Every essay — page 3
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Cloth doing a job
What happens when a fabric has to meet a specification rather than merely be described. A reinforcement that must hold fibre and still admit resin, a filter that must retain the soil and still pass water, a seam that must grip harder than the load without perforating the cloth, a pattern cut for a shape it does not yet have. Every requirement here is a pair of inequalities on quantities the earlier fields compute — and the useful answer is sometimes that no fabric satisfies both.
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.
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.
A drying cloth cannot lift what a sealed tube can
Every rise on this account is an equilibrium in a sealed tube, and a garment is neither sealed nor at equilibrium. Balancing the supply up a strip against the loss from its faces gives a quadratic whose width cancels exactly: in an ordinary room the fine system between the fibres stands at 500 millimetres instead of 6.37 metres, keeping 7.9 per cent of what a tube would give it, while the coarse system loses a third of a per cent. The forty-sixfold advantage becomes 3.7, and in a drying wind it reverses.
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 wick reaches its ceiling in the time its cloth takes to dry
A drying cloth lifts water to a steady height and no further, and the question left was how long it takes to get there. The answer has no permeability and no surface tension in it. Where gravity is small the front climbs as the ceiling times the root of one minus a decaying exponential, and the exponential's time is the cloth's own pore water divided by the rate its faces lose water — the time the room would take to dry it. In an ordinary room that is half an hour, and the front is nine tenths of the way up in fifty-three minutes.
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.
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 sweating cloth wicks as high as the room can dry it
Every wicking height so far has had its foot in unlimited water. Skin is not a reservoir: it supplies sweat at a rate, and a cloth fed at a rate stands where the supply equals what its faces lose — the supply over twice the evaporation, with no pore, fibre or thickness in it. The cloth decides only when it has had enough, and it says so all at once: the holes between its yarns stay empty until the fine pores are carrying ninety-nine per cent of what they can.
An air arch pays for its thrust out of its pressure
An inflated tube bent into an arch was expected to start with its inside wall pulled differently from its outside, and to wrinkle where that difference and the load's moment combined worst. It does not: a curved tube is pulled along its length at exactly the straight tube's pr/2, all the way round. What the arch spends its pressure on instead is the thing every arch exists to make — its own thrust — which takes a fifth of the wrinkling budget at the haunch of a shelter arch under snow and more than half on a tight one.
Wicking borrows drying area from the cloth that is not touching
A shirt is fed with sweat across the whole of the skin it touches, not at the foot of a strip. Through its own thickness the cloth could pass sweat a hundred thousand times faster than a body makes it, so a patch pressed flat never lacks capacity; what it lacks is drying area, because it can give water to the room only from its outer face. Pressed flat everywhere, it floods as soon as the sweat exceeds the room's evaporation — light work, in an ordinary room. With free cloth above the contact, wicking carries the surplus to faces that are not touching, and the sweat decides how much area it needs: three times the contact's at a run, whatever the contact's size.
A fibre end prickles only inside a window
A fibre end pressed against skin has two ways to fail to hurt. Too long, and the column lies over before it reaches the force a nerve answers to. Too short, and the tip cannot sink far enough into something as soft as skin to push that hard. The two limits run opposite ways with diameter and they cross, so a fibre below one diameter cannot prickle at any length at all — and that diameter has no protruding length in it. It also settles a sum left unweighed: a crop at 1.1 mm makes a wool cloth two and a half times worse, and one at 0.6 mm removes the prickle.
What cloth is
A fabric is a structure before it is a material. Interlacement, the draft as a matrix, and the question of whether a cloth holds together at all.
A fabric is a structure, not a material
Cotton is a material. Cloth is an arrangement, and almost everything a fabric does follows from how the threads are put together rather than from what they are made of.
The draft is a matrix
Point paper is not a diagram of cloth. It is a grid of yes-or-no decisions about which threads to lift, which makes a weave a binary matrix and almost every question about it arithmetic.
Does it hang together
A draft can interlace everywhere, have short floats, and describe two fabrics lying on each other rather than one. Nothing about the drawing says so, and the test that does is exact.
Braids and the third thread system
A braid is one set of strands interlacing with itself at an angle, so the question of whether it holds together is the same question a weave answers — and most crossing sequences answer it badly.
What the matrix cannot say
A weave is a binary matrix, and the whole of this collection rests on it. So it is worth setting down, in one place and precisely, what that encoding decides exactly, what it decides while appearing to decide something else, and the four quite different reasons a real fabric can fall outside it altogether.
Nonwovens, and what holds them together instead
A web of fibres laid down at random has no repeat, so the exact test for whether a fabric holds together has nothing to work on. What replaces it is a threshold, and the threshold is sharp.
Every cloth there is, at four by four
Sixty-five thousand matrices, twenty-two thousand weaves, and about a dozen with names. The complete census of the smallest interesting repeat is a map of a whole small world, and almost none of it has ever been woven.
What a repeat repeats
Point paper is ruled in squares, and the rectangle a draft is written on is a decision by whoever drew it. The unit a cloth actually has is smaller — two intersections for a plain weave, four for a 2/2 twill — and it is not a rectangle at all.
How many layers a draft can have
Of the 22,874 four-by-four drafts this site sweeps, 22,730 are one cloth and 144 are two. None is three, and none can be — a repeat of n ends holds at most n halved cloths, because every cloth needs two ends and two picks of its own before it interlaces at all.
Four ways to write a weave down
A weave's name in the trade is a pair of numbers — 2/2, 3/1, five-end satin — and the notation is so universal that it is easy to forget it is a notation. It has an image and the image is computable: of the 426 cloths at four by four, five have a name of that kind, and there are four names for the five.
Every crossing is a force
A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.
A thread is held one crossing at a time
Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.
A thread is gripped where it turns
The usual arithmetic for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.