Textile Structure
how cloth is put together
Theme

The thread: The float decides

How far a thread travels on the face before it goes under is the quantity behind lustre, drape, snagging, abrasion and how close the cloth can be set. Almost nothing else has that reach.
The 5-end satin. The 5-end satin on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it. Weaves

Plain, twill and satin

Three rules, and everything else in weaving is a variation on them. What separates the three is not appearance but one trade — how often a thread changes face against how far it runs when it does not.

 How much fibre a woven reinforcement holds. A unit cell of a woven reinforcement in section: flat tows, the thickness they add up to, and the wave the warp makes to cross them. The bar below is the fibre volume fraction against the same tows laid flat in two plies and against the packing factor of a tow, which is the most any cloth of them could be. The vertical scale is exaggerated so the interlacing is legible; the horizontal scale is the cloth's own. Cloth doing a job

The crimp is the price of being cloth

A woven reinforcement is bought for stiffness along its fibres, and the usual explanation of what it gives up — fibre content, lost to the crimp — has the sign wrong. At a given thickness a woven fabric holds slightly more fibre than two flat plies of the same tows. What the crimp costs is stiffness, and the float length is the knob.

 The 2/2 twill. The 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it. What cloth is

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.

 Where the floats are in the 5-end satin. Every longest run of warp on the face, marked. A float is smooth because nothing interrupts it, which is the same reason it snags: there is a length of thread lying on the surface with nothing holding it down. Weaves

The float decides

How far a thread runs on the face before it goes under is one number, and it sits behind lustre, drape, snagging, abrasion, tear strength and how densely the cloth can be set. Almost nothing else in the subject has that reach.

 How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it. After the loom

A cloth cannot shrink past its own crimp

However hard a loom held the warp, there is a limit to what relaxation can take back, and it is not a fitted constant or a measured one. It is the crimp itself, read as c over one plus c, and it is the only result in this field with nothing empirical in it at all.

 What an interlacing costs. Each weave's interlacing count beside the closest it can be set in the same yarn. The two run opposite ways, because a thread that changes face often has to bend often and a bend takes room. Weaves

Interlacings and firmness

Every time a thread changes face it has to bend, and a bend takes room. That one sentence decides how densely a cloth can be set, how firm it feels, and why the two run in opposite directions.

 A 2/2 twill reversed — the clean way. A twill reversed at intervals, generated from the rule rather than drawn. The count beside it is the number of adjacent ends that come out identical, which is the defect a weaver sees as a thick line at the seam and which no amount of looking at the grid announces. Weaves

Broken and herringbone twills

Reversing a twill is the cheapest way to turn a rule into a figure, and there are two ways to do it that look identical on paper. One of them leaves two adjacent ends doing exactly the same thing.

 Corduroy: floats of 6 ends, cut. 3 wales of an extra weft floating over 6 warp ends each, cut at their midpoints so that each half stands away from the ground. The pile height is half a float's length and nothing else decides it, so a finer sett at the same float count gives a shorter pile; the wale spacing is the float plus its binding ends. Compound and figured cloths

Corduroy is a cut float

An extra weft floats over some ends, a knife runs along the cloth, and each half of the float stands up. So the pile height is exactly half the float's length — which makes corduroy the one pile fabric whose surface is decided entirely by a quantity this site has been counting since its first essay.

 The highlights of a 8-end satin, move 3. Every unbroken run of warp on the face, drawn as the band of light it returns. A yarn is a cylinder and a cylinder reflects a line rather than a point, so a float's highlight is as long as the float — and how long that is, in inches, is the float divided by the picks per inch the weave can be set to. Weaves

Why satin shines

Lustre is usually filed under fibre, and silk gets the credit. It is a property of the weave: a cylinder reflects a line rather than a point, so the highlight a cloth returns is exactly as long as its longest float.

 What one cut costs. The length of thread set loose by a single abrasion cut through a warp float, with each weave measured at the densest setting that weave allows in the same yarn. The bars are inches of freed thread; the note beside each is the float in picks and the setting that float permits. Weaves

Floats and abrasion

A satin is said to wear badly. It does not wear quickly — its flat face spreads the rubbing over more thread than a plain weave's crowns do. What it does is fail badly, and those are different quantities moving in opposite directions.

 What the geometry says about a tear. A slit in a woven cloth, with the ends ahead of the tip gathered into the group that will break together. How many can gather is the opening divided by the slack in each gap, and the slack is the thread spacing less the thread diameter — an expression with no weave in it at all. Weaves

Does a loose weave tear better

The trade says a twill tears stronger than a plain weave of the same yarn, because fewer interlacings let the threads group. The geometry of that grouping has no weave in it at all — and where the geometry does speak, it predicts the opposite.

 single float as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here. Knits and other structures

The float in a knit

Five rungs of this anchor have taken the float to be a length of thread on the surface with nothing holding it down. A knit has one too, and it behaves the same way in the light and the opposite way in the hand — because a woven float lengthens its thread and a knitted one shortens its fabric.

 rib-float in section. A two-bed structure seen in section across the wales, over 3 repeats of course 1. The front bed's loops sit on the upper line and the back bed's on the lower one, offset by 0.5 of a needle pitch because the gating is rib. Of the 2 floats in the repeat, 2 lie in the gap between the beds and 0 on a surface. The upper panel is the fabric at the machine and the lower one is the same course with the beds closed up, which is what happens when the fabric is cast off — and neither panel is a relaxed fabric, because the loops are drawn as arches of one size and a real one settles wherever the yarn's bending leaves it. Knits and other structures

A second bed changes what a float is

Every float so far has been on a surface, because every knit so far has had one needle bed. Put a bed behind it and the yarn runs in the gap between them — and over the whole enumeration of two-bed structures, 1,248 floats of 1,272 lie inside the cloth, on no surface at all.

 What a float limit of 4 leaves. For each repeat, the fraction of the distinct twills on it whose longest float is within the limit. The constraint is usually stated as a rule about a drawing; it is really a statement about how much of the catalogue exists, and the catalogue shrinks as the repeat grows. Weaves

Designing to a float limit

Every jacquard designer works to a rule of the form nothing longer than four. It reads as a constraint on a drawing. It is really a statement about how much of the catalogue exists, and the catalogue shrinks as the repeat grows.

 Z twill, Z twist. A square of cloth with two directions on it. The broad lines are the twill, whose angle 45.0° from the warp comes from the setts alone — 24 ends and 24 picks per centimetre. The fine lines are the surface fibres of the warp ends, at 25.3° from the warp because the yarn is twisted 900 turns per metre. Between them is 19.7°, and that is the whole of the rule. Setting and geometry

Twist and the twill line

The oldest rule of thumb in weaving says to weave a Z twill from S-twist warp for a bold line and from Z-twist warp for a subdued one. Both halves of it are geometry: the twill's angle comes from the two setts, the fibres' angle comes from the twist factor, and the rule is their difference.

 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

Only a float can be raised

A raising machine drags wire teeth across a cloth and pulls fibre ends up into a nap. The teeth need something to catch, and what they catch is a float — so which fabrics can be napped at all is a question about the matrix, decidable exactly, and the answer over the whole four-by-four census is two.

 What holds a thread in a seam. A cloth thread inside a seam allowance, drawn past the crossings that grip it, with the capstan's factor at each. The tension falls by that factor at every crossing, so the grip is exponential in the crossing count and the count is what the weave decides. Beyond the thread's own strength the thread breaks rather than slides, and the rest of the allowance holds nothing. Cloth doing a job

What holds a thread in a seam

A seam fails in two quite different ways, and the one the trade worries about is not the stitches breaking. It is the cloth's own threads sliding out of the weave beside the seam — and what resists that is friction at the crossings, accumulating multiplicatively, so a satin gives a seam a thousandth of the grip a plain weave gives it.

 What a raising machine can catch in a 3/1 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

Raising spends the cloth's strength

Fibre standing up on the surface is fibre no longer in the load path. A nap is warmth bought with tensile strength, and the exchange rate runs along the same axis as everything else the float decides — which means a fabric cannot be optimised for both ends of it.

 The strain across a harness. The warp strain each shaft of a 24-shaft harness puts into its own ends, from 0.460 per cent at the front to 1.728 per cent at the back. Shafts within a budget of 1.0 per cent are drawn in one colour and those outside it in another; the budget is reached at shaft 13. Mechanics and drape

The back shaft works hardest

How hard the loom is on a warp end is the product of two numbers from different worlds — how often its column changes sides, which is a property of a binary matrix with no millimetre in it, and the strain of the shaft it happens to be on, which is a property of a machine with no weave in it. Neither knows about the other, and the threading that joins them is a decision nobody makes on structural grounds.

 Float lengths in a honeycomb. The same draft twice. On the left, filled where the warp is on the face — which is all point paper says. On the right, every intersection shaded by the length of the float it belongs to, from one at the palest to 6 at the strongest. The gradient on the right is the whole mechanism of a relief weave and it is invisible on the left. Weaves

A honeycomb gets its cells in the wash

The obvious mechanism is take-up on the loom, and the arithmetic says it is wrong: every end of a diamond passes through the long floats and the tight ones alike. What is left is finishing, and a cell is a region that wanted to shrink less than the cloth around it.

 Where a stitch can hide on a 5-end satin face. The face weave on point paper, with a mark on every gap between two ends at which a stitch would be covered. 15 of the 25 positions in the repeat pass the cover rule, which is 60 per cent of them, and 5 of those can be used at once without two stitches sharing a pick or a gap. 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.

 Plain weave, doubled. Plain weave, then the same weave with its picks grouped in 2, its ends grouped in 2, and both — which are a warp rib, a weft rib and a hopsack — and a 2/2 twill beside them for comparison. The rules under the drafts bracket the threads that lift together on every pick and so lie touching. Each doubling multiplies the cloth's own unit by 2: 2, then 4, then 8 intersections. The hopsack and the twill interlace equally often and are drawn from different rules. The setts under each draft are the densest that weave may be set at with a 0.25 mm yarn, and a rib's two are 0.750 apart where every other weave here is square. Weaves

A cord is a stripe with no colour in it

Warp rib, weft rib and hopsack are plain weave with its threads doubled, in the warp, the weft, or both. Six of the nine measures this site takes off a matrix cannot tell a 2/2 hopsack from a 2/2 twill, and the three that can are not the ones a weaver quotes.

 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 crossover length in four weaves. The gripped length at which a pick of a sheeting breaks rather than slides, in four weaves at a friction coefficient of 0.30. It is a millimetre or two for a plain weave and 8.0 mm for an eight-end satin, which is what a cut edge of each does. The interlacing rate appears in the answer only as a factor outside the logarithm, so the ratios between the four are exactly the ratios of their interlacing rates — in this model and in the sum of independent contacts alike. The rung below expected the float correction to change that ordering; it does not. It changes the size, by a factor of 3.7. What the rows cannot show is that all four use one cloth's crimp, so a satin's genuinely gentler turns are not in them. 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.

 A 3-pick fold across a 8-end satin, move 3. A 8-end satin, move 3 on point paper with a fold 3 picks wide drawn across it at pick 3. A warp end's crimp is made where it turns from over to under, and the dots mark the turns that fall inside the fold. Averaged over the ends there are 0.75 of them, and 4 of the 8 ends in the repeat have none at all — those ends are marked with a line, and each of them crosses the fold dead straight with no crimp to give up. Over every fold position, 50.0% of end-and-place pairs are like that. What the drawing cannot show is what happens to those ends instead, which is that the whole of the fold's length difference goes into their fibres. Weaves

How sharply a weave lets a cloth fold

A fold's length difference is paid for out of crimp, and crimp is not spread evenly along a thread — it is made at the interlacings and nowhere else. So what a fold has to spend is not the weave's average crimp but whatever is inside the few picks the fold crosses, and in an eight-end satin half the warp ends have nothing there at all.

 A slot and a square of the same area do not pass the same air. A hole of 62500 square micrometres, drawn out from a square to a slot twenty times longer than it is wide, at constant area throughout. The open area is unchanged by construction and the flow is not: it falls to 39 per cent of the square's. Two things move the same way and neither is a correction to the other — the hydraulic diameter falls as the rectangle is drawn out, and the shape factor rises from 14.23 for a square towards 24 for an infinitely thin slit, which is Shah and London's result quoted rather than derived. This is why a weave's float matters to what it passes even where its cover does not: a float lays parallel threads side by side and the hole beside it is a slot. Weaves

A satin's hole is a slot

Two cloths at the same cover have the same open area, exactly, and do not pass the same air. A float lays parallel threads side by side and the hole beside them is long rather than square, and drawing a hole out at constant area cuts what it passes to two fifths — because the hydraulic diameter falls and the shape factor climbs, and both of them move the same way.

 The surface of an 8-end shading. The height of the cloth's own surface at each tone of an 8-end shading, for the two chains, on a sheeting at 0.50 N in the end. A thread presses on the thread it crosses only where it turns, so a region that turns more often is pressed more often and finishes thinner — and firmness rises towards the midtone of a shading. The spread chain therefore sinks 84 µm between its ends and its midtone and the consecutive chain 43 µm, with 2 of its steps at exactly one thickness against the spread chain's 0. The tone scale is level by construction and the surface under it is not. What the plot cannot show is the light: a step in the surface reads as a line under a raking beam whatever the tone is doing, which is why a relief nobody specified is visible at all. Pattern and colour

A tone ramp is a valley, and the satin digs it

A shading's tone is exact and its lustre is measured; its thickness is neither, and nobody specifies it. A firmer weave is pressed harder at every crossing and finishes thinner, so an eight-end shading sinks eighty-four micrometres between its ends and its midtone — about a third of the cloth's whole thickness, on every cloth tried. Build the same chain on a twill instead of a satin and the sag is exactly nothing.

 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.

 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

Raising moves the surface onto the hairs

A raising machine pulls fibre ends up out of the floats, and what a finger or a light then meets is not the cloth's surface at all but a layer of loose fibre standing above it. The criterion for which cloths can be raised is the same criterion, exactly, that decides which cloths have crown line — so raising spends the surface that would have made the fabric shine.

 A cloth loses its strength long before it loses its mass. Rubbing a 2/2 twill in sheeting down, plotted against how much of its own solid volume has gone. The lower curve is the fraction of the plan the rubbing is touching; the upper is the fraction of the warp's section that has been cut away. They are wildly different because the wear is spread and the damage is concentrated: material comes off the whole surface, but it comes off every thread at the same place, and a thread breaks at its thinnest place. At one per cent of the mass gone the section is already 5% smaller. That is why a fabric that looks barely worn fails a strength test, and why abrasion resistance measured as mass loss and abrasion resistance measured as residual strength are two different quantities that are quoted as one. Cloth doing a job

A cloth loses its strength before its mass

Rub a fabric and it sheds material from all over its surface, but it sheds it from every thread at the same place — and a thread breaks at its thinnest place. So the strength gone is always several times the mass gone, the ratio is computable from the bearing curve, and it is worst for the weave whose crowns are points.

 A round thread reflects a beam into a fan. A round thread seen end-on, with a beam arriving from straight above. Every point across the thread has its own normal, tilted by the angle it sits at, and mirrors the beam through twice that angle — so a single direction in becomes a whole fan out, spread across the thread and not at all along it. Seen from the front that fan is a highlight lying along the thread, exactly as long as the length of thread that is straight, which is the float. Nothing about this depends on the fibre. A cylinder of any material returns a fan, and the only way to narrow it is to stop the section being a cylinder — which is what a calender does. Weaves

A float reflects into a line

A cylinder cannot return a beam to a point. Its normals sweep the whole half-turn across it and nothing at all along it, so a straight thread throws light into a fan — seen as a highlight lying along the thread, exactly as long as the length of thread that is straight. The lobe of a satin is twice as narrow along the thread as across it; the lobe of a plain weave is exactly round.

 plain over 5-end satin: what each layer's warp does. A warp end of each layer of a double cloth, in section over the same span of cloth, with the length of warp each eats drawn beneath. The crimps come from Peirce's geometry asked about a thread that bends at its own average rate rather than at every crossing, and they are 14.35% for the plain and 2.03% for the 5-end satin. They are not the same, so the two layers consume warp at different rates: a hundred metres of cloth takes 114.4 m of warp from one and 102.0 m from the other. A beam delivers one rate, so the difference has nowhere to go and accumulates with the length woven — one pick spacing after 4 mm. What the sections cannot show is the yarn's thickness, which is exaggerated so the path is legible; the crimps beside them are computed at the real diameter and are not read off these drawings. Compound and figured cloths

Two layers need two beams

A layer weaving a metre of cloth eats one plus its crimp metres of warp, and a beam delivers one rate. A plain face over a five-end satin back differs by twelve percentage points of crimp, which is twelve metres of warp over a hundred-metre piece and one pick spacing of slack after four millimetres of weaving. The difference has nowhere to go and does not settle — so the only double cloth that can share a beam is two layers of the same weave at the same sett.

 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.

 An 8-end weft-faced satin through a point tie on an end. The 8-end satin on a move of 3, weft-faced, woven through a point tie of 12 hooks that turns on a single end, so 22 ends carry one repeat. The strip above is the tie, hook number against end. Inside the satin the weft floats 7 ends; across the turn it floats 13, outlined, because on some pick the nearest interlacing is 7 ends away on both sides. What the drawing cannot show is how a long float at one line down the cloth behaves in use, which depends on the yarn and the finish rather than on the count. Compound and figured cloths

A point tie nearly doubles the float at the turn

A point tie halves the hooks a symmetrical figure needs by driving every hook's end twice, out and back. It mirrors the ground as well, and a satin is symmetrical about no end: across the turn its weft floats 2n − 3 ends, or 2n − 2 between ends, at every hook count and wherever the satin starts. At eight ends that is thirteen or fourteen against seven — and the turn's other promise, more repeats that fit the width, is not kept either.

 A lancé weft bound where the 8-end satin interlaces. A warp-faced 8-end satin on a move of 3, with an extra weft thrown after every ground pick and laid on the face over figures 12 ends wide and 28 ends apart. Between figures it is bound only on ends the ground already drops on an adjacent pick, 56 places in the drawn repeat, which leaves back floats of at most 4 and no binding on the face. The drawing is 40 ends by 16 picks and counts as 1 cloth. What it cannot show is whether a binding point counted as hidden is hidden in a real cloth, which depends on the yarns' contrast and on how closely the ground's floats cover it. Compound and figured cloths

A brocade weft floats as far as the next figure

A brocade's pattern weft is a third thread, laid on the face only where the figure wants it. Thrown from selvedge to selvedge it floats on the back across the whole gap to the next figure, unless an end is dropped under it — and on a satin ground a dropped end hides only where the ground already interlaces. That fixes the shortest float that shows nothing, at the larger of the move and its complement less one, and puts the smoothest satin ground at odds with the best-bound brocade.

 A warp rib's cord, at doublings of 2 and 4. One warp end drawn in section along the cloth, over the pick groups of a warp rib, at doublings of 2 and 4 and at one scale. The cord's crests are the pick groups and its two dimensions come from different places. The height is the weft's own crimp amplitude and Peirce's closure condition caps it at the two yarn diameters together — 500 µm here — so it runs 304 µm at a doubling of 2 and 361 at 4, which is 61 and 72 per cent of the ceiling. The pitch is the doubling times the pick spacing and has no ceiling at all. So a larger doubling gives a taller cord and a wider one, and wider faster: the aspect falls from 0.40 to 0.29. What the section cannot show is what a finish does, which flattens the cord without changing either the pitch or the ceiling. Weaves

A cord's height has a ceiling and its width has none

A warp rib's cord is a wave, and its two dimensions come from two different places. The height is the weft's own crimp amplitude, and Peirce's closure condition caps it at the two yarn diameters together — 500 µm for a quarter-millimetre yarn, of which a 2/2 rib reaches 304 and a 6/6 rib 391. The pitch is the doubling times the pick spacing and has no cap at all. So a bolder rib is taller and wider, and wider faster: the aspect falls from 0.41 to 0.22.

 Every 6-end decomposition, by the float its best chain holds the middle tones to. All 1,128,960 Latin squares of order 6 with their first row in order — every way of splitting a 6-end repeat into 6 parts with one mark in every end and every pick — each asked for its chains of tone steps. 2,816 can hold every tone between the extremes to a float of 2, and they fall into 64 classes once the repeat's starting corner and reading direction are set aside, the cyclic square a twill writes among them; 800,658 cannot do better than 4. 576 can put a plain weave at the midtone, and every chain that does floats three on either side of it. Every one of the 434,540 distinct tone steps met is one cloth. What the rows cannot show is which of the classes a designer would choose, since the float profile is one criterion among several. 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.

 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.

 How hard each grouping holds its own threads. The grip a cloth has on one of its own threads inside a 10 mm seam allowance, for six members of the doubled family at 24 threads per centimetre and a friction coefficient of 0.3. Grip accumulates multiplicatively at every crossing — the capstan equation on Peirce's own weave angle — so it is exponential in the crossings, and the crossings are the interlacing rate times the intersections in the allowance. The family's interlacing rate has a closed form, (a + b)/2ab, which is half the sum of the two reciprocals — so the two groupings enter symmetrically and each one saturates. The dashed line is the thread's own strength: a cloth whose grip falls short of it lets the thread slide out rather than break, which is seam slippage. A 2/2 hopsack is below it at this allowance and a 1/4 warp rib is above, on cloths whose firmness differs by an eighth. What the bars cannot show is the friction coefficient, which is measured and is not a constant of cloth; the ordering holds at every value anybody reports and the sizes do not. Weaves

What nothing separates comes out together

The doubled family's interlacing rate has a closed form — half the sum of the two groupings' reciprocals — so a seam's grip on its own threads is the exponential of a harmonic mean, and it saturates in each grouping separately. At a ten-millimetre allowance a 2/2 hopsack holds a thread at seventy-four times the applied tension and a 1/4 warp rib at two hundred and eighteen, on cloths whose firmness differs by an eighth.

 Every thread's interlacings in an eight-end satin stripe on a plain ground. An eight-end satin stripe on a plain ground on point paper, with a bar under every end and beside every pick for the share of its crossings at which it changes face. The warp's fewest is 0.25 a crossing against an average of 0.88, and the weft's 0.83 against 0.86; the draft's single firmness number is 0.87. What the bars cannot show is the friction at each crossing, which turns a count into a grip. 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.

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