Textile Structure
how cloth is put together
Theme

The thread: Measured, not claimed — page 5

Page 5 of 6 of the essays on this thread.
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. Knits and other structures

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

 A 4-layer stack round a 180° fold. 4 layers of 260 µm cloth taken round a 180 degree fold, drawn as concentric arcs at their own separation. The outer layer runs round a larger radius than the inner one, so it must be longer — by the angle times the separation, which is 2.45 mm here and 0.82 mm at every interface. In a loose stack that length is found by the layers sliding over one another at the fold. In a stitched seam they cannot: the stitches pin them every 3 mm, so the slip a fold needs is 82 per cent of the distance between two stitches and has nowhere to come from. What the arcs cannot show is what happens instead, which is that the fold opens out to a radius the stack can manage. After the loom

A crease cannot cross a seam

The outer layer of a folded stack has further to go than the inner, by the fold's angle times the stack's own thickness. A four-layer seam of quarter-millimetre cloth taken through a half turn needs its outer layer to be 2.45 millimetres longer than its inner — and a stitch line every three millimetres has pinned them. So the fold opens out where it crosses the seam, which is what a trouser crease visibly does, and the arithmetic gives the radius it opens to.

 What each figure-and-ground pairing costs in differential take-up. For each pairing of a figure weave with a ground weave, the difference between the two regions' warp crimps and the length of figure that difference allows before an end goes slack, taking the loom to absorb 1.2 millimetres. a damask: satin on its own complement: 0.00 per cent apart, no bound; an eight-end satin figure on a five-end satin ground: 0.31 per cent apart, a figure up to 392 millimetres; a five-end satin figure on a 3/1 twill ground: 0.90 per cent apart, a figure up to 133 millimetres; an eight-end satin figure on a 2/2 twill ground: 3.02 per cent apart, a figure up to 40 millimetres; an eight-end satin figure on a plain ground: 14.15 per cent apart, a figure up to 8 millimetres; a 2/2 twill figure on a plain ground: 11.14 per cent apart, a figure up to 11 millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were and the crimps are identical rather than close. What the bars cannot show is the slack, which is an input and which every length is proportional to. Compound and figured cloths

A damask is the only figure that costs its beam nothing

Figure and ground consume warp at different rates, and the difference accumulates down the length of the figure. An eight-end satin figure on a plain ground puts its two regions fourteen per cent apart, which on a loom absorbing a millimetre of slack bounds the figure at eight and a half millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were — and their crimps are identical rather than close, to the last bit of a double.

 Where a wet cloth keeps its water. For four cloths of this collection's own table, the share of the water a saturated cloth holds that sits inside the fibre as regain, between the fibres inside the yarn, and between the yarns in the cloth's own holes. muslin at 99 grams a square metre holds 183 per cent of its own weight, 4.6 per cent of it in the fibre; sheeting at 155 grams a square metre holds 113 per cent of its own weight, 7.5 per cent of it in the fibre; poplin at 100 grams a square metre holds 165 per cent of its own weight, 5.2 per cent of it in the fibre; duck at 207 grams a square metre holds 139 per cent of its own weight, 6.1 per cent of it in the fibre. The fibre's own water — the property cotton is sold on — is a twentieth to a thirteenth of the total, and the other nineteen twentieths are geometry. What the bars cannot show is the hair layer, which holds water outside all three of these and which this arithmetic has no place for. What cloth is

A cotton's own water is a twentieth of what a cloth holds

A wet cloth keeps water in three places and only one of them is the fibre. A sheeting saturated holds 113 per cent of its own dry weight: 7.5 per cent of that inside the cotton as regain, 39 per cent in the channels between the fibres of its yarns, and 54 per cent in the holes four threads bound. The same construction in polyester, whose regain is a fortieth of cotton's, holds 105 per cent — an eight-point difference from a fortyfold one, because absorbency is a geometry with a fibre in it rather than a fibre with a geometry round it.

 How many cloths any one cloth derives into. The 426 four-by-four cloths sorted into the orbits the manuals' derivations cut them into. 12 orbits hold 1 cloth; 83 orbits hold 2 cloths; 62 orbits hold 4 cloths. The largest orbit in the whole catalogue holds 4, so no cloth derives into more than 3 others by any sequence of the named operations, however long. The derivations generate a group of 256 elements and it cuts the catalogue into 157 pieces. What the bars cannot show is which cloths are in which orbit, which is the next figure. Weaves

No cloth derives into more than three others

Every weaving manual opens by saying the three basic weaves generate the rest. This collection counted the reach and found nine of 426, and left the nine as a count. It is not a count: every derivation the manuals name is a relabelling of the grid or a complementation of it, both invertible, so they generate a group — and that group cuts the 426 cloths into 157 closed pieces of which the largest holds four. The claim is not merely wrong about how much derivation reaches; derivation cannot reach more than four cloths from anywhere, by any sequence of operations, however long.

 How little eccentricity a curl needs. The radius a jersey would curl to, against how far the fabric's neutral surface fails to bisect its loops. The model's own loop is bisected exactly — its crest sits half a diameter behind the mid-surface and its trough half a diameter in front — so it carries no curling moment at all, and the curve here is what any departure from that would buy. The moment is proportional to the eccentricity, so one solve settles the whole curve. A radius of three millimetres, which is about what a fine jersey rolls to, needs 5.6% of a yarn diameter — 9.4 micrometres on a yarn 167 micrometres across. That is why curl is easy to see and hard to model: the whole of it lives inside a rounding error on the geometry. Only the course-wise edges are drawn, because bending about the other axis turns the thread's end tangents out of the fabric and a free thread given turned ends buckles clean out of it — 3.2 times the fabric's own thickness, for 37% off its energy. That configuration is not available to a thread with neighbours, so the second moment is refused rather than computed. Mechanics and drape

How little asymmetry a curl needs

A model with a thickness can finally be asked why stockinette rolls. It answers that it does not — its curling moment is exactly zero, by a symmetry — and the useful part is what that costs to break: five per cent of a yarn diameter buys the whole of the curl anybody has ever seen.

 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. Setting and geometry

What a high-twist yarn costs a cloth

Twist buys strength up to a point and then loses it, and everything else it does is a cost. A crepe twist is chosen knowing that, and the trade's twist limits are a balance among five quantities that this collection can now put beside one another.

 A tube of an elastic band knitted 157 mm round, up one leg. A tube knitted 157 mm round from an elastic band, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 40%, 20.0 mmHg; lower calf 29 cm, stretched 85%, 32.1 mmHg; calf 36 cm, stretched 129%, 39.4 mmHg; below the knee 34 cm, stretched 116%, 37.6 mmHg. The calf is pressed 1.97 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by. Cloth doing a job

A tube of one size presses the calf harder than the ankle

A band presses a limb with its tension over the limb's radius, so it is easy to conclude that a band grips hardest where the limb is thinnest. That is true of a band held at one tension, and no knitted tube is. A tube knitted to one size is stretched further wherever the leg is thicker, and its tension rises faster than the radius does: an elastic tube pressing an ankle at twenty millimetres of mercury presses the calf at thirty-nine, and a cotton jersey tube presses its calf three and a half times as hard as its ankle. A stocking graduated the other way has to pull hardest where it presses less.

 Which satin orders are row-free, from 5 ends to 40. Every satin order from 5 to 40, marked where the best move's lattice has two shortest steps of equal length rather than one — which is the condition under which the interlacings do not line up into a row. The row-free orders are 5, 10, 13, 15, 17, 24, 25, 26, 29, 34, 35, 37: twelve of the 35 orders that admit a regular satin at all. An n-end satin floats over n − 1, so a float limit is a ceiling on the order, and the ceilings for limits of 8, 12, 16 are drawn. What the strip cannot show is the spread, by which the orders are ranked and which decides which move is best within each. Weaves

A float limit leaves one row-free satin

A satin is chosen so that no diagonal forms, and an earlier essay found that most of them fail: the interlacings lie on a lattice, every lattice has a shortest step, and the marks line up along it unless two steps tie — which happens at twelve of the thirty-five orders from five to forty. The other constraint was named and not applied. An n-end satin floats over n − 1, so a yarn that will not carry a float longer than eight admits four orders in all, and exactly one of them is row-free: the five-end satin, which is the one everybody already weaves.

 The suction water supplies, against the pressure twist supplies. The pressure pressing a yarn's fibres together, against its twist, with the suction inside the menisci of a damp yarn's own pores drawn as a level. The suction is 0.0624 newtons a square millimetre — 2γ over the pore radius of 2.3 micrometres — and it does not depend on the twist at all. The twist's pressure crosses it at about 186 turns a metre, so water is worth as much as the first 186 turns and nothing more: at 600 turns it is 11 per cent of what the twist already supplies. What the chart cannot show is the saturation, at which the suction vanishes because a full yarn has no meniscus in it. What cloth is

A wet fibre is stiffer and a wet yarn is not locked

The intuitive reason a damp cloth stiffens is that water pulls the fibres together — a meniscus is curved, the pressure inside it is below atmospheric, and the suction presses the assembly exactly as twist does. Computed, that suction is worth the first 186 turns a metre of twist and nothing after them: six per cent of what an ordinary yarn's twist already supplies, and nowhere near enough to stop the fibres sliding. What wetting actually does is fatten the fibres, and a fibre's bending rigidity goes as the fourth power of its diameter — so a wet cotton fibre is 2.07 times as stiff with no contact in the argument at all.

 The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres. Compound and figured cloths

A heddle eye lets the kink through

A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

 A three-direction net over a square voile, averaged each way. The light passing through a 1.55 mm net of three thread directions in front of a 0.3 mm voile of two, 50 mm apart and seen from 3.0 m, sampled on a fine grid over 160 mm, averaged along one direction and smoothed over two net pitches. Down the voile the profile rises and falls with the 6.1 mm family one net set makes; across it, with the 27.4 mm family two sets together make; the vertical rules are the predicted spacings. Each panel is scaled to its own range, and the second family is roughly a tenth the strength of the first. What the profiles cannot show is the fringes' look in two dimensions, where both families cross. Pattern and colour

A net of three directions beats a voile one way at a time

A tulle's threads run three ways at sixty degrees and a voile's run two ways at ninety, so a net hung over a voile could show one family of fringes, three, or a lattice of them. It shows two, at right angles, and they are nothing alike. Along the voile threads that lie parallel to one of the net's, the net beats exactly as a one-directional net does: six-millimetre fringes from three metres. Across them no set of the net lies anywhere near, and the only slow beat comes from a line of points two sets make together at √3 over the net's pitch — fringes four times wider and a fifteenth as strong, with a null at 8.8 metres where the strong family has none.

 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%. Mechanics and drape

What a knit gives up when it is pressed

The woven half of this collection has had a compression curve for several rungs — a thickness that falls under load, a bearing area that grows, a pressure at every point. The knitted half had a plan and no depth. It has a relaxed thickness and an initial slope now, and the two fabrics turn out to resist for different reasons.

 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. Setting and geometry

What a sett is when the yarn is not round

A jamming condition says how close threads can be set, and it says it in terms of a diameter. A thread in a cloth does not have one diameter: it has a wide one and a narrow one, and which of them a jam is about depends on which way the threads are jamming.

 The course helix of a 30-inch machine with 96 feeders, unrolled. A knitted tube from a 30-inch machine with 96 feeders unrolled flat, one round wide, with its wales drawn straight along it and its courses climbing across it. The machine lays 96 courses in each turn, so each course climbs 96 course spacings — 48 mm of fabric — in one round of 2,262 wales, 162 cm: an angle of 1.70° from the tube's cross-direction. The course drawn heavy is followed round one turn. The vertical scale is exaggerated 6 times. What the drawing cannot show is the hand of the helix, which is set by the direction the cylinder turns. Knits and other structures

A circular machine leans its courses whatever the yarn

Spirality is blamed on the yarn, and the yarn is most of it. The rest is the machine's. A circular knitting machine's needles make wales that run straight along the tube, and its feeders lay one course each per turn, so every course climbs its feeder count in every round: on a 96-feeder machine, 48 millimetres round 162 centimetres of tube, an angle of 1.7 degrees. The helix has no machine size in it, its hand is set by the way the cylinder turns, and it survives every remedy aimed at the yarn — a steamed yarn, a plied one, S and Z on alternate feeders, and a rib.

 An inflated tube in section at 0.5, 1, 1.6 times its wrinkling moment. The cross-section of an inflated tube of 100 mm radius at 50 kPa, bent by 0.5 times, 1 times, 1.6 times its wrinkling moment of 78.5 N·m, with the outside of the bend at the top. Each stroke is the wall's axial tension at that point: the pressure's even 2.5 kN/m with the bending's cosine added. At half the wrinkling moment the inside is still in tension; at the wrinkling moment it falls to nothing; past it the inside has gone slack over a dashed arc and the rest carries both the pressure's end force and the moment. What the drawing cannot show is the wrinkles themselves, whose wavelength a cloth's bending stiffness decides and this section leaves out. Cloth doing a job

An inflated beam wrinkles at a moment with no cloth in it

An air-filled tube can be used as a beam because the pressure pulls its cloth taut along its length, and a cloth that cannot carry a push can carry a bending moment for exactly as long as that pull outweighs it. The inside of the bend goes slack at πpr³/2 and the tube folds at πpr³ — seventy-nine and a hundred and fifty-seven newton metres for a tube a fifth of a metre across at half a bar — and there is no property of the cloth in either. The cloth decides how far the beam bends on the way, and how much pressure it can be pumped to.

 What an irregular satin buys, order by order. For each order, the best regular satin's and the best irregular satin's scatter at the order's own best spread — the largest share of the closest pairs that point in one direction, where one is a line and less is a scatter. 5 ends: regular 0.50, irregular none at the best spread; 6 ends: regular none exists, irregular 0.25; 7 ends: regular 1.00, irregular 0.33; 8 ends: regular 1.00, irregular none at the best spread; 9 ends: regular 1.00, irregular 0.25; 10 ends: regular 0.50, irregular none at the best spread; 11 ends: regular 1.00, irregular none at the best spread. Irregularity buys something at 6, 7, 9 and nothing at the rest, and where it buys it scatters over four directions with no more than a third in any one. What the bars cannot show is whether a reader sees the difference, which is a question about a visual system. Weaves

An irregular satin scatters where a regular one lines up

A regular satin's marks lie on a lattice, so its closest pairs all run along one vector and make a row. An irregular satin has no lattice at all, so its closest pairs may point several ways at once — and at seven and nine ends, where every regular satin at the best spread has a row, an irregular one reaches the same spread with its closest pairs scattered over four directions and no more than a third in any one. At eight and eleven ends there is no such satin: the best spread is reached by regular satins alone, and irregularity has nothing to offer.

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

A nap is paid for by the taper of the pattern

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

 Which specifications a 30 tex yarn can be woven into. Warp sett across against weft sett down, in threads a centimetre, for a 30 tex cotton yarn. Each cell is one specification. 220 are left empty because no cloth in the catalogue can be woven at those two setts; 16 are filled and outlined in the warp's colour because every one of the 426 can; and 20 are filled and outlined in the float's colour because some can and some cannot. The admissible region is a rectangle because the warp sett and the weft sett are bounded by two different counts. What cloth is

A specification can name a cloth that is not there

The three notations measured so far write drafts, and every draft is a cloth somebody could weave. A mill works from none of them: it works from a count, two setts and a weave quoted together, and that is the first notation whose image has holes in it. Of 2,560 specifications across the trade's own working range, 2,097 name no cloth at all — and in the 240 where the weave field decides anything, both numbers the trade quotes to decide it rank the catalogue wrongly.

 A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it. Mechanics and drape

A thread has a second stiffness

Every mechanical number here came from one material constant: how hard a thread is to bend. A thread also resists being twisted, nothing here has ever used that, and the ratio between the two turns out to be the only stiffness number about a yarn that can be known 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. Setting and geometry

What a flattened yarn does to its cover

A cover factor is a sett times a diameter, and it decides how much of a cloth is thread and how much is hole. A flattened yarn is a third wider than a round one of the same area, so a cloth of flattened yarn covers more at the same sett — and every opacity, permeability and shade computed from a round diameter is wrong in one direction.

 What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 900-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 12 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve. Cloth doing a job

A grade charges by the length and a cutter pays by the panel

The four-point system scores a fault by how far it runs — one point to three inches, four beyond nine — and caps a linear metre at four points however many faults it holds. A cutting room pays by how many panels the fault lands in. Below a panel's length every fault costs exactly one panel while its score runs from one to four; above it the panels grow without bound and the score does not move at all. Two fifty-metre pieces built to the same 267 points a hundred square metres lose 33 per cent of their panels and 92.

 Imbalance against the torque a tuck carries, for six named structures. For six named two-bed structures, the imbalance of front-bed loops against back-bed loops in the worst fabric, as the share of a knit loop's torque a tuck carries runs from nought to one: single jersey from 1.000 to 1.000; a one-by-one rib from 0.000 to 0.000; a half-cardigan from 0.333 to 0.000; a full cardigan from 0.000 to 0.000; a rib that both tucks and floats from 0.333 to 0.143; a half-milano from 0.333 to 0.333. Structures without tucks are flat lines; a half-cardigan falls from a third to nought and a rib that tucks and floats from a third to a seventh. What the lines cannot show is where along them a real tuck sits, which is not measured. Knits and other structures

A tuck decides whether a third of two-bed fabrics lean

A jersey leans because every loop is on one bed and a rib does not because its loops are mirrored across two, and the count that says so treated a tuck as nothing. A tuck is a loop of the same lively yarn wrapped round a needle of one bed, and nobody here has measured how much of a knit loop's lean it carries. It matters. Of the 1,135 two-bed fabrics a two-needle, two-course frame can make, 135 are balanced whatever a tuck carries and 612 lean whatever it carries — and 388, a third, are balanced under one answer and lean under the other. A half-cardigan leans a third of a jersey if a tuck carries nothing and not at all if it carries a full loop's torque, which makes it the instrument that would settle the question.

 What one wrong symbol reaches. Five kinds of symbol, with how many of them a document holds and how many intersections of a 1800-by-1800 piece one of them decides. A threading digit reaches 810,000; a jacquard card's hole reaches one. The product of the two columns is 3,240,000 in every row, because each set of symbols decides every intersection of the piece exactly once. What cloth is

The shortest notation has the longest mistakes

Four essays of this account have asked what each notation can express. None has asked what happens when one is written down wrong, and the answer is an identity rather than a tendency: a notation's symbols partition the cloth, so the count of symbols times the reach of one is the piece, in every notation, with nothing to trade. One threading digit decides 810,000 intersections of a square metre and one of a jacquard's holes decides one — and only the threading can be wrong while the cloth is right.

 What each further binder takes off a snag. How far a snag drags thread through a bouclé, in binder points, against how many binders the yarn carries: 1, 7.43 points; 2, 3.72 points; 3, 2.48 points; 4, 1.86 points; 6, 1.24 points. The relation is exactly inverse, because the grip is the exponential of the wrap angle and the reach is a logarithm of it, so the second binder removes 3.72 binder points and the third a further 1.24. Compound and figured cloths

The second binder buys half of everything

A bouclé's loops are held by their binder, and a snag drags thread from one loop to the next until the thread breaks. The essay before it left that reach at seven binder points on one binder and said a second would square the grip. It does — and because the grip is exponential in the wrap and the reach is a logarithm of it, the reach is exactly inverse in the binder count: 7.43 points, then 3.72, then 2.48. Half of everything any number of binders can buy is bought by the second one, which is how many the trade uses.

 Where a calender works, and where recovery has been measured. Each fibre's measured elastic-recovery span drawn against the shape strains a calender imposes, on one axis. The settings run from 11.2 per cent shape strain at a flattening of 1.25 to 54.9 per cent at 3. cotton is measured to 5 per cent; wool is measured to 20 per cent; silk is measured to 5 per cent; flax is measured to 2 per cent; viscose is measured to 5 per cent; nylon is measured to 8 per cent; polyester is measured to 8 per cent. Only wool reaches any setting at all, and only the lightest. After the loom

A calender works where nothing has been measured

This account has twice recorded that the missing piece is plasticity — what fraction of a flattening survives the nip. The piece is missing for a sharper reason than nobody having written it down. A calender's flattening is a shape strain, and at the lightest setting in this account's own series that is 11 per cent while cotton's elastic recovery is measured from 2 to 5. Six fibres of seven have no data at any setting the machine has, and wool reaches only the lightest. The law cannot be had from the measurements; what can be had is the bracket, and it is a factor of twenty-four.

 Twist is not torsion: a straight rod, twisted. A rod drawn with a cross painted along it, and the cross turned through 1.5 full turns from one end to the other. The rod is straight, so it has no Frenet frame at all — a straight line has no osculating plane to turn. The two discs above are the rod's ends seen down its own axis, which is where the twist is an angle rather than a foreshortened wiggle. The twist is the rotation of the painted cross about the rod's own tangent, and it is 1.5 turns whatever the centre line does. That is the quantity a torsional rigidity resists, and it is why a collection that solved a plane curve had not thereby dealt with torsion. Mechanics and drape

Twist is not torsion

A curve has a torsion and a material has a twist, they share a word, and only one of them is what a torsional rigidity resists. Getting them the wrong way round would have made this collection conclude that a knitted loop carries no twist, on the strength of a theorem that says nothing of the kind.

 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. Setting and geometry

The count that decides how flat

A knitted fabric's demanded flattening is a function of its tightness factor, and a tightness factor is the square root of a count over a loop length. So a coarser yarn at the same loop is flatter — which is a prediction about a spinner's choice that nobody has framed as one.

 The first 12 metres of a bolt, cut three ways. A bolt of 50 metres carrying 40 faults, with its first 12 metres drawn: the fault positions above, and below them the 900-millimetre panels a cutter gets cut blind, cut with the same rigid tiling slid to its best offset, and cut around the faults with the map in hand. Over the whole bolt the three yield 28, 30 and 40 sound panels of 55. Sliding the tiling buys 2; breaking it buys 12, which is 43 per cent more cloth from the same roll. What the strip cannot show is the width, across which the same argument runs again with a different panel dimension. Cloth doing a job

A fault map is worth most where the grade is worst

A cutter who knows where the faults are can slide the marker or break it, and only one of those is worth anything: sliding a rigid tiling to its best offset recovers two panels of fifty-five, and letting the tiling break recovers twelve — two panels in five more cloth from the same roll. The gain has a maximum in the middle of the range, because there is nothing to recover on a clean bolt and nothing to be done on a ruined one. And the prediction the grading essay made, that a map is worth most on a bolt whose faults are bunched, is false: bunching leaves clear runs for the blind cutter too.

 What two layers of one yarn buy. For each cloth in the table here, the thickness and the thermal resistance of a two-layer cloth carrying exactly the same yarn per unit area, as ratios to the single cloth. The thickness ratio is √2 everywhere, to three figures. The warmth ratio runs from 1.588 on the openest cloth to 1.717 on the closest, because dividing the yarn also divides the fibre fraction and the mixture conducts less. Compound and figured cloths

Two layers are warmer than they are thick

Six essays have taken a double cloth apart from the draft's side. None has asked what the reader gets. Divide one cloth's yarn into two layers and the fabric is 41 per cent thicker — √2, which is the account's own law — and about 70 per cent warmer, because dividing the yarn also divides the fibre fraction and the mixture conducts less. The gap widens with every further layer and never closes, and a shaft loom stops at two.

 What a pair of colour orders does to the catalogue. Dark ends in the warp across against dark ends in the weft down, for the sixteen two-colour orders at four ends. Each cell gives the blind intersections of sixteen and the number of distinct surfaces the 22874 drafts collapse onto. Every cell stands for between one and thirty-six colour orders and they all behave identically, so the arrangement of the colours does not matter and only their counts do. The corner at four against nought is blind nowhere and separates every draft. Pattern and colour

The colour order that hides least

A caption on this account's top essay left a question: whether a warp of long runs against a weft of short ones hides less than either would against itself, and called it a lever nobody uses. Sweeping all sixteen orders against all sixteen says the lever is real and the caption named the wrong variable. Run length has nothing to do with it — the blind count, the separation and the largest confused class all depend on the two orders' colour counts and on nothing else, and what a mixed pair saves is exactly the square of the difference between them.

 One wrong threading digit against two. What becomes of a threading when one digit is wrong and when two are, over every draft in the four-by-four sweep: one digit wrong, 252,968 cases, 45.0 per cent refused by the drawdown, 54.8 per cent a different cloth, 0.228 per cent the same cloth; two digits wrong, 1,074,972 cases, 59.1 per cent refused by the drawdown, 39.6 per cent a different cloth, 1.285 per cent the same cloth. What cloth is

A second wrong digit is silent only by cancelling the first

One wrong threading digit leaves the cloth exactly as it was 576 times in 252,968, all on three-shaft drafts. The obvious guess about two was that silence would become the rule, since two changed digits can swap two shafts and swapping shafts is the harness's own freedom. Counted over all 1,074,972 pairs, two wrong digits are silent 1.29 per cent of the time — five and a half times as often, on fifteen times as many cloths — and never by adding one silent slip to another. Every silent pair is two audible errors cancelling, and at two errors point paper and lifting plans, which can never be wrong silently by one symbol, are silent almost as often.

 A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it. Mechanics and drape

The one fibre whose answer is known

A table of measured constants is worth what its worst row is worth, and nobody can tell which row that is. This one has a member whose answer was known before anybody measured it, and the ordering of the other nine turns out to say something about how fibres are made.

 What the count moves at 150 grams. Plain cotton cloths that all weigh 150 g/m², from 20 tex to 200 tex, with sett and crimp solved together. Thickness, which is two yarn diameters, rises 3.16 times across the line; the cover factor of each thread system falls 2.75 times; their product with one plus the crimp is the same number at every count, because the weight has fixed the volume of fibre and the count only decides whether it is laid out flat or stacked up. Setting and geometry

A weight fixes the fibre and not the drape

Every plain cotton cloth of 150 grams a square metre contains the same fibre, and the count decides only how it is arranged. Across the counts that can make that weight, thickness rises threefold and cover falls in step, so their product holds still. The bending length does something stranger: at the bound a woven yarn actually sits near, it depends on neither the count nor the weight, only on the fibre.

 What the crossing end asks for, against what a cam gives it. The length a leno's crossing end needs at each fraction of the crossed shed's opening, against an easer driven in proportion to the shed. The demand is nothing for the first eighth of the opening and then rises steeply, because the kink's extra length is a hypotenuse less its run and grows as the square of the climb. The proportional easer runs ahead of it everywhere, worst at 0.375 of the way open, where it has given 16.5 millimetres more than the end can use. Compound and figured cloths

A cam easer gives its slack too early

The essay before it left the crossing end with fifteen millimetres it could not use at half the shed, and three things it might do with them. All three are decidable. It cannot reach the shuttle, because the same eyes that trapped the kink keep the slack behind the harness — and the margin is negative, so that friction is not a nuisance here but the reason a leno weaves at all. It cannot snarl, because the span is under half this account's own threshold. It sags seventy-two millimetres where the easer gives it, and the repair is a cam cut to the demand rather than to the shed.

 Specular area against the size of the source. The share of a sheeting's face that reflects, for plain, 2/2 twill, 5-end satin, 8-end satin, against the angular half-width of whatever is lighting it. Plain weave's line has twice the slope of the others, because a specular area is a length times a width and both are proportional to the angle for a weave with no float — while a float contributes a crown line that does not shrink with the source at all. Weaves

No cloth shines under a sky

Every specular figure on this account is drawn at a tolerance of two degrees, which is a stand-in for how wide the source is. Sweeping it says something the account has not: the contrast between a floated weave and a plain one is inversely proportional to the source's angular size, exactly, because a float keeps a crown line as the source narrows and a plain weave has only turns. A satin outshines a plain weave by 177 under the sun's disc, by 22 under a small lamp, and by exactly nothing at sixty degrees.

 The depths an even eight-end shading can sink. Eight-end shading chains with every middle tone held to a float of three and the centre to two, sorted by the interlacing rate of their worst tone, with the depth that tone sinks below the extremes on a sheeting pressed at 0.50 N: 0.5000, 42.8 µm, 27,904 chains; 0.5625, 50.1 µm, 7,296 chains; 0.6250, 56.7 µm, 35,680 chains; 0.6875, 62.6 µm, 8,960 chains; 0.7500, 68.1 µm, 18,624 chains; 0.8125, 72.7 µm, 1,536 chains. The shallowest is the float argument's floor. The walk stopped after 100,000 chains, so the list is a lower bound on the depths the family has. Pattern and colour

An eight-end shading is pinned at three everywhere but its centre

At six ends every even shading holds its middle tones to a float of two, and every one sinks to the same depth. The question left was whether eight ends does the same. It cannot even start: at eight ends no shading can hold its second tone to two, a float of three there needs the second mark of every pick exactly opposite the first, and then the third mark cannot undo it. Tones two, three, five and six are pinned at a float of three, only the centre is free, and a family with a free centre does not sink to one depth — it sinks to at least six.

 What a film 80 µm deep touches on a sheeting. The part of a sheeting that a film reaching 80.3 µm below the crowns touches, in plan over two repeats each way, for a 2/2 twill and a 2/2 hopsack. On the 2/2 twill the film is separate patches, and joins at 161 µm, an add-on of 116 g/m² of the 186 that flattens the face; on the 2/2 hopsack the film is separate patches, and joins at 161 µm, an add-on of 116 g/m² of the 186 that flattens the face. The twill's crowns join edge to edge in its matrix and not on the cloth, because two neighbouring ends are separated by the gap the sett leaves. What cloth is

A ridge in the matrix is not a ridge in the cloth

A 2/2 twill's crowns join into a ridge across its matrix and a 2/2 hopsack's are islands, so a thin film on the twill should be continuous from the first gram and on the hopsack a scatter of patches. Laid on the cloth's own surface rather than its matrix, both films are patches — the twill's ridge crosses from one end to the next, and between two ends lies the gap the sett leaves. On a sheeting both join at the same depth, 116 grams into the 186 that flatten the face, and the weave's whole influence is a window of up to 27 grams at the setts where it opens at all.

 The cloth a calender nip exactly fills. The share of a cloth's plan a calender leaves specular, per unit of crown line per crossing, against the cover of the lustrous system, for nips that flatten the yarn by ×1.25, ×1.5, ×2, ×3. Each curve rises while the nip binds and falls once the threads meet their neighbours; the peak is at the cover the nip exactly fills, 0.919 for ×1.25, with a share of 0.200; 0.853 for ×1.5, with a share of 0.333; 0.754 for ×2, with a share of 0.500; 0.628 for ×3, with a share of 0.667. The yarn diameter, 0.2 mm here, appears in neither the peak's position nor its height. After the loom

A calender's best cloth is the one its nip fills

A calender's lustre is a length of crown line times a width of plateau. A closer cloth has more crown line and less room to widen each thread into, and the product was expected to peak at some interior cover that would move with the weave. It does not peak at all on the cap alone — per unit of crown line it falls all the way from the most open cloth. The peak is made by the nip: a nip that flattens by f has one best cloth, the one it exactly fills, at a cover of d over the flattened width, and there the lustre is (f − 1)/f. No yarn, and no weave, moves it.

 How high each pore system stands while the cloth is drying. The height at which supply up the cloth balances loss from its faces, for the two pore systems, against the evaporation rate. The coarse system between the yarns barely moves — 137 millimetres at the dry end against 109 at the wet — while the fine system between the fibres falls from its sealed-tube height of 6375 millimetres to 52. The two cross at 1553 grams a square metre an hour. Cloth doing a job

A drying cloth cannot lift what a sealed tube can

Every rise on this account is an equilibrium in a sealed tube, and a garment is neither sealed nor at equilibrium. Balancing the supply up a strip against the loss from its faces gives a quadratic whose width cancels exactly: in an ordinary room the fine system between the fibres stands at 500 millimetres instead of 6.37 metres, keeping 7.9 per cent of what a tube would give it, while the coarse system loses a third of a per cent. The forty-sixfold advantage becomes 3.7, and in a drying wind it reverses.

 Linked: a knitted interlacing: one loop drawn through the next. Two closed curves and the Gauss linking integral taken over them, which returns -1.0002 at 200 segments a curve. A knitted interlacing: one loop drawn through the next. A linking number is an integer, so a value coming back at a few thousandths of one is the discretisation reporting itself rather than a fabric that is slightly linked. The two arrangements are the two ways of making cloth: a knitted fabric's courses link and a woven cloth's threads do not, at any crimp and for ever. Mechanics and drape

What a closed thread cannot choose

A thread whose ends are held has a quantity it cannot change without breaking: the total number of times its material winds about its own axis, plus the number of times that axis winds about itself. The two can trade, and everything a twisted yarn does when it is let go is that trade happening.

 Which end of its interval a construction falls to. For a balanced 20 tex, 24-thread reference with its weft count and weft sett scaled, the share of the crossing height the warp takes at the least-energy state. 25 of the 28 solvable constructions land at an end of their own feasible interval — nought, with the warp dead straight, or one, with the weft dead straight — and the few between them are the frontier where the two regions meet. Setting and geometry

The energy has no crimp ratio to give

The account left a gap at its top: a construction whose preferred crimp ratio lies outside the interval its geometry admits sits at a boundary, and boundary states had not been studied. Profiling the whole energy rather than its minimum says there are two regions and a frontier — six of the eight cloths in the table of cloths here fall to an end of their own interval with the weft dead straight, and the two that do not have wells 3.27 and 0.52 per cent deep, against a rigidity known to within a factor of 408.

 The pressure a band puts on a calf with a ridge down its front, point by point. The section of a calf with a ridge down its front modelled as the convex hull of 2 circles, 360 mm round, with a band on it tensioned so that a round limb of the same girth would be pressed at 20.0 mmHg. The band rides the hull: on each arc it presses with its tension over that circle's radius, and on the straight stretches between arcs it presses nothing. shin ridge 214 mmHg over 2.4% of the girth, calf muscle 21 mmHg over 68.9% of the girth; 29% of the girth carries no pressure at all. The spikes are drawn outward from the band with length proportional to the pressure. Cloth doing a job

A band presses where the limb turns

Every compression pressure is quoted as a band's tension over the limb's radius, and a limb has no radius. It has a curvature that changes all the way round, and a band presses each point with its tension times the curvature there. The round-limb number survives exactly — as an average over the band's length — and is the pressure almost nowhere. On a calf with a ridge down its front the ridge takes ten times it and the flat face beside it takes none; on an ankle, two bones and a tendon carry nearly three quarters of the force on under a fifth of the girth.

 What stitching takes back from a double muslin. The warmth a two-layer muslin of the same yarn gains beyond its √2 thickness, against the share of a layer's intersections that carry a stitch, with a 5-end satin face. Under the parallel rule the bonus is 18.8 per cent whatever the stitching, because that rule already treats every fibre as running through the cloth. Under Maxwell's rule for fibres lying across the heat's path the bare bonus is 8.4 per cent and each stitch takes some back: one stitch a repeat (4% of intersections) leaves 7.6, one a pick, scattered (20% of intersections) leaves 4.7, every hidden position (60% of intersections) leaves -1.5. Compound and figured cloths

A stitch takes back what the parallel rule cannot see

Dividing a cloth's yarn into two layers makes it warmer than its thickness, and the essay that found it computed the warmth by the one mixing rule that cannot see a stitch — the one that treats every fibre as though it ran straight through the cloth. That rule gives the largest bonus there is. Yarn in a cloth lies in its plane, and by the rule for fibres lying across the heat's path a double muslin's bonus is 8.4 per cent, not 18.8. A stitch is the one fibre that does run through the cloth, and at half the intersections it takes the whole bonus back.

 A lamp in front of a window: the contrast against the lamp's share of the light. The contrast an eight-end satin shows over a plain weave when a 1° lamp and a 16° window both lie in the direction the cloth reflects to the viewer, against the lamp's share of the light: 0.0% gives 3.01, 0.5% gives 3.22, 1.0% gives 3.43, 2.0% gives 3.85, 5.0% gives 5.11, 10.0% gives 7.21, 20.0% gives 11.39, 30.0% gives 15.57, 50.0% gives 23.88, 70.0% gives 32.16, 100.0% gives 44.48. The points lie on a straight line between the window alone and the lamp alone, to within 0.8 per cent: the room's contrast is its sources' contrasts averaged by power. The dashed curve is a single source as wide as the power-weighted average width, which gives only 3.30 at a tenth of the light where the room gives 7.21. Weaves

A room lights a satin at the harmonic mean of its lamps

A satin outshines a plain weave by a factor inversely proportional to the width of the light, which is a clean law for one source and says nothing about a room. A room has a lamp and a window at once. Sum each source's highlight and the average falls out by itself: the room behaves as a single source whose width is the harmonic mean of its sources' widths, weighted by the power each supplies. The harmonic mean is ruled by the narrowest source, so a small lamp carrying a tenth of the light more than doubles the contrast a large window gives.

 Every way to make 4 intersections blind. The number of distinct surfaces the 22874 four-by-four drafts collapse onto, for every shape a set of 4 blind intersections can take, in however many colours it needs. 1×1 + 1×1 + 1×1 + 1×1: 3,632 surfaces, touching 8 threads, 4 colours needed; 1×1 + 1×1 + 1×2: 3,352 surfaces, touching 7 threads, 4 colours needed; 1×2 + 1×2: 3,102 surfaces, touching 6 threads, 3 colours needed; 1×2 + 2×1: 3,100 surfaces, touching 6 threads, 4 colours needed; 1×1 + 1×3: 3,038 surfaces, touching 6 threads, 3 colours needed; 1×4: 2,744 surfaces, touching 5 threads, 2 colours needed; 2×2: 2,402 surfaces, touching 4 threads, 3 colours needed. The same number of blind intersections keeps more of the catalogue apart the more threads it is spread over. What the chart cannot show is whether an eye can tell the surfaces apart. Pattern and colour

A blind set hides less the thinner it is spread

Two colours showed that where a cloth's threads match in colour the draft underneath is invisible, and that how many intersections match is not what decides how much is hidden. More colours say what does. A third colour lets the blind count take every value from nought to ten; a fourth lets the blind intersections sit one to a thread. At a fixed count, the catalogue keeps more of its cloths apart the more threads the blind set is spread across — 3,632 surfaces for four scattered blind cells, 2,402 for the same four in a square.

 One dip, two yarns: the same depth of dye. The cross-section of two cotton yarns drawn to one scale after a 20-second dip in a bath whose dye the fibres take up 10 times as strongly as the liquor holds it, each fibre shaded by the dye that reached its distance from the surface. 20 tex: 118 fibres, 167.1 µm across, dyed to 19.7 µm — 24% of its radius and 42% of its section; 74 tex: 435 fibres, 321.4 µm across, dyed to 18.2 µm — 11% of its radius and 21% of its section. The depth in micrometres is the same for both because a dip dyes a depth; the share is not, because the coarse yarn has more section behind the same ring. The fibres are drawn on a lattice and the count is computed; the affinity is an assumed value and every depth moves with its square root. After the loom

A dip dyes a depth, not a share

A yarn dyed after spinning is a cylinder the dye has to diffuse into through the liquor between its fibres, and the fibres slow it by taking dye out of the liquor as it passes. A twenty-second dip therefore dyes to a depth, some tens of micrometres, and that depth is almost the same in a fine yarn and a coarse one. So the same dip that colours two fifths of a shirting yarn's section colours a fifth of a denim warp's; eight dips deepen the shade eightfold and leave the ring where it was; and a tighter yarn rings more thinly, because packing closes the pores faster than it narrows the yarn.

 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.

 Where a knitted disc is flat. The circumference a knitted disc grows per unit of radius, as a share of the 2π a flat disc needs, against the loop's aspect, for 4 wedges (flat at 1.2732), 10 every 2 rounds (flat at 1.2566). Wedges of short rows fall as the aspect rises and rounds with increases rise. The shaded band is the aspect of plain knit between its dry-relaxed value, 1.2500, and its wet-relaxed value, 1.2927. Knits and other structures

A knitted disc is flat at one shape of loop

A disc lies flat only if its circumference grows by exactly 2π per unit of radius, and a knitted disc's growth is a count of loops in one direction over a count in the other. So it is flat at one value of the loop's aspect and no other. Knitted sideways in wedges of short rows the growth falls as the loop gets wider; knitted outward in rounds it rises. Four wedges and ten increases every two rounds are each flat at an aspect inside the range plain knit moves through when it is washed — and they cross it in opposite directions.

 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.

 Every count that can make 150 g/m² as plain jersey. Areal weight against count for plain jersey in the fully relaxed state: the shaded wedge is every knit between a tightness factor of 1.3 and 1.6, whose weights are kₛ times the tightness times the root of the count. The rule at 150 g/m² crosses it between 15.8 and 23.9 tex, a ratio of 1.51. A plain woven cloth of the same weight can be made from anything between 20 and at least 200 tex, a range of at least 10 to one against the knit's 1.51. Setting and geometry

A knit's weight nearly names its yarn

A woven cloth's weight is one equation in four unknowns, and a hundred and fifty grams can be woven from anything between twenty tex and two hundred. A plain jersey's weight has the loop in it and nothing else to spare, and the loop is bounded by the yarn it is knitted from. Put the two together and the loop cancels: the weight is a constant times the tightness times the root of the count, so at one weight the count is fixed to within half again — and in each relaxed state it is fixed to a different half.

 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.

 Silent threading errors under three lifting plans. For every four-shaft threading of eight ends, under three lifting plans: 2/2 twill, 36,320 of 40,824 threadings can be mis-threaded silently by two digits, 0.97% of double errors and 0.20% of single errors are silent; 1/3 twill, 29,728 of 40,824 threadings can be mis-threaded silently by two digits, 0.55% of double errors and 0.20% of single errors are silent; unrelated rows, 864 of 40,824 threadings can be mis-threaded silently by two digits, 0.0084% of double errors and 0.0000% of single errors are silent. A twill's plan slides one row a pick at a time, so its shafts are interchangeable by a slide along the picks, which is a writing of the same cloth. What the chart cannot show is a plan with more shafts. What cloth is

Silence lives in the lifting plan

Two wrong threading digits leave a four-end cloth exactly as it was 1.3 times in a hundred, and the question left was whether a longer repeat makes that rarer or commoner. At eight ends the answer depends on something the question did not name. With a lifting plan of unrelated rows, silence all but vanishes; with a twill's plan — one row slid a pick at a time — nine four-shaft threadings in ten can be silently mis-threaded, and even single errors are silent one time in five hundred. The repeat hardly matters. The plan's symmetry does.

 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.

 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.

 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.

 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.

All themes