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.

 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.

 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.

 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.

 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.

 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 usual arithmetic 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.

 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 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 the standard 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.

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