Textile Structure
how cloth is put together
Generator

A knitted loop, solved rather than drawn

A knitted loop, solved rather than drawn
A knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings.

3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings.

13 essays call loop-shape. 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 knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings. Knits and other structures

A loop is nine tenths free run

Half the yarn in a knitted stitch is slack — the straight line between two interlacings is a little over half the thread available to span it. That is two orders of magnitude more room than a woven thread has, and it is why a knitted loop is a shape that can be solved rather than a shape that has to be constructed.

 What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve. Knits and other structures

What a knit gives when it is pulled

How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight.

 A jersey gets taller before it gets shorter. How much a 20 tex cotton jersey shortens along its wales as it is pulled along its courses, with the course spacing at every extension chosen to minimise the loop's energy rather than assumed. Over the first 81% it is negative — the fabric gets 2.0% taller as it is pulled wider — and only then does it start to contract, reaching 88% at the geometric limit. A material with a negative Poisson ratio is a curiosity; a knit has one over part of its range for a reason with no material in it at all, which is that widening a wale at a fixed loop length first lets the loop's tightest bends open and only later starts taking height away from it. Knits and other structures

A jersey gets taller before it gets shorter

Pull a knit along its courses and the first thing it does is grow along its wales — by two per cent, over the first eighty per cent of extension, before it turns round and contracts. The transverse response changes sign, and there is no material in the explanation at all.

 Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left. Knits and other structures

How far a knit could go if its yarn were the limit

The yarn in a stitch allows three hundred and twenty per cent course-wise extension before the straight line between two interlacings reaches the thread spanning it. A jersey jams at about a hundred. The factor of three is the finding: what stops a knit stretching is not the loop running out of yarn.

 Why a cuff is ribbed. The force a knit pulls back with, over the range a cuff is used across. It rises the whole way — 0.79 N per metre at 23% to 3.38 at 104% — and it is small throughout, which is the combination a cuff needs and almost nothing else supplies. A rib gets its extension by geometry, folding alternate wales to opposite faces so that its relaxed width is about half its opened one, and it gets its recovery from the loop reconfiguring. Neither is the yarn stretching, which is why a cuff made of a fibre with no elastic recovery at all still works. Knits and other structures

A rib pulls back on a force the loop supplies

A cuff has to give a great deal at almost no load and come back reliably, and no ordinary material does both. A rib gets its extension from folding, which is geometry, and its return from the loop reconfiguring, which is now a computable force — under four newtons a metre over the whole range a cuff works across.

 What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve. After the loom

A loop is set and not sprung

The way out of a model that predicts a jersey should spread is to stop treating its yarn as a straight rod bent into a loop. A yarn that has been wetted, heated and dried has taken the loop as its own natural shape — and once the natural shape is the loop, every force downstream becomes computable with the relaxed fabric as the origin.

 A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns. After the loom

Two knits with one tightness factor are one knit

The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does.

 Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.00, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not. Mechanics and drape

A thread between two crossings is an elastica

Every crossing force Peirce's thread path can give comes from a tension, and a cloth on a table has none. What presses its threads together there is their own unwillingness to be bent — and recovering that needs a shape nobody had, because a path assembled from an arc and a straight line has a bending moment that jumps.

 Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.18, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not. Mechanics and drape

A force is what an energy does when a crossing moves

The force a thread presses its neighbour with is the rate its bending energy changes as the crossing is displaced. Solved as a constrained minimisation, that number arrives with the shape rather than after it — and the same force, recovered a second time from the curve's own equilibrium, agrees to a tenth of a per cent.

 How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip. Mechanics and drape

What a loop presses with

A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

 What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve. Cloth doing a job

A seam must give what the knit gives

A knitted seam fails because it is too short, not because it is too weak: the thread in it is nearly two thousand times stronger than the load it carries. What decides whether it survives is one line of geometry — the extension a seam can reach is twice the fabric's thickness times the stitches per unit length.

 Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.14, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not. Mechanics and drape

What a loop model still cannot say

A planar rod with a natural curvature and point contacts gets a knit's forces, its modulus and its extension. It does not get torsion, it does not get the third dimension the interlacing actually needs, and it does not stop adjacent courses passing through one another — which is why its extension ceiling sits three times beyond any jersey.

 What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve. Mechanics and drape

The modulus a knit has instead of one

A fabric's stiffness in extension is usually its yarn's modulus with the geometry taken out. A knit's is not: dimensionally it can only be a bending rigidity over a length cubed, and the same yarn laid straight and parallel is thirty thousand times stiffer than the fabric made of it.

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