Field

Weaves

Plain, twill and satin, and what actually separates them — float length, interlacing count, and a satin that cannot exist on six ends.

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.

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.

Which satins exist. For each order, the moves that give a satin — a step coprime with the order, and not the steps of one and one less which give a twill instead. Four and six admit none, so no regular satin exists on four ends or on six.

There is no satin on six ends

Weavers have known it as a rule for centuries. It is a theorem with a one-line proof about common factors, and it rules out four ends as well.

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.

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.

The 2/2 twill both ways round. The same twill stepped one end to the right and one end to the left. Every quantity the matrix can produce is identical for the two, so the direction is real in the cloth and invisible in the arithmetic.

Twill direction, and how it is named

The two twills are mirror images, and every quantity the matrix can produce is identical for both. What separates them is the yarn, which has a handedness of its own — and the angle, which is forty-five degrees only in a square-set cloth.

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.

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.

A double cloth. Two complete fabrics woven in one repeat, and what happens when a single intersection is turned over. The layer count beside the draft is computed from the matrix drawn, and the census below it is every single-square change tried in turn.

Backed and stitched constructions

A double cloth is two fabrics woven at once, and the check that verifies it says exactly two. Turn one intersection over and it says one — which means one mistake is enough to destroy the construction, and the enumeration says half of them would do it.

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.

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.

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.

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

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.

Which move to use on 16 ends. For each satin move the order admits, the distance from an interlacing to its nearest neighbour, measured on the torus the repeat lives on. A move whose interlacings crowd gives the eye something to find; the one that scatters furthest is the one to weave.

Which satins are worth weaving

Manuals give the moves a satin admits as a list, as though the survivors were interchangeable. They are not. Measure how far apart the interlacings sit and the traditional counter turns out to be the best move at the orders a weaver mostly uses — and to fail first at thirteen, where the square root names four and five scatters further.

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.

How many shafts a draft needs

Every quantity counted so far is a property of the cloth. This one is a property of the loom — the number of distinct columns in the matrix — and it is very probably the strongest single predictor of which of the twenty-two thousand four-by-four drafts anybody ever wove.

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.

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.

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.

How much a draft agrees with itself. On the left the draft; on the right its correlation at every offset, one cell per offset, with the offset of nothing at the top left. Warp-up counts as plus one and weft-up as minus one, so the number in each cell is agreements minus disagreements out of 64. The correlations away from the origin sum to exactly -64, whatever the draft — structure can be moved about and not removed.

A crepe cannot be structureless

A crepe weave is designed to have no line in it anywhere. The correlations of a draft with itself sum to a number fixed by the repeat alone, so structure can be spread and never removed — and on eight ends the floor turns out to be half the repeat, set by a fact about binary words with nothing textile in it.

Mock leno, 3 threads to a bundle. Threads that interlace identically have no weft passing between them, so nothing holds them apart and they lie touching. The reed still sets the average spacing, so the space they leave collects at the bundle's edge — a hole 0.60 by 0.60 mm, made without one thread crossing another. Drawn to scale on a fixed 9 mm square of cloth at 20 threads per centimetre and a 0.3 mm yarn.

A hole with nothing crossing

A real leno holds its holes open by crossing one thread over another, which is a topological arrangement and cannot come undone. A mock leno makes the same holes by grouping threads that nothing separates, and everything about it is friction.

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.

The basic weaves at four by four. Plain weave and the three twills a repeat of four admits, each drawn with its longest float and its layer count computed from the matrix. The fifth frame is empty: a satin needs a move coprime with its order and neither one nor one less than it, and four ends admits 0 such moves. So the smallest interesting repeat contains two of the three weaves every manual begins with.

The three basic weaves do not generate the rest

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

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.

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.

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.

A seersucker is made at the loom

Every other relief weave in this collection gets its shape after the loom, from a difference of crimp between two regions of a few per cent. A seersucker's surplus is thirty per cent and is put in as the cloth is woven — an order of magnitude more, which the square root turns into a factor of four in depth and no more than that.

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.

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

Only a plain weave has one size of hole

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

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.

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.

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

What a missing end does to the weave

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

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

A mispick is one row in the wrong place

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

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.

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.

Z twist at 1600 turns per metre. A 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 40.0°, and it is the only quantity in this family: the yarn is 13.8% shorter than the fibre in it and carries 59% of the strength the same fibre would give lying straight.

The other crepe is in the yarn

A crepe weave puts the texture in the matrix. A crepe yarn puts it nowhere the matrix can see: the cloth is a plain weave, and the surface comes from a thread twisted so hard that it shortens by a seventh and spends the rest of its life trying to untwist.

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.

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.

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.

A calender multiplies the highlight by 32, and all of it is width. A 2/2 twill in sheeting pressed at increasing force, with the specular area recomputed at each state from the standard compression model. It rises from 0.91% of the plan to 28.9%, a factor of 32, while the cloth thins from 381.6 µm to 186.1 µm. The gain is not in the length of the crowns: that moves by 5 per cent. It is in their width, which moves by 31.8 times, because pressing puts a flat top on the section and a flat top has one normal rather than a fan of them. The finish does not polish the thread. It changes the dimension of the highlight, from a line to a band, and the arithmetic says so by refusing to put any of the gain in the other factor.

A calender buys the width

Press a cloth and its lustre multiplies by twenty-four. None of that comes from the length of its crowns, which moves by six per cent; all of it comes from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refuses to put any of the gain in the other factor.

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.

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 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 shot effect needs a fibre with no ends. The peak-to-trough contrast of an eight-end satin in sheeting as the cloth is turned in the light, against how much hair stands on it. Bare, the contrast is 37 to one, because a straight thread's normals lie in the plane across it and the warp and the weft therefore reflect a quarter turn apart. A hair layer does two things and only one of them matters: it blocks, which takes the same factor off the peak and the trough and changes no contrast at all, and it returns light of its own, which is added to both. A hair population points every way at once, so its return has no azimuth in it — and adding a constant to both ends of a ratio of 37 destroys the ratio. On an ordinary spun cotton the contrast is already down to 5.1 to one; singeing recovers it to 29; raising kills it outright at 1.00. The one fibre with no staple length is the one fibre with no fibre ends, and every shot fabric ever woven is made of one.

A hair layer veils a highlight

An eight-end satin's shine swings by a large factor as the cloth is turned, because a straight thread's normals lie in the plane across it. Put fibre ends on it and the swing disappears — not because the hairs block the light, which changes no contrast at all, but because they return light of their own that has no direction in it.

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.

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 woven cloth asked the same question

A knitted fabric's two adjacent courses occupy the same space by a fifth of a diameter. A woven cloth's two systems overlap by nothing at all in an open cloth and by four and a half per cent in a dense one — and the difference is that they cross rather than run alongside.

What a group of 2 threads behaves as. A group of 2 threads of 250 µm with nothing separating them, beside the single thread of the same width and the single thread of the same yarn, all drawn to one scale. The bars are the group's four readings as ratios. Cover is a width and adds, so the group covers exactly what a 500 µm thread would — with 50 per cent of its yarn. Bending rigidity is a second moment and does not add: 2 threads free to slide give 12.5 per cent of the thick thread's and the same 2 fused into one body give 62.5, so a real group is somewhere between and where depends on friction. Against the thread of the same yarn the two readings point opposite ways: the group covers 1.41 times as much and bends 0.50 times as stiffly if free. What the drawing cannot show is the friction that decides where between the two limits a finished cloth sits.

A group is one thread for cover and two for bending

The rung below leaves a limitation standing: two ends with nothing between them lie touching, and whether they behave as one thread of twice the diameter was said to depend on twist, hairiness and finish. Three of the four measures have exact answers with no friction in them and no two agree. A pair covers exactly what a double-diameter thread covers, with half its yarn, and bends at between an eighth and five eighths of its rigidity — and at equal yarn it covers forty per cent more and bends half as stiffly.

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.

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.

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.

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.

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.

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.

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 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.

How many cloths any one cloth derives into. The 426 four-by-four cloths sorted into the orbits the manuals' derivations cut them into. 12 orbits hold 1 cloth; 83 orbits hold 2 cloths; 62 orbits hold 4 cloths. The largest orbit in the whole catalogue holds 4, so no cloth derives into more than 3 others by any sequence of the named operations, however long. The derivations generate a group of 256 elements and it cuts the catalogue into 157 pieces. What the bars cannot show is which cloths are in which orbit, which is the next figure.

No cloth derives into more than three others

Every weaving manual opens by saying the three basic weaves generate the rest. This collection counted the reach and found nine of 426, and left the nine as a count. It is not a count: every derivation the manuals name is a relabelling of the grid or a complementation of it, both invertible, so they generate a group — and that group cuts the 426 cloths into 157 closed pieces of which the largest holds four. The claim is not merely wrong about how much derivation reaches; derivation cannot reach more than four cloths from anywhere, by any sequence of operations, however long.

Which satin orders are row-free, from 5 ends to 40. Every satin order from 5 to 40, marked where the best move's lattice has two shortest steps of equal length rather than one — which is the condition under which the interlacings do not line up into a row. The row-free orders are 5, 10, 13, 15, 17, 24, 25, 26, 29, 34, 35, 37: twelve of the 35 orders that admit a regular satin at all. An n-end satin floats over n − 1, so a float limit is a ceiling on the order, and the ceilings for limits of 8, 12, 16 are drawn. What the strip cannot show is the spread, by which the orders are ranked and which decides which move is best within each.

A float limit leaves one row-free satin

A satin is chosen so that no diagonal forms, and an earlier essay found that most of them fail: the interlacings lie on a lattice, every lattice has a shortest step, and the marks line up along it unless two steps tie — which happens at twelve of the thirty-five orders from five to forty. The other constraint was named and not applied. An n-end satin floats over n − 1, so a yarn that will not carry a float longer than eight admits four orders in all, and exactly one of them is row-free: the five-end satin, which is the one everybody already weaves.

An irregular satin scatters where a regular one lines up

A regular satin's marks lie on a lattice, so its closest pairs all run along one vector and make a row. An irregular satin has no lattice at all, so its closest pairs may point several ways at once — and at seven and nine ends, where every regular satin at the best spread has a row, an irregular one reaches the same spread with its closest pairs scattered over four directions and no more than a third in any one. At eight and eleven ends there is no such satin: the best spread is reached by regular satins alone, and irregularity has nothing to offer.

8/3 at 1.00 and 8/3 at 1.29, drawn in cloth. Satin marks drawn over two repeats at the spacings of a cloth, with the pick direction stretched by the sett ratio — ends per centimetre over picks per centimetre — and every closest pair of marks joined. 8-end satin, move 3 at a sett ratio of 1.000: closest pairs point one way: a row; 8-end satin, move 3 at a sett ratio of 1.291: closest pairs point 2 ways: no row. What the drawing cannot show is the float each mark interrupts, which is what a reader of the cloth actually sees.

A satin's row belongs to its sett

Point paper draws an end and a pick as equal squares, and every result about which satins have a row was taken there. A cloth is not square: it is set at so many ends and so many picks a centimetre, and in cloth the ties that made twelve satin orders row-free are ties between steps of different shape, which break at the first per cent of unequal sett. The reverse happens too. An eight-end satin, rowed on paper, has no row at exactly 1.291 ends per pick; an eleven-end at 1.265 and 1.528. And the only scatter that survives a range of setts is an irregular satin whose closest steps are mirror images — which nine ends has from the first per cent and ten only by 1.3.

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.

No cloth shines under a sky

Every specular figure on this account is drawn at a tolerance of two degrees, which is a stand-in for how wide the source is. Sweeping it says something the account has not: the contrast between a floated weave and a plain one is inversely proportional to the source's angular size, exactly, because a float keeps a crown line as the source narrows and a plain weave has only turns. A satin outshines a plain weave by 177 under the sun's disc, by 22 under a small lamp, and by exactly nothing at sixty degrees.

A room lights a satin at the harmonic mean of its lamps

A satin outshines a plain weave by a factor inversely proportional to the width of the light, which is a clean law for one source and says nothing about a room. A room has a lamp and a window at once. Sum each source's highlight and the average falls out by itself: the room behaves as a single source whose width is the harmonic mean of its sources' widths, weighted by the power each supplies. The harmonic mean is ruled by the narrowest source, so a small lamp carrying a tenth of the light more than doubles the contrast a large window gives.

A satin's contrast as the lamp moves off the mirror. The specular area of an eight-end satin over a plain weave, both as a sheeting, for a source 2° wide moved away from the direction the viewer sees mirrored in the cloth. Moved across the satin's floats, the contrast is 22.4 on the mirror and 21.7 forty degrees off. Moved along them it is 22.4 on the mirror and 0.25 once the source is more than twice its own width away: the satin is darker than plain. A calendered satin falls to 1.83 across its floats as well. What the plot cannot show is shadowing, which a lamp far enough off the mirror brings in.

A lamp off the mirror lights a satin only across its floats

Every source in the account of a room was a source the viewer sees mirrored in the cloth. A lamp off to one side lights the cloth too, and which facets it reaches depends on which way it is off. Displaced across a satin's floats it is caught by the float's own curve and adds contrast like any lamp on the mirror; displaced along them by more than twice its own width it reaches only the turns, and the satin comes out a quarter as bright as plain weave.

Floated weaves against plain under an overcast sky. The sky's radiance each weave's facets mirror into the eye, over the plain weave's, as the viewer's elevation above a sheeting falls from overhead to thirty degrees, under an overcast sky with a ground of albedo 0.2. 2/2 twill: 1.25 overhead and 0.99 at 30°; 5-end satin: 1.30 overhead and 1.01 at 30°; 8-end satin: 1.37 overhead and 1.01 at 30°. What the plot cannot show is the diffuse reflection, the same for every weave of one fibre, which a real cloth's sheen sits on top of.

A satin mirrors the top of the sky

A uniform source sixty degrees wide leaves every weave reflecting the same share of its face, and that was read as saying no cloth shines under a sky. A real sky is not uniform. An overcast one is three times as bright overhead as at the horizon, and it stands over a ground darker than itself. Every facet of a cloth mirrors one direction, and a satin's facets mirror the top of the dome while a plain weave's mirror the horizon and the ground — so seen from above, under cloud, an eight-end satin still sends back a third more sky.

The selvedge turns of a 2/2 basket, 8 ends wide, from the left. A strip of 2/2 basket 8 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. 4 of the 8 turns are caught, where the edge end is on the other face on the second pick, and 4 slip. Across all its edge placements the weave catches every turn at 0 of 16. What the drawing cannot show is how far a slipped loop travels, which the beat-up and the weft tension decide.

A weave holds its selvedge with two of its own columns, or not at all

A selvedge holds where its edge end changes face at every turn of the weft there, and the turns at one edge fall on alternate pick boundaries. So an end can hold the right edge only if its column changes face at every odd boundary, and the left edge only at every even one. Whether a weave can hold a selvedge is then whether its set of columns contains one of each kind, and nothing about its width matters. Two ends re-entered on the right shafts make 13,238 of the 22,874 four-by-four drafts hold at any width, where 3,262 did. The other 9,636 need a shaft of their own. A weave with an odd number of picks has no column of either kind, and never can.

A reed can dent a sateen out of its row and not a satin

A reed pulls the ends in each dent closer together than the average spacing, so a satin's marks stop lying on a lattice, and a step that fits inside one dent shortens. For the eight-end satin, four or five ends to a dent scatter its row once the ends in a dent sit 22.5 per cent closer than average. Five shares no factor with eight, so that denting leaves no reed mark. But a warp-faced satin's ends are set too close to close up that far, and what replaces the row is the reed's own stripe. Two ends to a dent never scatter any satin's row, and they give the one row-free satin the trade weaves a row.

Which window sizes name every derivation orbit, on every repeat small enough to walk. Every draft of eleven repeat shapes from three picks by three ends to five by five, every thread interlacing, cut into the orbits the derivations that keep a shape can reach — slides, reversals, the counterchange and, on a square repeat, the quarter turn — and tested with censuses of two-, three- and four-thread-square windows. 3 × 3: 3 orbits; window 2 names all, window 3 names all. 3 × 4: 19 orbits; window 2 confuses 2, window 3 names all. 3 × 5: 66 orbits; window 2 confuses 43, window 3 names all. 3 × 6: 365 orbits; window 2 confuses 328, window 3 confuses 12. 3 × 7: 1,680 orbits; window 2 confuses 1,649, window 3 confuses 211. 3 × 8: 8,988 orbits; window 2 confuses 8,949, window 3 confuses 2,324. 4 × 4: 157 orbits; window 2 confuses 49, window 3 names all, window 4 names all. 4 × 5: 2,875 orbits; window 2 confuses 2,153, window 3 names all, window 4 names all. 4 × 6: 36,380 orbits; window 2 confuses 34,439, window 3 confuses 204, window 4 names all. 4 × 7: 447,756 orbits; window 2 confuses 444,174, window 3 confuses 6,503, window 4 confuses 2. 5 × 5: 45,077 orbits; window 2 confuses 42,480, window 3 names all, window 4 names all. On every shape a window no wider than half the ends confuses some orbits, and a window wider than half names all of them but for one pair at four picks by seven ends. What the grid cannot show is a shape too large to walk, six by six among them.

The window that names a cloth is half its repeat

At four by four a census of three-by-three windows tells every derivation family from every other, and the question it left was whether three stays enough as the repeat grows. Walked whole over eleven repeat shapes, three is still enough at five ends — all 45,077 families of the five-by-five cloths apart — and fails as soon as the repeat reaches six. On every shape a window no wider than half the repeat confuses some cloths, and a window wider than half names them all, but for one pair of seven-end cloths that differ in a single end.

The tucked-in edges of a 2/2 twill, 8 ends wide. A strip of 2/2 twill 8 ends wide over 8 picks, woven with tucked-in edges: every pick is cut at both edges and each tail is laid back into the next shed, so between every pair of picks there is a turn at both edges. 8 of the 16 tucked turns are caught, where the edge end is on the other face in the next shed, and 8 are not. Across all its edge placements the weave catches every tucked turn at 0 of 16. What the drawing cannot show is the tail itself, which lies alongside the next pick for the tuck's length and doubles the weft there.

A tucked-in edge asks every weave the odd repeat's question

A shuttle weft turns at one edge per pick, alternately, and that alternation is what let a 2/2 twill hold both selvedges with two ends re-entered on its own shafts. A rapier or air-jet loom cuts every pick at both edges and tucks both tails into the next shed, so every boundary between picks is a turn at both edges — the question an odd repeat always asked. Only a plain column answers it. Of the 22,874 four-by-four drafts, 10,382 can still hold their edges on their own shafts, against 13,238 under a shuttle; the 2/2 twill is among the 2,856 lost. And the tucked zone carries two picks in every shed, which lies flat only below between a half and two thirds of a weave's maximum pick density.

A tucked edge stands proud because its warp goes straight

A tucked zone holds two picks in every shed, and the obvious account is that the pairs run out of room and bulge. The loom says otherwise: the warp is let off as one sheet and the cloth taken up at one speed, so the zone's ends have only the body's length to spend. They rise less. The weft takes up the height, so the edge is proud well before the pairs stop fitting, and they stand up well after the count says they must.

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