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The thread: Exactly this many — page 3

Page 3 of 4 of the essays on this thread.

How much every four-by-four draft can shine. All 22,874 four-by-four drafts in which every end and every pick interlaces, at sheeting's construction and a tolerance of 2°, counted by specular area. The range runs from 0.02% to 0.93%, a factor of 60.0, and the distribution is not smooth — it clusters, because the quantity behind it is a count of whole crossings and takes only certain values. The dullest drafts in the catalogue are the plain weaves, which have no plateau at all and shine only from the crowns of their turns; the brightest carry the most float on the face, with the fewest turns interrupting it. Lustre over the catalogue is a length census, and nothing about the yarn enters it.

Lustre is a length times a width

The specular area of a cloth factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply. Neither factor knows anything about the other, and over the four-by-four catalogue the first alone spans a factor of sixty.

The magnification a cloth will carry. The largest magnification a moiré can be read at, against the irregularity of the cloth making it. The points are measured: a grating whose spacings are drawn from a seeded lognormal is laid against a perfect one, the fringes are found from the phase difference, and the gain is recorded at which their count first departs from what the ideal beat predicts. The product of the irregularity and that gain comes out at 0.82 to 0.84 across every irregularity tried, so the ceiling is 0.84 divided by the coefficient of variation — the solid curve. The dashed curve is the accumulated-wander model, in which position errors random-walk and the ceiling goes as the inverse square; it is wrong by a factor of 42 at a two per cent irregularity. What the plot cannot show is what a cloth's spacing irregularity actually is: it has not been measured, and the curve is therefore a prediction with an unmeasured input.

Pattern and colour

A moiré is a vernier, and it magnifies the error too

Two gratings a per cent apart in pitch beat at a hundred pitches, so a moiré reads a pitch difference at a hundred times — which is what a vernier is. The magnification is free and its ceiling is not: the fringes split when one thread's own spacing error reaches 0.84 of the pitch difference the beat is built on, so the usable gain is 0.84 divided by the cloth's coefficient of variation, inversely and not inverse-squarely. At an ordinary yarn's spacing irregularity, a moiré carries a magnification of five.

How many layers the repeat, the harness and the beams each allow. Three ceilings on the number of layers a double cloth can have, for five layer weaves. The repeat's bound is half its ends and is a property of the notation. The harness's is the strain budget — 13 shafts on an ordinary broad loom at a 1.0% warp strain limit — divided by the shafts one layer of that weave costs. The beams' is how many warps the loom carries, which is 2. The shortest of each three is marked, and it is the beams at the coarse end of the table and the harness at the fine end; the repeat is never the binding one except at two-end layers, where it happens to coincide with the harness. A double cloth of eight-end satin layers needs 16 shafts and the budget is 13, so it is a jacquard construction by arithmetic rather than by choice. What the bars cannot show is the pick rate: a k-layer cloth needs k times the picks per centimetre of finished cloth and takes k times as long to weave, which is a cost rather than a ceiling and is the reason four-layer cloths are rare even where they are possible.

Compound and figured cloths

The repeat allows four layers and the loom allows two

A repeat of eight ends can hold four separable cloths, and there is a witness that reaches the bound exactly. No loom weaves four. The harness's strain budget buys thirteen shafts and a layer costs its own weave's shaft count, so five-end satin layers stop at two and eight-end satin layers cannot be doubled on a dobby at all — and two differing layers already want a beam each. The notation's ceiling is the only one of the three that is never binding.

Setting and geometry

A two-fold yarn is not twice a single

Folding halves nothing. It improves a yarn's evenness by exactly √2 and lowers the floor that evenness is measured against by exactly √2, so the index of irregularity comes out identical — folding does not make a better yarn, it makes a bigger one.

Knits and other structures

Where a two-bed fabric's yarn is

Thirteen named structures, and for each of them the share of its yarn that crosses between the beds — read off its own traverse rather than quoted. It separates the fabrics into three groups, and one of the groups turns out not to be a fabric at all.

A designed thin place does not get worse and an accidental one does. The thinnest place a yarn reaches, against how many gauge lengths of it are tested, for a slub yarn and for a randomly uneven yarn of the same 36.3% coefficient of variation. The slub's floor is its base count — 89% of its mean — and it is a horizontal line, because a designed variation has a stated minimum and never goes below it. The random yarn's minimum is an order statistic and falls without limit: 56% over 10 lengths and 22% over 30000. So the advantage is not a number but a function of how much yarn is being asked about, running from 1.6× to 4.0×. What the curve cannot show is the break itself: a yarn's strength at a thin place is not proportional to its linear density there, and the conversion needs a fibre model.

Compound and figured cloths

A designed thin place is kinder than an accidental one

A slub yarn and a badly spun one can carry exactly the same coefficient of variation, and the number tells a mill nothing about which it has. The designed variation has a floor — its base count, and it never goes below it — while the accidental one has a tail that falls further the more yarn is tested. At 36% CV the slub bottoms at 89% of its mean and the random yarn reaches 24%, and the gap widens from 1.6 to 3.7 times as the test grows from ten gauge lengths to ten thousand.

The warp floats across a tone edge on 8 ends. 2 strips of point paper, each one repeat of a tone on either side of a straight edge between picks, with the edge ruled and every warp float of the greatest length that crosses it drawn along its thread. Ground the exact complement: tones of 7 and 1 marks per end (cosets 1 to 7 against coset 0), not nested, and the longest warp float across the edge is 8 against 7 inside either tone. Ground one pick along: tones of 1 and 7 marks per end (coset 1 against cosets 1 to 7), nested, and the longest warp float across the edge is 7 against 7 inside either tone. A float drawn in the warning colour is longer than anything either tone has on its own. What the strips cannot show is the cloth: the edge here is one intersection wide, and in a woven piece the two tones take up yarn differently, so the change is spread over threads that point paper draws as belonging wholly to one side or the other.

Pattern and colour

A damask's edge floats further than its figure

Figure and ground in a damask carry the same longest float, which is true of both areas and false along the line between them. A float can cross the edge where two tones meet, and when one tone's marks lie inside the other's it can never be longer than a float either tone already has. A damask built as an exact complement is the one place in n that its ground can start which breaks this, and it floats n picks at its edge against n − 1 inside.

$Two exponents in a compression curve. How far a flat plate sinks into plain, 2/2 twill, satin 8 in sheeting, against the pressure it is applying, on logarithmic axes where a power law is a straight line. The measured slopes over the light end of the range are 0.50 for the plain, 0.67 for the 2/2 twill, 0.67 for the satin 8 — against two thirds predicted for any weave carrying a float and one half for a weave carrying none. The prediction is one line of algebra: pressure is a stiffness times a strain times a bearing fraction, the bearing fraction is a square root of depth for a plateau and linear in it for a point, so the pressure is the three-halves power in the first case and the square in the second. Nothing is fitted to produce it; the slopes are measured afterwards and compared. At the heavy end every curve bends, because the crowns have merged and the cloth has stopped being a surface and started being a solid — which is a different regime with a different law, and it belongs to the compaction of a fibre mass rather than to the geometry of an interlacement.$ Mechanics and drape

A cloth compresses along its own bearing curve

A fabric's pressure–thickness curve is always fitted with an empirical power law and the exponent is reported without explanation. It is not empirical. At light loads it is two thirds for any weave carrying a float and one half for a weave carrying none, and the two numbers come out of one line of algebra with nothing fitted in it.

Turn the cloth and the highlight changes hands. A 2/2 twill in sheeting turned under a light, with the specular area of each system counted separately at each angle. The warp peaks at 0° and the weft at 83°, a quarter turn apart, and neither returns anything worth seeing where the other peaks. The reason needs no dye and no interference: a warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. A cloth woven with one colour in the warp and another in the weft therefore shows one colour at one angle and the other a quarter turn away, which is the whole of shot silk — a geometric effect that has been sold as a mysterious one for three hundred years.

Weaves

Turn the cloth and the shine changes hands

A warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. Turn the cloth a quarter turn and the weft takes over. That is the whole of shot silk — a geometric effect with no dye that changes and no interference in it.

A rib is quietest at a gap of two diameters. The through-thickness force of a one-by-one rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 14.8 mN at 5 diameters against 7.0 mN at two.

Knits and other structures

A rib is quietest at two diameters

Open the beds of a rib and everything about it should get stronger. It does not. The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters and rises on both sides of it, because a crossing's climb and an interlacing's own climb cancel there.

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.

What cloth is

Why a knit runs and a weave frays

The two fabrics fail in two ways and everybody knows which is which. This collection has described both accurately for eighteen phases without being able to say what causes them, and the cause turns out to be one integer each: nought for a cloth, one per wale for a knit.

Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Pattern and colour

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

Weaves

A crepe is flat in its draft and not in its surface

A crepe weave is chosen by pushing the draft's correlations as flat as they will go. That criterion turns out to tie: on the base this collection uses, 4,416 of the 5,040 rearrangements reach the floor. Sixteen of them additionally spread their crown line perfectly evenly — and the crepe actually drawn is not one of the sixteen.

A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns.

After the loom

Two knits with one tightness factor are one knit

The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does.

Knits and other structures

Which knitted fabrics lie flat

Curl was explained here by counting face changes between courses, which works for stockinette and garter and reaches nothing else. The same question turns out to be a signed sum over a structure's own grid — and it answers for every fabric a two-bed machine can make, including the ones nobody has a rule for.

Setting and geometry

Hairiness goes as the root of the count

A coarse yarn is hairier than a fine one and everybody knows it. What nobody has said is that its hairs are no longer — the count and the length obey different laws, one rises as a square root and the other does not move at all, and the identity behind both was asserted on this site for an entirely unrelated reason.

What a raising machine can catch in a 2/2 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

After the loom

What holds a nap in a knit

A fibre buried in a woven cloth breaks rather than slides once about thirty-four millimetres of it is held, and a cotton staple buries about fourteen. In a knit the same figure is over a metre — seventy-five times the burial available — so nothing is ever close, and a raised knit sheds for the whole of its life.

Two weft colour orders thrown on a loom with boxes at one side. Weft colour orders thrown pick by pick on a shuttle loom that picks alternately from the two sides. For an order with runs of four, two, two and four, every throw finds a shuttle of its colour on the side it leaves from, so the order can be woven. For an order with runs of three and three, pick 4 has to be thrown from the right in a colour whose shuttle is on the other side, and the order cannot be woven. On a loom with boxes at one side the box opposite holds only the shuttle just thrown, which the next pick must throw straight back, so colour can change only between pairs of picks. What the drawing cannot show is the mechanism that drops the boxes, which decides how fast a change can be made but not which changes are possible.

Pattern and colour

A weft stripe is counted in pairs of picks

A warp's colour order is laid out once at warping and the loom never has to think about it. A weft's is thrown, one pick at a time, by shuttles that cross the cloth and stay where they land. On a loom with boxes at one side that makes every coloured band an even number of picks; with boxes at both sides it admits odd bands and pays for them in shuttles; and a tartan, which uses one order in both directions, is designed for its weft whether its designer knew it or not.

What holds a thread in, as two factors. The two quantities whose product is the grip on a buried thread, for a woven poplin and a jersey of the same yarn. Each contact in the knit is lighter by 8.0 times, and the contacts are further apart by 5.6 — one per half loop length against one per thread spacing. They multiply rather than competing, so the grip per millimetre of buried thread is 31 times weaker in the knit, and the crossover length at which a thread breaks rather than slides moves with it: 11.6 mm in the cloth and 461 in the knit. The two factors are drawn separately because their product is two orders of magnitude and a bar chart of it would put the knit's bar below the width of a line.

Cloth doing a job

The knitted pilling criterion gets its number

Knitwear pills and shirting does not, and the standing explanation here has been that a knit presents more exposed yarn under less pressure between its threads. The second half had no number. It has one now, and it is bigger than the argument needed: the grip on a buried fibre is thirty times weaker in a knit than in a woven cloth of the same yarn.

The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.

Knits and other structures

How thick a knit is

Two centre lines pass at a diameter and each has a radius on either side, so a plain jersey is two yarn diameters thick with nothing fitted. It does not depend on the gauge, it lands inside the band of this collection's woven cloths, and it is lower than any gauge will read.

A woven crossing, and the number that never changes. A warp end and a weft pick at 6% crimp, drawn with the thickness expanded three times so the interlacing can be seen. Each goes over its neighbour and comes back; neither passes through the other. The Gauss linking integral over the pair, closed far outside the crossing, returns 0.0000. It returns that at every crimp and for every weave, because crimp moves a thread up and down across its neighbour and a curve that goes over and comes back has done nothing a linking number can see.

Weaves

A woven cloth is not linked at all

Every thread in every woven cloth passes over its neighbours and comes back. None of them passes through. So the linking number of any two threads in any weave is zero, at any crimp, permanently — and almost everything a cloth does that a knitted fabric does not follows from that one number being nought.

$A warp pinstripe in a 2/2 twill, 1, 2, 3 threads wide. A light stripe of ends in a dark 2/2 twill, drawn as the face a reader sees at 1, 2, 3 threads wide over 3 repeats. A single stripe thread is on the face at 50% of the crossings and goes under for up to 2 at a time, so it draws a broken line. Adjacent threads of the same colour cover one another's gaps, and the line becomes unbroken at 3 — the fewest neighbours for which, at every crossing, at least one is on the face — though an unbroken line is not a solid one, and beneath each panel is how much of its width is light, which varies along it until the line is a whole repeat wide. What the drawing cannot show is distance: a broken line whose gaps are a fraction of a millimetre reads as a fainter unbroken one from arm's length, and how far that is depends on the sett and on the eye.$ Pattern and colour

No weave draws an unbroken line one thread wide

A pinstripe is drawn on point paper as a single coloured column, and in cloth a single end is on the face only where it is up — so in every weave that interlaces, a line one thread wide has gaps in it. Neighbours of the same colour fill each other's gaps, and the fewest that leave no gap is a property of the weave: two in a plain weave, three in a 2/2 twill, and in a warp-faced sateen two across the warp and eight across the weft. Unbroken is not solid either — an eight-pick bar in that sateen has no gap and is an eighth light.

Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.18, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not.

Mechanics and drape

A force is what an energy does when a crossing moves

The force a thread presses its neighbour with is the rate its bending energy changes as the crossing is displaced. Solved as a constrained minimisation, that number arrives with the shape rather than after it — and the same force, recovered a second time from the curve's own equilibrium, agrees to a tenth of a per cent.

Setting and geometry

A woven thread has no room to bend

Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.

What a run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.

Cloth doing a job

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

What the third dimension changes, and by how much. Every number the planar loop model produced, beside the same number with the climb in it, for a 20 tex cotton jersey at a 3.5 mm loop. Four of the five fall and none moves by as much as four per cent, which is the useful part of the answer: the planar model was not wrong about a jersey, it was a projection of the right curve. What it could not have at all is the quantity that is not on this list — the force through the fabric's thickness, 7.81 mN a stitch, which a model with no thickness has nowhere to put.

After the loom

The constants say nothing about thickness

Munden's two constants give a knitted fabric's wale and course spacings from its loop length alone, and the tightness factor collapses every fabric's shape onto one curve. Neither reaches the third dimension: two knits that are one knit in plan are two different thicknesses.

Knits and other structures

A jersey's course has no writhe

The mechanism everybody quotes for why a hard-twisted jersey leans is that the fabric relieves the yarn's twist by writhing. This collection's own solved course has a writhe of minus six parts in a million, and it is not small — it is exactly zero, by a symmetry, and the symmetry is a statement about what the model left out.

A repeat across a 1800-end warp. Four repeat widths laid across the same 1800-end warp, drawn at the warp's own scale. The pale bands at the two ends are the selvedge threading, 24 ends each, which weaves its own firmer weave and is not part of the design. Between them the body is ruled into whole repeats, alternating so they can be counted, and the marked bands at the two sides are the remainder — the part of a repeat that did not fit, split between the two selvedges because the trade centres the pattern. None of these four repeats divides the body exactly, and the leftovers run from 2 to 24 ends. What the drawing cannot show is what the break looks like: a quarter of a repeat at the selvedge reads as a border and half of one reads as a mistake, and where the line between those falls is a judgement.

What cloth is

A repeat has to fit the width

A repeat tiles the plane and a warp has two edges, so somewhere between them a repeat is cut through. The set of repeat widths that divide a warp exactly is the set of divisors of its body, and a body of a few thousand ends has a few dozen — two to eight per cent of the candidates. So a designer choosing a repeat for any reason except the width chooses one that does not fit, and the leftover averages half a repeat, split between the two selvedges.

How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip.

Mechanics and drape

What a loop presses with

A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

Cloth doing a job

A seam must give what the knit gives

A knitted seam fails because it is too short, not because it is too weak: the thread in it is nearly two thousand times stronger than the load it carries. What decides whether it survives is one line of geometry — the extension a seam can reach is twice the fabric's thickness times the stitches per unit length.

$A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.$ Setting and geometry

How dense a knitted fabric is

A fabric's areal weight is what the trade specifies and it says nothing about bulk. Divide it by a thickness and the answer is a density — 0.40 grams a cubic centimetre for a jersey, a quarter of the fibre it is made of — and that quarter, the share of the volume that is not air, is the number every other property follows.

What links what, in the two ways of making cloth. The linking number between two adjacent courses, for a knitted tube of 12 wales, for the same tube as this collection's model draws it, and for a woven cloth's two thread systems. The fabric's is 12 — one for every needle loop drawn through the loop below. The model's is -0.0000, because it places the interlacing at a point where two centre lines pass a diameter apart and two curves passing beside one another are not linked. The woven cloth's is -0.0000 and always will be, at any crimp and for every weave. That last row is not a defect of any model: a woven cloth really is unlinked, and it is the reason it frays where a knitted fabric runs.

Knits and other structures

A point cannot link

A knitted fabric of n wales has a linking number of n between every pair of adjacent courses. This collection's model of the same fabric has zero, and it has zero because the interlacing was declared to be a point where two centre lines pass a diameter apart — which is a near miss, and a near miss is not a knot.

What matching a pattern costs a cutting room. The cloth a matched panel needs beyond its own length, against the pattern repeat, for 6 panels of 70 cm. Every panel of a patterned cloth must start at the same phase of the repeat or the pattern breaks at the seams, so a panel's cut length is rounded up to a whole number of repeats. The bar is the allowance a cutting room budgets — a whole repeat a panel, because a panel's length is not a multiple of anything — and the mark is the waste actually expected, which is half a repeat. The two differ by (L + r)/(2L + r), which is between a half and two thirds and is nearer two thirds the larger the repeat. The rows marked in the second colour are the repeats that happen to divide the panel exactly and waste nothing at all, which is what makes the real cost jagged rather than smooth. What the bars cannot show is nesting: a cutting room lays many panels on one length and a short panel can sometimes be taken from another's waste.

What cloth is

A repeat has to fit the panel, and the panel is cut

The warp's width is fixed at warping and a piece's length is not, so the fitting problem in the two directions is not the same problem. Along the length a repeat has to fit a *panel*, because every panel of a patterned cloth must start at the same phase or the pattern breaks at the seams — and the allowance is a whole repeat per panel. A ten-centimetre repeat costs a seventy-centimetre panel twelve and a half per cent and a sixty-four-centimetre repeat costs it forty-eight.

A 1.55 mm net 50 mm behind a 0.3 mm voile, from 0.6 m, 1.5 m, 4 m. A 1.55 mm net 50 mm behind a 0.3 mm voile, drawn across 40 mm of the near layer at true pitch as seen from 0.6 m, 1.5 m, 4 m. Two grids this different beat through a harmonic: the net's k-th against the voile's first, for the k nearest the ratio of their pitches as the eye sees them. From 0.6 m that is the fifth, in register every 6.2 mm; From 1.5 m that is the fifth, exactly in register, with no fringe; From 4 m that is the fifth, in register every 14.9 mm. What the strips cannot show is how strong each family is, which falls with the harmonic, nor the net's second family of threads at right angles.

Pattern and colour

A net over a voile beats through a harmonic

Two identical sheers hung apart make a moiré by perspective alone. A net in front of a voile is not two identical sheers — its mesh is five times the voile's pitch — and it beats anyway, through the net's fifth harmonic, which is a grid 3.3 per cent coarser than the voile. With the net behind, that is a pair of sheers with its coarser layer at the back, and the fringes vanish at exactly 1.5 metres. Closer in, the harmonic changes, the fringes dissolve into a texture twice the net's pitch and re-form, and they vanish again at 17 centimetres.

What cloth is

A profile draft is a notation whose alphabet is weaves

The rung below measured four notations for a single weave and left open the notations for something larger. A profile draft is the first of them: a grid of blocks, each carrying a figure weave or a ground weave. Its image is enumerable and it is tiny — every pair of two-by-two weaves against every assignment of two-by-two blocks reaches 306 of the 22,874 interlacing four-by-four drafts, which is 1.34 per cent. And the 306 are the ones anybody weaves.

Pattern and colour

A six-end shading can be even or have a plain centre, not both

A six-end repeat can be split into the parts a shading is built from in 1,128,960 ways, and every one of them has now been walked. Only 2,816 — sixty-four distinct shadings — hold every tone between the extremes to a float of two, and seven in ten cannot do better than four. Five hundred and seventy-six can put a plain weave at the midtone, and not one of those can keep twos beside it: taking a part out of a plain weave, or adding one, always leaves a float of three.

Mechanics and drape

The modulus a knit has instead of one

A fabric's stiffness in extension is usually its yarn's modulus with the geometry taken out. A knit's is not: dimensionally it can only be a bending rigidity over a length cubed, and the same yarn laid straight and parallel is thirty thousand times stiffer than the fabric made of it.

Grouping the two systems by different amounts. Plain weave with its ends grouped by one number and its picks by another, over a grid of both. The four weaves the trade names are the corners of this space — plain at one and one, a warp rib down the first column, a weft rib along the first row, a hopsack on the diagonal — and the interior is the oblong matt, which has a name and no literature. Every cell weaves on two shafts, so the harness cannot tell any of them apart; the fundamental domain is exactly 2ab, so the notation's cost is the product; and the longest float is the larger of the two groupings. The two densest setts move with the two groupings separately, so the sett ratio is one exactly on the diagonal and nowhere else — a 3×1 matt sets at 1.50 and its transpose at the reciprocal. What the grid cannot show is the cord: the diagonal has no directional relief at all and everything off it does, in the direction of the larger grouping.

Weaves

The four named weaves are corners of a family

Plain, warp rib, weft rib and hopsack are one construction with two knobs, and the trade turns both together or neither. Group the ends by two and the picks by three and the result is an ordinary cloth with a name, no literature and a fundamental domain of twelve intersections on two shafts — and the family's three quantities all have closed forms: two shafts everywhere, a unit of exactly 2ab, and a longest float of the larger grouping.

The draft for 2/2 twill, as a loom holds it. The 2/2 twill written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

What cloth is

A lifting plan says nothing without a threading

The second of the notations for something larger than a weave is a pair, not a notation: a threading and a lifting plan, and neither alone expresses anything. The pair's image is exactly the drafts with no more distinct columns than there are shafts — 98 at two shafts, 5,282 at three, all 22,874 at four — and it is not nested with the profile draft's in either direction. The profile reaches 192 drafts that need all four shafts, and misses 64 of the 98 a two-shaft loom weaves.

What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

Cloth doing a job

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

Mechanics and drape

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

Setting and geometry

A snarl comes in one size

The radius a twisted thread coils to is twice its bending rigidity over its torque. Write the torque out and the bending rigidity cancels completely, leaving a number that depends on the twist and on the ratio of two stiffnesses — and on nothing else about the yarn at all.

Pattern and colour

A curtain is gathered so that it is seen edge-on

A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

Mechanics and drape

What friction has to hold in a relaxed knit

A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

Where a fold's two moments cancel, and where the trade folds. The two moments about a fold's own axis, for 2 singles of 20 tex cotton at 800 turns a metre. The falling curve is what the singles' own residual twist supplies, which the folding takes out of them; the rising one is what bending each single onto its helix costs. They cross at 161 turns a metre, a ratio of 0.201, and the closed form for that crossing is C/(B+C) — the ratio of the two stiffnesses and nothing else. The shaded band is where the trade actually folds, 0.6 to 0.75 of the singles twist. The balance point is nowhere near it, by a factor of three.

Setting and geometry

The folding rule is not a torque balance

Fold a two-fold yarn at about two thirds of its singles twist. This collection has carried that as a bracket copied from the trade and derived nowhere. It is now derivable, the derivation gives a fifth rather than two thirds, and reaching two thirds would need a fibre stiffer in torsion than in bending.

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.

Knits and other structures

A flattening that follows the tightness factor

Eighteen solved fabrics — five loop lengths, three relaxation states, three counts — and the flattening each one's geometry demands falls on a single curve against one dimensionless group. Nothing about the fibre or the count survives except through that group, which is what turns an arithmetical result into a structural requirement.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 24.2 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.184 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.140 mm — 0.761 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 76% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

After the loom

What wetting does to the bending limit

A rod cannot be bent to a radius below its own. A relaxed knitted loop sits at twice that limit, and a wet yarn is a tenth thicker in the same loop — so wetting moves a fabric a tenth of the way towards a bend it cannot physically take.

Weaves

Most satins still have a diagonal

A satin is chosen so that no diagonal forms, and its move is ranked by how far apart its interlacings sit. But the interlacings of a regular satin lie on a lattice, every lattice has a shortest step, and the marks line up along it. Only when two shortest steps tie is there no row to follow — and between five and forty ends the best move manages that at twelve of the thirty-five orders.

Three rules for how hard to fold a yarn, and which one the trade uses. For 2, 3, 4 folds of 20 tex cotton at 800 turns a metre: the ratio that makes the fold's surface helix angle equal its singles', the ratio that sets the fold's net moment to zero, and the ratio the trade actually folds at. The surface rule is 1/√n exactly, because a fold of n singles is √n times the diameter. The torque balance is C/(B+C), which for this fibre is 0.201 and is the same at every count, every twist and every number of folds. The trade's brackets contain the surface rule in all three rows and the torque balance in none of them.

Setting and geometry

The folding rule is a surface angle

Fold at two thirds for two singles, six tenths for three, a half for four. Those are one over the square root of the fold count, they are what makes a fold's surface twist angle equal its singles', and all three of the trade's brackets contain the number exactly.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

Knits and other structures

A loop bends at twice its own radius

A rod of radius r cannot be bent to a centre-line radius below r without occupying its own space. A knitted loop's tightest bend is at 2.04 yarn radii — twice the hard limit, and falling as the fabric tightens. That is a ceiling on how tight a knit can be, from contact alone.

Weaves

A cloth slips at its least-interlaced thread

A weave's firmness is quoted as one number, the interlacings per crossing averaged over the whole repeat. A cloth does not fail on average. A thread pulled through a seam or out of a cut edge is held by its own crossings, the grip is exponential in them, and the thread with fewest goes first. In every four-by-four draft but plain weave some thread interlaces twice a repeat — the fewest possible — whatever the average says, and a satin stripe on a plain ground averages 0.87 while its satin ends grip at a seventh of the average thread.

The edge of a warp line in a 2/2 twill, at 3 and 4 threads. A light line in a dark 2/2 twill, drawn at 3 and 4 threads wide over 3 repeats with both of its boundaries traced crossing by crossing. The line has no gap at either width, and neither boundary is straight: at the crossings where the outermost thread of the band is under the ground, the edge retreats to the next thread in. At 3 it swings 2 threads with a period of 4; At 4 it swings 2 threads with a period of 4. What the drawing cannot show is distance, at which a swing of one thread width is below what an eye separates and a swing of three may not be.

Pattern and colour

An unbroken line is not a clean one

A line of colour has two boundaries and neither is straight, in any weave there is. The thread at the edge must go under somewhere, and where it does the edge retreats to its neighbour — so the boundary steps, and by exactly one thread less than the narrowest unbroken line the weave draws. Over 22,874 drafts there are three widths and three swings and no draft anywhere else, and widening the line past its narrowest unbroken width leaves the edge precisely where it was.

The four-by-four catalogue's crown line, counted three ways. Every one of the 22,874 interlacing four-by-four drafts, binned by how much horizontal crown line it carries, under three counts: both systems summed, which is what the published census reports; the warp alone; and the weft alone. A bearing curve sees one system, because a plate meets whichever crown stands higher and meets nothing else until it has sunk past the step between them. Under the summed count 2 drafts carry none; under the warp alone 494 do, and under the weft alone 494. What the histogram cannot show is which drafts moved, which is most of them.

What cloth is

The census counted two systems and a surface has one

Two drafts of twenty-two thousand touch at points, and the two are the plain weave. That is a count of the crown line both systems carry, and a bearing curve sees one: a plate meets whichever crown stands higher and meets nothing else until it has sunk past the step. Counted the way a surface is read, 494 drafts touch at points rather than two — and which 494 depends on a crimp division already called a convention rather than a measurement.

Mechanics and drape

What leaving the plane costs

Every number the flat loop model produced, beside the same number with the climb in it. Four of the five fall, none moves by four per cent, and the estimate that priced the third dimension beforehand had the sign the wrong way round for a reason worth naming.

The selvedge turns of a 2/2 twill, 4 ends wide, from the left. A strip of 2/2 twill 4 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. 0 of the 8 turns are caught, where the edge end is on the other face on the second pick, and 8 slip. Across all its edge placements the weave catches every turn at 8 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 selvedge holds only where its edge end changes face

A shuttle weft goes out on one pick and back on the next, and between them it turns round the end at the edge. The turn is caught only if that end is on the other face on the second pick; otherwise the loop has nothing to wrap and slides off. Plain weave catches every turn at every width. A 2/2 twill catches them at half its widths, and only if the first pick is thrown from the right side. A 3/1 twill, a hopsack and every satin catch them nowhere, and of the 22,874 four-by-four drafts, 9,636 cannot hold a selvedge at any width at all.

The named colour-and-weave effects, sorted by the loom their weft needs. Each named colour-and-weave effect with the shortest band in its colour order and the cheapest loom that can throw that order in the weft, on a loom with 4 boxes a side. end-and-end, runs of 1 and 1, needs picking at will; hairline, runs of 1 and 1, needs picking at will; log cabin, runs of 1 and 1, needs picking at will; tattersall, runs of 1 and 9, needs picking at will; birdseye, runs of 2 and 2, needs boxes at one side; crow's foot, runs of 2 and 2, needs boxes at one side; step pattern, runs of 2 and 1, needs boxes at both sides; three-and-one, runs of 3 and 1, needs picking at will; houndstooth, runs of 4 and 4, needs boxes at one side; shepherd's check, runs of 6 and 6, needs boxes at one side; gun club, runs of 4 and 4 and 4 and 4, needs boxes at one side; glen check, runs of 4 and 4 and 4 and 4 and 2 and 2 and 2 and 2, needs boxes at one side. The warp costs nothing, because a colour order in the warp is laid out once at warping; the weft is thrown one pick at a time by shuttles that stay where they land, so an effect's price is its weft order alone. What the bars cannot show is the pattern, which is in neither the order nor the weave but in what they make of each other.

Pattern and colour

The finest colour-and-weave effects need the rarest loom

A houndstooth, a shepherd's check and a gun club check are thrown by the cheapest shuttle loom there is. A hairline, an end-and-end and a log cabin are thrown by none — a colour on every other pick is thrown from the same side every time, so its shuttles pile up at the far end of the loom and never come back. The dividing line is the parity of the bands and nothing else, which is why it survives the fact that no two sources agree about how wide a shepherd's check is.

A comber board for 60 ends a centimetre, in side elevation. A jacquard's comber board seen from the side, with the fell of the cloth at the left and the back rest at the right. The board carries one hole per end; at 60 ends a centimetre the ends are 0.167 millimetres apart and a cord with a mail on it needs 0.9, so the holes are ruled in 6 rows staggered fore and aft, 6 millimetres apart — a harness 30 millimetres deep. Each row's ends are strained by its own distance from the fell: 0.460 per cent at the front and 0.524 at the back, a spread of 0.0636. What the elevation cannot show is the sideways fan of the cords above the board, which is a separate and much larger geometry.

Compound and figured cloths

A jacquard's harness has a depth after all

A jacquard was said to escape the shaft loom's depth entirely, because every mail hangs at the same distance from the fell. Every mail does, if the comber board has one row of holes — and it cannot. At sixty ends a centimetre the ends are a sixth of a millimetre apart and a cord with a mail on it wants most of one, so the holes are ruled in six rows thirty millimetres deep. The escape is real and it is a factor of eight rather than a release, and it closes as the cloth is set finer.

The bearing crowns of 2/2 twill and 2/2 hopsack, over 3 repeats. The cells at which the warp is on the face, drawn over 3 repeats of each draft — which is the surface a plate meets, since the other system is a step below it. 2/2 twill has 1 component in its repeat and a path that runs the whole way across the cloth, in both directions; 2/2 hopsack has 2 components in its repeat and no path across the cloth at all. Both carry the same length of crown line by the bearing count, and one is a ridge while the other is a field of islands. What the drawing cannot show is the depth of the gaps between them, which is the step to the second system and is a few micrometres.

What cloth is

Four drafts in five have no path along their own crowns

A 2/2 twill and a 2/2 hopsack carry exactly the same length of bearing crown line, which the surface census noted and could not explain. One of them is a ridge running diagonally across the cloth without a break; the other is a field of square islands with no path between them. Counted over the whole catalogue, 4,016 of 22,874 drafts have a crown path that reaches the far side, 1,616 have one in both directions, and 130 have crowns with no neighbour at all.

Mechanics and drape

The force that holds a knit open

Resolving a knitted loop's contact force out of the fabric's plane leaves a fifth of it pointing through the thickness. That fifth is 7.8 millinewtons a stitch, fifteen kilopascals over the area a stitch occupies, and it is the whole reason a jersey has a thickness rather than a plan.

Setting and geometry

A cabled yarn is a fold of folds

The rule that sets a fold's twist is one over the square root of the number of components. Apply it twice and a cabled yarn's three twist levels are fixed by two integers — which is a prediction with no free constants, about a class of yarn the trade quotes no rule for at all.

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