Generator

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.

37 essays call tally. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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 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.

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 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.

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.

Two sections, one yarn. The same yarn given a circular cross-section and a racetrack one of equal area, both jammed. The spacing the two models allow is nearly the same; the cover and the cloth thickness they predict are not.

Setting and geometry

Peirce against the racetrack, measured

Two models of a yarn's cross-section, given the same yarn and the same closure condition, agree on how densely the cloth can be set and disagree by nearly half on how thick it is. Which quantity is being asked about decides whether the choice of model matters at all.

1×1 rib — alternate wales to the back. A knitted fabric seen in section across the wales. Alternate wales pulled to the back fold the fabric like a concertina, so its relaxed width is a projection; pulling it wide unfolds the section and no yarn changes length while it happens.

Knits and other structures

Rib and interlock

A rib fabric is a plain knit folded like a concertina, and its enormous widthwise stretch is the fold opening out. Nothing in it is elastic, and the extension available is a cosine.

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

Four ways a fabric gets longer without stretching. Extension available from each mechanism, computed from the geometry that provides it. None of these numbers involves a yarn changing length; every one of them is a shape changing, and they differ by an order of magnitude.

Knits and other structures

Why a knit recovers and a woven does not

Four fabrics get longer without a single yarn stretching, and the four mechanisms are worth wildly different amounts. Three of them are recoverable and one is very nearly not, and which is which follows from where the extension came from.

What a tuft's wrap is worth at μ = 0.3. The two classical ways of binding a cut pile into its ground, with the holding force each provides as a multiple of the tension applied to the free end. The model is the capstan equation and the friction coefficient is measured rather than derived, so the ratio between the two matters and neither absolute number should be quoted alone.

Compound and figured cloths

How a tuft is held

A V-fastened tuft and a W-fastened tuft are both attached, and the integrity criterion returns the same verdict for both. One of them is specified for hotel corridors and the other sheds. What separates them is not a topological property at all — it is an angle, and it needs a coefficient somebody has to measure.

The ratchet a wool fibre is. A fibre with its scales, and the two strokes of one cycle of agitation. The push is the same in both directions; the distance is not, because the scales resist tip-first motion more than root-first. Every cycle therefore nets a displacement in one direction, and no amount of further agitation undoes it.

After the loom

The ratchet that makes wool felt

A wool fibre is covered in scales pointing one way, so it slides more easily root-first than tip-first. Agitate it and the motion is symmetric while the result is not — every cycle nets a displacement in one direction, and no amount of further agitation undoes it.

What a second guide bar buys. Independent fabrics left by each lapping across 12 wales, counted by walking the wale graph and checked against the greatest common divisor of the shogs with the width. A bar that leaves more than one is not making cloth; a second bar can put right what the first could not.

Knits and other structures

What a second guide bar is for

One warp-knit bar leaves the wales in as many independent fabrics as its shog shares factors with the width. Two bars leave the greatest common divisor of both — so a pair of shogs that each fail alone can succeed together, and a pair that share a factor cannot.

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.

$What a balanced cloth wastes under pressure. The fraction of a balanced fabric's fibre that is along for the ride, in a stress field of each ratio. A closed cylinder is exactly two to one — the ratio of the two areas the pressure acts on — so a balanced cloth reaches its limit around the circumference with the axial system at half its capacity, and a quarter of the fibre is doing nothing.$ Cloth doing a job

An inflated cylinder wants an unbalanced cloth

Balance is a virtue in almost every other cloth. Under pressure it is a defect with a size — a closed cylinder carries exactly twice the stress around its circumference as along its axis, so a balanced fabric reaches its limit in one direction with a quarter of its fibre doing nothing at all.

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

Compound and figured cloths

Corduroy is a cut float

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

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.

The angle a pressurised hose wants. A fixed length of yarn wound on a cylinder at a stated angle, with the radius and length the geometry gives, beside the volume it encloses as the angle varies. The maximum is at arctan √2 — 54.74° — with no material constant anywhere in it, and the same angle comes out of balancing the hoop and axial stresses, which is a different calculation with the same answer.

Cloth doing a job

The angle a hose wants

A braided hose has one angle at which pressure neither lengthens it nor shortens it, and the angle is arctan √2 — 54.74° — with no friction coefficient, no modulus and no fitted constant in it. Two arguments that share no algebra arrive at the same number, and which side of it a hose was braided on decides which way it moves.

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.

Pull it lengthways and it narrows. The same cloth before and after a small extension along the warp. No thread has stretched: the extension came out of the warp crimp, that crimp went into the weft, and the fabric is narrower for it.

Setting and geometry

What crimp interchange actually conserves

Pull a cloth lengthways and it narrows, because the crimp moves from one system to the other. Inextensibility says that much and no more — it is one equation short of an answer, and the second equation everybody uses is an assumption with a name.

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

Weaves

Floats and abrasion

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

What a shrink-resist treatment has to do. Net displacement per cycle of agitation as a treatment closes the gap between the two friction coefficients. The chemistry is sold as gluing the scales down; what it has to achieve is arithmetic — make the fibre slide equally well both ways and the ratchet has nothing to rectify.

After the loom

Shrink-resist is one number

Machine-washable wool is sold as a coating that glues the scales down. What the treatment has to achieve is narrower and more exact — make the fibre slide equally well in both directions, and the ratchet has nothing left to rectify, whatever the friction happens to be.

Terry at a let-off of 5 to one. Towelling has two warp beams. The ground warp is held at ordinary tension; the pile warp is let off 5 times as fast, and the excess has nowhere to go but up. The loop height follows from the ratio and the pick spacing, and no float length appears in it.

Compound and figured cloths

Terry needs two beams

A towel's pile is not woven longer. It is fed longer — from a second warp beam let off several times faster than the first — so the loop height follows from a ratio and a pick spacing, and the float length that decides a corduroy does not appear in it anywhere.

Yarn per needle, by structure. Wale spacings of yarn per needle position, at one loop length. A knitted loop is 4.30 of them — Munden's constant, measured rather than derived — a float across one needle is exactly one, and a tuck is taken as a stated multiple of a loop. Everything else follows by counting, so the percentages beside the bars are not estimates.

Knits and other structures

What a tuck costs

A knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop. That much is a count. What it does to the fabric's size is not, and this essay is careful about which is which.

Where a bias cut's waste actually is. A bolt of cloth with panels placed at a stated angle, the ones that fit drawn and the rest of the cloth left shaded. Identical panels at a single angle tile the plane exactly, so the interior of the bolt loses nothing at all and the whole of the waste is at the two selvedges. The inset is the single-panel bounding box, which is the picture the usual account of bias cutting draws.

Cloth doing a job

The bias cut and the selvedge

A bias-cut square costs exactly twice its own area, and every other shape costs more. That is an exact result about one panel — and it is not where a cutting room's waste comes from, because identical panels at one angle tile the plane. The loss is at the two selvedges, so it falls as the cloth gets wider, which no account in terms of the diagonal can explain.

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

Designing to a float limit

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

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

Setting and geometry

Twist and the twill line

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

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.

Every cloth at 150 grams. The counts and setts that all weigh 150 g/m² at a crimp of 7 per cent and a balance of 1. Hollow marks are past the jam — arithmetic rather than cloth. Among the 10 that can be woven the cover factor runs from 0.28 to 0.77, a factor of 2.77, and every one of them is the fabric the specification asked for.

Setting and geometry

What a fabric weighs

Every fabric is sold by its weight in grams per square metre, and the number is a sum of four products in which no term appears alone. One equation, four unknowns: a hundred and fifty grams describes an open coarse cloth and a close fine one, and the cover factors differ by a factor of nearly three.

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 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.

What a raising machine can catch in a 3/1 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

After the loom

Raising spends the cloth's strength

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

After the loom

A knit's dimensions come from its loop

A relaxed plain knit's courses, wales and stitch density depend on the loop length and on nothing else — not the yarn count, not the fibre, not the machine gauge. The constants are measured rather than derived, and the interesting thing about the published set is that it does not quite satisfy its own arithmetic.

Heddles per shaft: stripe. The threading of a narrow satin stripe on a broad plain ground, over a warp of 1,200 ends. Each bar is one shaft and its length is the heddles on it. The draft needs 10 shafts however they are loaded; the heaviest carries 500 and the lightest 25, a factor of 20.0. Spending 20 shafts instead brings the heaviest down to 100.

Compound and figured cloths

Where the heddles go

A draft says how many shafts it needs and says nothing about how the ends divide between them. On a satin stripe over a plain ground the two ground shafts carry twenty times what the stripe shafts do — and the only cure is to spend shafts, which turns the threading into an allocation problem with an exact answer.

After the loom

Why felting needs water

Wool felts in a wash and not in a drawer, and the usual explanation is that water lubricates the scales. It does the opposite of that. Water lowers one of wool's two friction coefficients and raises the other, so it widens the gap the ratchet rectifies — and what follows is a saturating function of the ratio, not of either coefficient.

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