Textile Structure
how cloth is put together
Theme

The thread: A weave is a matrix

Warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not an encoding of a weave — it is the weave, and it makes almost every question about it decidable.
The 5-end satin. The 5-end satin on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it. Weaves

Plain, twill and satin

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

 A draft is a two-coloured pattern. For each weave, the symmetries that leave warp-up as warp-up and those that exchange the two. Only a balanced weave has any of the second kind, because exchanging warp and weft is only a symmetry when there is as much of one on the face as the other. Pattern and colour

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

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

A fabric is a structure, not a material

Cotton is a material. Cloth is an arrangement, and almost everything a fabric does follows from how the threads are put together rather than from what they are made of.

 Colour and weave — 2/2 twill. One interlacement, threaded in two colour orders. What a reader sees is the colour of whichever thread is on the face, so the visible pattern can be something the draft gives no hint of. Pattern and colour

Colour and weave

Thread the same weave with a different order of coloured ends and a pattern appears that is nowhere in the draft. Houndstooth is an ordinary two-and-two twill, and nothing about its interlacement is unusual at all.

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

The draft is a matrix

Point paper is not a diagram of cloth. It is a grid of yes-or-no decisions about which threads to lift, which makes a weave a binary matrix and almost every question about it arithmetic.

 A 2/2 threading, run out to three widths. The same four shafts threaded for repeats of increasing width. The threading line runs up and back down; the harness never grows, and the number of ends it carries is bounded by the loom's width rather than by its shafts. Compound and figured cloths

The harness does not grow

Nearly every account of a loom says the shaft count limits the repeat. It limits the repeat of a straight draw, and of nothing else — a reversed twill on ninety-six ends weaves on the same four shafts as the twill it was made from, and the ratio grows without bound.

 Whether the cloth is one cloth. Two drafts. Both interlace everywhere, both have short floats, and both look like perfectly ordinary weaves. One is a single fabric and the other is two fabrics lying on each other, and the bars beside each strand say which layer it belongs to. What cloth is

Does it hang together

A draft can interlace everywhere, have short floats, and describe two fabrics lying on each other rather than one. Nothing about the drawing says so, and the test that does is exact.

 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. Pattern and colour

The diagonal is not a thread

A twill's diagonal is the most looked-at feature in weaving and the most misdescribed. No thread in the cloth runs diagonally. What runs diagonally is a pattern of which thread is on top, which is a different sort of object.

 The draft for herringbone, as a loom holds it. The herringbone 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. Compound and figured cloths

What a dobby stores

A pattern chain does not store picks. It stores lifts — the distinct sets of shafts a draft ever raises — and for most drafts worth weaving that number is very much smaller than the number of picks, which is the second half of the reason a wide repeat is affordable.

 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. Weaves

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.

 The plait braid. A braid written as a word of crossings and drawn from it. Each strand is coloured by the component of the above-and-below relation it belongs to, so a word describing two independent braids rather than one shows as two colours. This one is 1 braid. What cloth is

Braids and the third thread system

A braid is one set of strands interlacing with itself at an angle, so the question of whether it holds together is the same question a weave answers — and most crossing sequences answer it badly.

 The draft for 8-end satin, as a loom holds it. The 8-end satin 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. Compound and figured cloths

A jacquard is every end its own shaft

The machine is usually described as the one that weaves anything. What it actually does is abolish one of the loom's two budgets and make the other proportional to the width of the repeat — which turns a constraint on the design's kind into a constraint on its size, and those are very different things to be short of.

 The twelve groups a draft can have. Every four-by-four draft in which each thread interlaces, classified by its plane symmetry group. Twelve of the seventeen groups occur; the five that do not are the ones needing a three-fold rotation, which no grid of warp and weft admits at any size. Pattern and colour

The seventeen groups a draft can have

Twelve of them, in fact. The five missing ones all need a three-fold rotation, and no grid of warp and weft admits one at any size — which makes it a theorem rather than an artefact of the census.

 A preform through its thickness, and whether it is one piece. An orthogonal three-dimensional preform in section, with the binder's path, beside the digraph the integrity criterion consumes: one node per level, an arrow from each level to the one above it, and the binder's own contacts. The warp and weft here are straight and do not interlace at all, so the whole of the connectivity is the binder — and the count of separable pieces is what the criterion returns, unchanged from the weave it was written for. Cloth doing a job

The third index is not a repeat

The fancy weaves left three-dimensional weaving open as a possible fifth escape from the binary matrix. It is not one. The criterion that decides whether a plain weave is one cloth decides a five-layer preform unchanged — and what a third dimension actually takes away is periodicity, because a thickness has a top and a bottom and a repeat does not.

 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. What cloth is

What the matrix cannot say

A weave is a binary matrix, and the whole of this collection rests on it. So it is worth setting down, in one place and precisely, what that encoding decides exactly, what it decides while appearing to decide something else, and the four quite different reasons a real fabric can fall outside it altogether.

 Three different weaves, one cloth. Drafts drawn from one collision class, with the blind intersections marked, and the single surface all of them produce. Where the two crossing threads are the same colour the weave leaves no trace, so the cloth cannot report what it is. Pattern and colour

Colour and weave as a two-colour problem

Where the two threads crossing are the same colour, the intersection looks identical whichever is on top. Half of them are, so 22,874 drafts collapse onto 256 surfaces — eighty-nine weaves apiece, and no way to tell them apart.

 The hole between four threads. Three millimetres of a woven filter cloth in plan, with one clear opening dimensioned and a grain of the soil it must retain drawn at the same scale. Every hole in the repeat is this size — a woven cloth's pore distribution is a single value, which is why it is specified by an opening size while a nonwoven needs a curve. Cloth doing a job

The hole between four threads

A woven cloth's holes are all the same size. That is not an approximation — it is what a repeat means — and it is the whole reason a woven filter is specified by one number while a nonwoven needs a curve. The number is the spacing less the diameter, which this site has been computing since its first questions about setting.

 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. Weaves

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.

 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. Compound and figured cloths

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.

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

Broken and herringbone twills

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

 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. What cloth is

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.

 How many four-by-four weaves there are. The same census counted four ways. A draft is a notation; shifting the repeat's origin, turning the cloth over and turning it end for end all change the matrix and not the fabric. Each bar is the number of distinct objects left once those identifications are made, counted by canonical form and checked against Burnside's lemma. Pattern and colour

How many cloths are there

Twenty-two thousand eight hundred and seventy-four four-by-four drafts. Four hundred and twenty-six four-by-four cloths. And the site's own separation rate — the fraction of drafts that look like fabric and are not — more than doubles when fabrics are counted instead of notations, which is the opposite of what was expected.

 A V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it. Compound and figured cloths

Pile is a third thread system

Velvet, corduroy, plush and carpet all carry threads that are not woven into the ground in the ordinary sense. Two of the three systems make a perfectly good matrix and the third is not in it — so the encoding this whole site rests on has nothing to say about the part of the fabric a hand actually touches.

 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. Setting and geometry

The weave decides the sett, and two models disagree about it

Ashenhurst's rule and Peirce's geometry both answer how densely a cloth can be set, and for a plain weave they are fifteen per cent apart. Only one of them reaches the other weaves at all, and it is the cruder one.

 2/2 twill: written at 4×4, repeating on 4. On the left the draft as written, with the smallest rectangle that tiles it outlined. On the right the same draft with every translation that leaves it unchanged marked from the top left corner: there are 4 of them, so a fundamental domain holds 4 intersections rather than the 16 the rectangle claims and the 16 the point paper carries. Nothing in the drawing on the left says so. What cloth is

What a repeat repeats

Point paper is ruled in squares, and the rectangle a draft is written on is a decision by whoever drew it. The unit a cloth actually has is smaller — two intersections for a plain weave, four for a 2/2 twill — and it is not a rectangle at all.

 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. Weaves

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.

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

Knit, tuck and miss

A weave is a matrix over two symbols and a weft knit is a matrix over three. The site's central question survives the translation intact: a weave falls apart when its above-and-below relation is disconnected, and a knit falls apart when a needle never knits.

 A stripe of a satin stripe on a plain ground. a satin stripe on a plain ground, written as one draft of 8 picks and 20 ends. Each band is generated from its own rule and the whole matrix is measured as one: 10 shafts, which is the number of distinct columns in the union of the two bands, against 8 if every column were shared and 10 if none were. The bands have 0 columns in common. Pattern and colour

A stripe is a partition of the warp

A striped cloth is not a weave with decoration on it — it is one matrix in which different bands of ends obey different rules. The shaft count is a set union rather than a sum, and of forty-five pairs of standard weaves only three share a single column.

 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. What cloth is

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.

 A tartan sett in 2/2 twill. A pivoted sett of 70 threads used in both systems, which is what makes a tartan a tartan rather than a check. On the left, the cloth at block scale: the squares on the diagonal have one colour in both systems and the rest are mixtures. On the right, one mixture rectangle at thread scale, where the weave decides. In 2/2 twill the reflection survives on 50.0 per cent of a mixture's intersections, against a ceiling of 75 per cent that no weave can reach. The three tints stand for the three colours of the sett and carry no other meaning here. Pattern and colour

A check is two stripes and a tartan is one

A tartan is quoted as one sett of thread counts because the warp and the weft carry the same order. That is said to make it symmetric about its diagonal, and the arithmetic says it cannot be — the ceiling is three quarters, and a 2/2 twill reaches one half.

 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. Pattern and colour

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.

 A leno and the open plain weave it is not. Two warp ends and the picks they hold. On the left the doup end passes under its partner between picks and comes up the other side; on the right it never crosses, which is an open plain weave at the same sett. The layer count under each panel is computed by the same criterion that decides every other draft, and it returns the same verdict for both. Compound and figured cloths

Leno is not a matrix

A doup end crosses under its partner between picks and comes up on the other side, so the end at position three on pick two is a different end on pick three. The entry W[i][j] does not name anything. The encoding is not wrong about leno; it is undefined for it.

 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. What cloth is

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.

 tubular, as a graph of its wales. Every wale of tubular as a node — 2 on the front bed and 2 on the back — with one chain per course joining everything that course takes yarn on, because a course is one traverse of one yarn. The nodes are filled by which component they fall into. This structure comes out as 2 fabrics: F0+F1 and B0+B1. The same answer is obtained a second way, by walking every partition of the wales and asking whether any course straddles it, and the two are required to agree. Knits and other structures

Does a double jersey hang together

Two beds knitting with nothing passing between them are two fabrics that happen to have been made at once. Of 6,561 two-bed arrays, 1,135 are fabrics and 50 of those are two fabrics — and 28 of the 50 split across the beds rather than along them, so neither half is a layer.

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

Only a float can be raised

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

 What a figure costs in shafts. Two families of block designs in 8-end satin on 8-end sateen at blocks of 8. One repeats three block-columns however wide it gets and costs 24 shafts at every width from 24 to 120 ends. The other gives every block-column a different pattern and costs 16, 24, 32, 40, 48 shafts as it grows — exactly 8 more per new column. Shafts are counted as distinct columns of the composite matrix. Pattern and colour

What a figure costs the loom

A block design's shaft count does not depend on how wide the figure is, or how deep, or how many blocks it has. It depends on how many *different* block-columns it has, and each new one costs exactly one repeat of the ground weave — 8 shafts, then 16, then 24, in a straight line with no slope to fit.

 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. Weaves

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 staircase a woven outline is. Four straight edges on a block grid, stepping 1 across in 1, 1 across in 2, 2 across in 1, 1 across in 4. Each tread is one repeat of 8-end satin, which at 24 by 22 threads per centimetre is 3.33 mm across and 3.64 mm up. A jacquard hook at 140 cm width is 1.17 mm, so the machine resolves 2.9 times finer than the cloth can use. Pattern and colour

A woven outline is a staircase

A jacquard's resolution is quoted as its hook count, and on a 1,200-hook machine 140 cm wide that is a step of 1.17 mm. The cloth cannot use it. A figure's smallest feature is one repeat of the ground weave, which on an eight-end satin is 3.33 mm — so the machine resolves nearly three times finer than the fabric can hold, and a finer machine buys nothing at all.

 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. Weaves

How many shafts a draft needs

Every quantity this site has 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. Weaves

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.

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

The back shaft works hardest

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

 Flexes per end in the same check, ground a pick along. One bar per warp end above the draft of the same check, ground a pick along, each the number of times that end changes sides in one repeat — which is the number of times the shed drags it through its heddle eye. The counts run from 6 to 10, and they are the same numbers that give the cloth its interlacing rate of 0.500 per intersection. Mechanics and drape

A figure is harder on its warp

Every basic weave flexes all its ends exactly as often as each other — plain, twill, satin and sateen alike, and it is a one-line theorem. Figure one of them on another and that evenness goes, or does not, depending on where the ground weave was started relative to the figure. The same invisible offset that decides whether a fine figure holds together decides, at a coarse one, how unevenly the loom works the warp.

 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. Weaves

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.

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

A 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.

 A tube. A two-layer fabric 6 ends wide in each layer, with the weft's path drawn across the section below the draft. 1 piece of cloth, 0 free selvedges, 1 shuttle, and a developed width of 12 ends however the edges are joined. The connectivity of the infinite repeat is 2 for this draft and cannot tell this construction from the others. Compound and figured cloths

A tube and two cloths are the same draft

Two separate fabrics, a cloth twice the width of the loom and a seamless tube are the same draft to the last square. The check this site is built on reports two layers for all three and is right every time — because the thing that separates them is four free selvedges, or two, or none, and a repeat has no selvedges in it.

 What the two setts can be set to. Two rules on one scale from 8 to 40 threads per centimetre. The upper carries the 49 warp setts a metric reed catalogue reaches at one to four ends per dent; the lower carries the 4825 pick densities a change-wheel take-up reaches. The mean spacing is 0.656 threads per centimetre in the warp and 0.0066 in the weft, a ratio of 99. The widest gap in the reed's range is 2.10 threads per centimetre. Setting and geometry

The setts a loom can reach

Transposing a draft gives a perfectly good draft, and every count this site takes off a matrix either is symmetric under exchanging warp and weft or has a mirror twin. The loom is not symmetric at all: over the range ordinary cloth is woven in, it can choose a pick density 99 times more finely than a warp sett — and the warp sett cannot be changed once the warp is drawn in, at any granularity whatever.

 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. Pattern and colour

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.

 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. Pattern and colour

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.

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

A cord is a stripe with no colour in it

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

 A loop's cell, dry and wetted. One stitch of a 20 tex cotton jersey at a 3.50 mm loop, drawn inside the rectangle of one wale by one course that its own dimensions give. The thread is drawn at its own width, and it already fills 1.129 of the cell dry — more than the whole of it, which is what an opaque jersey looks like from above. Wetting takes it to 1.355. So the reason a knit does not build a swelling pressure is not that it has room; it is that its dimensions are a loop length times a constant with no yarn diameter in them, so there is no closure condition to fail. What the drawing cannot show is the third dimension: the legs lie over one another rather than overlapping in the plane, which is exactly why an occupancy above one is possible. Knits and other structures

A loop has no closure condition

A woven cloth can run out of room: its two systems must supply its whole thickness between them, and past a certain swelling they cannot. A knit has no such equation, so no critical swelling and no pressure. The obvious explanation — that a knit is open and has somewhere to put the swelling — is false, and the arithmetic refuses it.

 The channel through a muslin in plain. A cut across a muslin woven plain, at 373 µm per hundred pixels, showing one hole between two warp ends. The ends sit at the same level here, so the clear width between them is 250 µm at the waist against 250 µm straight through. The profile on the right is that width at every height in the 334 µm the threads occupy: the passage is an hourglass, its narrowest section is 250 µm across, and the band that every level contains — what a straight line of sight can use — is 250 µm. The two differ by 0.0 per cent, and the difference is the weave and nothing else. What cloth is

A hole is a channel, not an opening

Every hole drawn so far has been drawn from above, and the sentence underneath every one of them says the same thing: the hole between four threads is the spacing less the diameter, and every hole is the same size. That is a picture of a cloth's shadow. A cloth has a thickness, so its hole has a length, and the narrowest place along it is not the place a person looking through can see.

 A 40-by-40 motif on a poplin, drawn and finished. A motif 40 ends wide and 40 picks tall, at the setts the reed and the take-up were set to, and the same motif measured on the finished cloth. Point paper has one cell per end and per pick, so a motif's proportions are the ratio of the two setts — and both setts move in the finishing, in opposite directions. On the poplin the aspect changes by a factor of 0.8429, so a circle drawn as a circle at the loom's numbers comes back 15.7% out of round and a designer who wants a circle must draw an ellipse of 1.1863. What the drawing cannot show is that the correction is not a property of the design: it belongs to the cloth, so the same card woven on a different construction is a different shape. Pattern and colour

A motif is drawn at the wrong shape on purpose

Point paper has one cell per end and per pick, so a design's proportions on the cloth are the ratio of the two setts. Both setts move in the finishing and they move in opposite directions, so a circle drawn as a circle comes back out of round — by three per cent on a balanced cloth and by sixteen on a warp-dense one.

 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. Weaves

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. Weaves

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 crimp ratios a dense shirting can have. Poplins in 15 tex warp and 20 tex weft at 22 picks, from 32 ends per centimetre to 52. Each bar is the interval of warp-to-weft crimp ratios the construction admits at all: the warp must supply at least the thickness the weft cannot reach, and at most what it can reach itself. The rule at one is this site's standing default. It sits inside the interval up to 44.18 ends per centimetre and outside it beyond — so for a dense shirting an equal division of the crimp is not merely the wrong assumption but a geometric impossibility. The 44-end poplin this site's own cloth table called impossible for a long time sits a fifth of an end below that limit, which is why the solver failed on it: its feasible interval was real and narrow, and a bisection on the whole range walked away from it. Setting and geometry

The cloth that was called impossible

This site's own table of fabrics carries a note saying a real 44-end poplin has no solution in its geometry at all, and that the poplin row was therefore set at 32 ends. The cloth solves. What had no solution was the search — a bisection that treated a state it could not reach as evidence of having gone too far, and walked away from the answer every time.

 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. Weaves

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.

 The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 221.0 µm from the highest point to the lowest, the root-mean-square roughness is 74.9 µm, and 30% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 190.8 µm and the weft at 182.8 µm, a step of 8.0 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far. What cloth is

A cloth has an outside

Every quantity in this collection is a property of the inside of a fabric — a crimp, a cover, a hole, a fibre count. None of them says where the cloth stops. The outside is a height field the draft computes, its crowns stand at two different levels, and which of the two is higher decides what the cloth touches the world with.

 Seven weaves at one construction, and what each of them really passes. Every weave here is drawn at the same filter — the same yarn, the same sett, the same cover and therefore the same open area, equal to twelve decimal places. Each one shows an opening of 263.7 µm to anybody looking straight through it, and that is the number a specification quotes. The bar is what each will actually let past, which is the narrowest section anywhere along the channel rather than the narrowest view down it. The plain weave passes exactly what it shows and it is the only weave here that does: its ends transit at every gap, so the four threads round every hole are level in pairs and the waist is at the middle. A float leaves two ends together at the top of the cloth, their neighbours at the bottom, and a passage between them that is wider than its own mouth — up to 12.8 per cent over the rating, for the basket. Cloth doing a job

A filter is rated by the hole it does not show

A woven filter cloth is sold on two numbers pulling opposite ways: an opening small enough to hold the soil and an open area large enough to pass the water. Both are computed from the same holes, and they are not computed from the same statistic of them. One reads the maximum and the other reads the mean, so a change that improves either can worsen the other without moving a single measurable property of the cloth.

 Three places a mistake can be made, and three shapes it leaves. A loom holds a design in three separate objects, and a single mistake in each of them produces a fault of a completely different size — not because the mistakes differ, but because of how many intersections each object controls. On a 50 m piece of muslin 1500 mm wide, holding 396,000,000 intersections: one end drawn on the wrong shaft is wrong at every pick for the whole length, 110,000 of them; one pick made in the wrong shed is wrong across the whole width once, 3,600; and one shaft tied wrongly to one treadle is wrong wherever that shaft's ends meet that treadle's picks, which is everywhere — 6,187,500, or 1.6% of the cloth. The ratio between the extremes is 1719 to one, and the largest of the three is the one nobody sees happen, because every thread is exactly where it should be. Compound and figured cloths

Three mistakes and the shape each one leaves

A loom holds a design in three separate objects, and one error in each of them is the same size of error. What they cost differs by a factor of seventeen hundred — and the largest of the three is the one nobody can see happening, because every thread is exactly where it should be.

 The bearing curves of 4 weaves in one cloth. How much of the plan is within a given depth of the highest point, for plain, 2/2 twill, satin 5, satin 8 — all in sheeting, all at the same sett, the same counts and the same thickness. They differ only in their drafts. At a hundredth of the cloth's thickness the last of them is touching 9 times the area of the first, and the gap widens as the depth shrinks, because the curves do not merely differ by a factor — they have different exponents. A crown that is a line opens as the square root of the depth and a crown that is a point opens in proportion to it. What cloth is

The curve that says what a cloth touches with

Take a cloth's surface and ask what fraction of the plan lies within a given depth of its highest point. The answer is one curve, it answers every question of the form what does this touch, and its behaviour at the top is decided by a single bit of the draft — whether the longest float is one crossing or more than one.

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

How sharply a weave lets a cloth fold

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

 The crown line of every four-by-four draft there is. All 22,874 four-by-four drafts in which every end and every pick interlaces at least once, at sheeting's construction, counted by how much horizontal crown line each carries per square millimetre. The bar at zero holds 2 of them — the two plain weaves, and nothing else in the catalogue. Every other draft has a float somewhere, and a float is a plateau, and a plateau is a line of constant height. So the whole catalogue divides into two drafts that touch at points and 22,872 that touch along lines, with no intermediate case, because a float is either present or it is not. What cloth is

Two drafts of twenty-two thousand

Every four-by-four draft there is, measured by how much horizontal crown line its surface carries. Two of them carry none — the plain weave and its translation, and nothing else in the catalogue — and the quantity turns out to be smallest for the most balanced cloths and largest for the most one-sided, which is the opposite of what a float count suggests.

 A damask's figure and ground trade places when the cloth is turned. A satin 8 figure on a sateen 8 ground in sheeting — one cloth, one set of threads, one sett, and the ground is the figure's own complement. Their total specular areas are within a few per cent of one another, so neither is intrinsically the brighter. What differs is the direction: the figure's crowns run with the warp and the ground's with the weft. So the contrast between them is 2.0-to-one with the light coming from 8° and 0.47-to-one from 90° — it reverses, exactly, a quarter turn apart. That is what makes a damask visible in one colour, and it is not the step in its surface: the step is fifty micrometres and returns no light at all under a diffuse illumination, while this contrast is a factor of 2.0 and is present whenever there is a direction in the light. Pattern and colour

A figure shows by its shine, not its step

A damask is one cloth in one colour and its pattern is plainly visible. This collection attributed that to the step in its surface — fifty micrometres of relief, computed from the interlacing rates. The step is real and returns almost no light. What makes the figure visible is that its crowns run at right angles to the ground's, so the two trade places when the cloth is turned.

 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. Weaves

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 V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it. Compound and figured cloths

A chenille is a yarn that is already a fabric

A chenille yarn is a woven gauze cut into strips. Every structural question this collection asks about cloth can be asked about it — does it hang together, how many systems has it, what holds the pile in — and the answers are the same answers one level down.

 The presser foot sinks 1.8 µm into a 2/2 twill. A thickness gauge presses a flat foot onto the cloth at 1 kPa and reads the gap. It does not read the geometric thickness. The foot sinks until the area it is touching can carry the load, and that is 2.57% of the plan at a depth of 1.8 µm — so a 2/2 twill in sheeting whose outside stands 381.6 µm apart measures 379.8 µm. How far the foot sinks is a property of the draft, because the bearing area near the top is, and a weave with plateaux stops the foot in a fraction of the distance a plain weave lets it travel. The transverse stiffness used here is fitted to measured fabric thickness rather than predicted: across its published range the reading moves between 377.0 µm and 380.6 µm. What cloth is

A thickness gauge reads the draft

A presser foot does not stop at the top of a cloth. It sinks until the area it is touching can carry the load, and how far that is depends on the shape of the bearing curve near the top — which is a property of the weave. So there is a weave term inside a measurement nobody thinks of as a weave measurement, and it is worth about one per cent.

 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. Weaves

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.

 Hair, nap and pile are one construction at three settings. 12 mm of sheeting at one pair of scales, carrying each of this site's three protruding surfaces — 50 pixels to the millimetre along the cloth and 29 off it. They are the same object, fibre standing off a cloth with a density, a length and an anchor, and they differ by 9-fold in density and 4.8-fold in length. What actually separates them is the third line under each name: hairs are held by whatever the twist happened to leave, a nap by the float it was pulled from, and a pile by a W or a V through three picks. Only the last two were chosen. The pile's tufts are drawn in the float colour because that is what a woven pile is, and they are all one length because a blade cut them; the hairs and the nap are drawn as measured quantities and their lengths come from the population's own exponential. Each panel clips to the room it has, and the densest of the three is drawn at the model's own number rather than thinned. Compound and figured cloths

Hair, nap and pile are one construction

This site now has three surfaces made of fibre standing off a cloth, arrived at from three different directions and in three different fields. They are the same object at three settings, and what separates them is not what they are but how much of them anybody decided.

 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. Weaves

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

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

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

 The float a point fault gives a weave, over every draft there is. Every four-by-four draft that describes one cloth — 22,730 of them — laid out as 8 ends by 8 picks, with each intersection of its repeat reversed in turn and the longest float that produces kept. There are two answers and no others. 90 drafts hold the fault to a float of 3; 22,640 — 99.6% of every weave there is — hand it a float of 7. The mechanism behind the second number is that a reversal at a binding point joins the two floats on either side of it, so a weave with runs separated by single binding intersections gives a single mistake the sum of two of its own floats. The 90 that escape are the drafts whose floats are short and evenly spaced — the plain weave and the ribs — which is not the class of weave the trade recommends for hiding a fault. Weaves

Which weave hides a fault

The trade says a busy weave hides a mistake. Every four-by-four draft there is was laid out as cloth and given every wrong lift it could have, and ninety-nine in a hundred hand that single mistake a float of seven — because a reversal does not lengthen a float, it joins two.

 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. Pattern and colour

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.

 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. Pattern and colour

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.

 A two-layer interchange, 1 block by 2. A two-layer cloth whose layers change places from block to block, drawn as the draft and as the section a weaver would draw. Each block is one repeat of the stack, 4 picks by 4 ends, and the design has 2 boundaries in it. Every strand in the draft is coloured by the cloth this site's criterion puts it in, and there is one colour, because there is one cloth — with no intersection reversed anywhere in the repeat. A stitch joins two layers at a point; an interchange joins them along a line, and the line is the design's own block boundary rather than anything added to it. The section below is schematic: it draws each ply as a line rather than as its threads, because what has to be seen is that the two lines cross, and a section at thread level over 2 blocks is a picture nobody can count. Compound and figured cloths

An interchange joins what a stitch would have had to

The rung below spent a whole essay on where a stitch may be put in a double cloth, and found face weaves with nowhere to put one at all. A design in which the two layers change places from block to block needs no stitch anywhere: the boundary is the join. Two blocks side by side make one cloth with not a single intersection reversed, and the same two blocks with no boundary between them make two.

 A colour order against a 2/2 twill. The visible face of a 2/2 twill under 2 colour orders, drawn at the repeat the divisor arithmetic allows and outlined at the repeat the surface has. A filled cell is a dark thread on the face, which is the warp's colour where the warp is up and the weft's where it is not — so none of these patterns is in the draft, and the draft is the same in all of them. The colour period and the weave repeat beat exactly as a reed's grouping beats against a weave: the surface repeats on the least common multiple of the two, which here is 8×8 and 4×4. What the panels cannot show is colour: the two threads are drawn as filled and empty, and two colours of similar value make a pattern far weaker than this. Pattern and colour

A colour order beats the weave it is threaded on

The reed's grouping beats against the weave repeat and the arithmetic is a least common multiple. A colour order is a second grouping of the same warp and the arithmetic is identical — but where the reed's beat is a fault to be dented out of a cloth, the colour order's beat is the pattern the cloth is sold for. Across 472 colour orders on four weaves the divisor bound is the surface's exact repeat in 470 or more, and the handful that beat it have no pattern left at all.

 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. Weaves

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.

 How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted. Knits and other structures

Where a two-bed fabric's yarn is

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

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

A damask's edge floats further than its figure

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

 A crepe's search has 4,416 winners and the surface separates them. All 5,040 rearrangements of the base this collection's crepe is built on, scored by how unevenly their crown line is spread over the repeat. 4,416 of them reach the correlation floor, which is the criterion the crepe was chosen by — so that criterion is not choosing, it is tying, and the search takes the first of a very large set. 28 of the rearrangements have a perfectly even surface, the bar at zero, and 16 of those are also at the correlation floor. The crepe actually drawn, marked, sits at 0.236 — the thirty-eighth percentile, better than most and not at the floor. The improvement is available, it costs nothing, and no criterion this collection had could see it. Weaves

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

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

 Which knitted fabrics lie flat, counted from the structure matrix. The curl balance of every named two-bed structure this site holds, with a tuck counted in full on the bed that took its yarn: the yarn a repeat puts on the front bed minus the yarn it puts on the back, over the total. A fabric lies flat exactly when it is zero, and the criterion has to put single jersey at one end and a one-by-one rib at the other or it is worth nothing — which it does, at 1.00 and 0.00. What it is for is the rest: a tubular fabric balances because it is two jerseys facing opposite ways, both cardigans balance because a tuck holds yarn on the bed that took it, and half-milano and a three-by-one rib come out front-heavy — which is what they are and what they do. 6 of the 13 structures curl. Knits and other structures

Which knitted fabrics lie flat

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

 How much of a fabric's yarn crosses between the beds. The share of half periods that cross from one bed to the other, read off each structure's own traverse rather than quoted. It is the mechanical difference between these fabrics in this account: a half period that crosses climbs the whole bed gap and one that does not climbs a yarn diameter. Single jersey and a tubular fabric come out at zero — the tubular one because its two faces are made on separate courses and never meet — and a one-by-one rib comes out at one, with every sinker loop crossing. A two-by-two rib is at a half, which is the number a reader would guess and is here counted. Knits and other structures

A tube and an interlock balance for different reasons

Two structures come out identically flat on a signed count and are as unalike as two knitted fabrics get. One is two jerseys that curl in opposite directions and are joined at the edges; the other is a fabric whose every course crosses. Telling them apart needs a different question asked of the same grid.

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

A weft stripe is counted in pairs of picks

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

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

A woven cloth is not linked at all

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

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

No weave draws an unbroken line one thread wide

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

 A woven cloth asked the same question, and the answer is nearly one. The closest approach two crossing threads make, for four cloths from an open voile to a dense duck, in units of the separation they have where they touch. A value of one means the closest approach is exactly at the crossing and the cloth fits together; anything below one is an overlap. The values run from 1.000 to 0.955, so the worst overlap in the table is 4.5% of a contact separation — against 22% for a knitted fabric. The overlap rises with the crimp, which is what identifies the mechanism: the vertical gain from moving away from a crossing is the crimp, and a cloth that barely crimps has nothing to gain by moving. Weaves

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.

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

A repeat has to fit the width

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

 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. Weaves

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.

 What a profile draft can reach at 4 by 4. The share of the 22,874 interlacing 4-by-4 drafts that a profile draft can express, at two block sizes. A profile is a grid of blocks each carrying a figure weave or a ground weave, so its image is every draft reachable by any choice of the two weaves and any assignment — which is enumerated here rather than argued: 4,096 combinations at the larger block, and the distinct results counted. With two-by-two blocks it reaches 306 drafts, which is 1.34 per cent. With one-by-one blocks the profile is the draft and it reaches all of them, which is the control. What the bars cannot show is that the reachable drafts are the useful ones: every figured cloth ever woven is in the small set, and the notation is narrow because designs are. What cloth is

A profile draft is a notation whose alphabet is weaves

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

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

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

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

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

The four named weaves are corners of a family

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

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

A lifting plan says nothing without a threading

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

 The lattice under 8/3 and 10/3. Satin marks drawn as points over two repeats, the 8-end satin on a move of 3, whose closest marks are √8 apart and next √10, so its marks line up 45° off the weft; and the 10-end satin on a move of 3, whose closest marks are √10 apart and next √10, two equal directions at right angles and so no single diagonal. For a regular satin the blue arrow is the shortest lattice vector and the red the next, and the faint lines run along the shortest through every mark — the diagonal the marks make. What the drawing cannot show is whether an eye finds that diagonal in woven cloth, where the marks are not points but short interruptions of a float, and where the yarn's own twist lies across them at an angle of its own. Weaves

Most satins still have a diagonal

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

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

An unbroken line is not a clean one

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

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

The census counted two systems and a surface has one

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

 The selvedge turns of a 2/2 twill, 4 ends wide, from the left. A strip of 2/2 twill 4 ends wide over 8 picks, the first thrown from the left, with the weft's turn between every pair of picks drawn at the edge it reaches. 0 of the 8 turns are caught, where the edge end is on the other face on the second pick, and 8 slip. Across all its edge placements the weave catches every turn at 8 of 16. What the drawing cannot show is how far a slipped loop travels, which the beat-up and the weft tension decide. Weaves

A selvedge holds only where its edge end changes face

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

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

Four drafts in five have no path along their own crowns

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

 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. Weaves

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. Weaves

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.

 What an irregular satin buys, order by order. For each order, the best regular satin's and the best irregular satin's scatter at the order's own best spread — the largest share of the closest pairs that point in one direction, where one is a line and less is a scatter. 5 ends: regular 0.50, irregular none at the best spread; 6 ends: regular none exists, irregular 0.25; 7 ends: regular 1.00, irregular 0.33; 8 ends: regular 1.00, irregular none at the best spread; 9 ends: regular 1.00, irregular 0.25; 10 ends: regular 0.50, irregular none at the best spread; 11 ends: regular 1.00, irregular none at the best spread. Irregularity buys something at 6, 7, 9 and nothing at the rest, and where it buys it scatters over four directions with no more than a third in any one. What the bars cannot show is whether a reader sees the difference, which is a question about a visual system. 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.

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