Textile Structure
how cloth is put together
Concept

Float — where it appears

A length of thread lying on one face without interlacing, measured by how many opposite threads it passes over. Its length is decidable from the matrix by counting, with no yarn or dimension in it, and it is the quantity with the widest reach here.

Named by 49 essays across 7 fields — each of them below, with the objects they name alongside it.

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.

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.

weaves · Foundation weaves 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.

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.

cloth · What cloth is 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.

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.

weaves · Float 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.

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.

finishing · Shrinkage Figure and ground on 8 ends. A damask is a satin and its own complement. The figure is warp-face and the ground is weft-face, they carry the same longest float and need the same shafts on the same threading, and the whole of the pattern is carried by which system is on top.

A damask is its own complement

The pattern is carried by direction alone. Figure and ground are the same satin, one warp-face and one weft-face, with the same longest float, the same shaft count and the same threading — so the cloth's most famous effect costs it no structural difference whatever between the two areas.

fancy · Jacquard How often a draft falls apart. Every four-by-four draft in which each end and each pick interlaces at least once, sorted by its longest float, with the fraction that describe more than one cloth. The counts are produced by running the enumeration rather than by recalling it.

Every cloth there is, at four by four

Sixty-five thousand matrices, twenty-two thousand weaves, and about a dozen with names. The complete census of the smallest interesting repeat is a map of a whole small world, and almost none of it has ever been woven.

cloth · What cloth is 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.

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.

fancy · Pile 3 cloths on 6 ends. A repeat of 6 ends and 6 picks holding 3 complete cloths, each with 2 ends and 2 picks of its own. The bars beside the strands say which cloth each belongs to; the ceiling at this size is 3, and the longest float is 5 because the face warp passes over every pick below it.

How many layers a draft can have

Of the 22,874 four-by-four drafts this site sweeps, 22,730 are one cloth and 144 are two. None is three, and none can be — a repeat of n ends holds at most n halved cloths, because every cloth needs two ends and two picks of its own before it interlaces at all.

cloth · Integrity 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.

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.

weaves · 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.

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.

weaves · Float What four notations can say. Four ways of writing a weave down, with how many of the 426 four-by-four cloths each can express and whether expressing one identifies it: point paper, 426, identifies; a four-part draft on 3 shafts and 4 treadles, 110, identifies; a fraction name, 5, does not identify; a longest-float specification, 426, does not identify. A longest-float specification names every cloth and separates them into only 3 classes.

Four ways to write a weave down

A weave's name in the trade is a pair of numbers — 2/2, 3/1, five-end satin — and the notation is so universal that it is easy to forget it is a notation. It has an image and the image is computable: of the 426 cloths at four by four, five have a name of that kind, and there are four names for the five.

cloth · Notation 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.

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.

weaves · Float 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.

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.

knits · Float 8-end satin figured on 8-end sateen. A 3 by 3 block profile, drawn above at one square per block, and the cloth it produces below at one square per intersection. The figure weave is 8-end satin and the ground is 8-end sateen, both single cloths on their own; each block is 8 ends and 8 picks. The composite is 1 cloth, with a longest float of 8 and 0 threads lying loose, all counted from the matrix that drew the picture.

A figure is not a stripe

Two sound weaves side by side always make sound cloth — that is a theorem, and it was proved here. Put one of them inside a *region* instead of a band and it stops being true, and whether it is true or not turns out to depend on a number that appears nowhere in either draft: where the two weaves start relative to each other.

pattern · Blocks 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.

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.

knits · Two-bed Where a thread stops sliding and starts breaking. A pick of sheeting gripped over a length of cloth, drawn one crossing at a time. The resistance is 0.30 times the 0.599 N each crossing presses with, so it rises with the length held; the breaking load of 3.74 N does not. The two are equal at 7.4 mm. What the drawing cannot show is that μ is a range rather than a constant, so the mark is a band and its position is exactly inversely proportional to the friction.

A thread is held one crossing at a time

Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.

cloth · Contact 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.

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.

weaves · Float A thread withdrawn from a plain weave and from a satin. One pick of a sheeting being pulled from 1.60 mm of cloth, in a plain weave and in a five-end satin, at a friction coefficient of 0.30. The drawn width of each thread is the tension in it at that point, to a common scale; the faint rules are where the thread actually turns, which is at its interlacings and nowhere else. A plain weave turns 2.80 times per millimetre and a satin 1.12, so the plain weave's tension compounds 2.78-fold over this length against the satin's 1.46. What the drawing cannot show is that the wrap angle is taken from a plain-weave geometry in both panels: a satin's crimp is genuinely smaller, so its real turns are gentler than these and its grip weaker still.

A thread is gripped where it turns

The arithmetic this site has used for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.

cloth · Contact 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.

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.

finishing · Nap 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.

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.

applied · Seams How many twills there are. Every way of writing a twill on each repeat, reduced by the two operations that leave the cloth unchanged: starting on a different pick, and turning it over. What is left is the number a designer actually chooses between.

How many twills a repeat admits

Sixty-four ways of writing a twill on eight ends, and twenty-one twills. The difference is two operations that leave the cloth unchanged, and a catalogue that does not quotient by them is counting notations rather than fabrics.

weaves · Foundation weaves 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.

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.

finishing · Nap The block condition, one half at a time. Every block shape from 8×8 down, for 8-end satin figured on 8-end sateen, each an exhaustive census of all 65,536 four-by-four profiles. The bar is split: the dark part is figures that separate with a thread left loose on the face, and the light part is figures that separate with nothing visible wrong. Shapes satisfying the block condition in one direction only are marked, and none of them has a light part at all. What the chart cannot show is why: the census is exhaustive and the theorem behind it is not proved here.

A rectangular block is not half a rule

The block rule was proved for square blocks and the rectangular case was recorded as not run, on the grounds that a block a repeat wide and half a repeat deep satisfies only half the condition. Running it turns up two things: half the condition rules out the failure nobody can see, and a block turned through a right angle is a different design — which a square census cannot notice, because a square block is its own transpose.

pattern · Blocks 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.

A dimension without a state

Every essay in this field turned on the same omission, and collecting them produces a result none of them had on its own — the quantities a cloth's geometry determines divide cleanly into those a finishing works can move and those it cannot, and the division is not the one the vocabulary suggests.

finishing · State Everything the relative origin decides. Each of six quantities computed from a weave matrix, swept over all 64 relative origins of 8-end satin figured on 8-end sateen. A quantity that takes one value across the sweep is a gauge freedom of the pair; one that takes several is something a designer chooses without knowing it. longest float takes 2 values; interlacing rate takes 2 values; weft-face fraction takes 3 values. The origins fall into 3 different partitions, so the moving quantities are not all decided by the same number. What the table cannot show is the census over all 65,536 profiles, which is a seventh column, and the one that showed the origin decides whether the cloth holds together.

What else the relative origin decides

Where two weaves start relative to each other decides whether the cloth holds together, and it has a second consequence beside that. Two is what a search finds when it looks twice. Sweeping every origin against every measure of the cloth turns up a third — the longest float, which is the one property a weaver actually looks at — and turns up an earlier figure caption that says the opposite.

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

A figured cloth has a step in its surface

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

pattern · Blocks 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.

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.

fancy · Double cloth 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.

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.

weaves · Rib weaves 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.

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.

pattern · Shading Which seams slip and which break. For each cloth, the grip a 10 mm seam allowance offers divided by the thread's own breaking load. Above one the fabric or the thread gives first and the seam holds until it does; below one the threads slide out and the seam opens with the cloth intact. voile, batiste, poplin, sheeting break; cheesecloth, muslin, duck, filter slip. The allowance that would save every cloth in the table is 74 mm, which is set by the openest of them alone. What the chart cannot show is the stitching itself, which has its own strength and its own way of cutting the threads it passes through.

A seam slips before it breaks

A sewn seam fails in one of two ways and the trade names them separately: the threads pull out of the cloth beside the stitching, or something breaks. Which one a given cloth does is decided by whether its seam allowance clears the length at which grip beats strength — and at the ordinary ten millimetres, the table divides four against four.

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

What combining two weaves reaches

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

weaves · Derivation From a calender's line load to a force at one crossing. A sheeting through a nip loaded at 30 N per millimetre of bowl width, with the cloth in contact over 5.0 mm. The pressure is the first divided by the second, 6.00 N/mm², and the force at one crossing is that pressure times the area a crossing owns — the product of the two thread spacings, 0.1374 mm². So the crossing carries 0.824 N, the sections flatten to 2.42 and 2.63, and the cloth thins from 0.388 mm to 0.218 mm. What the drawing cannot show is that two cloths through the same nip are not given the same treatment: the area a crossing owns varies fivefold across this site's table, and it is a factor in the force.

A calender spends the compression for good

Calendering was described on this site as moving a cloth from one thread-section model to another, which was right and had no number in it because the amount of the move was a free parameter. It is a pressure now — and the same nip setting turns out to give two cloths quite different treatments, because the force at a crossing is the pressure times the area a crossing owns.

finishing · Lustre finish The size of the capstan correction. The ratio of the capstan crossover to the crossover a sum of independent contacts gives, for every cloth in the table at a friction coefficient of 0.30. It is exactly ln(1 + z)/z, where z is the thread's breaking load times the wrap angle, over the contact force — a quantity with no friction coefficient in it at all. That is why the earlier result that μ·L* is exactly constant survives this correction to twelve figures: μ was only ever in the factor outside the logarithm. The correction is largest for the duck, whose coarse strong yarn makes z large, and smallest for the batiste. What the rows cannot show is that a real cut edge frays at a friction nobody measured on that particular cloth.

How far a cut edge frays

A seam allowance, a fray width and a tuft's bound length are the same number wearing three hats, and the earlier estimate of it was four times too long. Correcting it moves the whole table across the boundary an ordinary allowance sits on — from four cloths holding and four slipping, to all eight holding — and turns a specification argument into a different one.

applied · Seams 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.

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.

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

The six-end satin that does exist

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

weaves · Satin 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.

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.

weaves · Float 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.

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.

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

A tone step does not need a satin

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

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

A weave is a halftone screen with n greys

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

pattern · Shading 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.

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.

fancy · Double cloth 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.

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.

pattern · Shading 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.

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.

fancy · Jacquard 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.

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.

fancy · Jacquard 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.

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.

pattern · Stripe 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.

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.

weaves · Rib weaves 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.

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.

pattern · Stripe 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.

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.

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

A jersey has no float for a wire to catch

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

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

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

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

weaves · Derivation

Named alongside it

The objects these essays reach for when they reach for this one.

SatinInterlacingCensusPoint paperRepeatSettCloth integrityDamaskContact forceFrictionJacquardTwill

All concepts