What the matrix cannot say

The claim this collection was built on is stated in its first essays and has not been softened since: a weave is a binary matrix, and that is not a convenient encoding of a weave — it is the weave. Warp up or warp down at every intersection, over a repeat that tiles the plane, with no tolerance to choose and no residual to interpret.

The claim has earned its keep. It is why float length, interlacing count, plane group, jammed sett and cloth integrity are decidable rather than debatable, and why this site can count rather than quote.

An encoding that commits like that has an edge, and after a great many essays’ worth of structures it is worth walking round the edge and describing it properly. Not as a list of limitations — as a map of where the commitment holds, where it holds while seeming to say more than it does, and where it stops.

Four ways out, and they are genuinely different #

The structures this site has met that are not weave matrices escape for four distinct reasons, and running them together as “not a weave” loses the whole point.

Leaving the plane. A pile fabric has a third thread system that departs from the sheet between its anchor points. There is no intersection to record because the threads are not in the same place. What is lost: float length degenerates, interlacing count measures the wrong system, cover measures the ground.

Losing the order. A leno permutes its warp ends between picks, so the column index stops naming a thread and W[i][j] ceases to be a proposition. What is lost: the plane group outright, since there are no squares; and the interlacing count undercounts, because it has no cell for an end-to-end crossing.

Having only one system. A knit is one thread system interlooping with itself, and a braid is one system interlacing with itself obliquely. There is no warp and weft to index by. What is lost: everything that presupposes two systems, which is most of the site’s measurements — but the structures are periodic and have their own exact descriptions.

Having no structure to speak of. A nonwoven has no repeat, no periodicity and no deterministic contacts at all. What is lost: exactness itself. Coherence there is a percolation threshold, measured with replicates and error bars, and it is the one place on this site where a question about integrity gets a statistical answer.

Four escapes, four different casualties. The useful generalisation is not “the matrix handles woven cloth” — it is that each structure loses precisely the properties that depended on the feature it gave up, and keeps the rest.

What survives all four #

One thing survives every one of them, and identifying it is the most valuable single result of this whole exercise.

The integrity criterion never needed the matrix. It consumes a list of contacts with a verdict about which strand is above at each; the matrix was a convenient way of generating that list for two systems in a plane. Hand it any structure whose contacts can be enumerated and it works, and it works from the same implementation.

So one code path decides a plain weave, a satin, a herringbone, a double cloth, a braid word, a warp-knit lapping, a pile fabric, a double plush and a leno gauze. That is not tidiness. It is the difference between one criterion applied everywhere and several that could drift apart, and this site has a worked example of what having one is worth.

The nonwoven is the exception that proves the shape of the rule: its contacts are not enumerable in advance because they are random, so the criterion is applied to sampled realisations and the answer is a probability. Even there the question transfers; only the exactness does not.

Inside the encoding: three things that look decided and are not #

The edge is not the only place to be careful. There are quantities the matrix appears to settle and does not, and these are more dangerous than the outright boundary because nothing announces them.

Handedness. A Z twill and an S twill are reflections, so every count taken from the matrix — floats, interlacings, layers, plane group — is identical. What distinguishes them is the yarn’s own twist direction, which is not in the matrix at all. The site has said so plainly since its earliest essays, and it is worth repeating here because the matrix returns a confident answer to a question it cannot see.

Face and back. Which side of the cloth is the face is a convention applied to the matrix, not a property of it. A rising shed and a sinking shed reading the same tie-up produce complementary cloth.

Everything about the yarn. The matrix decides floats, interlacings and integrity exactly, and it has no notion of thickness, twist, friction or stiffness. Two cloths with the same draft in different yarns behave completely differently, and a figure that implies otherwise is over-claiming — which is this site’s third invariant and the one it is easiest to break by accident.

The last of these is the reason the setting field exists at all: Peirce’s geometry, cover, crimp and jamming are all attempts to say something about the fabric that the matrix has no vocabulary for, and every one of them is a named model with assumptions rather than a count.

A notation is not a fabric either #

There is a fifth gap and it sits inside the encoding rather than at its edge, which is why it went unnoticed for a long time.

A matrix is a notation, and several notations describe one fabric. Shift the repeat’s origin, turn the cloth end for end, turn it over — the matrix changes and the fabric does not. Counting the census as fabrics rather than as notations turns 22,874 drafts into 426 cloths, and it turns the 144 drafts that fall apart into six.

The reason that matters here is what it did to a fraction. The separation rate — this site’s central claim, quoted in a dozen essays — is 0.63 per cent by draft and 1.41 per cent by cloth, because the drafts that fall apart are unusually symmetric and therefore over-represented once symmetry is quotiented out.

So an encoding can be exact, and every computation on it correct, and a rate quoted from it still be answering a different question from the one a reader takes it for. That is not a boundary of the encoding. It is a question about what the objects being counted are, and it is easy to skip precisely because everything on both sides of it is exactly right.

The rule this site keeps, and why it is this rule #

All of the above is one invariant applied repeatedly: never claim the machinery went where it did not.

It is worth saying why that is the invariant rather than something more ambitious like “be right”. Being right about a fabric requires knowing about yarn, finish, wear and the person wearing it, and no site is going to manage that. What is achievable is being exact about a stated question and explicit about which question it was — and the failure mode that produces is a gap, which somebody can notice and fill.

The alternative failure mode is a number that appears to answer the question a reader has and answers a neighbouring one. That is what a thread count is, what a cover factor past its domain was until this site found it, and what a layer count would be if it were read as a durability measure.

The encoding is a commitment, and that is why it works #

It is worth closing the argument the other way, because a long list of limitations reads as a case against the encoding and is meant as the opposite.

An encoding that could describe every fabric would decide almost nothing about any of them. The weave matrix is productive precisely because it commits: two systems, in a plane, crossing at every intersection, over a repeat that tiles. Each of those four commitments buys a class of decidable questions, and each of them is exactly what one of the four escapes gives up.

• Two systems buys the warp-and-weft indexing that float length and interlacing count are defined on. A pile gives it up by adding a third.
• A plane buys the crossing itself. A pile gives that up too, which is why it loses three measurements rather than one.
• A fixed order buys the column index. A leno gives it up.
• A repeat buys periodicity, and therefore symmetry, and therefore the plane group. A nonwoven gives it up and loses exactness altogether.

Read that way the four escapes are not four accidents; they are the four commitments, each declined in turn. And that is a considerably more satisfying account of the boundary than a list, because it says why there are four and what a fifth would have to give up.

It also says what the criterion is. The integrity check survives all four because it uses none of the four — it needs only contacts and a direction at each, which is less than any of the commitments above. The most portable thing in the toolkit is the one that assumes least, which is not surprising once stated and was not obvious before the boundary was walked.

The fifth commitment, which is that a crossing has two participants #

The list above names four commitments and asks what a fifth would have to give up. There is one, it is the least visible of the five, and a real fabric declines it.

Every cell of a matrix records which of two strands is on top. That is not implied by any of the other four — a structure can be planar, periodic, fixed in order and still have three strands meeting at a point, which is exactly what a triaxial weave is: three families at sixty degrees, crossing in threes rather than in pairs.

Count the states. Separate the three crossings slightly and each pair has a verdict, so a node carries 2³ = 8 states rather than 2. Six of those are consistent stacking orders — one family on top, one in the middle, one beneath. The remaining two are cyclic: A over B, B over C, C over A, with no topmost family at all.

That is not a curiosity. A cyclic node is a three-cycle in the above-and-below digraph, so it is strongly connected on its own, and the criterion this essay has been praising reports it as locked without being told anything new. A transitive node has a topmost family that can in principle be lifted away. The two triaxial basic weaves the trade distinguishes are exactly these two cases, and the distinction is a connectivity one.

So the criterion crosses the fifth boundary as easily as the other four, for the same reason: it consumes contacts pairwise and never asked how many met at a point.

What does not cross is the symmetry. This collection has shown that five of the seventeen plane groups are unreachable by any draft, because a square grid admits no three-fold rotation. A triaxial fabric sits on a triangular lattice, so those five groups are precisely the ones it can have and the matrix cannot — the boundary that was proved as a theorem about drafts turns out to be a description of the fabrics on the other side of it.

Boundaries found by walking into them #

Every item on the map above was found the same way, and the method is worth naming because it is repeatable and because none of these was found by thinking about the encoding in the abstract.

By a figure asserting something and a check disagreeing. The cover factor running past its own domain was found because an essay printed two numbers and claimed a factor of two between them, and the arithmetic said 1.19. Nothing about the model announced that it had a domain; a figure went outside it and a caption made a claim the model forbids at every setting.

By writing an essay about a construction and finding no machinery for it. The pile and the leno both arrived that way. The intention was to describe a fabric; the encoding had nothing to say, and the interesting part turned out to be exactly which things it had nothing to say about.

By doing a computation somebody had decided was unnecessary. The orbit count was recorded as skippable on the argument that no fraction would move. One did, and by a factor of two and a quarter.

By parameterising a generator that had only ever run at its default. Two figures on this site had fixed canvases that fitted their defaults and nothing else, and both were found on the first build after being asked for something different.

The common shape is that a boundary is invisible from inside and obvious the moment something crosses it. So the productive move is not to reason about where the edge might be — it is to keep asking the machinery questions it has not been asked before, and to make sure it can refuse.

Where the boundary earns its keep #

Two results in this collection came out of standing at the edge rather than working inside it, and neither would have been visible from within.

The criterion is portable and the measurements are not. That was invisible while every construction had a matrix; it became a fact about the criterion the moment something did not. It is the reason the fancy-weaves field could be written at all.

The criterion is topological, so it cannot see friction. A tuft under one pick and a tuft under three are both attached; a gauze and an open plain weave are both one cloth. Stating that boundary exactly is what made it possible to bring in a second model — the capstan — and keep it visibly separate rather than folding an estimate into an exact count.

Both are results about the tools rather than about cloth, and both were produced by asking what the tools do not do. That is the argument for writing an essay like this one rather than treating the boundary as an embarrassment to be worked around.

What was counted, and how #

This essay computes nothing new; every number in it is quoted from the essay that produced it, and each of those says how.

What the figures here do is exercise the claim about portability directly. The pile graph and the leno panels are drawn by the two families that handle constructions with no matrix, and the layer counts printed on them come from the same stronglyConnected implementation that decides a twill. If that implementation were forked — if the pile had its own criterion — the claim of this essay would be false and the figures would not reveal it. That it is not forked is a property of the code and is checked by the code being read, which is a weaker check than an assertion and is the honest description of it.

Where the model stops #

Four escapes is a list, not a classification. Nothing here proves there are exactly four ways out of the encoding, and a construction escaping in a fifth way would be a discovery rather than a contradiction. Woven three-dimensional preforms, for instance, are arguably a fifth: they are periodic and have two systems and a third dimension, and this site has not worked out what happens to the matrix there.

The nonwoven’s inclusion is generous. Calling a random fibre web a structure that has “escaped” the encoding implies it was ever a candidate. It was not, and it is listed because the comparison is instructive rather than because it is the same kind of case.

And the inside-the-encoding cautions are a sample. Handedness, face and yarn are the three that have caused trouble on this site. There are certainly others, and they will be found the way these were: by a figure claiming something and a check disagreeing.

Who found it, and when #

Nothing in this essay is a discovery, and the pieces belong to the essays that made them. What it does is collect them, which is a different activity and is worth doing at the point where a collection acquires enough structures to have a shape.

That point is now. The foundation had one encoding and two solvers. A great many essays later there are five kinds of structure, four of which are outside the encoding, and one criterion that decides all of them. A map of the edge is only possible once there is enough on both sides of it to draw.

Where this goes next #

The field that follows leaves the loom entirely and takes up what happens to cloth afterwards — relaxation, shrinkage, milling, and why a finished fabric is not the one that came off the loom. Everything in this essay is about a structure as woven, and corduroy already showed that the surface a hand judges can be decided after the loom by a process the draft does not contain. That is a fifth kind of gap and it deserves its own field.