Four drafts in five have no path along their own crowns

Two drafts of twenty-two thousand ordered the whole four-by-four catalogue by how much crown line each surface carries, and noticed in passing that the ordering puts two very different cloths in the same place:

A two-and-two twill carries 1.350 counting both systems, and 0.700 counting only the one that bears. A hopsack of the same order carries the same 0.700, which is worth noticing — two drafts that look nothing alike and behave differently in almost every other respect have identical surfaces by this measure, because both give every thread runs of exactly two.

That observation is correct and it names its own limit. By this measure. A length is a quantity and a surface is a shape, and two shapes of equal length can be arranged completely differently — which is what those two cloths do.

A ridge and a field of islands #

Draw the cells where the bearing system is on the face, over several repeats, and the difference is immediate.

The twill’s crowns are a ridge and the hopsack’s are islands. Both carry 0.700 millimetres of crown line per square millimetre; both give every thread runs of exactly two; and anything travelling along the top of the cloth — a coating blade, a printing squeegee, a film of finish spreading, a thread being drawn across — meets a continuous path on one and a series of gaps on the other.

That is a property of the matrix and nothing else. It needs no yarn, no sett and no crimp division, so unlike the length census’s own ordering it carries no convention with it at all.

Most of the catalogue has no path #

Running the question over every draft gives a number that is not what the twill suggests.

4,016 of 22,874 span — a little under a fifth. The other four in five present a bearing surface that anything travelling along it has to leave and come back to, and the drop is the step between the two systems, which on a sheeting is eight micrometres.

The 130 wholly isolated drafts are the extreme case: every face cell has four neighbours of the other system, so there is not one pair of adjacent crowns anywhere. The two plain weaves are among them, and this is their third appearance as an exception — no run of two means no adjacency, in exactly the same sentence as no crown line and one size of hole.

A single component is not a ridge #

The distribution of component counts is the part that would mislead anybody reading it without the wrap test.

9,920 drafts have exactly one component and only 4,016 span, so more than half of the single-component drafts are a single island — one connected blob of crowns that does not reach out of its own repeat and therefore repeats as a lattice of separate blobs across the cloth.

That is the distinction the count of components cannot make and the wrap test can. A component is a set of crowns that touch; a span is a component that touches its own translate, which is a statement about the pattern rather than about the repeat, and it is the one the physical question needs. A blade travelling across a cloth does not care how many pieces the crowns come in within one repeat; it cares whether it can keep going.

The surface percolates at balance, and the interlacing condition says why #

Sorting the census by how many of the sixteen intersections the bearing system holds produces a threshold, and the threshold is at exactly half.

No draft with fewer than half its intersections on the bearing face has a path along its crowns, however they are arranged. That is a hard combinatorial bound rather than a statistical tendency: 8,000 drafts sit below the line and not one of them spans.

The reason is the interlacing condition, and it is worth setting out because it is the one place in this account where the catalogue’s admission rule does real work.

The cheapest possible path across the cloth is a straight column. To wrap the repeat in the pick direction a component has to reach through all four picks, and four face cells stacked in one end would do it with the fewest cells any path could use.

That column is a thread that never interlaces, and the catalogue does not admit it — every end must go under somewhere, which is the integrity criterion’s own first requirement. So the cheapest legal path has to leave its column and come back, and every jog sideways costs a cell in a second column while the column it left still needs its own down.

Counting the cheapest legal arrangement out gives eight, and the census finds eight. The bound on a woven surface’s connectivity is set by the rule that makes it a cloth, which is an unusually direct connection between the two halves of this collection: the criterion that decides whether a draft is a fabric at all also decides whether its outside is a path or a scatter.

It also settles what kind of quantity this is. Percolation on a random lattice has a threshold that is a limit rather than a law — a share of sites above which a path is likely — and this one is exact and is at exactly a half because the constraint is exact. Nothing here is approximate and nothing depends on the repeat being large.

The two systems span exactly as often, and that is not obvious #

The crown-line census has an asymmetry in it: at twenty-eight ends and twenty-six picks the warp’s maximum is 1.400 and the weft’s is 1.300, and the ratio is the sett ratio exactly. The earlier essay traced the whole lopsidedness of its U to that one fact.

Connectivity has no such asymmetry, and the reason is that no spacing enters it. Whether two face cells are adjacent is a question about the matrix; complementing the matrix — turning the cloth over — maps the warp’s face cells to the weft’s one for one and preserves adjacency exactly, so it is a bijection between the warp-spanning drafts and the weft-spanning ones.

So the two counts are equal to the last draft: 4,016 either way, 1,616 both ways either way, 130 isolated either way. A census of lengths could not have that property at any construction but a square one, and a census of shapes has it at every construction.

That makes connectivity the more portable of the two quantities. The length census is a statement about one cloth’s surface; the connectivity census is a statement about the catalogue, and it is the same statement for every cloth ever set at any sett.

What it is for #

A quantity is worth a census when something depends on it, and three things here do.

A coating. A coating fills the crowns before it bridges the holes computes the volume above the bearing curve that an add-on has to bury before it can span a hole. That volume is the same whether the crowns join or not — and what happens at low add-on is not. A film laid on a spanning surface has a continuous path of support from the first gram; a film laid on islands is a set of separate puddles until the add-on is deep enough to bridge, and a discontinuous film is not a barrier at all. So the same add-on gives a coated cloth on one draft and a spotted one on the other.

A printed edge. A print is only as sharp as the hairs are long puts the blur in the hair layer. Underneath the hairs the ink is carried by the crowns, and a spanning crown set gives the ink a route along the cloth that an island set does not — which should make a spanning draft’s printed edge bleed further along the ridge direction and less across it, a directional blur that the hair-layer account has no mechanism for.

And a thread drawn across. A thread pulled over a cloth’s surface rides the crowns, and on an island draft it falls between them at every gap. That is the same population question friction is two surfaces, not one asks of two cloths face to face, one level down: the arrangement of the contacts decides what happens between them, and the convolution of two height fields is exactly a question about arrangement.

None of those three has been computed here. What this account supplies is the property they would each need and the census of it.

The named weaves, and the one that is both #

Reading the named weaves off the census puts the result in a form a designer could use, and it separates them into two groups with nothing in between.

The twills span. A 2/1 twill, a 2/2 twill and a 3/1 twill each form one component that wraps both ways, because a twill’s face cells step diagonally and a diagonal on a torus closes on itself. Their crowns are one continuous ridge running across the whole cloth in the twill’s own direction, which is the thing a reader can see on a piece of gabardine held to the light.

The rest do not. A plain weave is isolated; a four-end sateen is four isolated points; a 2/2 hopsack is two-by-two squares that touch nothing; a 1/3 twill’s warp is four isolated points, and it is the same cloth as the 3/1 twill turned over.

So the ordinary twills are the exception rather than the rule, which is the opposite of the impression the catalogue’s best-known members give. Four drafts in five have no path, and every weave a reader is likely to have looked at is a twill.

That is worth saying plainly because it is the shape of a sampling error, and the same error has turned up before. The named weaves are not a sample of the catalogue; they are the ones somebody found worth naming, and the properties they share are properties of what gets named. The census is what the catalogue actually contains, and what it contains is mostly islands.

What it says about the two censuses together #

This account now has two censuses of the same catalogue, and they answer different questions about the same cells.

The length census says how much. It is a sum over runs, it carries a thread spacing, and it is therefore a statement about a particular cloth. It orders the catalogue and the ordering depends on which system bears, which depends on a crimp division that is a convention.

The connectivity census says where. It is a property of the matrix, it carries nothing, and it is the same for every cloth at every sett. It splits the catalogue into a fifth that has a path and four fifths that do not, and the split is the same on both faces.

Put together they make four classes rather than two, and the four are genuinely different surfaces. Much crown line and a path is a twill: a heavy ridge that runs. Much crown line and no path is a hopsack: a heavy field of islands. Little crown line and a path is the thin case — a draft whose few face cells happen to chain — and it is rare, because a path needs eight cells and eight cells at a balance carry crown line. Little crown line and no path is the plain weave and its 129 relatives.

The classes that matter for a given question are decided by the question, and the useful observation is that the two quantities are nearly independent above the percolation threshold. A designer choosing among the eight-or-more class can have a great deal of crown line with or without a path, and the length census cannot say which they are getting.

That is a better statement of the earlier essay’s own worry than the one it made. It concluded that the catalogue is narrow — that a designer choosing among twenty-two thousand drafts is choosing among surfaces that mostly differ by twenty per cent in the quantity that decides what the cloth touches with. The quantity is narrow and the surfaces are not, and what separates them is a property no length can carry.

What was counted, and how #

Each draft’s bearing face cells are taken as a set in the plan, two cells adjacent when they share an edge, and the repeat treated as a torus — a draft tiles, so a crown leaving one side of the repeat arrives at the other. The components are found by union-find, and each cell carries a displacement from its component’s root in repeats; an edge that closes a cycle whose net displacement is not nought is a path that wraps, which is the statement that a route exists across the whole cloth. The two directions are tested separately.

Five things are checked. A plain weave’s crowns are isolated and no path runs along them, which is the case known by hand and is the check on the adjacency. Most of the catalogue has no path, under half, which is the finding stated so that a monotone error in the wrap test would fail it. Some drafts have none at all, more than nought and fewer than all. More drafts fail to span than are wholly isolated, which is the statement that spanning is a stronger condition than having a neighbour and would fail if the two tests had been confused. And the two systems span exactly as often, to the last draft, which is the bijection checked rather than argued.

The catalogue is the four-by-four enumeration and nothing else enters: there is no sett, no yarn and no crimp division anywhere in this census.

Where the census stops #

Adjacency is by an edge and a crown is not a cell. Two face cells meeting at a corner only are counted as not adjacent, which is right for a blade that has to stay in contact and wrong for a film that can wet a corner. The diagonal-adjacency census would give a larger spanning count and has not been run.

The repeat is four and four is small. A spanning path in a four-by-four repeat needs a component to reach across four cells, which is a demanding condition at a small repeat and much less demanding at a large one. An eight-end satin’s crowns, on a repeat twice as wide, would be tested against a different bar — so the fifth of the catalogue that spans here is a fact about small repeats, exactly as the narrowness of the crown-line distribution was.

And a path is not a plane. A blade riding a spanning ridge is riding a line, not a surface: it is supported along the ridge and unsupported either side of it, so what it actually does is a bending problem the census does not pose. What the census supplies is whether the line exists.

Still open: whether a coating on an island draft is discontinuous #

The whole essay ends at a prediction the coating arithmetic could test and has not.

Lay a stated add-on on two cloths of identical crown line, one spanning and one not — a 2/2 twill and a 2/2 hopsack, at the same sett in the same yarn. The volume above the bearing curve is identical, so the two should take the same add-on before they bridge their holes. What should differ is the film at intermediate add-ons: continuous on the twill from the first gram, and a set of separate patches on the hopsack until the add-on is deep enough to reach across the gaps between islands.

The gap on a hopsack is one thread spacing wide and the step is a few micrometres deep, so the add-on at which the patches join is computable from the same geometry the coating essay already uses. If it comes out well below the add-on at which the holes bridge, the distinction does not matter and the coating account is right as it stands. If it comes out comparable, there is a range of add-ons in which two cloths of identical surface quantity give a barrier and a sieve — and the census would have earned its place in the coating arithmetic rather than beside it.

Who found it, and when #

Connectivity of a binary pattern is elementary and the union-find on a torus is standard. The observation that two four-by-four drafts of identical crown line are differently arranged is in the earlier census’s own text, where it is offered as a curiosity; measuring the arrangement, and finding that four drafts in five have no path along their crowns, is new here.