cloth being different between the two tests.

 The crossings under a 25 mm² presser foot. A 25 mm² foot on a muslin covers 12 ends and 11 picks, which is 132 crossings — and a first guess treats those as 132 chances of finding a thick place. They are not independent chances. Every crossing along one end shares that end's diameter, so the largest crossing is the largest end plus the largest pick, and the number of tries is 23: the threads. Each cell here is shaded by its own two diameters, and the darkest is at the meeting of the darkest row and the darkest column, which is what that identity looks like. The gauge rests on it and reads 0.431 mm, against 0.342 for the cloth's mean crossing — 26% over. Mechanics and drape

A thickness is a maximum, not a mean

A presser foot rests on whatever is highest beneath it, so the thickness of a fabric is an extreme value — and an extreme grows with how much cloth is asked. The standard specifies the foot's area because the foot's area is in the answer.

 Where a muslin's air goes, as the cloth is set closer. A permeability computed from the average hole says nothing about which holes the air uses, and once the holes have a spread the answer is: not many of them. Two curves, both over the same population of holes — the share of the flow carried by the widest tenth, and how few of the holes carry half of it. At 16 ends per centimetre the cloth is nearly democratic: the widest tenth takes 12% and half the air needs 45% of the holes. At 44 the widest tenth takes 53% and half the air goes through 8.8%. Both inputs move together as the cloth closes — the spread in the holes rises because the spacing is fixed and the diameter is not, and the exponent rises because the pressure drop stops being inertial — so the concentration rises faster than either. Setting and geometry

Half the air goes through a tenth of the holes

A permeability computed from the average hole says nothing about which holes the air uses. Once the threads have a spread, the answer is: not many of them — and in a close cloth, half the flow leaves through less than a tenth of the openings.

 A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 15% is 0.7380 of the mean thread, reached with 93% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.753, 0.744, 0.741 at the last three sizes. The scatter falls the other way, from 14.5% at one thread to 0.7% at 1600. Mechanics and drape

A bundle is weaker than its threads

Threads pulled together do not break together. The weakest goes first and hands its load to the rest, so a bundle carries its maximum well before every thread is at its own limit — and the shortfall is a quarter, decided by the spread and by nothing else.

 Where a muslin's warp jams, over 40 ends. 40 ends drawn at their own diameters, with every neighbouring pair's combined width plotted beneath. A cloth cannot be set closer than its threads will lie, and the pair that decides that is not the average pair — it is the widest one anywhere across the warp, which here is ends 8 and 9 at 205 µm apiece against a mean of 167 µm. Over the 2000 ends of a real warp rather than the 40 drawn here the worst pair is 42% above the mean, and it goes on growing with the width of the cloth: the same yarn in a wider loom jams sooner. The naive estimate that treats every window as an independent try overstates it by 0.48%, which is small enough to say that the overlap between neighbouring windows is not what is going on here. Setting and geometry

A warp jams where its threads are thickest

The closest a cloth can be set is decided by its worst pair of neighbours, not its average thread — and the worst pair depends on how many pairs there are. The same yarn in a wider loom jams sooner, which makes a jammed sett a property of the machine as well as of the yarn.

 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 5 tex there are 29 fibres in the section and the floor is 19.86 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. Setting and geometry

A finer yarn is a worse yarn

Fineness is the thing a yarn is priced for, and it is bought with irregularity at a fixed exchange rate. Once the spread is a function of the count, every correction this collection computes becomes a function of the count too — and the cheapest yarn on the shelf is the one the arithmetic describes best.

 A designed thin place does not get worse and an accidental one does. The thinnest place a yarn reaches, against how many gauge lengths of it are tested, for a slub yarn and for a randomly uneven yarn of the same 36.3% coefficient of variation. The slub's floor is its base count — 89% of its mean — and it is a horizontal line, because a designed variation has a stated minimum and never goes below it. The random yarn's minimum is an order statistic and falls without limit: 56% over 10 lengths and 22% over 30000. So the advantage is not a number but a function of how much yarn is being asked about, running from 1.6× to 4.0×. What the curve cannot show is the break itself: a yarn's strength at a thin place is not proportional to its linear density there, and the conversion needs a fibre model. Compound and figured cloths

A designed thin place is kinder than an accidental one

A slub yarn and a badly spun one can carry exactly the same coefficient of variation, and the number tells a mill nothing about which it has. The designed variation has a floor — its base count, and it never goes below it — while the accidental one has a tail that falls further the more yarn is tested. At 36% CV the slub bottoms at 89% of its mean and the random yarn reaches 24%, and the gap widens from 1.6 to 3.7 times as the test grows from ten gauge lengths to ten thousand.

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