A ridge in the matrix is not a ridge in the cloth

Four drafts in five have no path along their own crowns took the cells of a draft where one system is on the face and asked whether they join, edge to edge, across the repeat. A 2/2 twill’s do — its crowns climb diagonally into one ridge that runs off the edge of the repeat and on into the next. A 2/2 hopsack’s do not; they are two-by-two squares with nothing beside them.

It ended on a prediction. A coating fills the crowns before it bridges the holes had priced the volume a film must bury, and that volume is the same for a twill and a hopsack of one yarn and one sett. What should differ is the film at small add-ons: continuous on the twill from the first gram, and a set of separate patches on the hopsack until the film is deep enough to reach across between the islands. If that window is large, two cloths of identical surface volume give a barrier and a sieve.

The window turns out to be small, and the reason is that the twill’s ridge is not there.

A film lives on the surface, not on the matrix #

A film laid on a cloth — by a blade riding the crowns, by a transfer sheet pressed onto them, by a finish wetting down from the top — first touches the cloth’s highest points and then, as it thickens, reaches further down. The part of the cloth it touches at any thickness is every point within that depth of the top, and the question the prediction asks is the depth at which that region first becomes one connected piece across the whole cloth.

That region is not a set of matrix cells. It is a region of the cloth’s height field, the surface built from the threads’ paths from each thread’s centre line and section: warp and weft rising and falling over one another, with a gap between neighbouring threads wherever the sett leaves one. The matrix says which thread is on top at each crossing. The height field says how high it is there, and how far it is to the next one.

Where the matrix and the cloth disagree #

They agree about one kind of neighbour and disagree about the other.

Two face cells in one column of the matrix are one end floating over two picks, and on the cloth they are one continuous plateau along that end. The matrix and the surface agree: those cells are joined.

Two face cells in one row are two neighbouring ends, both on the face at one pick. In the matrix they share an edge. On the cloth they are two separate threads with the gap between them that the sett leaves — at a sheeting’s 28 ends to the centimetre, a gap of about a sixth of a millimetre — and a film at the crowns’ level meets nothing across it until it has sunk far enough to reach the pick running underneath.

A 2/2 twill’s ridge is built entirely out of the second kind of step. It climbs one pick and moves one end, again and again; every move to the next end crosses a gap. The ridge the census counted is a staircase whose risers are empty.

Both films join at the same depth #

On the sheeting, the film on the twill and the film on the hopsack become one connected film at the same depth.

Both join 161 micrometres below the crowns, at an add-on of 116 grams a square metre. The face is flattened at 186. So a thin film on either weave is a scatter of patches for its first six tenths, and at the moment it becomes continuous it is continuous for a reason that has nothing to do with the crowns: it has reached down to the threads that run underneath the gaps, and those threads join everything.

The census’s distinction between a ridge and a field of islands has not appeared at all. At this sett the two weaves are the same surface for the one question the distinction was meant to answer.

That is worth dwelling on because the two surfaces are not the same in other respects. Their crown line per unit area is identical, their float lengths are identical, and their bearing curves differ only in how the crowns are arranged — which is exactly the property the census was built to isolate. It is isolated, and at this depth it does nothing, because the film that could use it is still sitting on the separate threads when the arrangement would matter.

The weave matters at some setts, and by little #

That is at the sheeting’s own sett. Vary the warp sett and the two films part company, a little.

At 16 ends to the centimetre the twill’s film joins at 143 grams and the hopsack’s at 170; at 20 ends, 131 and 146; at 24 and 28, together; at 32, 113 and 124. The twill is never later than the hopsack, which is the census’s direction, and the window between them is between nothing and 27 grams — a tenth of the flattening add-on at its widest.

In every case both films join between 57 and 73 per cent of the way to a flat face. The window the prediction imagined — continuous from the first gram on one weave, patches on the other until the holes bridge — does not exist. What exists is a narrow band well past half the flattening add-on, inside which the twill is whole and the hopsack is not.

Why the open sett is where the weave shows #

The window opens where the gap between ends is wide, which is the opposite of what a reader would guess.

At an open warp sett the picks carry more of the surface: the ends are far apart and the picks between them rise nearly as high as the ends’ crowns. A film then reaches the picks early, and the picks join the ends — but only where an end is on the face at the pick next to the one it crosses, which a twill’s diagonal arranges at every step and a hopsack’s squares arrange only at their corners. So the twill’s staircase gets its risers filled by the picks at a shallower depth than the hopsack’s islands do.

At a close sett the ends crowd the picks down, the picks sit far below the crowns, and both films have to sink to the same depth before the picks can join anything. The weave’s arrangement stops mattering because the thing that would exploit it is out of reach.

A warp-faced cloth, where the weave shows most #

A poplin, set close in the warp and open in the weft, is the construction in the table where the ends crowd hardest and the picks lie lowest.

Here the twill leads at every warp sett in the sweep, by 8 to 18 grams, and both films join between 53 and 70 per cent of the way to a flat face. It is the closest the cloth comes to the census’s picture, and it is still a matter of grams in a hundred and twenty, not of a barrier against a sieve.

What a coater buys below the join #

The finding has a plain practical reading, and it is the same for both weaves.

Below the join, a film is patches whatever the weave: a set of separate pieces sitting on the crowns, each surrounded by bare cloth. It changes the crowns’ handle and their lustre, it may glue fibres at the surface together, and it is not a barrier to anything, because the bare cloth between the patches passes air and water exactly as it did. On a sheeting that is the whole range up to 116 grams a square metre — more than half the add-on that makes the face flat, and more than most of the cloth’s own weight.

Above the join the film is one piece and the holes are what is left. A coated cloth fails at its holes: the film that spans a hole between four threads is the weak point under pressure, and the join is the first add-on at which there is a spanning film to fail. So the join is where a coated cloth begins, and the flattening add-on is where its surface stops showing the weave.

Between the two lies the range in which a coating is a membrane over a textured surface, and the weave’s only influence on where that range starts is the narrow window the sweep found — up to 27 grams at an open sett, nothing at the sheeting’s own.

The same film, the other way round #

The argument runs in both directions, and the second is worth stating because it applies to every finish that is removed rather than added.

A cloth that has been coated and is then worn, rubbed or washed loses its film from the top down: the crowns wear first, then the film between them thins. Coated is a state found that a film changes a cloth’s mechanics by bonding its crossings, and a film worn back to patches bonds only the crossings under the patches; the joining depth is where the cloth passes from one state to the other. A worn film becomes patches again at the join, and it does so at the same depth on a twill and a hopsack of one sett — so a cloth that stays barrier-tight for longer under wear has a thicker film, not a better-connected weave.

What the matrix census is still right about #

None of this says the census was wrong about the matrix. The twill’s face cells do join edge to edge and the hopsack’s do not, and the census was careful to say it counted cells and not crowns. What this essay adds is the step from one to the other, and it is not free: an edge between two matrix cells in the same row is a gap on the cloth.

So the census answers a question about paths that stay on the face one end at a time — a thread drawn along the cloth’s length, riding one end’s floats — and for that question a column step is the only step that matters, and the twill and the hopsack differ as the census says. For a film, which has to cross between ends, the census’s edges are the wrong edges.

The prediction about prints and threads, revisited #

The census essay named two other things its distinction should decide, and the height field says something about each.

A printed edge was predicted to bleed further along a twill’s ridge than across it. Ink carried by the crowns has the same problem a film has: to travel along a twill’s diagonal it must cross the gap between ends at every step. So any directional bleed a twill shows is carried by the picks beneath the gaps, and the ridge direction is not the path. A print is only as sharp as the hairs are long puts the blur in the hair layer, and nothing here moves it out of there.

A thread drawn across the cloth rides one end’s floats for as long as it runs along that end. Drawn along the warp, it meets exactly the census’s column steps, and there the twill and the hopsack genuinely differ. Friction is two surfaces, not one is where that question lives.

The model named #

The surface is this collection’s height field: every end and pick as a centre line rising and falling over the threads it crosses, with its own section, sampled sixteen times per thread spacing over the repeat. A film at depth δ touches every sample within δ of the highest point. The film is one film when the set of samples it touches wraps the repeat in either direction — a cyclic connectivity test on the samples, with a union-find that carries each sample’s offset so that a cycle round the repeat is told from a cycle within it. The add-on at each depth is the coating fill: the volume between the top and the level, at a film density of 1.2.

Depths are read on the fill’s own grid, 120 steps from a ten-thousandth of the cloth’s thickness to all of it, so a join depth is the first grid depth at which the film wraps. Two weaves reported as joining together join within one step of each other.

What was counted #

Two weaves on two cloths: the sheeting at five warp setts from 16 to 32 ends a centimetre, and the poplin at eight from 16 to 44. At each, the depth and add-on at which each weave’s film first wraps the repeat, and the add-on that flattens the face. Two plan views at the sheeting’s own sett, one before and one at the join, and one at 16 ends.

Where the model stops #

A film has a thickness of its own and a surface tension. A real liquid finish spreads by wetting as well as by depth, and it can bridge a small gap by capillarity before its depth reaches the thread beneath. Where a gap between ends is narrow enough to bridge that way — at close setts, and for low-viscosity finishes — the twill’s staircase would regain its risers and the census’s distinction would return. Nothing here models a meniscus.

The threads are smooth. A light touch never reaches the crowns found the hair layer standing above them, and a film spreading through the hairs is connected long before the crowns are reached. On a singed cloth the height field is nearly the whole story; on a raised one it is not.

And the repeat is four. An eight-end satin has longer floats and fewer steps between ends, and its film’s joining depth is a separate calculation.

Nor does the film penetrate. A real paste is driven into the yarns as well as between them, and a thin coat can soak the crowns without lying on them. Penetration moves material from the film into the cloth, which lowers the add-on at every depth by the share that went inside; it cannot connect two crowns across an empty gap, since the gap has no yarn to soak into. So a penetrating coat joins later in grams than the figures here, and at the same depth — which makes the prediction’s window narrower still rather than wider.

Who found it, and when #

Coating add-on and its dependence on the fabric’s surface are the coater’s daily arithmetic, and the idea that a thin film is continuous only above a percolation threshold is borrowed from the physics of disordered media. The matrix census of crown connectivity and the height field are this collection’s.

What is new here is putting the two side by side, and finding that the edge a matrix draws between two neighbouring ends is a gap on the cloth — which turns the census’s ridge into patches for a film and leaves its distinction standing only for a thread drawn along the warp.

Still open: when a meniscus bridges the gap #

The one mechanism that could restore the ridge is capillarity. A wetting liquid on a twill’s crowns meets a gap between ends of a sixth of a millimetre and a pick lying some tens of micrometres below; whether it bridges the gap by its own surface tension, before its depth reaches the pick, depends on the liquid’s contact angle and the gap’s width, and both are quantities the wicking account already carries.

At what sett a gap between ends closes to a meniscus — below which the twill’s film is whole from the first gram and the hopsack’s is not — would say whether the census’s prediction is right for low-viscosity finishes on close cloths, which is exactly where a thin, continuous film is most often wanted.