Textile Structure
how cloth is put together
Theme

The thread: Exactly this many — page 4

Page 4 of 4 of the essays on this thread.
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.

 The surface of every even six-end shading, against the twill's. The height of the cloth's surface at each tone of a six-end shading on a sheeting at 0.50 N, for every chain of the 64 even classes, drawn once per distinct shape — 5 of them — and for the cyclic square's twill read one step at a time. The twill is level to the micrometre and floats 5, 4, 3, 4, 5. Every even chain sinks to exactly 43.3 µm at its midtone; the shapes differ only at the second and fourth tones. What the plot cannot show is which of these a raking light would reveal, which depends on the finish. Pattern and colour

An even shading cannot keep its surface level

Sixty-four six-end shadings hold every middle tone to a float of two, and the question left over was whether any of them keeps a tone ramp's surface level the way a twill read in order does. None does, and all of them sink by exactly the same depth — 43.3 micrometres on a sheeting, a sixth of the cloth. The reason is a counting argument a recording engineer would recognise: a limit on how long a thread may float is a limit on run length, and a run-length limit forces a floor on how often the thread changes face. A level ramp needs every tone at the extremes' rate, which needs a float of half the repeat at the midtone and more beside it — exactly the floats the twill in order has.

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

A tuck is the one stitch that links twice

Knitting has three stitches and only one of them makes a new link. A knit stitch links a loop to the loop below; a miss links nothing; and a tuck holds two loops in one head — which is why a tuck stops a run and why the three cannot be described by one number.

 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.

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

A jersey has no float for a wire to catch

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

 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.

 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.

 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.

 The beams a disc 12 blocks across needs, column by column. A disc 12 blocks across, 12 blocks by 12, figure shaded, with the beam feeding each column of ends written above it, for a figure floating 8 on a ground floating 1, whose warp crimps are 14.15 per cent apart, at 4 mm blocks over a 100 m piece with 1.2 mm of slack. It has 4 distinct columns and 4 distinct shares of the repeat in figure, and needs 4 beams. What the grid cannot show is the cloth's own crimp interchange, which would move tension between the columns instead. Compound and figured cloths

A figured warp needs a beam for every share of its figure

Figure and ground take up warp at different rates, so a figured cloth on one beam is bounded in how long its figure may run, and a second beam is the obvious escape. It escapes only for ends that live wholly in one region. An end that crosses the figure for part of the repeat consumes warp at its own rate, and two ends can share a beam only if they spend the same share of the repeat in the figure and never drift a slack apart inside it. So a round figure twelve blocks across needs four beams, ninety-six blocks across needs twenty-nine, and any crimp difference at all — a third of a per cent will do — costs every one of them over a piece.

 A 2/2 twill with warp 1/1 and weft 2/2, as drawn and woven across. A 2/2 twill coloured 1/1 in the warp and 2/2 in the weft, drawn as the face of the cloth, and the same cloth turned through a right angle, which is what the loom makes if the two colour orders are exchanged and the weave turned with them. As drawn the weft order is 2/2 and needs boxes at one side; woven across the weft order is 1/1 and needs a loom picking at will. What the drawings cannot show is whether the cloth's two systems can be exchanged, which depends on their yarns and setts. Pattern and colour

A colour-and-weave look costs its cheaper order

The finest colour-and-weave effects need the rarest loom because a weft order is thrown pick by pick and a warp order is laid out once, so an effect's price was said to be its weft. That is true of a construction and false of a cloth. The same cloth can be woven lying across the loom, with its warp order thrown as weft and its weft order laid in the warp, and then it costs the other order. Over every two-colour look twelve small weaves make with orders up to six threads — 4,036 of them — 55 per cent need a loom picking at will as drawn and 31 per cent need one either way round. The looks turning rescues are the ones fine in one direction only, and not one of the trade's named effects is among them.

 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.

 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.

 8/3 at 1.00 and 8/3 at 1.29, drawn in cloth. Satin marks drawn over two repeats at the spacings of a cloth, with the pick direction stretched by the sett ratio — ends per centimetre over picks per centimetre — and every closest pair of marks joined. 8-end satin, move 3 at a sett ratio of 1.000: closest pairs point one way: a row; 8-end satin, move 3 at a sett ratio of 1.291: closest pairs point 2 ways: no row. What the drawing cannot show is the float each mark interrupts, which is what a reader of the cloth actually sees. Weaves

A satin's row belongs to its sett

Point paper draws an end and a pick as equal squares, and every result about which satins have a row was taken there. A cloth is not square: it is set at so many ends and so many picks a centimetre, and in cloth the ties that made twelve satin orders row-free are ties between steps of different shape, which break at the first per cent of unequal sett. The reverse happens too. An eight-end satin, rowed on paper, has no row at exactly 1.291 ends per pick; an eleven-end at 1.265 and 1.528. And the only scatter that survives a range of setts is an irregular satin whose closest steps are mirror images — which nine ends has from the first per cent and ten only by 1.3.

 8×2 and 2×8 over every origin. Two grids of the 16 relative origins of 2/2 twill under 8-end satin, the row being how many picks the ground is started along and the column how many ends. The left grid is the census at a block 8×2, the right at 2×8; a square is filled where some of the 65,536 profiles separate. 8×2 fails at 8 origins and 2×8 at 8; both fail at 0 and neither at 0. Turning the cloth over and through a right angle sends each origin to another, and the letters mark where: every letter lands on a square with the same answer, so the two shapes are one census read at relabelled origins. Pattern and colour

A turned block is a moved origin

An eight-end satin figured on a 2/2 twill fails on 55,536 profiles at a block eight picks by two ends and on none at two by eight, and that was read as a property of the block's shape. Sweep the relative origin as well and the two shapes trade places: at every one of the sixteen origins exactly one of them fails, and over the sixteen they fail equally often. A shape asymmetry that no origin removes exists, and it needs a satin whose move squared is not one.

 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.

 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.

 How many derivation orbits each measure separates. For the 426 four-by-four cloths in 157 derivation orbits, the number of classes each invariant measure cuts the orbits into: marks, up to exchanging face and back, 5; interlacings, 8; layers, 2; plane group, 12; floats of both faces, both systems, 60; changes of face per end and per pick, 12; distinct ends and distinct picks, 4; all seven familiar measures, 120; census of two-by-two patches, 127; all seven, and the two-by-two patches, 153; census of three-by-three patches, 157. Only the census of three-by-three patches reaches 157. What the bars cannot show is which orbits a measure confuses, which the pair figure draws for the seven measures together. Weaves

A cloth's derivation class is its census of small patches

The manuals' derivations cut the 426 four-by-four cloths into 157 orbits, and an orbit is found by searching a group of 256 operations. The question left was whether a short list of numbers read off a draft could do the same job. The familiar ones cannot. Marks, interlacings, layers, plane group, float lengths, crossings per thread and distinct ends and picks are all invariant, and all seven together tell 120 of the orbits apart. A census of the two-by-two patches a draft contains tells 127 apart. A census of its three-by-three patches tells all 157 apart — a complete invariant of derivation, computed by counting windows rather than by searching operations.

 How much plateau a cloth of stated cover can be given. The flat run a calender can put on a 200 micrometre yarn, against the cover the cloth already has. A thread may widen until it meets its neighbours, so the flattening is capped where the section's width equals the pitch; at a cover of 0.3 that is a flattening of 13.9 and 619 micrometres of plateau, and at 0.9 it is 1.32 and 53 micrometres. At full cover both are nothing. After the loom

A covered cloth cannot be calendered

A calender widens a section at conserved area, and a thread may widen until it meets its neighbours. So the flattening is capped by the cover the cloth already has — ×30 on a scrim, ×1.32 at a cover of nine tenths, and exactly one at full cover. The plateau a calender buys falls to nothing along with it, which makes the lustre a nip can add and the cloth's opacity the same constraint read twice: a cloth that cannot be seen through is a cloth a calender cannot help.

 What a second bar wants that the first does not. How much more yarn each guide bar needs than a tricot bar, per wale, over a 100 metre piece at 14 courses a centimetre — which is 140000 courses. A cord bar wants 111.3 metres more and a satin bar 232.3, both of which are more than the piece is long. A shared beam is one course spacing out after 0.28 courses. Knits and other structures

Two bars cannot share a beam

Two guide bars each make one overlap a course and one underlap, and the overlap is the same object on both — the same needle, the same loop. So the difference in what they consume is exactly the difference in their underlaps, with no model of a loop in it: a cord bar wants 111 metres more yarn per wale than a tricot bar over a hundred-metre piece, and a satin bar 232. A shared beam is one course spacing of slack out after less than half a course.

 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.

 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.

 The locus a warp knit's underlaps put it on. Width against length for each shog, at 28 needles an inch and 14 courses a centimetre. An underlap keeps its own length, so the two spacings lie on a circle and every fabric moves along an arc of it. A tricot can be pulled 27.3 per cent wider and 62 per cent longer; a cord 7.5 and 173. The shog decides which. Knits and other structures

The shog is the anisotropy

An underlap is a straight run from one needle to another on the next course, so it is a hypotenuse — and its length is fixed while the two spacings are not. Pulling the fabric wider turns it towards the horizontal and pulls the courses together, on a circle. Every state a warp knit's underlaps allow lies on that circle, its as-knitted point sits where the shog puts it, and the ratio of the length it can give to the width it can give runs from 2.3 at a tricot to 218 at four needle spaces.

 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.

 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.

 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.

 A short-row heel on 64 needles, laid flat. The 32 heel stitches of a 64-needle sock, each column drawn as the courses it was worked in during the short rows: nought at the heel's two edges, rising by four courses a stitch to 42 across the 11 stitches left at the turn. The cells are drawn at the knit's own aspect, a wale 1.2791 times as wide as a course is tall. The instep's 32 columns beside it are worked in none of the short rows. Down the back line the extra length turns the tube through 92.4 degrees, and the whole trapezoid is 65.6 per cent of the fabric a true bend of that angle would need. Knits and other structures

A heel turns a right angle because of the loop

A sock's heel is knitted in short rows on half the needles until a third are left, then back out again. Down the back line that adds a length of courses, and a tube's back line longer than its front by that much has turned through π times two thirds over the loop's own aspect — 93.8 degrees, on any needle count, in any yarn. The trade's third is the loop's shape in disguise. What the rule does not do is make a bend: the heel supplies two thirds of the fabric a true right angle needs, and every missing stitch is on its sides.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 What flattening a section does to the ratio of the two stiffnesses. C/B is 2G/E for a circular section, because a circle's polar second moment is exactly twice its flexural one. A flattened section has two different flexural moments — easy about the long axis, hard about the short one — and the polar moment is still their sum, which is the perpendicular axis theorem and holds for any section whatever. So a flattened thread has three constants rather than two, and the multiplier on C/B depends on which way it is being bent. At the 0.78 a knitted fabric's own geometry demands, the easy direction multiplies the ratio by 1.322 and the hard one by 0.804. A knitted loop bends in the easy direction, so the collection's headline ratio is a lower bound for a yarn in cloth. Mechanics and drape

The section that changes both stiffnesses

A thread's two rigidities are in the ratio 2G/E, and that is a fact about a circular section: a circle's polar second moment is exactly twice its flexural one. A yarn in cloth is not circular, so a yarn in cloth has three constants rather than two — and the ratio a whole ladder rests on is a lower bound.

 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.

 A ball knitted in rounds, count by count. The stitch count of each round of a ball knitted outward from one pole in 40 rounds, from the pole to the equator, in the fully relaxed state, against the count a sphere asks for. Increasing 5 at a time it reaches 65 at the equator, increasing on rounds 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14, 16; Increasing 10 at a time it reaches 60 at the equator, increasing on rounds 3, 5, 8, 11, 15. What the chart cannot show is where round the ball the increases fall, which spirals or stacks them and is a choice of the pattern. Knits and other structures

A ball knitted in rounds washes the other way

A ball can be knitted sideways in gores or outward from one pole in rounds, and both have to follow the same sine. But a gore counts its meridian in stitches and its circumference in rows, and a round counts them the other way, so the loop's aspect sits underneath one construction and on top of the other. A wash raises the aspect — and closes a gored ball's ruffled poles while it opens a round-knitted ball's.

 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.

 The selvedge turns of a 2/2 basket, 8 ends wide, from the left. A strip of 2/2 basket 8 ends wide over 8 picks, the first thrown from the left, with the weft's turn between every pair of picks drawn at the edge it reaches. 4 of the 8 turns are caught, where the edge end is on the other face on the second pick, and 4 slip. Across all its edge placements the weave catches every turn at 0 of 16. What the drawing cannot show is how far a slipped loop travels, which the beat-up and the weft tension decide. Weaves

A weave holds its selvedge with two of its own columns, or not at all

A selvedge holds where its edge end changes face at every turn of the weft there, and the turns at one edge fall on alternate pick boundaries. So an end can hold the right edge only if its column changes face at every odd boundary, and the left edge only at every even one. Whether a weave can hold a selvedge is then whether its set of columns contains one of each kind, and nothing about its width matters. Two ends re-entered on the right shafts make 13,238 of the 22,874 four-by-four drafts hold at any width, where 3,262 did. The other 9,636 need a shaft of their own. A weave with an odd number of picks has no column of either kind, and never can.

 The advantage of spreading a blind set, against the repeat. The number of distinct surfaces a blind set keeps when its cells are spread one to a thread, over the number it keeps when they are gathered into a block, against the size of the repeat: 4 blind, 1.512 at 4, 1.254 at 5, 1.129 at 6, 1.064 at 7, 1.032 at 8, 1.008 at 10, 1.002 at 12; 8 blind, 1.080 at 8, 1.020 at 10, 1.005 at 12. Sampled from forty thousand surfaces at each point. The dashed curves are the closed form — each thread the set misses must interlace on its own, which fails once in 2^(n−1) — and the sampled points sit on them from six threads up; at four and five the form overstates, because a thread the set does touch can still fail there. At four by four spreading keeps half as much again; at eight, a few per cent; at twelve, nothing to speak of. What the chart cannot show is whether an eye can tell the surfaces apart. Pattern and colour

A blind cell costs half the cloths at any real repeat

At four by four, spreading a blind set over more threads kept half as many cloths apart again as gathering it. At eight by eight the same rule holds, and it is almost nothing: three per cent for four blind cells, eight for eight. The reason is one factor. A blind set's surfaces are the count's bound, 2^(n² − b), times the chance that every thread it misses interlaces on its own, and at eight threads a thread lies all on one face once in 128 tries. So shapes that touch the same number of threads tie exactly, whatever their rectangles; every blind cell costs very nearly half the catalogue; and by twelve threads the arrangement is free.

 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.

 Every even eight-end shading, by the depth it sinks. All 1,001,574,400 eight-end shading chains whose middle tones float at most three and whose centre floats at most two, counted exactly and sorted by the interlacing rate of their worst tone, with the depth that tone sinks below the extremes on a sheeting at 0.50 N. 0.50000: 10.6 per cent, 42.8 µm; 0.53125: 0.9 per cent, 46.4 µm; 0.56250: 4.3 per cent, 50.1 µm; 0.59375: 1.0 per cent, 53.3 µm; 0.62500: 17.1 per cent, 56.7 µm; 0.65625: 2.3 per cent, 59.7 µm; 0.68750: 13.3 per cent, 62.6 µm; 0.71875: 2.3 per cent, 65.2 µm; 0.75000: 37.8 per cent, 68.1 µm; 0.78125: 0.2 per cent, 70.4 µm; 0.81250: 3.0 per cent, 72.7 µm; 0.87500: 3.0 per cent, 76.7 µm; 0.93750: 2.1 per cent, 80.5 µm; 1.00000: 2.1 per cent, 83.6 µm. The largest class is 0.7500, with 37.8 per cent of the family. Beside each share is the share the walk's first hundred thousand chains gave it, which reached 6 of the fourteen classes and put 27.9 per cent at the floor against a true 10.6. Pattern and colour

Only one start in six can reach the shallowest shading

The even eight-end shadings — middle tones floating three, centre floating two — were too many to walk, and a walk stopped at a hundred thousand chains found six depths. Split at the centre, the family can be counted whole: 1,001,574,400 chains in fourteen depths, every one of them decided by the centre alone. Only 816 of the 5,040 ways to begin a shading can still reach the shallowest ramp, and a designer who wants it has to choose the first two parts before anything else.

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