Generator

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.

16 essays call bundle-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.

What cloth is

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.

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

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.

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

The pressure a 30° twist puts on its own fibres. A twisted yarn squeezes itself. Every fibre under tension at a radius pulls inward with sin²θ of its tension per unit length, and integrating outward to the surface — where the pressure is zero by definition, because there is nothing outside to push against — gives p(r) = ½σ(cos²θ(r) − cos²α) in closed form. At a surface angle of 30° the pressure on the axis is 12.5 per cent of the core fibre's own axial stress and the mean over the section is 5.65 per cent of it. The shading is that closed form and the curve is the same function plotted; the shape is what matters, and its one uncompromising feature is the zero at the edge. The outermost fibres are held by nothing, which is why a yarn is hairy, why a surface fibre is the one that comes away on a finger, and why singeing changes a yarn's behaviour out of all proportion to the mass it removes.

Compound and figured cloths

A tuft is set so it cannot untwist

A cut pile tuft has a free end, and at a free end the pressure holding the twist is zero. So the twist runs out over a computable length, the tuft opens, and a carpet loses its appearance long before it loses any material.

Singeing is the cheapest change to a surface

Pass a cloth through a flame fast enough and it loses under one per cent of its mass. What it loses is the part of itself that was doing most of the touching, and lustre, friction, printability, pilling and measured cover all move at once.

$20 tex, counted. The cross-section of a 20 tex cotton yarn, with every fibre in it drawn. The count is a division and nothing else: a 20 tex yarn spun from 0.17 tex fibre has 117.6 fibres crossing any plane through it, and the yarn is 14.0 fibre diameters across because n fibres packed at 0.6 fill a circle √(n/φ) times as wide. The arrangement is drawn on a lattice and is not claimed: real fibres are not on one, they migrate between the core and the surface as they run, and everything this collection says about a yarn's strength turns on their doing so.$ Setting and geometry

How many fibres make a thread

Every number in this collection began with a diameter, and a diameter is not a measurement — it is a count of fibres, divided. Once the division is written down, three quantities that had nothing to do with each other turn out to be the same number.

$Where a 20 tex yarn breaks, against how much was clamped. A tensile test clamps a length of yarn and pulls until the thinnest section between the clamps gives. So a yarn's strength is a minimum, and a minimum depends on how many independent tries the sample contains. The tries are not sections — a plane can be taken anywhere — but staple lengths, because two planes closer together than one fibre share most of their fibres. At 28 mm staple a 100 mm specimen holds 3.6 independent tries and a 500 mm one holds 17.9, and the longer test reads 11% lower. The spread is not fitted either: it is the evenness floor at 118 fibres times an index of 1.35, which is 13.4%.$ Mechanics and drape

A yarn breaks at its thinnest place

A tensile test does not measure a yarn. It measures the worst section between the clamps — so evenness and strength are one measurement taken twice, and the number of independent tries in a specimen is set by the length of a fibre.

Folding, and what it takes out of the singles. Two 20 tex singles spun at 800 turns per metre and folded the other way at 460 — a ratio of 0.575, which is the trade's own and is a measurement rather than a derivation. A single held at its ends and wound round its neighbour turns about its own axis once for every turn of the fold, so it is left with 340 turns per metre of its own: its surface fibres lie at 10.1° to its axis rather than the 22.8° they were spun at. Folding untwists. The short strokes are drawn at that residual angle; the two long curves are centre lines and are not the yarn — each strand is itself a bundle of 118 fibres, and the residual angle is what holds them.

Cloth doing a job

A sewing thread is a different animal

It is folded, balanced, lubricated and finished, and every one of those is an answer to a requirement no weaving yarn has. The lubricant is the interesting one: it makes the thread sewable by lowering the friction that was holding its own fibres together.

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.

Where the strain goes, at a 25° twist. Extend a twisted yarn and the fibres in it are not all strained alike. If each stays at its own radius — the affine model — a fibre at helix angle θ takes cos²θ of the yarn's strain, so the fibre on the axis takes all of it and the fibre at the surface takes 82.1% of it. The core reaches its breaking extension first and the yarn cannot realise its own fibres. If instead every fibre migrates between the core and the surface, every fibre has the same mean strain and they all break together. The two are exactly computable — 0.8214 against 0.9509 of what the fibres could give — and the ratio between them, 2/(cos α(1 + cos α)) = 1.158, is what migration is worth. The shading is that arithmetic and the two panels use the same scale.

Mechanics and drape

A straight fibre cannot share the load

Extend a twisted yarn and its fibres are not all strained alike: the one on the axis takes the whole of it and the one at the surface takes cos²α. So they do not break together, and what a wandering fibre is worth comes out as one expression with nothing fitted in it.

Z twist at 1600 turns per metre. A 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 40.0°, and it is the only quantity in this family: the yarn is 13.8% shorter than the fibre in it and carries 59% of the strength the same fibre would give lying straight.

Weaves

The other crepe is in the yarn

A crepe weave puts the texture in the matrix. A crepe yarn puts it nowhere the matrix can see: the cloth is a plain weave, and the surface comes from a thread twisted so hard that it shortens by a seventh and spends the rest of its life trying to untwist.

The pressure a 23° twist puts on its own fibres. A twisted yarn squeezes itself. Every fibre under tension at a radius pulls inward with sin²θ of its tension per unit length, and integrating outward to the surface — where the pressure is zero by definition, because there is nothing outside to push against — gives p(r) = ½σ(cos²θ(r) − cos²α) in closed form. At a surface angle of 23° the pressure on the axis is 7.6 per cent of the core fibre's own axial stress and the mean over the section is 3.61 per cent of it. The shading is that closed form and the curve is the same function plotted; the shape is what matters, and its one uncompromising feature is the zero at the edge. The outermost fibres are held by nothing, which is why a yarn is hairy, why a surface fibre is the one that comes away on a finger, and why singeing changes a yarn's behaviour out of all proportion to the mass it removes.

Mechanics and drape

What grips the end of a fibre

A twisted yarn squeezes itself, and the squeeze holds the fibre ends in. Write the slip and the break out side by side and the fibre's own strength cancels — and so does the load on the yarn — leaving a gripped length that depends on fineness, friction and the twist and on nothing else.

Both halves of the twist curve, at 28 mm staple. The falling curve is obliquity and is exact — the affine end of the bracket, computed from the helix and nothing else. The rising curve is cohesion and is the half this collection had declined: a fibre end is gripped by friction under the twist's own radial pressure, the fibre's strength and the yarn's load cancel out of the comparison, and what is left is a critical length that depends on the twist through a pure function of the angle. Their product has a maximum at a twist factor of 3101 — 693 turns per metre at 20 tex, a surface angle of 20° — which is inside the range spinners use. The scale of the rising curve is fitted, through a contact efficiency of 0.05, and moving it moves the optimum; what it cannot move is the ordering between two staples or two fibres, which is what the two claims made from this figure are about.

Setting and geometry

The other half of the twist curve

This collection computed the falling half of the strength–twist curve exactly and declined the rising half as being out of reach. It is not out of reach. With the grip derived rather than assumed, the optimum comes out at a twist factor — and the same twist factor at every count, which is why the trade quotes twist factors at all.

Which end of the bracket a thread in cloth is at. A yarn bends as a solid rod while its fibres cannot slide and as a loose bundle once they can, and the crossover is a curvature: the coherent state demands an axial force in the outer fibre that has to be built up by friction under the twist's own radial pressure. The curves are the crossover radius against twist at four contact efficiencies, the top one being 1.0 — the claim that fibres touch along their whole length, which nobody makes. The rule at the bottom is the radius a thread is bent to by its own crimp in a cloth, about 0.25 mm. Every curve is above the rule by at least 19-fold, so a thread in cloth is at the free end of its bracket at every twist and every efficiency, and the collection's habit of using the lower bound is a result rather than a convention.

Mechanics and drape

Twist decides where in the bracket

A yarn's bending rigidity can only be bracketed, and the bracket is three hundred wide. What decides where a yarn sits in it is whether its fibres can slide — and for a thread bent by its own crimp in a cloth, the answer is not close.

One yarn, two packing factors. The same 20 tex cotton yarn — the same fibres, the same count, the same mass per metre — drawn at a packing factor of 0.45 and of 0.75. Its diameter is 192.9 µm in one and 149.5 µm in the other, a difference of 29.1%, because a diameter goes as the inverse square root of the packing. Every cover factor, every jammed sett and every hole in this collection went through that number, and the site's value of 0.6 was obtained by inverting a rule published for cotton yarns at one particular twist. Nothing here models how packing moves with twist; the figure is here to show the size of the thing that has been held constant.

Setting and geometry

The diameter was quoted at one twist

Every diameter in this collection came from a packing factor of 0.6, and that number was got by inverting a rule published for cotton yarns at one particular twist. Here is what moves if it is wrong by the width of the range real yarns occupy — and which single quantity does not move at all.

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