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The thread: A mechanism, not a material — page 4

Page 4 of 6 of the essays on this thread.

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.

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 rib is quietest at a gap of two diameters. The through-thickness force of a one-by-one rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 14.8 mN at 5 diameters against 7.0 mN at two.

Knits and other structures

A rib is quietest at two diameters

Open the beds of a rib and everything about it should get stronger. It does not. The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters and rises on both sides of it, because a crossing's climb and an interlacing's own climb cancel there.

A knitted interlacing, and the number that makes it a fabric. Two loops idealised to rings, one drawn through the other, which is what a needle does. The Gauss linking integral returns 1. A knitted fabric of n wales has that between every pair of adjacent courses n times over, and it is the whole of why a knitted fabric can be made from one thread and taken apart by pulling it.

What cloth is

Why a knit runs and a weave frays

The two fabrics fail in two ways and everybody knows which is which. This collection has described both accurately for eighteen phases without being able to say what causes them, and the cause turns out to be one integer each: nought for a cloth, one per wale for a knit.

A bouclé at an overfeed of 0.80. A core of 20 tex running straight, an effect thread of 30 tex delivered 80 per cent faster than it, and a binder of 15 tex over the top. The surplus effect thread has nowhere to go but into loops, and their size is not a free choice: a semicircular loop of radius r consumes r(π − 2) of surplus, so at one loop every 3 mm the radius is 2.10 mm — 0.70 of the spacing. The yarn's resultant count is 89 tex and the sum of its three components is 65, because the effect enters multiplied by its own overfeed. What the drawing cannot show is the load path: the core and the binder are at the yarn's own length and the effect thread is longer than the yarn, so a pull on the yarn is carried by 39 per cent of its mass and the loops stay slack.

Compound and figured cloths

A fancy yarn has its crimp in the wrong thread

A bouclé is made by feeding an effect thread eighty per cent faster than the core it wraps, so the yarn contains more thread than it is long — which is crimp by this collection's own definition. A cloth's crimp is in the thread that carries the load, so removing it is the cloth's first extension. A bouclé's surplus is in a thread that carries nothing, so pulling the yarn stretches the core at once and the loops never straighten. Thirty-nine per cent of the yarn's mass is on the load path, and the other sixty-one is decoration.

Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Pattern and colour

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

Setting and geometry

The sett owns the pitch and the yarn owns the height

Set a cloth twice as close and its surface does not get smoother. The crowns come twice as often, because that is what a sett is, and they stand at very nearly the same height, because the closure condition pins the amplitude to the yarn — so a fine cloth is finer-grained rather than flatter, and the two are confused in every description of handle.

A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

Mechanics and drape

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

Thirty micrometres is a buckling load. The wool fibre diameter at which a protruding end reaches the measured threshold for prickle, against how far it protrudes. A fibre end pressed against skin is a column held at the cloth and free to slide at its tip, so it buckles at 20.19EI/ℓ² and carries no more load than that; below the threshold it bends away and is felt as touch, above it the load stands and is felt as pain. At a two-millimetre protrusion — which this site's own hair model puts in the top fifth of a wool's population — the threshold diameter is 32 µm, bracketed at 29–34 by the range of the measured force. Neither the thirty micrometres nor the two millimetres was put in. The band is the force bracket, and the fourth power in I = πd⁴/64 is what makes the boundary sharp: at a mean of 21 µm and a spread of 24%, 3.2% of the fibres are over it, and it is that few per cent that decides whether a garment can be worn.

Cloth doing a job

Prickle is a buckling load

The wool trade specifies comfort against skin by the percentage of fibres coarser than thirty micrometres, and the thirty is a measured boundary with no derivation attached. It is a column formula: solve for the diameter at which a protruding fibre end stops bending away and starts standing its ground, and thirty micrometres falls out.

An 8-end weft-faced satin through a point tie on an end. The 8-end satin on a move of 3, weft-faced, woven through a point tie of 12 hooks that turns on a single end, so 22 ends carry one repeat. The strip above is the tie, hook number against end. Inside the satin the weft floats 7 ends; across the turn it floats 13, outlined, because on some pick the nearest interlacing is 7 ends away on both sides. What the drawing cannot show is how a long float at one line down the cloth behaves in use, which depends on the yarn and the finish rather than on the count.

Compound and figured cloths

A point tie nearly doubles the float at the turn

A point tie halves the hooks a symmetrical figure needs by driving every hook's end twice, out and back. It mirrors the ground as well, and a satin is symmetrical about no end: across the turn its weft floats 2n − 3 ends, or 2n − 2 between ends, at every hook count and wherever the satin starts. At eight ends that is thirteen or fourteen against seven — and the turn's other promise, more repeats that fit the width, is not kept either.

Seeing through a white voile, from the street and from the room. How much of a scene's contrast survives a white voile, whose open area is 49%, looked through from the street into the room and from the room out to the street, against how much brighter the street is than the room. The scene comes through the clear lines of sight and nothing else; everywhere else the viewer sees thread lit from the viewer's own side, which returns light with no image in it. On a bright day, a hundred times brighter outside, 0.5% of the room's contrast reaches the street and 65% of the street's reaches the room. At equal light both are 28% — the curves cross at a ratio of one whatever the cloth — and with the lamps on after dark the room is the side on show, at 63%. What the plot cannot show is the threads' own optics: their reflectance and transmittance are assumed values for a white sheer rather than measurements, and only the crossing point is independent of them.

Pattern and colour

A sheer hides whichever side is darker

A net curtain hides a room by day and shows it at night, and the cloth has nothing to do with which. What a viewer sees through a sheer is an image through its clear lines of sight against a veil of lit thread, and the only thing that decides their balance is how much brighter one side is than the other. At equal light the two views are identical for every cloth there is. And a black sheer of the same openness shows twelve times more of the room by day than a white one.

Warmth is a canopy, and a canopy grows as a logarithm. Thermal resistance of sheeting against how much its hair population has been multiplied by raising, both faces counted. The unraised cloth is given no still air at all, because its hairs cannot reach one another — n_A λ² is under one and there is no canopy to hold air still. Past the threshold the canopy's depth is λ·ln(n_A λ²), so every doubling of the hair population adds the same depth of nap and no more: the steps here are 434 µm apiece, all the way up. At 256× the nap is 3.07 mm deep and worth 39 times the cloth it grows on, which is the whole reason a flannel is warm and a poplin of the same yarn is not — a canopy is two parts in ten thousand fibre, so its conductivity is air's, while the cloth itself is a quarter fibre. What is claimed is a conduction resistance across a depth of nearly still air; whether the air is still is a question about flow and is not asked here.

Mechanics and drape

Warmth is mostly the hairs

This collection established that a fabric's thermal resistance is its thickness and not its fibre, and that twice the thickness is twice the warmth. It never asked what the thickness was made of. On a raised cloth almost none of it is cloth.

A shot effect needs a fibre with no ends. The peak-to-trough contrast of an eight-end satin in sheeting as the cloth is turned in the light, against how much hair stands on it. Bare, the contrast is 37 to one, because a straight thread's normals lie in the plane across it and the warp and the weft therefore reflect a quarter turn apart. A hair layer does two things and only one of them matters: it blocks, which takes the same factor off the peak and the trough and changes no contrast at all, and it returns light of its own, which is added to both. A hair population points every way at once, so its return has no azimuth in it — and adding a constant to both ends of a ratio of 37 destroys the ratio. On an ordinary spun cotton the contrast is already down to 5.1 to one; singeing recovers it to 29; raising kills it outright at 1.00. The one fibre with no staple length is the one fibre with no fibre ends, and every shot fabric ever woven is made of one.

Weaves

A hair layer veils a highlight

An eight-end satin's shine swings by a large factor as the cloth is turned, because a straight thread's normals lie in the plane across it. Put fibre ends on it and the swing disappears — not because the hairs block the light, which changes no contrast at all, but because they return light of their own that has no direction in it.

$A woven filter catches what its rating says it cannot. What fraction of a particle stream is intercepted by the hair layer of a filter cloth, against particle size, at three levels of raising. The cloth's own largest opening is 290 µm, so by geometry it stops nothing smaller than that at all — and the hairs catch a few per cent of particles ten and a hundred times finer, because a particle whose path passes within its own radius of a hair touches it. On a bare cloth the numbers are small; the point is that they are not zero, because a cake grows from the particles that stop, and once a cake exists the cloth is no longer doing the filtering. Raising the same cloth 32-fold takes a ten-micrometre capture from 0.8% to 23%, which is why a napped filter cloth exists. Interception is taken as the bare geometric ratio of the two diameters with no flow model behind it, so every number here is a lower bound.$ Cloth doing a job

A woven filter beats its own rating

A filter cloth's rating comes from the largest channel through it, and by geometry it stops nothing smaller. It stops a few per cent of particles ten times smaller anyway, on the fibre ends standing in its holes — and a few per cent is not filtration. It is exactly enough to start a cake, and after that the cloth is not filtering.

What a raising machine can catch in a 2/2 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

After the loom

What holds a nap in a knit

A fibre buried in a woven cloth breaks rather than slides once about thirty-four millimetres of it is held, and a cotton staple buries about fourteen. In a knit the same figure is over a metre — seventy-five times the burial available — so nothing is ever close, and a raised knit sheds for the whole of its life.

What cloth is

Where a yarn is thinnest

A yarn in a fabric is pressed where it crosses and free where it does not, so its section changes along its own length. Every flattening this collection has ever quoted is a single number for a profile that runs from four fifths of a diameter to nearly four.

A lancé weft bound where the 8-end satin interlaces. A warp-faced 8-end satin on a move of 3, with an extra weft thrown after every ground pick and laid on the face over figures 12 ends wide and 28 ends apart. Between figures it is bound only on ends the ground already drops on an adjacent pick, 56 places in the drawn repeat, which leaves back floats of at most 4 and no binding on the face. The drawing is 40 ends by 16 picks and counts as 1 cloth. What it cannot show is whether a binding point counted as hidden is hidden in a real cloth, which depends on the yarns' contrast and on how closely the ground's floats cover it.

Compound and figured cloths

A brocade weft floats as far as the next figure

A brocade's pattern weft is a third thread, laid on the face only where the figure wants it. Thrown from selvedge to selvedge it floats on the back across the whole gap to the next figure, unless an end is dropped under it — and on a satin ground a dropped end hides only where the ground already interlaces. That fixes the shortest float that shows nothing, at the larger of the move and its complement less one, and puts the smoothest satin ground at odds with the best-bound brocade.

Two weft colour orders thrown on a loom with boxes at one side. Weft colour orders thrown pick by pick on a shuttle loom that picks alternately from the two sides. For an order with runs of four, two, two and four, every throw finds a shuttle of its colour on the side it leaves from, so the order can be woven. For an order with runs of three and three, pick 4 has to be thrown from the right in a colour whose shuttle is on the other side, and the order cannot be woven. On a loom with boxes at one side the box opposite holds only the shuttle just thrown, which the next pick must throw straight back, so colour can change only between pairs of picks. What the drawing cannot show is the mechanism that drops the boxes, which decides how fast a change can be made but not which changes are possible.

Pattern and colour

A weft stripe is counted in pairs of picks

A warp's colour order is laid out once at warping and the loom never has to think about it. A weft's is thrown, one pick at a time, by shuttles that cross the cloth and stay where they land. On a loom with boxes at one side that makes every coloured band an even number of picks; with boxes at both sides it admits odd bands and pays for them in shuttles; and a tartan, which uses one order in both directions, is designed for its weft whether its designer knew it or not.

Compacting a spinning triangle moves one instrument and not the other. What happens to each hairiness reading when a 20 tex cotton yarn is spun compact instead of ring, as a percentage of the ring value. The total falls by 8% and the long hairs by 65%, a ratio of 8.3. The asymmetry is a prediction rather than a fit. Compaction removes ends that were unbound over a long stretch of the spinning triangle, which is the long population and nothing else; the short population is untouched, and it carries about 88% of the length the integrating instrument is adding up. So the instrument that sees everything barely moves and the one that sees only the tail collapses. The model under-states the fall in the total, because compaction certainly does something to the short population too and nothing here models it — the direction of that error is stated and it is the conservative one.

Setting and geometry

The spinning triangle decides the hair

A ring frame converges a flat ribbon of fibres to a round yarn, and for the length of that convergence the fibres at the ribbon's edges are held by nothing. Everything a spinner can do about hairiness is done in that triangle, and the two hairiness instruments respond to it by wildly different amounts.

What holds a thread in, as two factors. The two quantities whose product is the grip on a buried thread, for a woven poplin and a jersey of the same yarn. Each contact in the knit is lighter by 8.0 times, and the contacts are further apart by 5.6 — one per half loop length against one per thread spacing. They multiply rather than competing, so the grip per millimetre of buried thread is 31 times weaker in the knit, and the crossover length at which a thread breaks rather than slides moves with it: 11.6 mm in the cloth and 461 in the knit. The two factors are drawn separately because their product is two orders of magnitude and a bar chart of it would put the knit's bar below the width of a line.

Cloth doing a job

The knitted pilling criterion gets its number

Knitwear pills and shirting does not, and the standing explanation here has been that a knit presents more exposed yarn under less pressure between its threads. The second half had no number. It has one now, and it is bigger than the argument needed: the grip on a buried fibre is thirty times weaker in a knit than in a woven cloth of the same yarn.

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.

After the loom

A state is a thickness too

This collection's rule is that a fabric dimension quoted without its relaxation state is not a measurement. A knitted fabric has three dimensions and only two of them obey the rule: its thickness is the same in every state, because the interlacing that sets it does not relax.

What cloth is

A fabric is a population of contacts

Only a fifth of a knitted fabric's yarn is inside a diameter of its neighbour. So a fabric's friction lives in a fifth of its length, and every calculation this collection makes about withdrawal, slippage and fraying has assumed it lives everywhere.

A jacquard harness 130 cm wide under a 150 cm fall. A jacquard harness in front elevation, to scale: hooks spread over 40 cm, cords fanning down 150 cm to a comber board 130 cm wide, and hanging straight from the board to their mails. The centre cord is vertical and lifts its mail by the whole 10 cm hook lift; the edge cord leans 16.7° and lifts its mail 9.60 cm, 96.0% as far, and its bend at the board adds 9% to the hook's load at a friction coefficient of 0.3. What the drawing cannot show is the tie that decides which hook feeds which hole, which in a real mount is not the simple spread assumed here.

Compound and figured cloths

A jacquard harness needs three half-spans of height

A jacquard has no front shaft and no back one, so every end takes the same shed — as long as every cord hangs straight. Across the width they cannot: the hooks sit in a machine a few tens of centimetres wide and the comber board is as wide as the cloth, so an edge cord leans and its mail rises by the difference of two hypotenuses rather than by the hook's lift. Holding the edge shed within five per cent of the centre's takes a fall of about three times the edge cord's sideways reach, and that is a height a room has to have.

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.

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.

Setting and geometry

A woven thread has no room to bend

Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.

What a run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.

Cloth doing a job

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

Knits and other structures

A jersey's course has no writhe

The mechanism everybody quotes for why a hard-twisted jersey leans is that the fabric relieves the yarn's twist by writhing. This collection's own solved course has a writhe of minus six parts in a million, and it is not small — it is exactly zero, by a symmetry, and the symmetry is a statement about what the model left out.

Pattern and colour

A figured sheer is a negative from one side

A net curtain with a pattern in it carries two patterns, one for each side, and by day they are opposites. A more open figure in a white voile is a dark figure a quarter below its ground from the street and a light one five per cent above it from the room, and after dark the two views trade places exactly. The figure vanishes from the street at one light ratio and from the room at another. And a figure can be made that the room cannot see at all while the street sees it nearly black — along one line of openness and thread tone, and never from both sides at once.

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.

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.

What leaving the plane costs, which is less than nothing. Bending energy a stitch against how far the yarn climbs between one interlacing and the next, for a 20 tex cotton jersey at a 3.5 mm loop. The curve falls. A climb is not an extra bend added to a curve that was otherwise unchanged; it is part of the straight line the thread has to span, so a longer climb leaves less slack to be spent on curvature. At a jersey's own climb of one diameter the energy is 24395 nJ against 25060 for the planar model — 2.7% lower, where an earlier estimate on this site put the ride at three per cent of the loop's bending and had the sign the other way round. At a rib's climb of three or four diameters the fall is 33%.

After the loom

A rib's relaxation is not its bending either

A jersey does not settle where its bending energy is least, and this collection has said so for several rungs. A rib does not either, and the model now says how far from least it would have to go: sixteen yarn diameters of bed gap before the yarn runs out, against the two or three a machine is set to.

What matching a pattern costs a cutting room. The cloth a matched panel needs beyond its own length, against the pattern repeat, for 6 panels of 70 cm. Every panel of a patterned cloth must start at the same phase of the repeat or the pattern breaks at the seams, so a panel's cut length is rounded up to a whole number of repeats. The bar is the allowance a cutting room budgets — a whole repeat a panel, because a panel's length is not a multiple of anything — and the mark is the waste actually expected, which is half a repeat. The two differ by (L + r)/(2L + r), which is between a half and two thirds and is nearer two thirds the larger the repeat. The rows marked in the second colour are the repeats that happen to divide the panel exactly and waste nothing at all, which is what makes the real cost jagged rather than smooth. What the bars cannot show is nesting: a cutting room lays many panels on one length and a short panel can sometimes be taken from another's waste.

What cloth is

A repeat has to fit the panel, and the panel is cut

The warp's width is fixed at warping and a piece's length is not, so the fitting problem in the two directions is not the same problem. Along the length a repeat has to fit a *panel*, because every panel of a patterned cloth must start at the same phase or the pattern breaks at the seams — and the allowance is a whole repeat per panel. A ten-centimetre repeat costs a seventy-centimetre panel twelve and a half per cent and a sixty-four-centimetre repeat costs it forty-eight.

Compound and figured cloths

The doup end pays for the crossing

A leno holds its picks because its doup end crosses under its partner and comes up on the other side — half a turn at every crossing, whatever the sett. That half turn is also a length. When 0.25 mm ends cross every millimetre, the doup end travels 22.5 per cent further than its partner: twenty-two metres of extra warp over a hundred-metre piece, and a shared beam a pick short within four and a half millimetres of cloth. The extra falls with the square of the crossing interval, which is why a leno is woven from two beams and its crossings are spaced as far apart as the cloth allows.

A 1.55 mm net 50 mm behind a 0.3 mm voile, from 0.6 m, 1.5 m, 4 m. A 1.55 mm net 50 mm behind a 0.3 mm voile, drawn across 40 mm of the near layer at true pitch as seen from 0.6 m, 1.5 m, 4 m. Two grids this different beat through a harmonic: the net's k-th against the voile's first, for the k nearest the ratio of their pitches as the eye sees them. From 0.6 m that is the fifth, in register every 6.2 mm; From 1.5 m that is the fifth, exactly in register, with no fringe; From 4 m that is the fifth, in register every 14.9 mm. What the strips cannot show is how strong each family is, which falls with the harmonic, nor the net's second family of threads at right angles.

Pattern and colour

A net over a voile beats through a harmonic

Two identical sheers hung apart make a moiré by perspective alone. A net in front of a voile is not two identical sheers — its mesh is five times the voile's pitch — and it beats anyway, through the net's fifth harmonic, which is a grid 3.3 per cent coarser than the voile. With the net behind, that is a pair of sheers with its coarser layer at the back, and the fringes vanish at exactly 1.5 metres. Closer in, the harmonic changes, the fringes dissolve into a texture twice the net's pitch and re-form, and they vanish again at 17 centimetres.

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.

$A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.$ Cloth doing a job

A knit is warm because of where its yarn is not

Warmth is a thickness of still air, and until a knitted fabric had a thickness there was nothing to compute. It has one now, and the answer is that a rib's warmth is a machine setting: opening the beds from two diameters to five nearly trebles the fabric's resistance without changing a gram of yarn.

Two numbers that stay at zero however large the fabric gets. The linking number of two adjacent courses, and the writhe of one course per wale, for tubes of 6 to 20 wales. Both sit at zero and stay there: the largest departure anywhere on the plot is 1.5e+1, which is the sampling. A quantity that should grow with the fabric and does not is the cleanest kind of null result: the model has the geometry of knitting and none of its topology, and making the fabric bigger does not make the topology appear.

Knits and other structures

Five symptoms of one omission

Five things this collection recorded as unexplained, found in four different ladders over three years of work. They are the same defect seen from five directions, and the defect is one sentence written for good reasons with no visible cost at the time.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

Setting and geometry

Why a slack yarn snarls

Let go of a twisted thread and it wraps on itself. That is not the yarn being badly behaved: it is a buckling, it has a criterion, and the criterion turns a nuisance into an instrument for measuring the one constant this collection cannot pin down.

How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

After the loom

A yarn that has been set has no torque

Every torque on the torsion ladder assumes a yarn is elastic in twist for ever. It is not: a steamed yarn's residual torque is gone, its twist is unchanged, and the process that removes one without the other is the trade's whole answer to liveliness.

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.

Setting and geometry

How much yarn has to hang

The tension a thread needs to stay straight, converted into the only unit anybody has an intuition for: the length of the yarn's own weight. One bound says two metres and the other says six hundred, and everybody who has handled thread already knows which.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

Knits and other structures

The fabric that does not fit

Every solve in this collection minimises an energy over a centre line, and a centre line has no thickness. Nobody had checked whether the fabric that comes out of it can be built. It cannot: two adjacent courses of the relaxed jersey approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere.

After the loom

The twist a fabric gives back

A T-shirt that hung straight in the shop has its side seam round the front after three washes. The torque was there all along, the setting had hidden it, and the water gave it back — which makes spirality a finishing failure rather than a knitting one.

The draft for 2/2 twill, as a loom holds it. The 2/2 twill written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

What cloth is

A lifting plan says nothing without a threading

The second of the notations for something larger than a weave is a pair, not a notation: a threading and a lifting plan, and neither alone expresses anything. The pair's image is exactly the drafts with no more distinct columns than there are shafts — 98 at two shafts, 5,282 at three, all 22,874 at four — and it is not nested with the profile draft's in either direction. The profile reaches 192 drafts that need all four shafts, and misses 64 of the 98 a two-shaft loom weaves.

What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

Cloth doing a job

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

The crossed shed, with the back standard 4 shafts behind the doup. A leno's crossing end in section from the fell to the back rest of an ordinary broad loom, in the crossed shed. The doup 300 mm from the fell lifts it 50.0 mm; its back standard 64 mm further back leaves it 60.7 mm down, beneath its partner. Between the two eyes the end climbs 110.7 mm in 64 mm, and the whole path is 70.2 mm longer than the straight line, 5.85% of the free warp, of which 63.8 mm is that one climb. The same end in the open shed, lifted at its back standard, is 7.2 mm longer, and an ordinary end on the doup's shaft 5.5 mm: the crossed shed is 12.7 times the ordinary shed. Vertical scale exaggerated 3 times. What the drawing cannot show is friction at the eyes, which would keep the extra length in the short span between them.

Compound and figured cloths

An easer gives back the kink the crossed shed puts in

A leno loom gives its crossing ends slack at every crossed shed, and the obvious reason — the doup carrying its end sideways — is worth a hundredth of an ordinary shed. The length is one heddle further back. In the crossed shed the doup lifts the end on the far side of its partner while the end's own back heddle, a few shafts behind, holds it down, and between those two eyes the end climbs eleven centimetres in six and a half. On a loom that stretches an ordinary end 0.46 per cent, that is 5.85 per cent — nearly thirteen times as much — and no position of the back heddle takes it below four.

A white thread 70% open figure behind a net 60% open, from both sides. A lozenge figure in the inner curtain of a pair, drawn as the street sees it and as the room sees it through a plain net 60% open, with the street 30 : 1 as bright as the room. From the street the figure's contrast against its ground is −7.0% where a single curtain would have given −24.6%; from the room it is +6.9% where a single curtain would have given +5.4%. Each panel is shaded relative to its own brighter region, both at one gain, so the two steps are to scale against each other, and the contrast is stretched 9 times to be visible at all — the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the light between the two curtains, which is 19.80 times the room's illuminance from the street and 0.57 from the room, nor the thread optics, which are assumed.

Pattern and colour

A net in front gives the figure to the room

The commonest double curtain is a plain net outside a patterned one, and the plain net does not merely dim the pattern. It weakens the figure from the street by a factor of three and a half and strengthens it from the room by a quarter, so a pattern four and a half times stronger outside than in becomes one the room sees slightly better. A figure designed to be invisible from indoors reappears at nearly three per cent, and the day-and-night exchange a single curtain obeyed exactly stops holding at all.

How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

Mechanics and drape

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

Cloth doing a job

A run cannot cross a bed

A dropped stitch unroves because its neighbour above can pull it out along a path that costs almost nothing. In a rib the neighbour above is on the other bed, and the path goes through the gap — so a run in a two-bed fabric has to pay for a climb before it can take a single loop.

Setting and geometry

A snarl comes in one size

The radius a twisted thread coils to is twice its bending rigidity over its torque. Write the torque out and the bending rigidity cancels completely, leaving a number that depends on the twist and on the ratio of two stiffnesses — and on nothing else about the yarn at all.

The section the fabric asks for, beside the one the model drew. A 20 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve here has assumed: a diameter of 0.167 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 78% of it, which is the closest the fabric's own adjacent courses come to one another — 0.130 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which has been carried here as a free parameter ever since it first put a number on a sett.

Knits and other structures

The flattening nobody fitted

A yarn in cloth is not round, everybody knows it, and nothing has ever predicted how flat. This collection's own knitted geometry turns out to require a flattening of four fifths — from a solve that knew nothing about flattening, made no allowance for it, and would have been written the same way if the idea had never occurred to anybody.

The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

After the loom

A wet knit's yarn is flatter

A knitted fabric's own geometry demands a flattened yarn, and how flat depends on how much room the fabric leaves. A wet cotton yarn is a tenth thicker than a dry one at the same length, so a wet fabric leaves less room — and asks its yarn to be flatter by an amount the geometry gives.

Two lots of one cloth matched at the top over a 250 cm drop. Two lengths of the same patterned cloth, woven with the warp at tensions of 0.6 and 0.7, which relax by 6.80% and 7.94% along the length, hung side by side over a 250 cm drop and matched at the top, with a mark at every 32 cm repeat. The whole drop is drawn to scale on the left and the last mark at six times the scale on the right: lot B's repeat is 31.61 cm, and 7 repeats down its mark is 27.3 mm from lot A's. A 3 mm match holds for the first 25 cm. What the drawing cannot show is how large an offset an eye accepts across a seam, which the tolerance stands in for.

What cloth is

Two lots of one cloth drift apart down a drop

A pattern repeat is a count of picks, and its length in the finished cloth is whatever that count's loom length became after the cloth gave back its crimp. Two lots woven with the warp held a tenth harder give back 1.13 points more of their length — so matched at the top, two lengths from the two lots are thirty millimetres apart at a 250 cm hem, and visibly apart a quarter of a metre down. The size of the repeat does not enter, and no cutting allowance can take it out.

Pattern and colour

A curtain is gathered so that it is seen edge-on

A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

Mechanics and drape

What friction has to hold in a relaxed knit

A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

Where a fold's two moments cancel, and where the trade folds. The two moments about a fold's own axis, for 2 singles of 20 tex cotton at 800 turns a metre. The falling curve is what the singles' own residual twist supplies, which the folding takes out of them; the rising one is what bending each single onto its helix costs. They cross at 161 turns a metre, a ratio of 0.201, and the closed form for that crossing is C/(B+C) — the ratio of the two stiffnesses and nothing else. The shaded band is where the trade actually folds, 0.6 to 0.75 of the singles twist. The balance point is nowhere near it, by a factor of three.

Setting and geometry

The folding rule is not a torque balance

Fold a two-fold yarn at about two thirds of its singles twist. This collection has carried that as a bracket copied from the trade and derived nowhere. It is now derivable, the derivation gives a fifth rather than two thirds, and reaching two thirds would need a fibre stiffer in torsion than in bending.

$A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot.$ Cloth doing a job

A knot is nothing but contact

A knot has no fastening in it. Nothing is glued, hooked, sewn or threaded through a hole: a thread is bent round itself until the friction where it presses on itself is more than the load. That makes a knot the purest contact problem in the subject, and the place to look first for what contact does.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 24.2 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.184 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.140 mm — 0.761 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 76% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

After the loom

What wetting does to the bending limit

A rod cannot be bent to a radius below its own. A relaxed knitted loop sits at twice that limit, and a wet yarn is a tenth thicker in the same loop — so wetting moves a fabric a tenth of the way towards a bend it cannot physically take.

3 widths of cloth at 2% skew and 0% bow, hung level. 3 widths of a 140 cm patterned cloth hung side by side and joined at their selvedges, with three of its pattern rows drawn across each. The cloth's weft is skewed by 2% of the width and bowed by 0%. Hung level, every width starts its rows at the same height, so each seam shows the skew's 28 mm step. The rows' displacement is drawn 8 times its true size against the width. What the drawing cannot show is the ceiling, pole and hem a real hanging is seen against, which is what a tilt is read by.

What cloth is

A skew steps at every seam and a bow at none

A finished cloth's weft rows are seldom square to its selvedges, and on a patterned cloth the weft rows are the pattern rows. A skew — rows straight but slanting — puts its whole rise across every seam where two widths meet: two per cent on a 140 cm width is a 28 mm step, and matching the steps out tips the whole hanging instead, 112 mm across four widths. A bow — rows curving between level selvedges — matches perfectly at every seam and scallops each width in between. A three-millimetre match needs a skew under a quarter of a per cent, and down the seam a lot difference overtakes it.

What a passer-by is looking at, for rooms of four reflectances. The street's view of a window behind a sheer white thread 49% open at 30 : 1, split into the part that is an image of the room, arriving through the holes, and the part that is the cloth's own threads lit by the street. a room reflecting 8% contributes 0.4% of the view; a room reflecting 20% contributes 1.0% of the view; a room reflecting 35% contributes 1.8% of the view; a room reflecting 60% contributes 3.1% of the view. The veil is identical in every row because the cloth and the street are: only the furniture changes. What the chart cannot show is where in each room that reflectance sits, and a dark room with a lit lamp in it is not its own average.

Pattern and colour

What a sheer hides is decided by the furniture

Every result these essays have produced assumed the room and the street reflect the same three tenths of the light on them. That was never a fact about cloth. A room reflecting a twentieth is hidden two hundred and thirty-eight times better than the street it faces and one reflecting four fifths only fifteen times, with the same curtain at the same window — and the day-and-night exchange they rested on, exact for equal scenes, is out by more than half the glow for an ordinary pair of unequal ones.

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