Theme

The thread: Does it hang together

A draft can look entirely reasonable on point paper and describe cloth that falls into two layers, or has threads lying loose on the surface. The failure is invisible and the test is exact.

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.

One break, two outcomes. The same single break in a knit and in a weave. In the knit nothing holds the loop above the break, so the failure climbs the wale; in the weave every other thread is still held by the threads crossing it, and one thread comes loose.

Knits and other structures

Ravel, fray and run

Cut a woven cloth and one thread comes loose. Break one loop in a knit and every loop above it follows. The two structures fail in opposite ways, and the reason is topology rather than strength.

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.

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 tricot lapping. A warp-knit lapping drawn from the guide bar's movement. Each thread is coloured by the group of wales it belongs to, so a lapping that leaves the wales independent shows as several colours. This one joins them into 1 group.

Knits and other structures

Warp knitting, which is a different thing entirely

Every wale has its own thread, and if the thread never leaves its wale the fabric is a set of independent cords. Whether a lapping makes cloth is decided by a coprimality condition — the satin theorem, in a knit.

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.

A web at 6 fibre lengths squared per unit area. Straight fibres dropped at random positions and random angles. The largest connected group is drawn solid; everything not joined to it is drawn faintly. Whether that group reaches both edges is what decides whether this is a sheet or a heap.

What cloth is

Nonwovens, and what holds them together instead

A web of fibres laid down at random has no repeat, so the exact test for whether a fabric holds together has nothing to work on. What replaces it is a threshold, and the threshold is sharp.

Setting and geometry

Balance, and what an unbalanced cloth does

How much of the surface each thread system takes is the mean of the weave matrix, and it is one of the very few quantities in this subject that can be read straight off. Almost nothing that follows from it can.

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.

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 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.

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.

3 cloths on 6 ends. A repeat of 6 ends and 6 picks holding 3 complete cloths, each with 2 ends and 2 picks of its own. The bars beside the strands say which cloth each belongs to; the ceiling at this size is 3, and the longest float is 5 because the face warp passes over every pick below it.

What cloth is

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.

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.

Milling, on the nonwoven's own scale. Migrated fibre accumulating with agitation, measured against the stick-percolation threshold this site uses for nonwovens. Above the line the fibre network holds on its own, which is why a milled cloth can be cut without fraying — the weave is no longer the only thing keeping it together.

After the loom

Milling holds the cloth a second time

This site's integrity criterion returns exactly the same answer for a melton as for the loose twill it was woven as, and it is right both times. What has changed is that a second network now holds the fabric together, made of migrated fibre rather than of thread crossings — and it is the one that decides whether a cut edge frays.

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.

The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica this site does not have.

What cloth is

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

One cloth, before the knife. Double plush: two ground cloths woven face to face with a pile warp shuttling between them. Uncut, the pile is the only thing either ground touches the other through, and the integrity criterion says one cloth. Cut down the middle, every pile end is severed and the same criterion says two. The whole manufacture is the deliberate destruction of an integrity the criterion otherwise exists to confirm.

Compound and figured cloths

Velvet is cut apart

Two ground cloths are woven face to face with a pile warp shuttling between them, and while it does the whole thing is one cloth — the criterion says so, and the pile is the only reason it is true. Then a knife runs down the middle and the same criterion says two. The manufacture is the deliberate destruction of cloth integrity.

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.

Where a thread stops sliding and starts breaking. A pick of sheeting gripped over a length of cloth, drawn one crossing at a time. The resistance is 0.30 times the 0.599 N each crossing presses with, so it rises with the length held; the breaking load of 3.74 N does not. The two are equal at 7.4 mm. What the drawing cannot show is that μ is a range rather than a constant, so the mark is a band and its position is exactly inversely proportional to the friction.

What cloth is

A thread is held one crossing at a time

Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.

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.

Compound and figured cloths

The criterion cannot see friction

This site's central check is exact, decidable in linear time, and structurally incapable of distinguishing a carpet from a fabric that sheds. That is not a defect to be repaired — a separation exists or it does not, and there is no margin in it — and the fancy weaves are precisely the constructions that live in the gap.

The block condition, one half at a time. Every block shape from 8×8 down, for 8-end satin figured on 8-end sateen, each an exhaustive census of all 65,536 four-by-four profiles. The bar is split: the dark part is figures that separate with a thread left loose on the face, and the light part is figures that separate with nothing visible wrong. Shapes satisfying the block condition in one direction only are marked, and none of them has a light part at all. What the chart cannot show is why: the census is exhaustive and the theorem behind it is not proved here.

Pattern and colour

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.

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.

$Everything the relative origin decides. Each of six quantities computed from a weave matrix, swept over all 64 relative origins of 8-end satin figured on 8-end sateen. A quantity that takes one value across the sweep is a gauge freedom of the pair; one that takes several is something a designer chooses without knowing it. longest float takes 2 values; interlacing rate takes 2 values; weft-face fraction takes 3 values. The origins fall into 3 different partitions, so the moving quantities are not all decided by the same number. What the table cannot show is the census over all 65,536 profiles, which is a seventh column, and the one that showed the origin decides whether the cloth holds together.$ 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.

$A spread shading on 8 ends. The 7 tone steps of a spread shading on an 8-end repeat, each built by adding one more coset of the 8-end satin with move 3. Every coset has exactly one mark in every end and every pick, so the fraction of warp on the face is k over 8 exactly at every step, with no averaging in it. The numbers beneath are the longest float, and they run 7, 3, 3, 1, 3, 3, 7 — so the lustre scale is not the tone scale. The midtone is a plain weave and the extremes are satins, which is why a spread shading is at its most matt exactly in the middle. What the drafts cannot show is the surface: a long float stands proud of a short one, so an evenly toned shading is not an even surface either.$ Pattern and colour

A shading changes two things at once

The tone steps of a shaded damask are exactly even — each adds one satin coset, so the fraction of warp on the face is k over n with no averaging in it. The lustre steps are not even at all: on eight ends the longest float runs 7, 3, 3, 1, 3, 3, 7, so a series that grades smoothly in tone is at its most matt exactly in the middle.

What a coating does to the rest of the site. Seven quantities this site computes for uncoated fabric, with what each becomes once a film bonds the crossings: the shear a cloth will take — changed in kind; tear strength — changed in kind; the loss from a hole — reversed; wicking — halved; air permeability — reversed; the sett's effect on strength — unchanged in sign; areal weight — added to. Two of the seven reverse outright. The table is a collection rather than a computation, and each row points at the essay whose result it qualifies.

After the loom

Coated is a state

Every mechanism on this site assumes threads that can move relative to one another — the bias is a mechanism because the crossings rotate, a tear runs because threads gather, a cloth takes a hole without minding because the neighbours pick the load up. A film bonds the crossings. Two of those results reverse outright, the rest change in kind, and no number on this site has ever said which state it belongs to.

A coating over the hole it has to bridge. Four warp ends of 40 tex seen end-on at 20 per centimetre, with a 120 micrometre PVC film over them. The film is supported everywhere it lies on a thread and unsupported over the 252 micrometre clear span between them, which is the same span at every hole in the repeat. The film's thickness is drawn to scale against that span; the bulge is exaggerated. At 20 MPa the film will hold at most 381 bar, which is a bound and not a prediction.

Cloth doing a job

A coated cloth fails at its holes

Spread a film over a fabric and it is supported everywhere it lies on a thread and unsupported over every hole. A woven cloth's holes are all the same size — exactly, because a repeat is a repeat — so the film has one span to bridge and the fabric has a burst pressure rather than a distribution. A nonwoven gives it a distribution, and a membrane fails at the largest hole it meets.

Which seams slip and which break. For each cloth, the grip a 10 mm seam allowance offers divided by the thread's own breaking load. Above one the fabric or the thread gives first and the seam holds until it does; below one the threads slide out and the seam opens with the cloth intact. voile, batiste, poplin, sheeting break; cheesecloth, muslin, duck, filter slip. The allowance that would save every cloth in the table is 74 mm, which is set by the openest of them alone. What the chart cannot show is the stitching itself, which has its own strength and its own way of cutting the threads it passes through.

Cloth doing a job

A seam slips before it breaks

A sewn seam fails in one of two ways and the trade names them separately: the threads pull out of the cloth beside the stitching, or something breaks. Which one a given cloth does is decided by whether its seam allowance clears the length at which grip beats strength — and at the ordinary ten millimetres, the table divides four against four.

How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

Compound and figured cloths

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

Knits and other structures

A jersey leans because its yarn still turns

A single-jersey T-shirt comes back from the wash with its side seam spiralling round the body, and a rib does not. The difference is not the yarn: it is a count. Loops on opposite beds are mirror images, so their torques oppose, and a fabric that knits equally on both nets to zero whatever the yarn is doing.

Compound and figured cloths

A wetting supplies the force the criterion needs

This collection's criterion decides whether a cloth is one cloth, exactly, and cannot see friction — so pricing what it misses needed a contact force, and the only one available had a measured fabric thickness inside it. A wetted close cloth generates one from geometry alone, and it lands within seven per cent of the measured route's answer.

Mechanics and drape

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

A sheeting's crossing, dry and wetted. One crossing of a sheeting in section at three swellings: dry, at the swelling where its geometry has its last state, and fully wetted at 20%. The closure condition is that the two systems' crimp heights add to the cloth's thickness, and each can supply at most the height it reaches when its straight portion has just vanished. Swelling raises the demand in proportion and the supply more slowly, so the margin closes and then goes negative — at 9.29% for this cloth against cotton's 20%. The bottom panel is drawn as far as the threads reach and no further, because there is no state to draw. What the drawing cannot show is what happens instead, which is that the yarn is compacted.

Mechanics and drape

The swelling a cloth cannot take

Three of the eight cloths in this collection's table have no wet state at all. A thread of fixed length cannot wrap a partner that has grown by a fifth, so the closure condition fails and the geometry has nothing to offer. What happens instead costs megapascals, and the alternative route is not merely dearer but unavailable — the thread would break first.

What holds a pick in. The holding force on one weft, as a multiple of the tension applied to its free end. For a plain weave it is the capstan on twice the weave angle, which Peirce's geometry gives at each sett and which falls towards nothing as the cloth opens out. For a leno it is the capstan on a half-turn, which the sett does not enter at all.

Compound and figured cloths

A leno's hole cannot drift

A filter cloth is rated by its largest hole, and the largest hole grows one micrometre for every micrometre an end moves sideways. What holds an end in place in an ordinary weave is friction at its crossings, and that friction falls smoothly to nothing as a cloth opens — with no threshold to warn anybody. A leno's crossing does not: its ends are wrapped through half a turn by construction, so the grip has no sett in it, and at an open cloth it holds eighteen times what a plain weave manages.

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.

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 fold the trade actually makes. 2 singles of 20 tex cotton at 800 turns a metre, folded at 566 — a ratio of 0.707. That is 1/√2, the ratio at which the fold's surface helix angle equals its singles' — 22.8° — because a fold of 2 singles is √2 times the diameter. The trade's own bracket for 2 folds is 0.6 to 0.75, and it contains this number.

Compound and figured cloths

A leno twists what a weave only crosses

Every woven cloth's threads have a linking number of zero, and that is why an open cloth slips. A leno is the one woven structure whose warp ends wind about one another, so it is the one whose threads are linked — and it is famously the structure that holds at setts where nothing else does.

Compound and figured cloths

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.

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.

How many layers the repeat, the harness and the beams each allow. Three ceilings on the number of layers a double cloth can have, for five layer weaves. The repeat's bound is half its ends and is a property of the notation. The harness's is the strain budget — 13 shafts on an ordinary broad loom at a 1.0% warp strain limit — divided by the shafts one layer of that weave costs. The beams' is how many warps the loom carries, which is 2. The shortest of each three is marked, and it is the beams at the coarse end of the table and the harness at the fine end; the repeat is never the binding one except at two-end layers, where it happens to coincide with the harness. A double cloth of eight-end satin layers needs 16 shafts and the budget is 13, so it is a jacquard construction by arithmetic rather than by choice. What the bars cannot show is the pick rate: a k-layer cloth needs k times the picks per centimetre of finished cloth and takes k times as long to weave, which is a cost rather than a ceiling and is the reason four-layer cloths are rare even where they are possible.

Compound and figured cloths

The repeat allows four layers and the loom allows two

A repeat of eight ends can hold four separable cloths, and the site has a witness that reaches the bound exactly. No loom weaves four. The harness's strain budget buys thirteen shafts and a layer costs its own weave's shaft count, so five-end satin layers stop at two and eight-end satin layers cannot be doubled on a dobby at all — and two differing layers already want a beam each. The notation's ceiling is the only one of the three that is never binding.

A knitted interlacing, and the number that makes it a fabric. Two loops idealised to rings, one drawn through the other, which is what a needle does. The Gauss linking integral returns 1. A knitted fabric of n wales has that between every pair of adjacent courses n times over, and it is the whole of why a knitted fabric can be made from one thread and taken apart by pulling it.

What cloth is

Why a knit runs and a weave frays

The two fabrics fail in two ways and everybody knows which is which. This collection has described both accurately for eighteen phases without being able to say what causes them, and the cause turns out to be one integer each: nought for a cloth, one per wale for a knit.

Two courses as centre lines, and their closest approach. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, as centre lines, with the closest approach marked. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

What cloth is

The closest approach is not the crossing

Two wavy curves that touch at a point are not necessarily closest at that point. Whether they are depends on one thing: whether they run alongside one another or cross. That distinction decides which of this collection's two fabrics fits together and which does not.

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.

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.

What cloth is

A fabric is a population of contacts

Only a fifth of a knitted fabric's yarn is inside a diameter of its neighbour. So a fabric's friction lives in a fifth of its length, and every calculation this collection makes about withdrawal, slippage and fraying has assumed it lives everywhere.

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

What a run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.

Cloth doing a job

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

What links what, in the two ways of making cloth. The linking number between two adjacent courses, for a knitted tube of 12 wales, for the same tube as this collection's model draws it, and for a woven cloth's two thread systems. The fabric's is 12 — one for every needle loop drawn through the loop below. The model's is -0.0000, because it places the interlacing at a point where two centre lines pass a diameter apart and two curves passing beside one another are not linked. The woven cloth's is -0.0000 and always will be, at any crimp and for every weave. That last row is not a defect of any model: a woven cloth really is unlinked, and it is the reason it frays where a knitted fabric runs.

Knits and other structures

A point cannot link

A knitted fabric of n wales has a linking number of n between every pair of adjacent courses. This collection's model of the same fabric has zero, and it has zero because the interlacing was declared to be a point where two centre lines pass a diameter apart — which is a near miss, and a near miss is not a knot.

Two numbers that stay at zero however large the fabric gets. The linking number of two adjacent courses, and the writhe of one course per wale, for tubes of 6 to 20 wales. Both sit at zero and stay there: the largest departure anywhere on the plot is 1.5e+1, which is the sampling. A quantity that should grow with the fabric and does not is the cleanest kind of null result: the model has the geometry of knitting and none of its topology, and making the fabric bigger does not make the topology appear.

Knits and other structures

Five symptoms of one omission

Five things this collection recorded as unexplained, found in four different ladders over three years of work. They are the same defect seen from five directions, and the defect is one sentence written for good reasons with no visible cost at the time.

Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

Knits and other structures

The fabric that does not fit

Every solve in this collection minimises an energy over a centre line, and a centre line has no thickness. Nobody had checked whether the fabric that comes out of it can be built. It cannot: two adjacent courses of the relaxed jersey approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere.

A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

Cloth doing a job

A run cannot cross a bed

A dropped stitch unroves because its neighbour above can pull it out along a path that costs almost nothing. In a rib the neighbour above is on the other bed, and the path goes through the gap — so a run in a two-bed fabric has to pay for a climb before it can take a single loop.

$A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot.$ Cloth doing a job

A knot is nothing but contact

A knot has no fastening in it. Nothing is glued, hooked, sewn or threaded through a hole: a thread is bent round itself until the friction where it presses on itself is more than the load. That makes a knot the purest contact problem in the subject, and the place to look first for what contact does.

Where a knot breaks

It breaks at the entry, before the knot has done any gripping at all. Two quantities run along a knot's path and only one of them rises; the other falls from the first millimetre; and their sum is largest where the thread arrives.

The extension ceiling, with the yarn given a thickness. As a jersey is pulled along its courses the wale spacing grows and the course spacing has to fall, because the yarn between two interlacings is a fixed length. The upper curve is how small the course spacing may be before the yarn runs out; the lower is how small it may be before two adjacent courses occupy the same space. The geometric ceiling is 322% and the contact one 299% — 7% lower. That is the result and it is a negative one: a measured jersey extends by about a hundred per cent, so contact between courses is not what puts the computed ceiling three times beyond a real one. The candidate this ladder was written to test is ruled out.

Knits and other structures

Contact is not why a jersey stops

The model says a jersey can be pulled to three hundred and twenty per cent along its courses. Real ones stop at about a hundred. The recorded diagnosis was that nothing stops adjacent courses passing through one another — and giving the yarn a thickness closes seven per cent of a gap of two thirds.

The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

Knits and other structures

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

Knits and other structures

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.

8×2 and 2×8 over every origin. Two grids of the 16 relative origins of 2/2 twill under 8-end satin, the row being how many picks the ground is started along and the column how many ends. The left grid is the census at a block 8×2, the right at 2×8; a square is filled where some of the 65,536 profiles separate. 8×2 fails at 8 origins and 2×8 at 8; both fail at 0 and neither at 0. Turning the cloth over and through a right angle sends each origin to another, and the letters mark where: every letter lands on a square with the same answer, so the two shapes are one census read at relabelled origins.

Pattern and colour

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.

Mechanics and drape

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one.

Mechanics and drape

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

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