Theme

The thread: Exactly this many — page 2

Page 2 of 4 of the essays on this thread.

What the two setts can be set to. Two rules on one scale from 8 to 40 threads per centimetre. The upper carries the 49 warp setts a metric reed catalogue reaches at one to four ends per dent; the lower carries the 4825 pick densities a change-wheel take-up reaches. The mean spacing is 0.656 threads per centimetre in the warp and 0.0066 in the weft, a ratio of 99. The widest gap in the reed's range is 2.10 threads per centimetre.

The setts a loom can reach

Transposing a draft gives a perfectly good draft, and every count this site takes off a matrix either is symmetric under exchanging warp and weft or has a mirror twin. The loom is not symmetric at all: over the range ordinary cloth is woven in, it can choose a pick density 99 times more finely than a warp sett — and the warp sett cannot be changed once the warp is drawn in, at any granularity whatever.

The step between a figure and its ground. Three figures on a plain ground, all in one sheeting's threads at its own setts. A thread presses on the thread it crosses only where it turns, and it turns at its interlacings — so the pressing a region receives per unit area is the contact force at one turn times the turns per unit area, and the second factor is a property of the matrix exactly. A five-end satin turns two fifths as often as a plain weave, is pressed two fifths as hard, flattens less, and stands 55 µm proud of it. What the rows cannot show is that both regions are given a plain weave's weave angle: a satin's crimp is genuinely smaller and its turns genuinely gentler, so the real step is larger than this, by an amount not computed here.

Pattern and colour

A figured cloth has a step in its surface

A damask is one cloth in one set of threads at one sett, and it is not flat. A thread presses on the thread it crosses only where it turns, so a region that turns less often is pressed less often, flattens less and stands thicker — and the step is a ratio of interlacing rates, read off the matrix with no yarn property in it.

The weight against the constant nobody has measured. Areal weight against the stitch-density constant, for four two-bed structures at 20 tex on a 0.35 cm loop with a tuck taken as 1.15 of one. Every line is exactly straight through the origin, because the weight is tex times yarn per repeat times k_s divided by the area of the repeat and there is no fitted constant anywhere in that division. The three vertical rules are the only measured values this site has — Munden's published k_s for plain single jersey in three relaxation states — and none of them is the right value for any structure drawn here. Where each line should be read is the whole of what is missing.

After the loom

The constants do not compose

The yarn in a knitted structure is a sum of what each element takes, exactly, over structures nobody has measured. The size the structure relaxes to is not a sum of anything — and the whole gap between the two is one number per structure, which would cost forty-five fabrics each to obtain.

What a shed costs, in newtons. The tension the shed puts into one end at each shaft of a 24-shaft harness, for a 25 tex cotton yarn. The strain is set by the loom's geometry alone; the tension is that strain times the yarn's modulus. The front shaft holds 0.52 N and the back 1.95 N, which is 14 and 52 per cent of the yarn's breaking load. What the chart cannot show is the rest of the warp tension, which the let-off adds on top of all of these and which no geometry decides.

Mechanics and drape

What the shed costs, in newtons

The shed's strain has been computed here and could not be priced: a strain is a length over a length and says nothing about how hard a thread is being pulled. A modulus turns it into a tension — and the back shaft of a twenty-four-shaft harness turns out to hold its ends at half their breaking load, all day, from the geometry alone.

Compound and figured cloths

Where a stitch can hide

One reversed intersection turns two cloths into one, and half the intersections in the repeat would do it. Almost none of them may be used — a plain-faced double cloth has nowhere at all to put a stitch, a five-end satin has fifteen places or none depending on which rule is asked, and two satins of the same order differ by a factor of two.

$A spread shading on 8 ends. The 7 tone steps of a spread shading on an 8-end repeat, each built by adding one more coset of the 8-end satin with move 3. Every coset has exactly one mark in every end and every pick, so the fraction of warp on the face is k over 8 exactly at every step, with no averaging in it. The numbers beneath are the longest float, and they run 7, 3, 3, 1, 3, 3, 7 — so the lustre scale is not the tone scale. The midtone is a plain weave and the extremes are satins, which is why a spread shading is at its most matt exactly in the middle. What the drafts cannot show is the surface: a long float stands proud of a short one, so an evenly toned shading is not an even surface either.$ Pattern and colour

A shading changes two things at once

The tone steps of a shaded damask are exactly even — each adds one satin coset, so the fraction of warp on the face is k over n with no averaging in it. The lustre steps are not even at all: on eight ends the longest float runs 7, 3, 3, 1, 3, 3, 7, so a series that grades smoothly in tone is at its most matt exactly in the middle.

The harness a strain budget buys. How many shafts stay inside a 1.0 per cent warp-strain budget, against the clear shed opening the loom needs at the reed: 12 mm gives 43, 16 mm gives 36, 20 mm gives 28, 24 mm gives 21, 30 mm gives 13, 36 mm gives 7, 44 mm gives 1. The shed's tangent enters the strain squared, so the opening is much the strongest thing a loom builder controls.

Compound and figured cloths

The harness has a depth

Why does a dobby carry sixteen or twenty-four shafts rather than two hundred? The usual answers are about the mechanism — how many jacks a box can drive, how many hooks a dobby has — and they are real limits that are not the binding one. A stated tolerance on warp strain is a stated distance from the fell, and a stated distance is a whole number of shafts. One per cent buys thirteen.

The load–extension curve, computed from a stiffness. The tension in one end of a sheeting against how far the cloth has been extended, computed as the slope of its bending energy along its own constant-thread-length locus. The curve passes through zero at the state of least energy, which is where an unloaded cloth sits, and rises either side of it. What the curve cannot show is what happens after the crimp runs out: past the end of the locus the load is carried by stretching yarn rather than by straightening it, and that is a modulus three orders of magnitude higher and a different figure.

Mechanics and drape

The locus gets a force

This site has drawn the set of states a cloth can reach without stretching any yarn, and has never been able to say which of them it is in or what it would cost to move. Both questions are one derivative of a bending energy — and the answer explains the flat start every fabric's load–extension curve has, which is not slack yarn but a symmetry.

The interchange budget against cover, at three yarn counts. The extension a plain cloth can reach with no thread stretching, plotted against its warp cover factor, for 10, 20, 40 tex yarn. The three curves coincide, because every length in Peirce's geometry is a multiple of the yarn diameter and a spacing measured in diameters is a cover factor — so the count divides out exactly and the maximum is at 0.407782 for all of them, at a budget of 7.2857%. That the curve has a maximum at all is the finding: crimp is what a cloth spends, so more of it should be better, and past this cover the weft has nowhere to put what the warp gives up. What the plot cannot show is the balance, which moves the height of the maximum a long way and its position hardly at all.

What cloth is

The most a cloth can give back

The extension a cloth can find without stretching a thread comes from crimp, so setting a cloth closer ought to give it more. It does, up to a point, and then takes it away again — and the point is a cover factor of 0.4078 at a budget of 7.2857 per cent, identical to six figures for every yarn count from five tex to a hundred.

A front on a thread with 8 per cent crimp. Three rows at one scale. The top row is the warp end laid out straight, with the wetted front marked at four equal quarters of its own length — which is where Washburn's law puts it at four times whose square roots are evenly spaced. The middle row is the same thread crimped at 8 per cent, so it covers 92.6 per cent of the paper the straight one did. The bottom row is the cloth, and the four fronts on it are the four above pulled back by 1.08. A coefficient is a length squared over a time, so it comes down by 1.1664 — exactly (1 + c)², with no property of the liquid or the fibre in it. The thread's thickness is not drawn and neither is the liquid: a meniscus in a 2.33 µm pore is finer than any line on this canvas.

Cloth doing a job

Wicking is slower along a crimped thread

A front travelling up a warp end travels the thread's path, which is longer than the cloth by exactly the crimp. So the wicking coefficient measured on the fabric is the yarn's own divided by (1 + c)² — 14.3 per cent lost at eight per cent crimp, whatever the liquid.

Selectable against reachable. How far up the take-up gear's catalogue a beat-up force reaches, on a sheeting. The catalogue holds 4,825 distinct pick densities between 8 and 40 per centimetre, which is the previous rung's count of what the machine can select. At 200 N per metre only 548 of them can be woven; at 2,000 it is 4,256. The fineness of the choice is untouched by the ceiling and the top of the range is cut off entirely, so the weft direction's advantage is resolution rather than reach. What the chart cannot show is the loom's own force, which depends on the beat-up mechanism and is not a property of the cloth.

Setting and geometry

A pick density is a force budget

The take-up ladder counted what a change-wheel take-up can select: 4,825 distinct pick densities between eight and forty threads per centimetre, against forty-nine warp setts a reed catalogue offers over the same range. That count assumed every setting is available. A beat-up force says otherwise, and cuts the top off the range without touching the fineness of the choice.

Weaves

The three basic weaves do not generate the rest

Every weaving manual opens with the same sentence: there are three basic weaves, and everything else is derived from them. The complete catalogue of the smallest interesting repeat is in hand, so the claim can be checked instead of repeated. Starting from plain weave, every twill and every satin, and applying every derivation the manuals name, reaches nine of the 426 cloths that exist there.

Two layers of one cloth, against how they happen to lie. Two identical muslins laid over one another and slid across each other by one thread spacing. In register the pair passes 37.9 per cent — as much as one cloth, because every hole is over a hole — and it falls linearly to 12.5 per cent before rising again at the next thread. The rule everybody uses is that two layers pass the product of their open areas, which is 14.3 per cent. That number is the average of this curve over all offsets, exactly — an identity, not a fit — and it is the answer at two points on it and nowhere else. Nothing about a real pair of layers is at its average, and the openness varying from place to place across a folded cloth is what a moiré is.

Pattern and colour

Two layers are the product on average and nowhere

Everybody knows what two layers of a cloth pass: the product of their open areas. That figure is exactly right — it is the mean of the true answer over every way the two layers can lie — and it is the answer at two registrations out of a continuum. In register a doubled cloth is as open as a single one; half a thread out it can be shut completely. The variation across a folded curtain is what a moiré is.

What combination adds to the manuals' reach. The three counts. The manuals' own operations on their own seeds reach 9 of the 426 four-by-four cloths. Admitting stripes, checks and figures of any two seeds — 17,787 constructions, of which 177 repeat inside four ends and four picks — takes it to 28, a gain of 19. That is a threefold rise and it leaves 398 cloths unreached, which is 93.4 per cent of the catalogue. What the chart cannot show is combinations of combinations, which are excluded on purpose: the closure's seeds have to be what a chapter actually teaches or the count measures something else.

Weaves

What combining two weaves reaches

The account before it found that the manuals' own operations on their own basic weaves reach nine of the 426 four-by-four cloths, and recorded one exclusion honestly: combination — striping, checking and figuring — was left out, because a combination of two four-end weaves is eight ends wide and so is not a four-by-four cloth at all. Admitting it triples the reach and leaves ninety-three per cent of the catalogue outside.

The crimp ratios a dense shirting can have. Poplins in 15 tex warp and 20 tex weft at 22 picks, from 32 ends per centimetre to 52. Each bar is the interval of warp-to-weft crimp ratios the construction admits at all: the warp must supply at least the thickness the weft cannot reach, and at most what it can reach itself. The rule at one is this site's standing default. It sits inside the interval up to 44.18 ends per centimetre and outside it beyond — so for a dense shirting an equal division of the crimp is not merely the wrong assumption but a geometric impossibility. The 44-end poplin the standard cloth table called impossible for a long time sits a fifth of an end below that limit, which is why the solver failed on it: its feasible interval was real and narrow, and a bisection on the whole range walked away from it.

Setting and geometry

The cloth that was called impossible

This site's own table of fabrics carries a note saying a real 44-end poplin has no solution in its geometry at all, and that the poplin row was therefore set at 32 ends. The cloth solves. What had no solution was the search — a bisection that treated a state it could not reach as evidence of having gone too far, and walked away from the answer every time.

The evenness a cotton yarn cannot be better than. Lay staple fibres down at random and count how many cross a plane: the count is Poisson, its variance is its mean, and the coefficient of variation of the mass per unit length is therefore 1/√n with no material and no machine in it. The lower curve is that floor. At 20 tex there are 118 fibres in the section and the floor is 9.93 per cent; at 5 tex there are 29 and it is 19.86. The upper curve is what a yarn spun at an index of irregularity of 1.35 actually measures, which is the floor times a constant — so the whole shape belongs to the counting and none of it to the spinning. The exponent is exactly −½ and is asserted as such rather than fitted to the curve.

What cloth is

The spread was never free

This collection has taken a yarn's irregularity off a delivery note and used it as an input. It is not an input. Counting the fibres in a cross-section puts a floor under it that no spinner can beat, and the floor is one over the square root of the count.

Weaves

A float presses on nothing

The rung below expected the float correction to change how a satin's hold compares with a plain weave's, by something like the ratio of their interlacing rates. It does not change the comparison at all — the interlacing rate leaves the answer outside the logarithm and divides straight out of any ratio. What it changes is the absolute answer, by a factor of four, for every weave alike.

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

Knits and other structures

A knit has no hole to lose

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

Pattern and colour

A random error hides and a periodic one shows

A warp whose threads vary by fifteen per cent looks perfectly even. One dent of the reed a tenth of a millimetre wide makes a streak that gets the piece rejected. The same amount of error, arranged two ways — and the ratio between them is √(2n/π), with nothing fitted in it.

The two halves of a beat-up force. The force the reed must apply per metre, for a sheeting, against the number of picks that are still sliding against the warp. The elastic half — the warp tension times the crimp's elasticity with respect to the pick spacing — is 573 N/m and does not depend on the zone at all; that cancellation is exact and is the result the rung below established. The frictional half is 1133 N/m per sliding pick and is nothing but zone. Against a reported 400–1500 N/m, that leaves room for at most 0.82 picks sliding — so the fell region a weaver can see, ten to fifty picks deep, is not the same quantity as the picks that are still moving. What the rows cannot show is that this is a static friction throughout, and a beat-up is a blow.

Setting and geometry

The half of the beat-up that is all zone

The elastic half of the beat-up force is exact and the length of the beat-up zone cancels out of it, which is this site's own result and disagrees with every practical account of weaving. The frictional half is nothing but zone — and requiring the total to match the force a loom is actually built to apply puts the number of picks still sliding at about one.

$The shear a dome demands, against how far round it the cloth reaches. A flat sheet of inextensible threads takes a double curvature only by shearing, and the shear it needs depends on how far round the dome it has to reach rather than on how big the dome is — a knee, a shoulder and a beach ball demand exactly the same at the same fraction of their own radius. The horizontal line is the locking angle for a sheeting, where the threads are touching side by side and the mechanism has nowhere left to go. Reaching one radius round takes the cloth to 85% of that, and reaching 1.2 radii passes it. What the plot cannot show is the frictional part: a shear well inside the locking angle is still a shear at crossings friction is holding, so a knee that is domed a thousand times keeps a little of each one.$ Cloth doing a job

A knee is a dome imposed a thousand times

A flat sheet of inextensible threads takes a double curvature only by shearing, and how much shear it needs depends on how far round the dome it has to reach — not on how big the dome is. So a knee and a beach ball demand the same, and a trouser knee covered to its own equator is at 85 per cent of the angle at which the threads touch side by side.

Cover, dry and wetted. Warp cover for every cloth in the table when its threads swell by 20% and its spacings are held, with the dry value and the sett at which the swollen threads touch beside each bar. Cover is a diameter over a spacing and only the diameter moves, so every cover is multiplied by exactly 1.20 and every jamming sett divided by it — an identity rather than a result, and the one statement in this ladder a reader can check by hand. No cloth here reaches a cover of one, so none of them jams laterally on wetting; the closest is the sheeting at 0.628. What the bars cannot show is the through-thickness condition, which the sheeting fails at a swelling of half this one.

Setting and geometry

A wet cloth is set closer than it was woven

Cover is a diameter over a spacing. Wetting moves the diameter and does not move the spacing, so every cover factor on this site is multiplied by exactly the swelling ratio and every jamming sett divided by it — which is an identity, and the one statement in this ladder a reader can check by hand.

The 6-end satin. The 6-end satin on point paper, drawn over 2 repeats with its marks joined in reading order inside the first. The joining segments are not parallel and not equal, because there is no number of picks the mark advances by at every end. That is what irregular means, and it is not visible in the squares alone. At this order there is one distinct satin and none is regular. The longest float is 1 in the warp and 5 in the weft, and the cloth is one cloth. What the drawing cannot show is that this is the only one: that is a statement about 36 arrangements and is made by the enumeration.

Weaves

The six-end satin that does exist

There is no six-end satin, and this collection proved it in its founding essays. The proof is about satins with a move number. Drop that word — keep one mark per end and no two marks touching — and six ends has exactly one satin, unique up to where the repeat is started, and four ends still has none at all.

The ratchet a wool fibre is. A fibre with its scales, and the two strokes of one cycle of agitation. The push is the same in both directions; the distance is not, because the scales resist tip-first motion more than root-first. Every cycle therefore nets a displacement in one direction, and no amount of further agitation undoes it.

After the loom

Why felting needs water

Wool felts in a wash and not in a drawer, and the usual explanation is that water lubricates the scales. It does the opposite of that. Water lowers one of wool's two friction coefficients and raises the other, so it widens the gap the ratchet rectifies — and what follows is a saturating function of the ratio, not of either coefficient.

The poplin's two budgets as it is held stretched. What is left of a poplin's interchange in each direction as it is held at more and more warp strain. The warp's budget falls to nothing at 3.93%, which is the point of the curve; the weft's rises, because the crimp the warp gives up is crimp the weft takes on. There is one locus and one position on it, so the two are not two quantities that happen to be related — they are the two distances from one point to the two ends of one curve. The consequence is that a pre-tensioned cloth has almost no warp recovery left and more weft recovery than it started with. What the plot cannot show is that the exchange rate between them is not constant: the curve is not a straight line, and its slope is the Poisson ratio this site computes elsewhere.

Mechanics and drape

A cloth has one budget for two directions

A woven cloth looks as though it carries two independent reserves of free extension, one along the warp and one across the weft. It carries one. There is a single curve of states and a single position on it, so every hundredth spent one way is refunded the other — which means a pre-tensioned cloth has not used its recovery up, it has moved it.

Wash-by-wash shrinkage, reported and modelled. The shrinkage an unfinished cotton cloth shows in each of five laundering cycles, beside what a model with no rate in it predicts. The model says a wash lets every crossing whose frictional barrier is below the cloth's current excess slip to the edge of its own band, and that is a distribution rather than a rate. Two numbers are fitted — the excess the cloth came off the loom with, 7.13%, and the spread of the barriers, 37.2× — against the first two washes. Washes three, four and five are predictions with nothing left to adjust and come out at 0.506%, 0.284%, 0.179% against reported 0.50%, 0.30%, 0.20%. What the bars cannot show is the finding underneath: the reported yarn-on-yarn friction range gives a spread of only 1.34×, which would have the tail over by the third wash.

After the loom

A cloth shrinks most the first time

A laundering test reports five numbers and they fall away like a geometric series. Nothing in a wash is slow — a cloth is agitated tens of thousands of times in half an hour — so a second wash that shrinks it again is direct evidence that its frictional barriers are spread, and the ratio between successive washes measures how far.

The stripe a knitting machine chooses. A circular machine takes its yarn from a fixed number of feeders arranged round the cylinder, and feeder k lays every F-th course. So any difference between packages — a shade, a count, an evenness — is reproduced in the fabric with a period of exactly F courses, and at 20 courses per centimetre that is a band every 48.0 mm on a 96-feeder machine. The machine chooses the period, not the yarn. The spacing is proportional to the feeder count and inversely proportional to the course density, as the turn of the cylinder requires whatever the bars show, and a one-feeder machine produces no band at all from the same packages.

Knits and other structures

Why a knit shows a thick place

A woven cloth has hundreds of separate warp ends and averages a yarn's faults among them. A knit has one thread and a machine that repeats — so a difference between two packages becomes a stripe, and the machine chooses its period.

The bearing curves of 4 weaves in one cloth. How much of the plan is within a given depth of the highest point, for plain, 2/2 twill, satin 5, satin 8 — all in sheeting, all at the same sett, the same counts and the same thickness. They differ only in their drafts. At a hundredth of the cloth's thickness the last of them is touching 9 times the area of the first, and the gap widens as the depth shrinks, because the curves do not merely differ by a factor — they have different exponents. A crown that is a line opens as the square root of the depth and a crown that is a point opens in proportion to it.

What cloth is

The curve that says what a cloth touches with

Take a cloth's surface and ask what fraction of the plan lies within a given depth of its highest point. The answer is one curve, it answers every question of the form what does this touch, and its behaviour at the top is decided by a single bit of the draft — whether the longest float is one crossing or more than one.

$How much of a 20 tex yarn's unevenness survives being woven. Weaving averages, and the averaging is a square root. A patch of cloth 3 mm across contains 7.2 warp threads and as many picks, each contributing its own mass independently, so the patch's coefficient of variation is 3.53% against the yarn's 13.4% — a reduction of 3.8-fold. The rule at the top is the yarn's own figure. The curve steepens past the staple length, where a patch starts to contain independent samples along each thread as well as across them, and the second regime is the one a large area of cloth is judged in.$ Pattern and colour

A cloth cannot be more even than its yarn

It can be very much more even than its yarn, and by exactly the square root of the threads in view. Which raises a question the fineness argument left open — and the answer is that at a fixed cloth weight the count cancels out entirely.

How far each cloth's sett moves between the loom and the finished state. A cloth on the loom is held: the warp is under beam tension and the picks are driven up at whatever density the take-up says. Let it go and it relaxes to the least-energy state of its own locus, which is a state at a different sett. The bars are how far each sett moves, and they always move in opposite directions because there is one locus: warp ends per centimetre fall as the cloth widens and picks per centimetre rise as it shortens. Seven of the eight move a little over one per cent; the poplin, whose two counts and two setts are the only unbalanced pair in the table, moves six and ten. What the bars cannot show is what a designer does with them, which is that the two numbers a specification quotes are not two free numbers — the finished construction is a point on a one-dimensional curve.

Setting and geometry

The construction a loom must be set to

A specification quotes ends and picks per centimetre in the finished cloth, and a loom is set to neither of them. The cloth relaxes to the least-energy state of its own locus, which is a state at a different sett — and because the locus is one curve, the two numbers a specification quotes are not two free numbers.

The crown line of every four-by-four draft there is. All 22,874 four-by-four drafts in which every end and every pick interlaces at least once, at sheeting's construction, counted by how much horizontal crown line each carries per square millimetre. The bar at zero holds 2 of them — the two plain weaves, and nothing else in the catalogue. Every other draft has a float somewhere, and a float is a plateau, and a plateau is a line of constant height. So the whole catalogue divides into two drafts that touch at points and 22,872 that touch along lines, with no intermediate case, because a float is either present or it is not.

What cloth is

Two drafts of twenty-two thousand

Every four-by-four draft there is, measured by how much horizontal crown line its surface carries. Two of them carry none — the plain weave and its translation, and nothing else in the catalogue — and the quantity turns out to be smallest for the most balanced cloths and largest for the most one-sided, which is the opposite of what a float count suggests.

$What a tensioned sheeting has left of its load. A sheeting pulled to a strain, clamped at that length and left. Its length does not change, and its load does: crossings rearrange locally until the load has fallen to what friction alone can hold, which is 0.0756 N per end and is the same number whatever the cloth was pulled to. So the fraction retained is that floor over the load applied, and it falls — a cloth tensioned to 4.94 per cent keeps 29 per cent of what it was given. Below the resting band's own edge nothing is lost at all, because the cloth was never outside what friction could hold. What the plot cannot show is time: nothing here says how long the rearrangement takes, only where it stops.$ Mechanics and drape

A tensioned cloth loses its load

Clamp a fabric at a fixed length and its tension falls overnight. Nothing crept and nothing flowed: the crossings rearranged locally until the load had dropped to what friction alone can hold, and that level is the same number whatever the cloth was pulled to — so the harder it was tensioned, the smaller the share it keeps.

The 14 drafts whose holes are all one size. Every four-by-four draft in which each end and each pick interlaces — 22874 of them — built and asked whether all sixteen of its holes pass the same thing. At a muslin's construction 14 of them do. Set the same yarn square and 170 do; turn the cloth over and it is 14 again. Only 2 drafts are in all three lists, and they are the plain weave and its complement, marked. The other 12 are uniform because this cloth's warp is set closer than its weft, so the gap across the ends is the smaller of the two and binds every hole whatever the picks are doing — a fact about the sett wearing a fact about the weave's clothes. Two of them carry floats of three, which is as long as this repeat allows.

Weaves

Only a plain weave has one size of hole

A weave's holes come in kinds, and the kinds are read off the matrix. Asking which weaves have only one kind looks like a question with an obvious answer and a one-line proof. Every draft at four by four was built and asked instead, and the count came back fourteen — of which twelve turn out to be telling the truth about the sett rather than about the weave.

The strain at a fold against the strain each fibre breaks at. For each fibre, the surface strain at the sharpest fold a 20 tex yarn of it can make — which is √(packing × fibre tex ÷ yarn tex), with no measurement of a crease in it — beside its own measured breaking extension. cotton and flax are strained past the low end of their breaking range, so some of their fibres break at the fold, and that is what a linen crease is. The rest survive, and among them viscose and wool return less than three fifths of what they were given, which is the other way a cloth creases. Both columns are needed: viscose survives with a factor of two to spare and is among the three worst by measurement. What the bars cannot show is wool, which the census puts in the wrong column because the recovery figures are the immediate ones and wool's is the most delayed of any fibre here.

After the loom

Which fibres crease, and why there are two answers

Run the sharpest fold a yarn can make against every fibre's breaking extension and its recovery, and the ranking that falls out is the trade's own — linen worst, cotton next, wool and the melt-spun filaments best. It falls out of two columns rather than one, because two fibres in the bottom three get there by different mechanisms, and a treatment that fixes one does nothing for the other.

How open a muslin is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this muslin it is 37.9 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 36.1° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 5.36 per cent open — 7.1 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none.

Setting and geometry

One minus the cover is a cloth with no thickness

The covering rule says a cloth's openness is one minus its cover factor, and this collection derived it and has used it ever since. It is the answer for a light directly behind the cloth. Move the light and a line of sight has to clear the hole at the top of the fabric and the same hole one thickness below, so the openness falls, and it reaches nothing at thirty-six degrees. Averaged over the whole sky a muslin is a seventh as open as the rule says.

Three kinds of surface, and only one of them starts open. The bearing curves of a 2/2 twill in sheeting, of a terry loop pile, and of a cut corduroy pile of 30 tex, over the first six per cent of a cloth's thickness. The woven curve opens as the square root of the depth and the loop pile's does the same, because a loop's top is a curved thread like any other. The cut pile does not open: it is a flat line at 5.2% of the plan, from a depth of nothing, because a blade severed every tuft in one pass and left every end in one plane. That is not a larger contact area, it is a different kind of contact area — one that does not vanish as the load goes to zero, which is a property no woven surface has.

Compound and figured cloths

A pile is the only surface with no crowns

Every other fabric touches on the tops of curved threads, so its contact vanishes as the load goes to zero and the pressure on what is touching rises without limit. A cut pile touches on flat ends left in one plane by a blade — so its contact is finite at no load at all, its pressure concentration is bounded, and it is the only fabric whose abrasion mass loss is an honest measure of its damage.

A knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings.

Knits and other structures

A loop is nine tenths free run

Half the yarn in a knitted stitch is slack — the straight line between two interlacings is a little over half the thread available to span it. That is two orders of magnitude more room than a woven thread has, and it is why a knitted loop is a shape that can be solved rather than a shape that has to be constructed.

Setting and geometry

Half the air goes through a tenth of the holes

A permeability computed from the average hole says nothing about which holes the air uses. Once the threads have a spread, the answer is: not many of them — and in a close cloth, half the flow leaves through less than a tenth of the openings.

A plain weave with one end missing. A plain weave on the left and the same cloth with one end broken and not pieced up on the right, drawn over 2 repeats so that the fault can be seen as the cloth has it: absent from every repeat, for the whole length of the piece. The picks that were held by the missing end are now held by whatever is on either side of it, so the longest float across the ends goes from 1 to 3 end widths — measured in the width the cloth had rather than in the narrower repeat, because the place the end used to occupy is still there. Every pick still changes side somewhere, so the cloth holds together — which is what happens in 55% of all the ways a four-by-four draft can lose an end.

Weaves

What a missing end does to the weave

Every four-by-four draft there is, with each of its four ends taken away in turn: ninety-one thousand cloths, and not one of them falls into layers. What happens instead is worse, and the criterion has never had to report it before.

Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left.

Knits and other structures

The relaxed knit is not at a minimum

Differentiate a loop's bending energy along the fabric instead of across it and the answer should be zero, because a relaxed fabric is one nothing is pulling. It is not zero. It is tens of newtons a metre, downhill in both directions at once — and the three relaxation states everybody measures run the wrong way up the slope.

$A bundle is weaker than the threads it is made of. Threads pulled in parallel do not break together. The weakest goes first and hands its load to the rest, which are now carrying more than they were, so the bundle's peak load is reached before every thread is at its own strength. With a load per surviving thread of x carried by the fraction that has not yet broken, the bundle's strength per thread is the largest value of x(1 − F(x)) — Daniels' maximum, which for a lognormal at CV 15% is 0.7380 of the mean thread, reached with 93% of the threads still unbroken. A simulated bundle that knows none of that arithmetic sits above the limit at every finite size — its strength is a maximum over the sample it happens to have drawn — and closes on it as the bundle grows: 0.753, 0.744, 0.741 at the last three sizes. The scatter falls the other way, from 14.5% at one thread to 0.7% at 1600.$ Mechanics and drape

A bundle is weaker than its threads

Threads pulled together do not break together. The weakest goes first and hands its load to the rest, so a bundle carries its maximum well before every thread is at its own limit — and the shortfall is a quarter, decided by the spread and by nothing else.

One pick of a 2/2 twill made in the wrong shed. Pick 2 of a 2/2 twill laid in the shed belonging to another pick. The thread is there, it is beaten up in its place, and it is simply not the pick the design asked for — so the fault is a bar the whole width of the cloth and one pick deep. This one leaves the cloth sound, with its longest float at 3. Across every four-by-four draft and every possible wrong shed — 1,372,440 substitutions — 63.2% leave a cloth that still hangs together, 36.6% leave a thread loose, and 0.25% split the cloth. The same wrong shed is harmless in one draft and destroys another, so nothing about the size of the mistake predicts the size of the fault.

Weaves

A mispick is one row in the wrong place

Every four-by-four draft, with each of its picks replaced by every shed the loom could have made instead: a million and a third substitutions. Two thirds leave a cloth that still hangs together, a third leave a thread held by nothing, and the same wrong shed is harmless in one draft and fatal in another.

Cloth doing a job

A missing end is a fault the length of the piece

A broken end and a mispick are the same size of accident — one thread — and they condemn areas that differ by a factor of thirty. The ratio is the aspect ratio of the piece and nothing else, which makes it a fact about how cloth is made rather than about what went wrong.

The fold the trade actually makes. 2 singles of 20 tex cotton at 800 turns a metre, folded at 566 — a ratio of 0.707. That is 1/√2, the ratio at which the fold's surface helix angle equals its singles' — 22.8° — because a fold of 2 singles is √2 times the diameter. The trade's own bracket for 2 folds is 0.6 to 0.75, and it contains this number.

Compound and figured cloths

A leno twists what a weave only crosses

Every woven cloth's threads have a linking number of zero, and that is why an open cloth slips. A leno is the one woven structure whose warp ends wind about one another, so it is the one whose threads are linked — and it is famously the structure that holds at setts where nothing else does.

Knits and other structures

What a knit gives when it is pulled

How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight.

The float a point fault gives a weave, over every draft there is. Every four-by-four draft that describes one cloth — 22,730 of them — laid out as 8 ends by 8 picks, with each intersection of its repeat reversed in turn and the longest float that produces kept. There are two answers and no others. 90 drafts hold the fault to a float of 3; 22,640 — 99.6% of every weave there is — hand it a float of 7. The mechanism behind the second number is that a reversal at a binding point joins the two floats on either side of it, so a weave with runs separated by single binding intersections gives a single mistake the sum of two of its own floats. The 90 that escape are the drafts whose floats are short and evenly spaced — the plain weave and the ribs — which is not the class of weave the trade recommends for hiding a fault.

Weaves

Which weave hides a fault

The trade says a busy weave hides a mistake. Every four-by-four draft there is was laid out as cloth and given every wrong lift it could have, and ninety-nine in a hundred hand that single mistake a float of seven — because a reversal does not lengthen a float, it joins two.

$20 tex, counted. The cross-section of a 20 tex cotton yarn, with every fibre in it drawn. The count is a division and nothing else: a 20 tex yarn spun from 0.17 tex fibre has 117.6 fibres crossing any plane through it, and the yarn is 14.0 fibre diameters across because n fibres packed at 0.6 fill a circle √(n/φ) times as wide. The arrangement is drawn on a lattice and is not claimed: real fibres are not on one, they migrate between the core and the surface as they run, and everything this collection says about a yarn's strength turns on their doing so.$ Setting and geometry

How many fibres make a thread

Every number in this collection began with a diameter, and a diameter is not a measurement — it is a count of fibres, divided. Once the division is written down, three quantities that had nothing to do with each other turn out to be the same number.

One hairiness reading does not fix the other. Every yarn on this curve has exactly the same total protruding fibre length — the quantity an integrating hairiness meter reports — and they differ in how that length is distributed. The count of hairs at least three millimetres long runs from 170 to 4354 per hundred metres, a factor of 26, across decay lengths real yarns actually have. The two instruments read two functionals of one population: the first moment N₀λ and the tail N₀e^(−3/λ). A correlation between them can exist only if λ is fixed across the yarns being compared, and λ is a fibre property, so it is not. That is the whole of why the trade's two hairiness numbers have never agreed, and it is arithmetic rather than instrumentation. What the figure cannot show is which of the two predicts anything: the tail does, because pilling, prickle and a printed edge all need reach.

What cloth is

Two hairiness meters read two moments

The trade has two instruments for yarn hairiness and thirty years of failing to predict either from the other. They are not measuring the same thing badly. One reports the first moment of a distribution and the other reports a tail probability, and two functionals of one curve are related only through a parameter neither of them reports.

Knits and other structures

A jersey gets taller before it gets shorter

Pull a knit along its courses and the first thing it does is grow along its wales — by two per cent, over the first eighty per cent of extension, before it turns round and contracts. The transverse response changes sign, and there is no material in the explanation at all.

Compound and figured cloths

A braid is a third way to hold threads

Weaving holds by friction and knitting holds by linking. A braid does neither: its strands travel across the structure and back, so no pair of them is linked and no pair of them returns to where it started — and it holds without a reed, a beat-up or a sett.

The cyclic decomposition of a 4-end repeat. A 4-end repeat split into 4 parts, each with exactly one warp mark in every end and every pick, drawn above with each intersection numbered by the part it belongs to. That object is a Latin square, and it is what a shading actually requires: a tone step is a union of parts, so it has exactly k marks in every end and pick and its tone is k over 4 exactly. This is the cyclic square, which is what a satin's cosets write — and at four and six ends there is no satin, so the same square has to be reached through a twill instead. The drafts below are the tone steps in the best order this square admits, whose longest floats run 3, 1, 3. What the drawing cannot show is that the numbering is arbitrary: relabelling the parts gives the same square and a different chain, which is exactly the freedom the order is chosen out of.

Pattern and colour

A tone step does not need a satin

Every account of shading builds its tone steps out of satin cosets, and at four ends and at six there is no satin to build them from. The construction was never about satins: what a tone step actually needs is that every end and every pick carry the same number of marks, which makes a chain of them a Latin square. A four-end repeat has twenty-four of those and a six-end repeat 1,128,960.

$Where a 20 tex yarn breaks, against how much was clamped. A tensile test clamps a length of yarn and pulls until the thinnest section between the clamps gives. So a yarn's strength is a minimum, and a minimum depends on how many independent tries the sample contains. The tries are not sections — a plane can be taken anywhere — but staple lengths, because two planes closer together than one fibre share most of their fibres. At 28 mm staple a 100 mm specimen holds 3.6 independent tries and a 500 mm one holds 17.9, and the longer test reads 11% lower. The spread is not fitted either: it is the evenness floor at 118 fibres times an index of 1.35, which is 13.4%.$ Mechanics and drape

A yarn breaks at its thinnest place

A tensile test does not measure a yarn. It measures the worst section between the clamps — so evenness and strength are one measurement taken twice, and the number of independent tries in a specimen is set by the length of a fibre.

Setting and geometry

A finer yarn is a worse yarn

Fineness is the thing a yarn is priced for, and it is bought with irregularity at a fixed exchange rate. Once the spread is a function of the count, every correction this collection computes becomes a function of the count too — and the cheapest yarn on the shelf is the one the arithmetic describes best.

Knits and other structures

How far a knit could go if its yarn were the limit

The yarn in a stitch allows three hundred and twenty per cent course-wise extension before the straight line between two interlacings reaches the thread spanning it. A jersey jams at about a hundred. The factor of three is the finding: what stops a knit stretching is not the loop running out of yarn.

One tone of an 8-end cell, arranged three ways. The same 32 marks in the same 8 × 8 cell, placed three ways, with the number of marks in each end printed beneath it. On the left the clustered dot a halftone screen makes: 4 of its threads carry every mark or none, so they never leave a face, and the criterion reports 16 separable layers rather than one cloth. In the middle a cloth built from the tone step by moving 2 marks sideways within their own picks — every pick still carries 4, the ends run 3 to 5, and that difference is a warp stripe of 25.0% contrast the design did not draw. On the right the tone step: every end and every pick at exactly 4. What the drawing cannot show is how visible the middle one's stripe is, which depends on the sett and on the viewing distance and is not computed here.

Pattern and colour

A weave is a halftone screen with n greys

An eight-by-eight cell of dots gives a printer sixty-five levels of grey. The same cell in cloth gives seven. Sixteen of the missing fifty-eight go to the requirement that every thread reach both faces and forty-two go to the requirement that every thread carry the same number of marks — so evenness, not interlacing, is what a weave pays for its tone scale.

A two-layer interchange, 1 block by 2. A two-layer cloth whose layers change places from block to block, drawn as the draft and as the section a weaver would draw. Each block is one repeat of the stack, 4 picks by 4 ends, and the design has 2 boundaries in it. Every strand in the draft is coloured by the cloth this site's criterion puts it in, and there is one colour, because there is one cloth — with no intersection reversed anywhere in the repeat. A stitch joins two layers at a point; an interchange joins them along a line, and the line is the design's own block boundary rather than anything added to it. The section below is schematic: it draws each ply as a line rather than as its threads, because what has to be seen is that the two lines cross, and a section at thread level over 2 blocks is a picture nobody can count.

Compound and figured cloths

An interchange joins what a stitch would have had to

The rung below spent a whole essay on where a stitch may be put in a double cloth, and found face weaves with nowhere to put one at all. A design in which the two layers change places from block to block needs no stitch anywhere: the boundary is the join. Two blocks side by side make one cloth with not a single intersection reversed, and the same two blocks with no boundary between them make two.

$Two cloths touch on a fraction of what one cloth does. A 2/2 twill in sheeting pressed against a flat plate, and the same cloth pressed against another piece of itself. At an approach of 24.9 µm the single surface is touching 15.87% of the plan and the pair 3.130% — a factor of 5. The reason is that a gap between two rough surfaces is the sum of two depths, so both surfaces have to be near their own maxima at the same place, and the chance of that is the product of two small numbers. The pair's curve is the convolution of the two height distributions, computed exactly on histograms rather than fitted to a Gaussian — because a woven surface is bimodal and nothing about it is Gaussian.$ After the loom

Friction is two surfaces, not one

A gap between two rough bodies is the sum of two depths, so two cloths face to face touch on the convolution of their height distributions rather than on either of them. At the approach a light touch produces that is twenty times less contact than the same cloth against a plate — which is why a fabric's friction against a plate and against another fabric are two different measurements.

Why a cuff is ribbed. The force a knit pulls back with, over the range a cuff is used across. It rises the whole way — 0.79 N per metre at 23% to 3.38 at 104% — and it is small throughout, which is the combination a cuff needs and almost nothing else supplies. A rib gets its extension by geometry, folding alternate wales to opposite faces so that its relaxed width is about half its opened one, and it gets its recovery from the loop reconfiguring. Neither is the yarn stretching, which is why a cuff made of a fibre with no elastic recovery at all still works.

Knits and other structures

A rib pulls back on a force the loop supplies

A cuff has to give a great deal at almost no load and come back reliably, and no ordinary material does both. A rib gets its extension from folding, which is geometry, and its return from the loop reconfiguring, which is now a computable force — under four newtons a metre over the whole range a cuff works across.

A colour order against a 2/2 twill. The visible face of a 2/2 twill under 2 colour orders, drawn at the repeat the divisor arithmetic allows and outlined at the repeat the surface has. A filled cell is a dark thread on the face, which is the warp's colour where the warp is up and the weft's where it is not — so none of these patterns is in the draft, and the draft is the same in all of them. The colour period and the weave repeat beat exactly as a reed's grouping beats against a weave: the surface repeats on the least common multiple of the two, which here is 8×8 and 4×4. What the panels cannot show is colour: the two threads are drawn as filled and empty, and two colours of similar value make a pattern far weaker than this.

Pattern and colour

A colour order beats the weave it is threaded on

The reed's grouping beats against the weave repeat and the arithmetic is a least common multiple. A colour order is a second grouping of the same warp and the arithmetic is identical — but where the reed's beat is a fault to be dented out of a cloth, the colour order's beat is the pattern the cloth is sold for. Across 472 colour orders on four weaves the divisor bound is the surface's exact repeat in 470 or more, and the handful that beat it have no pattern left at all.

Folding, and what it takes out of the singles. Two 20 tex singles spun at 800 turns per metre and folded the other way at 540 — a ratio of 0.675, which is the trade's own and is a measurement rather than a derivation. A single held at its ends and wound round its neighbour turns about its own axis once for every turn of the fold, so it is left with 260 turns per metre of its own: its surface fibres lie at 7.8° to its axis rather than the 22.8° they were spun at. Folding untwists. The short strokes are drawn at that residual angle; the two long curves are centre lines and are not the yarn — each strand is itself a bundle of 118 fibres, and the residual angle is what holds them.

Setting and geometry

Folding is untwisting

Wind two singles round each other and each one turns about its own axis once per turn of the fold. So a folded yarn's singles are not the singles that went into it, and at a folding ratio of exactly one half the two helices cancel.

All themes