Textile Structure
how cloth is put together
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The thread: A mechanism, not a material — page 6

Page 6 of 6 of the essays on this thread.
Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

Where a torsion model stops

A second stiffness was added because an earlier ladder named its absence as the first thing to disbelieve. It settled four things, refuted one trade explanation, and left the question it was built for exactly where it found it.

 The wicking front against time, in four rooms. The height of the wicking front in a sheeting-weight strip against time on a logarithmic clock. The fine system, the pores between fibres, rises to a steady height set by the room: 1064 mm at 20 g/m²·h, nine tenths of it after 4.1 h; 501 mm at 100 g/m²·h, nine tenths of it after 53 min; 255 mm at 400 g/m²·h, nine tenths of it after 14 min; 133 mm at 1500 g/m²·h, nine tenths of it after 3.7 min. The coarse system, the holes between yarns, stops at 137 mm after 17 s whatever the room, because gravity stops it rather than drying. Cloth doing a job

A wick reaches its ceiling in the time its cloth takes to dry

A drying cloth lifts water to a steady height and no further, and the question left was how long it takes to get there. The answer has no permeability and no surface tension in it. Where gravity is small the front climbs as the ceiling times the root of one minus a decaying exponential, and the exponential's time is the cloth's own pore water divided by the rate its faces lose water — the time the room would take to dry it. In an ordinary room that is half an hour, and the front is nine tenths of the way up in fifty-three minutes.

 A satin's contrast as the lamp moves off the mirror. The specular area of an eight-end satin over a plain weave, both as a sheeting, for a source 2° wide moved away from the direction the viewer sees mirrored in the cloth. Moved across the satin's floats, the contrast is 22.4 on the mirror and 21.7 forty degrees off. Moved along them it is 22.4 on the mirror and 0.25 once the source is more than twice its own width away: the satin is darker than plain. A calendered satin falls to 1.83 across its floats as well. What the plot cannot show is shadowing, which a lamp far enough off the mirror brings in. Weaves

A lamp off the mirror lights a satin only across its floats

Every source in the account of a room was a source the viewer sees mirrored in the cloth. A lamp off to one side lights the cloth too, and which facets it reaches depends on which way it is off. Displaced across a satin's floats it is caught by the float's own curve and adds contrast like any lamp on the mirror; displaced along them by more than twice its own width it reaches only the turns, and the satin comes out a quarter as bright as plain weave.

 What a sheer multiplies a street reading's errors by. The factor by which an error in reading the light from a window is multiplied on its way into the room's reflectance, against the street-to-room light ratio, for three sheers and a room reflecting three tenths. It is the reading over the room's own image, the reciprocal of the share of the view that is room. white voile: 67.3 at 30 : 1, 3.6 at equal light, 1.6 at night; grey voile: 28.7 at 30 : 1, 2.1 at equal light, 1.3 at night; black net: 2.1 at 30 : 1, 1.1 at equal light, 1.0 at night. What the chart cannot show is the thread optics, which are assumed values here and not measured ones. Pattern and colour

A sheer's privacy is the error it multiplies

Everything a passer-by sees of a room through a sheer is the room's image plus a veil the street lights, and the veil can be known without going in. So one subtraction and one division ought to read the room's reflectance from the pavement. They do not, and the reason is the curtain's whole purpose: every error in the veil arrives in the answer multiplied by the reading over the image — sixty-four through a white voile by day, three and a half at equal light. The privacy a sheer gives and the precision it denies are one number.

 The fibre pieces in one cut tuft. A cut-pile tuft 3 mm tall on a 1 mm base, spun from a 25 mm staple, with a seeded sample of its fibre pieces drawn along the yarn: 16 pass under the binding pick and 2 lie wholly in one leg, between a fibre end and the cut tip. Across a whole pile the loose share is 21.9 per cent of the pieces and 7.0 per cent of the fibre. What the drawing cannot show is the twist, which holds the loose pieces by friction and is what they escape from. Compound and figured cloths

A blade leaves loose fibre in every tuft

The account of hair, nap and pile ended on a clean claim: a blade collapses a population's length to one value, so a cut pile has no tail and nothing in it reaches past the rest. The tips are one length. The fibres are not. A tuft cut from staple yarn is a length of yarn with fibre ends scattered along it, and every fibre end that lands in a leg leaves a piece between it and the blade that nothing in the draft holds — a fifth of the pieces in an ordinary wool carpet, none in a filament one.

 A ring-dyed yarn worn flat, step by step. One cotton yarn dyed in a ring 20% of its radius deep, cut flat from the top by an abrader to five depths: worn to 10% of the radius, 1.9% of the yarn has gone and 0% of its width is white; worn to 20% of the radius, 5.2% of the yarn has gone and 0% of its width is white; worn to 30% of the radius, 9.4% of the yarn has gone and 39% of its width is white; worn to 45% of the radius, 16.8% of the yarn has gone and 58% of its width is white; worn to 70% of the radius, 31.2% of the yarn has gone and 74% of its width is white. The olive line is the cut face and the red bar above it is the part of the face that has reached the undyed core. Nothing is white until the cut passes the ring, and then the white arrives quickly. The section is a circle and the ring's edge is drawn sharp; a real yarn is neither, and its fibres migrate, so the core's edge is ragged. After the loom

A ring-dyed yarn whitens all at once

Rub a yarn that is dyed in a ring, and for a while nothing shows. The abrader takes off blue fibre, the cut face is blue, and the yarn looks as it did. Then the cut reaches the undyed core, and the white arrives with a vertical tangent, as the square root of the wear past the ring: one per cent of the radius more and an eighth of the width is white. A ring a fifth of the radius deep hides the first five per cent of the yarn's loss and shows half its width white by thirteen. A yarn dyed through never shows white at all. Denim's high-contrast fading is that threshold, drawn over a cloth.

 What flattening costs, against what the loop's bending is worth. The energy of squashing a 20 tex cotton to the 78% the fabric's geometry demands, as a multiple of the whole bending energy of one stitch, at three lateral rigidities. The free bound is exactly zero: a bundle of fibres free to slide resists a change of shape at constant area not at all, so flattening is free and the bracket on it has no floor. The coherent bound is 36 times the loop's bending, so a yarn that could not rearrange could not be knitted into this fabric at all. The fabric flattens, so it is reading the free end — which is the fourth ordinary observation on this site to land at that end of the bracket. Mechanics and drape

Flattening is free and impossible

The fabric demands a flattening and the yarn has to supply it. At one end of this collection's oldest bracket the deformation costs exactly nothing; at the other it costs thirty-six times the whole bending energy of a stitch. The fabric flattens — which is the fourth everyday observation in one phase to land at the same end.

 A bolt in plan with all three, cut three ways. The first 9 metres of a 50-metre bolt 1500 mm wide, drawn in plan with its length across the page. Over its whole length it carries 23 point faults, 8 weft bars across the width, 1 warp streak along the length, the streak at 911 mm across. Panels are 450 by 900 mm, 3 across with 150 mm to spare. Over the whole bolt the three markers yield 80, 95 and 145 sound panels of 165: rigid tiles; the same lanes cut around their faults along the piece; and lanes also placed across the width wherever the faults leave room. Cloth doing a job

The width of a bolt is worth what its faults leave it

A fault map lets a cutter move panels along a bolt and across it, and the length was expected to matter far more than the width, because a bolt is fifty metres long and three panels wide. That is true of two kinds of fault and false of the third. Against a fault across the whole width the width is worth exactly nothing; against a scatter of points it is worth between a sixth and two fifths of what the length is; and against a missing end, which runs the whole length, the length is worth nothing and the width is worth everything.

 One gore of a 4-gore knitted ball, laid flat. A gore of 30 stitches from pole to pole, knitted sideways in short rows, drawn flat with its rows stacked at the plain-knit loop's fully relaxed aspect of 1.2791. Turned on the sine, its 10 row pairs turn at stitches 0, 1, 2, 3, 4, 5, 6, 8, 9, 11 from each pole — one stitch apart near the pole and further apart towards the equator — and its edge follows the width a sphere asks for. Turned one stitch every pair, the same gore is a diamond of 15 pairs, 1.50 times as wide at the equator. What the drawing cannot show is the fabric smoothing its own staircase edge. Knits and other structures

A knitted ball's short rows have to slow down

A knitted disc needs one count to meet 2π; a sphere needs a count that follows a sine. Knitted sideways in gores, a ball's short rows must turn one stitch apart at the pole and ever further apart towards the equator — evenly spaced turns knit two flat discs joined at the rim. The pole is the disc again, so gores come in fours; the equator's row pairs have to come out whole, so only some sizes knit round; and a wash moves a four-gore ball from a ruffled pole to a round one.

 Floated weaves against plain under an overcast sky. The sky's radiance each weave's facets mirror into the eye, over the plain weave's, as the viewer's elevation above a sheeting falls from overhead to thirty degrees, under an overcast sky with a ground of albedo 0.2. 2/2 twill: 1.25 overhead and 0.99 at 30°; 5-end satin: 1.30 overhead and 1.01 at 30°; 8-end satin: 1.37 overhead and 1.01 at 30°. What the plot cannot show is the diffuse reflection, the same for every weave of one fibre, which a real cloth's sheen sits on top of. Weaves

A satin mirrors the top of the sky

A uniform source sixty degrees wide leaves every weave reflecting the same share of its face, and that was read as saying no cloth shines under a sky. A real sky is not uniform. An overcast one is three times as bright overhead as at the horizon, and it stands over a ground darker than itself. Every facet of a cloth mirrors one direction, and a satin's facets mirror the top of the dome while a plain weave's mirror the horizon and the ground — so seen from above, under cloud, an eight-end satin still sends back a third more sky.

 What a figured warp pays in tension to share one beam. For six pairings of figure and ground, the difference in warp tension that lets their ends share one beam: the force per end that takes the take-up difference out of the ground's crimp, along that region's own constant-length locus, with the yarn's rigidity at both ends of the band two tests place it in. a damask: satin on its own complement, 0.00% apart: none needed; an eight-end satin figure on a five-end satin ground, 0.31% apart: 0.006 to 0.020 N; a five-end satin figure on a 3/1 twill ground, 0.90% apart: 0.021 to 0.069 N; an eight-end satin figure on a 2/2 twill ground, 3.02% apart: 0.304 to 0.995 N; an eight-end satin figure on a plain ground, 14.15% apart: past what the ground's crimp can give; a 2/2 twill figure on a plain ground, 11.14% apart: past what the ground's crimp can give. The loom's front shaft holds 0.52 N an end. What the chart cannot show is how uneven a warp's tension may be before it shows in the cloth. Compound and figured cloths

A figured warp pays in tension before it needs a beam

Figure and ground take up warp at different rates, and counted against a fixed slack, any difference at all — a third of a per cent — made a round figure need four beams. A real let-off holds tension, not length. Ends that consume more pull harder, and an end pulled harder gives up crimp, until every end consumes alike. For an eight-end satin figure on a five-end satin ground that costs a hundredth of a newton an end, and one beam serves. For a satin on plain it costs more crimp than the ground has, and no tension will do.

 The crimp split of a muslin against the warp's tension. The warp's crimp and the weft's at the least-energy state of a muslin at its own sett, with the warp held at tensions from 0.0001 to 1.0 newtons, and the yarn's rigidity 4.0 times the free bound. With no pull the warp holds 28.5 per cent and the weft 0.0; they are equal at 0.019 N, and by the front shaft's 0.52 N the warp holds 0.00 per cent. What the plot cannot show is friction: a real cloth is held wherever its crossings stop it, not at the least state. Setting and geometry

The loom hands the crimp to the weft

Bending alone gave the fixed-sett energy nothing to say: six of eight cloths fell to the end of their interval with the weft dead straight. Put the warp's tension in and the answer is not a well but a switch. The crimp changes hands across a factor of two or three in tension, at a few hundredths of a newton — and the loom holds its warp at half a newton, so on the loom every cloth's warp is as straight as its geometry allows.

 What a pane of glass does to a sheer's multiplier. The factor by which an error in reading a window is multiplied on its way into the room's reflectance, against the angle a passer-by looks at the glass from, at 30 : 1, for three sheers behind a pane of window glass. The dashed level beside each is the same cloth with no glass. white voile: 64.0 bare, 59.8 behind glass straight on, 65.7 at 60°, 92.7 at 75°; grey voile: 27.3 bare, 28.7 behind glass straight on, 34.7 at 60°, 61.7 at 75°; black net: 2.0 bare, 5.2 behind glass straight on, 8.8 at 60°, 25.4 at 75°. The white voile is easier to read through glass up to 56.7°, because the pane dims the street on its threads by more than it mirrors; the black net is harder at every angle. The pane mirrors a street reflecting 0.3; a sky in the mirror would be brighter, and nothing here models one. Pattern and colour

Glass hides a black net and not a white voile

A window pane in front of a curtain mirrors the street, and a mirror is a veil with no thread in it. It also dims the street's light on the cloth. For a white voile the two nearly cancel, and straight on the dimming wins, so the voile is slightly easier to read through glass than without it. For a black net the mirror is most of what a passer-by sees: its multiplier goes from 2 to 5 head-on and to 25 from along the pavement. The glass makes the net private, not the voile. The angle at which it stops helping does not depend on the time of day, and a polarising filter at Brewster's angle takes the mirror out altogether.

 Two ring-dyed faces worn to 5% of their mass. The faces of 3/1 twill and plain woven from the same ring-dyed 50 tex warp and undyed weft, at 24 ends and 16 picks a centimetre, each worn flat by an abrader until 5% of its thread has gone, drawn over two repeats: dyed warp dark, exposed undyed core red, weft pale, holes blank; the 3/1 twill shows 30% of its covered face as exposed core; the plain shows 17% of its covered face as exposed core. A float's crown is a line and is cut along its length; a plain weave's crowns are points and are cut through as small patches. The geometry is the plain-weave solution for these counts and setts with the floats drawn over it, which is more open than a real denim; what the figure cannot show is a denim at its real density. After the loom

A float fades late and hard, a crossing early and soft

Wear a ring-dyed face flat and the white arrives where the cut first passes the ring at a crown, which is the same depth for every weave. What differs is how much cloth has gone by then and how fast the white spreads after. A plain weave's crowns are points, cut through by almost nothing: it shows white after a fifth of a per cent of its thread and never shows much. A twill's floats are lines, cut along their length: a 3/1 twill shows nothing until more than one per cent has gone, then whitens faster than any plain weave can, 39 per cent of its face by a tenth worn against plain's 24. Denim's high-contrast fade is a float's; chambray's soft one is a crossing's.

 Which lifting plan exposes a threading error another plan hides. For every four-shaft threading of eight ends and every double threading error silent under the row's lifting plan, the share that the column's plan exposes — weaving a cloth that is caught or visibly different — when the same threading is re-pegged. Silent under the 2/2 twill (99,872 pairs): 2/2 other way 0%, 1/3 twill 44%, 3/1 twill 44%, broken 70%, point 34%, plain 0%, hopsack 34%, unrelated 99%. Silent under the 1/3 twill (56,352 pairs): 2/2 twill 0%, 2/2 other way 0%, 3/1 twill 0%, broken 76%, point 50%, plain 0%, hopsack 47%, unrelated 98%. Silent under the broken twill (29,888 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 55%, 3/1 twill 55%, point 0%, plain 0%, hopsack 53%, unrelated 97%. Silent under the point twill (66,208 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 57%, 3/1 twill 57%, broken 55%, plain 0%, hopsack 47%, unrelated 99%. A zero means the column's plan hides every error the row's does; the plan of unrelated rows exposes all but the 864 pairs that are the threading itself started elsewhere. What the table cannot show is which plans a given mill pegs one warp with. What cloth is

A proof plan has no slide in it

A threading error that a 2/2 twill hides is invisible in the cloth and still wrong on the loom, and it shows only when the loom is re-pegged. Re-weaving every silent error under eight other plans says which do the showing. The twill run the other way and plain weave never expose one. A 1/3 twill exposes 44 per cent, a broken twill 70, and a plan of eight unrelated rows all but 864 — and those 864 are not errors at all, because each is the right threading started at another end. So the plans nest: whatever a broken or a point twill hides, the 2/2 hides too, and a threading proved under a 2/2 twill has passed the weakest test there is.

 The springs a leno easer can be. The stiffest spring, per end, that an easer can carry without any span of the crossing end rising above an ordinary end's working tension at any fraction of the crossed shed, against how far its preload sits above the warp's resting tension of 0.5 N. Behind the harness, where an easing bar acts, it is 1.1 N/m at a preload equal to the resting tension and falls to nothing by 15 per cent above it; in front of the harness it is 4.8 N/m. Everything under a curve works and nothing above it does. The window is a corner: a spring barely stiffer than a dead weight, with a preload matched to the warp's own tension. What the chart cannot show is the bar's mass, which the loom's speed makes matter and which the next views take up. Compound and figured cloths

A leno easer should be a light weight

A spring easer gives length when the crossing end pulls, so it cannot give it too early, which was the fault in a cam driven off the shed. The question left was its rate. The answer is that it hardly has one. Behind the harness the easer feels the back span, and two gripping eyes keep that span within fifteen per cent of its resting tension while the kink carries its load. So the spring must hold its span almost constant over a sixty-eight-millimetre stroke: at most 1.1 newtons a metre per end, a dead weight in all but name. At speed the bar's own mass is what limits it, and it falls with the square of the loom's speed: 2.3 grams an end at a hundred picks a minute, 0.6 at two hundred.

 A graduated stocking knitted in 8 equal steps. The pressure a stocking knitted as 8 steps of fixed girth puts on the illustrative leg from ankle to knee, against the smooth graduation from 20 mmHg at the ankle to 12 below the knee that it approximates. The steps are equal in height. The rings, the pressure change along each step, run 4.79, 3.96, 3.32, 2.82, 2.43, 2.12, 0.87, 0.92 mmHg from the ankle up; the largest is 4.79 mmHg. Cloth doing a job

A stepped stocking should step most at the ankle

A graduated stocking asks for a girth that rises smoothly up the leg, and a knitting machine gives it one girth, then the next. Over each step the tube is one size on a stretch of leg that is not, so it presses harder at the top of the step than at the bottom, and the pressure up the leg is a saw-tooth. Each tooth is the leg's change of girth across the step divided by the girth squared — so it is largest just above the ankle, where the leg is thinnest and widening fastest. Eight equal steps leave a ring of 4.8 mmHg there; eight steps spaced by the leg's own shape leave 2.7 everywhere.

 A jersey's bending length against its tightness. The wale-wise bending length of plain jersey in the fully relaxed state at 10, 20, 40 tex, across the whole tightness band from 1.3 to 1.6, at the two ends of the yarn's stiffness bracket. At the free bound every count and every tightness gives 10.13 mm. At the locked bound the lines separate by count — 55 mm at 10 tex, 70 mm at 20 tex, 88 mm at 40 tex — and are still flat. The weights along the lines run from 97 to 239 g/m². What the plot cannot show is friction between the loops, which is where a tight jersey's firmness has to come from. Setting and geometry

A jersey's drape does not know its loop

A woven cloth's bending length at the free bound turned out to hold neither its count nor its weight — only the fibre. A plain jersey goes further. Its stiffness per width and its weight per area are both a number of loops per millimetre times something about one loop, so the loop cancels at every stiffness the yarn could have; at the free bound the count cancels too, and what is left is the fibre and which relaxed state the fabric is in.

 Which loose pieces in a cut tuft are free at once. A cut-pile tuft 12 mm tall on a 2 mm base, spun from a 90 mm staple, with a seeded sample of its fibre pieces drawn along the yarn and the tip's untwisted run-out, 5.6 mm, shaded down each leg. 22 pieces pass under the binding pick; of the 9 that lie wholly in one leg, 3 lie wholly inside the run-out, where the twist presses on nothing, and are free at once; the other 6 reach down into twisted yarn and are held until wear opens the twist further. What the drawing cannot show is the run-out's edge, which is a fall in pressure rather than a line. Compound and figured cloths

A cut pile sheds in two stages

Every cut tuft of staple yarn holds loose fibre pieces that nothing anchors but the twist, and at the tip the twist holds nothing: it runs out over a length the grip arithmetic already gives, five and a half millimetres in a wool carpet yarn. A loose piece lying wholly inside that run-out is free the day the carpet is laid; one reaching below it is held until walking opens the tip's twist. So the reservoir drains in two stages, and the pile height decides the split. A velvet sheds everything it will ever shed at once. A twelve-millimetre wool carpet sheds a fifth of its reservoir at once and four fifths later. A shag sheds almost nothing at first and fourteen per cent of its fibre eventually.

 A crease along a 3/1 twill's floats and one across them, worn alike. A strip of 3/1 twill of ring-dyed 50 tex warp and undyed weft folded round a 2 mm radius, the ridge drawn across the page as a dashed line, worn by a flat abrader 50 µm into the ridge: above, with the ridge running along the warp and its floats; below, running across them. Dyed warp dark, bared core red, weft pale. crease along the warp: 83% of the crease's length white, in runs 1.72 mm long on average, the longest 2.08 mm; crease across the warp: 31% of the crease's length white, in runs 0.15 mm long on average, the longest 0.17 mm. On a twill the white along the warp lies in lines as long as a float's crown, across it in dashes one end wide; on a plain weave, with no floats, both are short. What the drawing cannot show is the fibre a real crease tears loose, which blurs every edge here. After the loom

A crease across a twill fades in dashes

A garment is not worn flat. It is worn along its folds, where the ridge of a crease stands out and is rubbed first, so the fade follows the crease. On a ring-dyed 3/1 twill a crease running along the warp bares the cores of the floats that lie under its ridge, float by float, and reads as a white line broken only where each float dives: at fifty micrometres of wear, 83 per cent of its length is white, in runs of 1.7 millimetres. The same crease across the warp meets every float at one point and reads as a row of dots, 31 per cent white in dashes of 0.15 millimetres, one end wide. It bares much the same core. A plain weave's creases are dashes whichever way they run.

 What a spin leaves in a cotton load, against the drum's speed. The water left in a 68 mm load of cotton sheeting in a drum of 250 mm radius, as a share of the load's dry weight, against the spin speed. Read with the fibre holding its swollen water, the spin leaves 51% at 1,000 rpm and 41% at 1,400, and no speed takes it below 30%, the volume the fibre grew by. Read with the fibre holding only its regain, as the earlier account of a wet cloth did, the same spin leaves 16% at 1,400 and the floor is 8.5%. What the chart cannot show is the water a drained channel keeps as a film and as rings at the fibre contacts, which the model takes as nothing and which lifts every point on both curves. What cloth is

A spin leaves the water a fibre swelled by

A washing machine's spin is a centrifuge, and it drains a cloth in the order its pores give up water: the holes between the yarns at a few g, the channels inside the yarns only far enough from the drum's wall, and the water inside the fibre never. That last reservoir is not the regain. A soaked cotton fibre holds the volume it swelled by — 30 per cent of its dry weight, three and a half times the regain — and it is the floor under every spin speed there is.

 The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways. Mechanics and drape

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

 How high a strip wicks a finite supply, in four rooms. The steady height of the fine pore system's wicking front against the supply of sweat at the strip's foot, both on logarithmic scales, for a sheeting-weight strip in four rooms. Below each room's capacity the height is the supply over twice the evaporation from each face, a line of slope one that no cloth property moves. It meets the room's drying ceiling at the capacity — 43 g/(m·h) at 20 g/m²·h, ceiling 1064 mm; 100 g/(m·h) at 100 g/m²·h, ceiling 501 mm; 204 g/(m·h) at 400 g/m²·h, ceiling 255 mm; 400 g/(m·h) at 1500 g/m²·h, ceiling 133 mm — and past it the strip takes no more. The marks along the bottom are four sweat rates from rest to hard work, fed from a contact patch 50 mm tall. Cloth doing a job

A sweating cloth wicks as high as the room can dry it

Every wicking height so far has had its foot in unlimited water. Skin is not a reservoir: it supplies sweat at a rate, and a cloth fed at a rate stands where the supply equals what its faces lose — the supply over twice the evaporation, with no pore, fibre or thickness in it. The cloth decides only when it has had enough, and it says so all at once: the holes between its yarns stay empty until the fine pores are carrying ninety-nine per cent of what they can.

 One crease at four angles to a 3/1 twill's warp, worn alike. Four strips of 3/1 twill of ring-dyed 50 tex warp and undyed weft, each folded round a 2 mm radius with the ridge drawn across the page, worn 50 µm into the ridge by a flat abrader, 24 mm of crease each. Dyed warp dark, bared core red, weft pale. along the warp (0° from the warp): 80% of the crease white, its longest white run 2.0 mm; along the twill's line (33.7° from the warp): 100% of the crease white, its longest white run 45.0 mm; two degrees off the line (35.7° from the warp): 79% of the crease white, its longest white run 15.7 mm; along the other diagonal (-33.7° from the warp): 66% of the crease white, its longest white run 1.0 mm. Along the twill's line the white never breaks, because the crease meets the crown of every end's float in turn; two degrees off, it runs as long strokes that drift off the line of crowns and come back. What the drawing cannot show is the fibre a real crease tears loose. After the loom

A crease along the twill fades as one line

A crease along the warp of a ring-dyed 3/1 twill fades as broken lines a float long, and across it as dashes one end wide. Turn the crease to the twill's own diagonal and it fades as a single white line with no break in it at all: the fold meets the crown of one end's float, then the next end's one pick along, then the next, and every crown it meets is at the same place in its float. It is the one direction in which the weave and the crease line up — and two degrees off it the line breaks into strokes whose length is a vernier's.

 A ball knitted in rounds with its increases stacked in 5 lines. The shape a ball of plain knit takes unstuffed when its increases, 5 at a time, are stacked in 5 lines from pole to pole, with the same stitches in every round as a sphere. Every stitch between two lines is flat knit, so the 5 panels are flat and the one shape they close up into without stretching has 5 flat sides in every round and 5-sided points at its poles: 2.42 sphere radii tall, 2.14 across its ridges and 1.73 across its flats, so 1.13 to 1.40 times as tall as it is wide. The knitted counts of a 40-round ball give 1.13 to 1.40, against 1.02 for the same counts with the increases spread round, whose rounding is its own. What the drawing cannot show is stuffing, which would stretch the panels towards round. Knits and other structures

Stacked increases knit a ball with flat sides

A ball knitted in rounds can put its increases anywhere in each increase round. Stack them in lines from pole to pole and every stitch between two lines has a flat knit's neighbours, so the ball is made of flat panels and all its curvature sits on the lines. Flat panels with straight rows close up without stretching in exactly one way: every round a regular polygon, each pole a point. At five lines that ball is 13 per cent taller than it is wide across its ridges and 40 per cent across its flats, and to make it round the stuffing has to stretch the middle of every panel by π²/4m² — nine per cent at five lines, less than a wash moves a course at ten.

 How likely one staple length of yarn is to be weaker than a tension, under three laws. The probability that one independent try — a staple length of a 20 tex cotton yarn — is weaker than a tension, on a logarithmic scale, with the tension as a fraction of the yarn's 500 mm breaking load. Three laws are drawn, a normal, a lognormal and a Weibull, each fitted to reproduce the same two tensile tests at 100 and 500 mm exactly. Near the breaking load, where a tensile test samples at about one chance in 18, they lie together. At one chance in 1.4×10⁸, where a warp of 4,000 ends by 1,000 metres samples, they are far apart: at the back shaft's 52 per cent the three give normal 1.3×10⁻⁷, lognormal 1.1×10⁻¹¹, Weibull 4.0×10⁻⁶ per try. What the chart cannot show is which of the three a real yarn follows, which no tensile test can decide. Setting and geometry

A warp breaks in a tail no tensile test reaches

A 500 mm tensile test is the weakest of about eighteen staple lengths of yarn. A warp of four thousand ends by a thousand metres is the weakest of a hundred and forty million, and an end breaks wherever one of them is weaker than the shed's tension. Three strength laws fitted to the same two tensile tests agree within five per cent at every gauge a tester can clamp, and disagree by a factor of 375,000 on how many ends a warp will break at the back shaft.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

A fabric reads its own bracket four ways

A yarn's stiffness is unknown to a factor of three hundred, and no laboratory measurement has closed it. Four unrelated everyday observations — a snarl, a knot, a flattened yarn and a cloth's own thickness — all say the same thing about which end of it a yarn sits at.

 Where an air arch first wrinkles under snow over the whole span. A semicircular air arch 4 m across, pinned at both feet, of a tube 200 mm across at 50 kPa, under snow over the whole span; beside it, unrolled from the left foot to the right, the share of the tube's wrinkling budget πpr³/2 spent at each section when the first wrinkle forms, with the thrust's part, N·r/2, shaded darker. The first wrinkle forms 25° above the right foot, at 167 N per metre of span, where the moment of 60.2 N·m compresses the inside of the curve and the thrust is 336 N; there the thrust has taken 21 per cent of the budget. Judged by its moment alone the arch would carry 218. Cloth doing a job

An air arch pays for its thrust out of its pressure

An inflated tube bent into an arch was expected to start with its inside wall pulled differently from its outside, and to wrinkle where that difference and the load's moment combined worst. It does not: a curved tube is pulled along its length at exactly the straight tube's pr/2, all the way round. What the arch spends its pressure on instead is the thing every arch exists to make — its own thrust — which takes a fifth of the wrinkling budget at the haunch of a shelter arch under snow and more than half on a tight one.

 A 50 tex yarn after 1, 4, 8 dips into a fibre that fills. The cross-section of a 50 tex cotton yarn after 1, 4, 8 twenty-second dips, each fibre shaded by the dye fixed at its distance from the surface, on one scale of shade for all three. The fibre's room for dye is finite and one dip at the bath's strength fills 20% of an empty fibre's, so each dip meets less affinity near the surface than the last and carries its dye further in. after 1 dip: ring 14% of the radius deep, the surface 20% full; after 4 dips: ring 18% of the radius deep, the surface 59% full; after 8 dips: ring 24% of the radius deep, the surface 83% full. With room to spare every dip would leave the first dip's 14%. The share is assumed; nothing here measures it. After the loom

A fibre that fills dyes deeper with every dip

Dipped eight times, a ring-dyed yarn is darker than after one dip but no deeper — if its fibres have room for all the dye they are offered. They do not: every dip's dye occupies some of the fibre's room, the next dip meets less affinity near the surface, and dye that is taken up less travels further before it is taken. How much further depends on one number the dip arithmetic never needed — the share of an empty fibre's room one dip fills — and eight dips turn that number into a ring 1.15 times deeper at five per cent and 1.68 times deeper at twenty.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

 How tall a skin contact can be before its cloth floods. The tallest patch of skin contact whose cloth keeps the holes between its yarns empty, against the sweat rate, on logarithmic axes, for a sheeting-weight cloth in rooms evaporating 20, 100, 400, 1500 g/m²·h from each face. The patch evaporates from its outer face only and hands its surplus to the free cloth above, which spreads it until the fine system's capacity; so a contact floods when it is taller than 98.9% of that capacity over the surplus. Below the room's own evaporation rate no contact of any size floods. In an ordinary room a run floods any contact taller than 247 mm and hard work any taller than 90 mm. Cloth doing a job

Wicking borrows drying area from the cloth that is not touching

A shirt is fed with sweat across the whole of the skin it touches, not at the foot of a strip. Through its own thickness the cloth could pass sweat a hundred thousand times faster than a body makes it, so a patch pressed flat never lacks capacity; what it lacks is drying area, because it can give water to the room only from its outer face. Pressed flat everywhere, it floods as soon as the sweat exceeds the room's evaporation — light work, in an ordinary room. With free cloth above the contact, wicking carries the surplus to faces that are not touching, and the sweat decides how much area it needs: three times the contact's at a run, whatever the contact's size.

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