Textile Structure
how cloth is put together
Generator

Where the holes are in a 2/2 twill, and how big each one is

Where the holes are in a 2/2 twill, and how big each one is
Where the holes are in a 2/2 twill, and how big each one is. One repeat of a 2/2 twill at a muslin's construction. The point paper is the draft; the marks between the squares are the 16 holes the repeat has, each shaded by what it would let past. They run from 274 µm to 274 µm, in 2 distinct sizes, against an opening of 250 µm that every one of them shows when looked straight through. A plain weave in the same cloth returns one size and one only, because its ends transit at every gap and are therefore level in pairs; a float leaves two ends side by side at the top of the cloth and their neighbours at the bottom, and a hole bounded by one of each is wider at its waist than at its mouth. The rating a filter cloth is sold on is the largest of these, which is 9.9 per cent over the figure the specification quotes.

One repeat of a 2/2 twill at a muslin's construction. The point paper is the draft; the marks between the squares are the 16 holes the repeat has, each shaded by what it would let past. They run from 274 µm to 274 µm, in 2 distinct sizes, against an opening of 250 µm that every one of them shows when looked straight through. A plain weave in the same cloth returns one size and one only, because its ends transit at every gap and are therefore level in pairs; a float leaves two ends side by side at the top of the cloth and their neighbours at the bottom, and a hole bounded by one of each is wider at its waist than at its mouth. The rating a filter cloth is sold on is the largest of these, which is 9.9 per cent over the figure the specification quotes.

8 essays call through-plan. 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.

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

A hole is a channel, not an opening

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

 Two layers of one cloth, against how they happen to lie. Two identical muslins laid over one another and slid across each other by one thread spacing. In register the pair passes 37.9 per cent — as much as one cloth, because every hole is over a hole — and it falls linearly to 12.5 per cent before rising again at the next thread. The rule everybody uses is that two layers pass the product of their open areas, which is 14.3 per cent. That number is the average of this curve over all offsets, exactly — an identity, not a fit — and it is the answer at two points on it and nowhere else. Nothing about a real pair of layers is at its average, and the openness varying from place to place across a folded cloth is what a moiré is. Pattern and colour

Two layers are the product on average and nowhere

Everybody knows what two layers of a cloth pass: the product of their open areas. That figure is exactly right — it is the mean of the true answer over every way the two layers can lie — and it is the answer at two registrations out of a continuum. In register a doubled cloth is as open as a single one; half a thread out it can be shut completely. The variation across a folded curtain is what a moiré is.

 How open a batiste is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this batiste it is 40.1 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 39.0° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 6.42 per cent open — 6.3 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none. Pattern and colour

Opacity is not cover

The covering rule counts a thread as a bar that stops everything, and a single fine cotton thread held to a window plainly does not. What a cloth transmits is the open area plus whatever comes through the threads, so the covered fraction is a lever rather than a barrier — and the lever is longest exactly where the rule says the cloth is most closed. Nothing here computes a thread's transmittance, and saying why is the useful half.

 When a knit has a hole between its loops, and when it has none. A loop's occupancy is its length times its diameter over the cell it sits in, and Munden's constants make that cell ℓ²/(k_c·k_w). The loop length cancels once and what is left is d·k_s/ℓ, where k_s is Munden's own stitch-density constant — so whether a knit has a hole between its loops depends on d/ℓ and on nothing else: no gauge, no count, no fabric dimension. The three curves are the three relaxed states of 20 tex cotton. Each crosses one at a loop length of 3.34, 3.63, 3.95 mm respectively, and a jersey is knitted at 2.63 to 3.44 mm — the shaded band. Every commercial jersey is therefore on the wrong side of the threshold in at least two of its three states: it closes its own holes as it relaxes, and its air goes through its threads rather than between them. Knits and other structures

A knit has no hole to lose

Every argument in this ladder is planar: threads at a spacing, a rectangle between four of them, a channel down it. Applied to a jersey it returns nothing at all, and the nothing is the finding. A loop's occupancy is its diameter times Munden's own stitch-density constant over its loop length, with no gauge and no fabric dimension in it — and the whole commercial range of tightness is on the wrong side of the threshold.

 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.

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

A filter is rated by the hole it does not show

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

 The 14 drafts whose holes are all one size. Every four-by-four draft in which each end and each pick interlaces — 22874 of them — built and asked whether all sixteen of its holes pass the same thing. At a muslin's construction 14 of them do. Set the same yarn square and 170 do; turn the cloth over and it is 14 again. Only 2 drafts are in all three lists, and they are the plain weave and its complement, marked. The other 12 are uniform because this cloth's warp is set closer than its weft, so the gap across the ends is the smaller of the two and binds every hole whatever the picks are doing — a fact about the sett wearing a fact about the weave's clothes. Two of them carry floats of three, which is as long as this repeat allows. Weaves

Only a plain weave has one size of hole

A weave's holes come in kinds, and the kinds are read off the matrix. Asking which weaves have only one kind looks like a question with an obvious answer and a one-line proof. Every draft at four by four was built and asked instead, and the count came back fourteen — of which twelve turn out to be telling the truth about the sett rather than about the weave.

 A slot and a square of the same area do not pass the same air. A hole of 62500 square micrometres, drawn out from a square to a slot twenty times longer than it is wide, at constant area throughout. The open area is unchanged by construction and the flow is not: it falls to 39 per cent of the square's. Two things move the same way and neither is a correction to the other — the hydraulic diameter falls as the rectangle is drawn out, and the shape factor rises from 14.23 for a square towards 24 for an infinitely thin slit, which is Shah and London's result quoted rather than derived. This is why a weave's float matters to what it passes even where its cover does not: a float lays parallel threads side by side and the hole beside it is a slot. Weaves

A satin's hole is a slot

Two cloths at the same cover have the same open area, exactly, and do not pass the same air. A float lays parallel threads side by side and the hole beside them is long rather than square, and drawing a hole out at constant area cuts what it passes to two fifths — because the hydraulic diameter falls and the shape factor climbs, and both of them move the same way.

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