Textile Structure
how cloth is put together
The collection

Every essay — page 7

Page 7 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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After the loom

The cloth that comes off the loom is not the cloth anybody buys. Shrinkage as the crimp coming back, the yarn flattened and swollen, the floats raised into a nap, and the fibres entangled until the weave is no longer the only thing holding the fabric together.

Which knitted structures present a float a raising wire could catch. Each of the named two-bed structures, with how many of its floats lie exposed on a face rather than inside the cloth. A plain jersey has no float at all; the ribs and the cardigans have none; the interlocks and milanos have floats and every one of them is interior, closed over by the other fabric. a single jersey with a float and a three-by-one rib with a float present an exposed float, on the back. So the criterion that decides which woven cloths can be napped decides the same question here and answers it for 2 of 13. What the bars cannot show is the hair layer, which a wire also catches and which no float census can see.

A jersey has no float for a wire to catch

Only a float can be raised, and the four-by-four census answers which woven cloths qualify: two. Asked of a knit the same question needs this collection's knitted float rather than a draft's, and the answer is that a plain jersey has no float at all — not a short one, none — while the ribs and cardigans have none either and the interlocks and milanos have floats every one of which is interior. Of 1,135 two-bed structures a machine could make, twenty present a float a wire could hook, and not one of them presents it on both faces.

6 figures · Nap 6 tapered panels laid across a 150 cm cloth three ways. 6 panels 14 cm across the top, 32 cm across the bottom and 75 cm long, laid across a cloth 150 cm wide, drawn to scale. Turned end for end alternately, 6 fit side by side and the 6 take 75 cm of cloth. Laid all one way in lanes, 4 fit and they take 150 cm. Laid all one way with alternate columns shifted half a length, 5 columns fit and they take 150 cm. What the drawing cannot show is a real marker's other pieces, which fill the gaps these leave.

A nap is paid for by the taper of the pattern

A raised cloth's fibres lean, so a panel turned end for end shows a different amount of fibre and every piece of a garment has to lie the same way along the bolt. What that costs is not a property of the cloth. A rectangle costs nothing laid one way; a tapered panel costs (1 − r)/(1 + r) of extra cloth in lanes, where r is its narrow width over its wide one; and the best any one-way lay can do is exactly half of that, because a trapezoid's difference body is a hexagon and hexagons tile. On a real width it arrives in whole panel lengths: six skirt gores take 75 centimetres two ways and 150 one way.

7 figures · Nap

Knits and other structures

Loops rather than crossings. Why a knit stretches without a bias, why stockinette curls, and why a dropped stitch runs while a woven cloth frays.

The knitted loop. One thread, bent into a course of loops, each of them drawn through the loop below. Nothing here is straight, which is why a knit extends in every direction while a woven cloth extends only on the bias.

The loop

A knit is one thread bent into loops, each drawn through the loop below. Nothing in it is straight, which is why it stretches in every direction while a woven cloth stretches only at an angle.

6 figures · Knit Why one curls and the other does not. A knitted loop is not symmetric front to back. Worked every course the same way, the asymmetries add along the edges and the fabric rolls; worked alternately, consecutive courses point opposite ways and cancel.

Why stockinette curls

A knitted loop is not symmetric front to back. Work every course the same way and the asymmetries add up along the edges; alternate them and they cancel. That is the whole difference between a fabric that rolls and one that lies flat.

6 figures · Knit One break, two outcomes. The same single break in a knit and in a weave. In the knit nothing holds the loop above the break, so the failure climbs the wale; in the weave every other thread is still held by the threads crossing it, and one thread comes loose.

Ravel, fray and run

Cut a woven cloth and one thread comes loose. Break one loop in a knit and every loop above it follows. The two structures fail in opposite ways, and the reason is topology rather than strength.

7 figures · Knit The tricot lapping. A warp-knit lapping drawn from the guide bar's movement. Each thread is coloured by the group of wales it belongs to, so a lapping that leaves the wales independent shows as several colours. This one joins them into 1 group.

Warp knitting, which is a different thing entirely

Every wale has its own thread, and if the thread never leaves its wale the fabric is a set of independent cords. Whether a lapping makes cloth is decided by a coprimality condition — the satin theorem, in a knit.

7 figures · Warp knit 1×1 rib — alternate wales to the back. A knitted fabric seen in section across the wales. Alternate wales pulled to the back fold the fabric like a concertina, so its relaxed width is a projection; pulling it wide unfolds the section and no yarn changes length while it happens.

Rib and interlock

A rib fabric is a plain knit folded like a concertina, and its enormous widthwise stretch is the fold opening out. Nothing in it is elastic, and the extension available is a cosine.

7 figures · Knit Four ways a fabric gets longer without stretching. Extension available from each mechanism, computed from the geometry that provides it. None of these numbers involves a yarn changing length; every one of them is a shape changing, and they differ by an order of magnitude.

Why a knit recovers and a woven does not

Four fabrics get longer without a single yarn stretching, and the four mechanisms are worth wildly different amounts. Three of them are recoverable and one is very nearly not, and which is which follows from where the extension came from.

6 figures · Knit What a second guide bar buys. Independent fabrics left by each lapping across 12 wales, counted by walking the wale graph and checked against the greatest common divisor of the shogs with the width. A bar that leaves more than one is not making cloth; a second bar can put right what the first could not.

What a second guide bar is for

One warp-knit bar leaves the wales in as many independent fabrics as its shog shares factors with the width. Two bars leave the greatest common divisor of both — so a pair of shogs that each fail alone can succeed together, and a pair that share a factor cannot.

6 figures · Warp knit 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.

8 figures · Stitch notation Yarn per needle, by structure. Wale spacings of yarn per needle position, at one loop length. A knitted loop is 4.30 of them — Munden's constant, measured rather than derived — a float across one needle is exactly one, and a tuck is taken as a stated multiple of a loop. Everything else follows by counting, so the percentages beside the bars are not estimates.

What a tuck costs

A knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop. That much is a count. What it does to the fabric's size is not, and this essay is careful about which is which.

7 figures · Stitch notation single float 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.

The float in a knit

Five rungs of this anchor have taken the float to be a length of thread on the surface with nothing holding it down. A knit has one too, and it behaves the same way in the light and the opposite way in the hand — because a woven float lengthens its thread and a knitted one shortens its fabric.

8 figures · Float rib-float in section. A two-bed structure seen in section across the wales, over 3 repeats of course 1. The front bed's loops sit on the upper line and the back bed's on the lower one, offset by 0.5 of a needle pitch because the gating is rib. Of the 2 floats in the repeat, 2 lie in the gap between the beds and 0 on a surface. The upper panel is the fabric at the machine and the lower one is the same course with the beds closed up, which is what happens when the fabric is cast off — and neither panel is a relaxed fabric, because the loops are drawn as arches of one size and a real one settles wherever the yarn's bending leaves it.

A second bed changes what a float is

Every float so far has been on a surface, because every knit so far has had one needle bed. Put a bed behind it and the yarn runs in the gap between them — and over the whole enumeration of two-bed structures, 1,248 floats of 1,272 lie inside the cloth, on no surface at all.

8 figures · Two-bed tubular, as a graph of its wales. Every wale of tubular as a node — 2 on the front bed and 2 on the back — with one chain per course joining everything that course takes yarn on, because a course is one traverse of one yarn. The nodes are filled by which component they fall into. This structure comes out as 2 fabrics: F0+F1 and B0+B1. The same answer is obtained a second way, by walking every partition of the wales and asking whether any course straddles it, and the two are required to agree.

Does a double jersey hang together

Two beds knitting with nothing passing between them are two fabrics that happen to have been made at once. Of 6,561 two-bed arrays, 1,135 are fabrics and 50 of those are two fabrics — and 28 of the 50 split across the beds rather than along them, so neither half is a layer.

7 figures · Two-bed A fashioned edge at 1 wales in 2 courses. A knitted panel narrowing by 1 wale every 2 courses, drawn at the fabric's own aspect: a wale is 1.2791 times as wide as a course is tall, which is Munden's ratio of the two published constants. The edge therefore stands at 32.60 degrees from the wale, and that angle is the same in every yarn, at every gauge and at every loop length. Marks show where the 8 transfers fall.

A fashioned edge has a quantised angle

A knitted panel is shaped by transferring loops, so its edge steps by whole wales at whole courses and its angle is the arctangent of a fraction. The available angles turn out to be the same for every plain knit there has ever been — in any yarn, at any gauge, at any loop length — because the constant they scale by cancels the loop out. There are eighteen of them, and 16.67° between the last two.

6 figures · Shaping How far a tube can be tapered by its loop. The same 10 wales by 10 courses of plain knit at the two ends of the usable loop range for a 20 tex yarn — 3.44 mm at the loose end and 2.80 mm at the tight one — drawn at a common scale in centimetres. On 240 needles the circumference falls from 192.0 cm to 156.0 cm, a taper of 18.8 per cent, and the fabric becomes 1.51 times as dense.

A tube can only be shaped by its loop

On a circular machine the needle count is the cylinder, so a seamless tube's circumference is its wale count times its wale spacing — and the wale spacing is the loop length over one constant. The loop is the only free quantity, the yarn bounds it at both ends, and what is left is a taper of 18.8 per cent bought at the price of a fabric half again as dense.

6 figures · Shaping The loop that costs nothing to extend. A plain knitted loop at rest and extended by 35 per cent, with the arcs marked. The arcs' radius is the diameter of the yarn the loop wraps, 0.167 mm, and it is set by contact rather than by the fabric's dimensions — so extending the fabric lengthens the legs and bends nothing further. The bending energy is 0.0176 N·mm at both, and the model therefore asks no force at all for an extension a woven cloth would refuse. What the drawing cannot show is what a real knit's first few per cent do cost, which is friction and yarn flattening and is not a bending property.

A knit is soft because it bends

Ask the same energy question of a woven cloth and a knitted one and the answers are not different by a factor — they are different in kind. A woven cloth's bending energy changes the moment it is extended. A knitted loop's does not change at all, exactly, over the whole of its extension, because its arcs are held to a radius by contact rather than by the fabric's dimensions.

6 figures · Knit A knit's restoring force, and the column that does not move. A plain knit of 20 tex cotton at a loop length of 3.5 mm, over the extension range its own geometry admits. The bending energy stored in one loop is the same number at every extension — the loop's arcs are held to a radius by the thread they wrap rather than by the fabric's dimensions, so extending the fabric does not bend anything more. The frictional resistance at the interlocks is not zero: it is μ times the force pressing there, times 2.34 interlocks per millimetre of width. So a knit's resistance to extension is dissipative rather than elastic, which is why it does not spring back and why its dimensions depend on how much it has been agitated. What the rows cannot show is the interlock force itself, which this site does not have for a knit and which is recorded as missing.

What stops a knit extending

A knitted loop's bending energy does not change as the fabric extends — exactly, over the whole range its geometry admits. Something resists, and it is not stiffness. It is friction at the interlocks, which is dissipative rather than elastic, and that single fact accounts for why a knit does not spring back, why a softener changes its dimensions and why the constants its size is quoted with contain no yarn property at all.

6 figures · Knit A loop's cell, dry and wetted. One stitch of a 20 tex cotton jersey at a 3.50 mm loop, drawn inside the rectangle of one wale by one course that its own dimensions give. The thread is drawn at its own width, and it already fills 1.129 of the cell dry — more than the whole of it, which is what an opaque jersey looks like from above. Wetting takes it to 1.355. So the reason a knit does not build a swelling pressure is not that it has room; it is that its dimensions are a loop length times a constant with no yarn diameter in them, so there is no closure condition to fail. What the drawing cannot show is the third dimension: the legs lie over one another rather than overlapping in the plane, which is exactly why an occupancy above one is possible.

A loop has no closure condition

A woven cloth can run out of room: its two systems must supply its whole thickness between them, and past a certain swelling they cannot. A knit has no such equation, so no critical swelling and no pressure. The obvious explanation — that a knit is open and has somewhere to put the swelling — is false, and the arithmetic refuses it.

6 figures · Knit Munden's states against the fibre swelling. What a plain knit does between its relaxation states, beside the swelling of the fibre it is made of. Going from dry-relaxed to wet-relaxed a jersey loses 5.66% along its courses and 2.44% across its wales, and going on to fully relaxed it loses 9.1% and 7.0%. The fibre swells 20%. Two things rule the swelling out as the cause: it is several times the whole change, and the change is markedly anisotropic while a swelling enters both of a loop's dimensions through the same loop length. What the bars cannot show is what does cause it, which is friction — water lets the loops move to where the yarn's own bending had been trying to put them.

A knit's change of state is not its swelling

A jersey is smaller wet-relaxed than dry-relaxed, by 5.7 per cent along its courses and 2.4 across its wales. Water is obviously involved, so the swelling is the obvious cause. Two things rule it out, and both are properties of the constants rather than measurements of a fabric.

6 figures · Knit A plain knit's two relaxation steps. Munden's three relaxation states are usually given as three sets of constants. Read as a path they are two steps, and the two compose to the whole exactly — which is a real check, because the three sets were measured independently. The first step is the larger in the course direction and the smaller across the wales, and the second is 0.64 of the first lengthwise. That is the shape of a laundering series and it is the same mechanism: a fully relaxed state is reached by tumbling rather than by waiting, so what the standard specifies is a quantity of agitation and not a duration. What the bars cannot show is the loop length, which cancels out of all four numbers because every dimension of a knit is a loop length times a dimensionless constant.

A knit relaxes for as long as it is allowed to

Munden's three states are usually given as three sets of constants. Read as a path they are two steps, they compose exactly, and the second is not a smaller version of the first — the fabric shrinks twice as much along its courses as across its wales on the first step and rather less than half as much on the second.

7 figures · Knit Which knitted structures spiral, at a twist factor of 4.0. A yarn leaves the spinning frame with a torque it has not been allowed to release, and a loop knitted from it leans. The lean per unit of twist factor above balance is measured; what is counted here is the structure. A loop on the front bed and one on the back are mirror images, so their torques have opposite signs, and a fabric that knits equally on both beds nets to zero whatever the yarn is doing — which is why 1x1-rib, 2x2-rib, interlock do not spiral and plain, half-cardigan, tubular do. The count has to be made per fabric and not per structure: an interlock and a tube both knit equally on the two beds, and they are opposite cases, because an interlock's two components each straddle the beds while a tube's are each wholly on one. The integrity criterion, which asks what nothing holds together, is what tells them apart. What the bars cannot show is the tube's second face, which leans the other way.

A jersey leans because its yarn still turns

A single-jersey T-shirt comes back from the wash with its side seam spiralling round the body, and a rib does not. The difference is not the yarn: it is a count. Loops on opposite beds are mirror images, so their torques oppose, and a fabric that knits equally on both nets to zero whatever the yarn is doing.

6 figures · Spirality When a knit has a hole between its loops, and when it has none. A loop's occupancy is its length times its diameter over the cell it sits in, and Munden's constants make that cell ℓ²/(k_c·k_w). The loop length cancels once and what is left is d·k_s/ℓ, where k_s is Munden's own stitch-density constant — so whether a knit has a hole between its loops depends on d/ℓ and on nothing else: no gauge, no count, no fabric dimension. The three curves are the three relaxed states of 20 tex cotton. Each crosses one at a loop length of 3.34, 3.63, 3.95 mm respectively, and a jersey is knitted at 2.63 to 3.44 mm — the shaded band. Every commercial jersey is therefore on the wrong side of the threshold in at least two of its three states: it closes its own holes as it relaxes, and its air goes through its threads rather than between them.

A knit has no hole to lose

Every argument in this ladder is planar: threads at a spacing, a rectangle between four of them, a channel down it. Applied to a jersey it returns nothing at all, and the nothing is the finding. A loop's occupancy is its diameter times Munden's own stitch-density constant over its loop length, with no gauge and no fabric dimension in it — and the whole commercial range of tightness is on the wrong side of the threshold.

6 figures · Knit geometry 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.

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.

6 figures · Knit