Census — where it appears
Named by 41 essays across 8 fields — each of them below, with the objects they name alongside it.
Every cloth there is, at four by four
Sixty-five thousand matrices, twenty-two thousand weaves, and about a dozen with names. The complete census of the smallest interesting repeat is a map of a whole small world, and almost none of it has ever been woven.
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.
How many layers a draft can have
Of the 22,874 four-by-four drafts this site sweeps, 22,730 are one cloth and 144 are two. None is three, and none can be — a repeat of n ends holds at most n halved cloths, because every cloth needs two ends and two picks of its own before it interlaces at all.
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 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.
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.
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.
A rectangular block is not half a rule
The block rule was proved for square blocks and the rectangular case was recorded as not run, on the grounds that a block a repeat wide and half a repeat deep satisfies only half the condition. Running it turns up two things: half the condition rules out the failure nobody can see, and a block turned through a right angle is a different design — which a square census cannot notice, because a square block is its own transpose.
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.
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.
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.
A pick density is a force budget
The take-up ladder counted what a change-wheel take-up can select: 4,825 distinct pick densities between eight and forty threads per centimetre, against forty-nine warp setts a reed catalogue offers over the same range. That count assumed every setting is available. A beat-up force says otherwise, and cuts the top off the range without touching the fineness of the choice.
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 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.
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 braid is a third way to hold threads
Weaving holds by friction and knitting holds by linking. A braid does neither: its strands travel across the structure and back, so no pair of them is linked and no pair of them returns to where it started — and it holds without a reed, a beat-up or a sett.
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.
Lustre is a length times a width
The specular area of a cloth factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply. Neither factor knows anything about the other, and over the four-by-four catalogue the first alone spans a factor of sixty.
A 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 is flat in its draft and not in its surface
A crepe weave is chosen by pushing the draft's correlations as flat as they will go. That criterion turns out to tie: on the base this collection uses, 4,416 of the 5,040 rearrangements reach the floor. Sixteen of them additionally spread their crown line perfectly evenly — and the crepe actually drawn is not one of the sixteen.
A 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.
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 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.
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.
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.
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 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.
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 finest colour-and-weave effects need the rarest loom
A houndstooth, a shepherd's check and a gun club check are thrown by the cheapest shuttle loom there is. A hairline, an end-and-end and a log cabin are thrown by none — a colour on every other pick is thrown from the same side every time, so its shuttles pile up at the far end of the loom and never come back. The dividing line is the parity of the bands and nothing else, which is why it survives the fact that no two sources agree about how wide a shepherd's check 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.
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.
An even shading cannot keep its surface level
Sixty-four six-end shadings hold every middle tone to a float of two, and the question left over was whether any of them keeps a tone ramp's surface level the way a twill read in order does. None does, and all of them sink by exactly the same depth — 43.3 micrometres on a sheeting, a sixth of the cloth. The reason is a counting argument a recording engineer would recognise: a limit on how long a thread may float is a limit on run length, and a run-length limit forces a floor on how often the thread changes face. A level ramp needs every tone at the extremes' rate, which needs a float of half the repeat at the midtone and more beside it — exactly the floats the twill in order has.
A tuck is the one stitch that links twice
Knitting has three stitches and only one of them makes a new link. A knit stitch links a loop to the loop below; a miss links nothing; and a tuck holds two loops in one head — which is why a tuck stops a run and why the three cannot be described by one number.
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.
A jersey has no float for a wire to catch
Only a float can be raised, and the four-by-four census answers which woven cloths qualify: two. Asked of a knit the same question needs this collection's knitted float rather than a draft's, and the answer is that a plain jersey has no float at all — not a short one, none — while the ribs and cardigans have none either and the interlocks and milanos have floats every one of which is interior. Of 1,135 two-bed structures a machine could make, twenty present a float a wire could hook, and not one of them presents it on both faces.
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.
A colour-and-weave look costs its cheaper order
The finest colour-and-weave effects need the rarest loom because a weft order is thrown pick by pick and a warp order is laid out once, so an effect's price was said to be its weft. That is true of a construction and false of a cloth. The same cloth can be woven lying across the loom, with its warp order thrown as weft and its weft order laid in the warp, and then it costs the other order. Over every two-colour look twelve small weaves make with orders up to six threads — 4,036 of them — 55 per cent need a loom picking at will as drawn and 31 per cent need one either way round. The looks turning rescues are the ones fine in one direction only, and not one of the trade's named effects is among them.
A tuck decides whether a third of two-bed fabrics lean
A jersey leans because every loop is on one bed and a rib does not because its loops are mirrored across two, and the count that says so treated a tuck as nothing. A tuck is a loop of the same lively yarn wrapped round a needle of one bed, and nobody here has measured how much of a knit loop's lean it carries. It matters. Of the 1,135 two-bed fabrics a two-needle, two-course frame can make, 135 are balanced whatever a tuck carries and 612 lean whatever it carries — and 388, a third, are balanced under one answer and lean under the other. A half-cardigan leans a third of a jersey if a tuck carries nothing and not at all if it carries a full loop's torque, which makes it the instrument that would settle the question.
A turned block is a moved origin
An eight-end satin figured on a 2/2 twill fails on 55,536 profiles at a block eight picks by two ends and on none at two by eight, and that was read as a property of the block's shape. Sweep the relative origin as well and the two shapes trade places: at every one of the sixteen origins exactly one of them fails, and over the sixteen they fail equally often. A shape asymmetry that no origin removes exists, and it needs a satin whose move squared is not one.
A cloth's derivation class is its census of small patches
The manuals' derivations cut the 426 four-by-four cloths into 157 orbits, and an orbit is found by searching a group of 256 operations. The question left was whether a short list of numbers read off a draft could do the same job. The familiar ones cannot. Marks, interlacings, layers, plane group, float lengths, crossings per thread and distinct ends and picks are all invariant, and all seven together tell 120 of the orbits apart. A census of the two-by-two patches a draft contains tells 127 apart. A census of its three-by-three patches tells all 157 apart — a complete invariant of derivation, computed by counting windows rather than by searching operations.
What a closed thread cannot choose
A thread whose ends are held has a quantity it cannot change without breaking: the total number of times its material winds about its own axis, plus the number of times that axis winds about itself. The two can trade, and everything a twisted yarn does when it is let go is that trade happening.
Named alongside it
The objects these essays reach for when they reach for this one.
FloatCloth integrityPoint paperSatinConnectivityEnumerationFloat lengthRepeatStripeCatalogueColour orderCrown line