What combining two weaves reaches

Every weaving manual opens the same way. Three basic weaves — plain, twill, satin — and everything else derived from them by operations the chapters name: reversing, turning, breaking, doubling, counterchanging.

The last rung took that claim literally and ran it. Seeds: plain weave, every twill a repeat admits, every satin that exists, the hopsacks, and the pointed, reversed, broken and diamond constructions, at frames of four and eight so that a derivation which widens a repeat and then repeats inside its own writing is not missed. Operations: reversal both ways, the quarter turn, counterchange, and the slides and turns that leave a cloth the same cloth.

The closure reached nine of the 426. The catalogue is the site’s own, from the enumeration of every cloth at four by four, and the seeds are as generous as the chapters allow: every twill a repeat of eight admits, every satin that exists on eight ends, and the constructions beside them.

It also recorded, in its own list of what was not done, a specific exclusion: the derivation census excludes combination. Stripes, checks and figures are derivations the manuals name, and a combination of two four-end weaves is at least eight ends wide and therefore not a four-by-four cloth at all.

That was fair, and it made the nine a lower bound rather than a measurement. This rung removes the exclusion.

The claim #

Admitting combination triples the reach and does not change the conclusion. Nine becomes twenty-eight; 417 unreached becomes 398; the manuals’ own constructions still miss ninety-three per cent of the catalogue of four-by-four cloths.

And there is a second result which is the more interesting one, because it explains the first. Only one construction in a hundred stays inside the frame. Of 17,787 stripes, checks and figures built from the chapter’s own weaves, 177 repeat inside four ends and four picks. The rest are genuinely wider cloths — perfectly good ones, but not members of the set being counted.

What was admitted, and what was not #

Three combinations are allowed and they are the three the trade names.

A stripe: one weave on the first half of the ends and the other on the second. A check: the same partition applied to the picks as well, so the four quadrants alternate. A figure: a two-by-two block profile in the diagonal arrangement, which is the smallest thing that is neither a stripe nor a check.

Every pair of seeds is combined in every one of those three ways, in both orders, and the results are then put through the same six operations as before — sliding, reversing both ways, the quarter turn, counterchange — so a combination that becomes something else when reversed is counted as both.

Two things are deliberately not admitted, and the reasons are different.

Combinations of combinations are excluded. A reader who has made a stripe can stripe it again, and the closure of that operation is enormous. It is excluded not because it is unreasonable but because the seeds of a closure have to be what a chapter actually teaches: once arbitrary iteration is allowed, the census stops measuring the manuals and starts measuring the operation.

Nothing is admitted that a chapter does not name. No blends of three weaves, no irregular block profiles, no partitions at a ratio other than a half. Each of those would raise the count and each would be answering a different question.

Why a combination almost never stays in the frame #

The one-per-cent figure is not an accident of the seeds chosen, and the reason is worth setting out because it is also what makes the census cheap.

A grid on eight ends and eight picks is a four-by-four cloth exactly when it repeats inside four in both directions. A stripe puts weave A on ends 0–3 and weave B on ends 4–7; for that to repeat with period four, the two halves must agree at every cell — which means A and B must be the same weave over that frame. So a stripe of two genuinely different weaves is never four-periodic, unless the two weaves happen to coincide on the frame.

The 177 survivors are therefore the near-degenerate cases: pairs whose members agree on the relevant half, or whose difference is itself four-periodic. The construction that makes new cloth is exactly the construction that leaves the frame, and there is no way to have one without the other.

That has a consequence the census exploits. All six operations preserve four-periodicity — a shift, a reversal, a quarter turn and a counterchange all carry a four-periodic grid to a four-periodic grid — so a combination that is not four-periodic has an orbit that is not either, and expanding it cannot change the answer. Filtering first takes the census from a million and a half grids to a thousand and ninety, and gives the same reach. The unfiltered version was the first one written and is what that claim was checked against.

What the nineteen new cloths are #

The gain is small in count and it is not degenerate. The nineteen cloths reached only by combining are single cloths with the same range of float lengths as the nine reached without it: they are ordinary weaves, not curiosities.

What they are not is a natural family. There is no description of them shorter than the list, which is itself worth noticing: a construction that reached a describable subset would be evidence that the manuals’ vocabulary carved the catalogue at some joint. This one reaches nineteen cloths that have nothing in common except the accident of being four-periodic combinations.

And the 398 that remain are not degenerate either. The previous rung established that for its own 417 — 411 of them are single cloths with the same float range as the reached ones — and removing nineteen does not change the character of the remainder.

What this settles about the manuals’ claim #

The previous rung’s finding was open to one obvious objection, and it is now closed.

The objection: of course a closure over reversal and counterchange misses most of the catalogue — those are the wrong operations. The manuals also teach combination, which is how real cloth is designed, and combination is enormously productive.

Both halves of that are true and the conclusion does not follow. Combination is enormously productive — 17,787 constructions from 77 seeds, and the great majority of them are perfectly good cloths. What it does not do is produce four-by-four cloths, because a combination of two different weaves is wider than either.

So the manuals’ claim splits into two claims that need separating.

As a claim about how to design cloth, it is sound. Reversing, breaking and combining the basic weaves is how the trade’s fabrics are actually made, and the products are real.

As a claim about generating the catalogue of weaves at a given size, it is false and stays false. A designer working entirely within the chapters reaches twenty-eight of the 426 cloths that fit in a four-by-four repeat. The other 398 exist, are single cloths, are weavable, and no operation the manuals name will produce them at that size.

The difference between those two claims is the difference between a vocabulary and a generating set, and the manuals do not distinguish them because there was never a reason to. The catalogue was not enumerated until it was enumerated.

It is also worth saying which of the two claims a reader of plain, twill, satin is being sold. The three weaves really are the foundation in the sense that matters practically: they are the shortest description of what an interlacement can do, and the count of how many cloths there are is a fact nobody needed until somebody asked it. The overselling is in the word derived, which promises a generating set and delivers a vocabulary.

What was counted, and how #

The catalogue is the site’s own: every four-by-four binary matrix that describes a single cloth with every thread interlacing, taken modulo the moves that leave a cloth the same cloth. It comes to 426 and is enumerated rather than quoted.

The seeds are rebuilt at the frame of eight: plain weave, every twill a repeat of eight admits, every satin that exists on eight ends, two hopsacks, and the reversed, pointed, broken and diamond constructions at two widths apiece — 77 in all.

Every ordered pair of seeds is combined in each of the three shapes, giving 17,787 constructions. Each is tested for four-periodicity; 177 pass. Those are pushed through the closure of the six operations, giving 1,090 distinct grids, and each grid is reduced to its minimal repeat and looked up in the catalogue.

The result is unioned with the previous rung’s closure, because a cloth reached either way is reached. Two assertions guard it: admitting a construction cannot reduce the reach, and the reach must still fall short of the catalogue — the second because a census that suddenly reached everything would be a census with a bug in it rather than a discovery.

The periodicity filter is the one step that could have changed the answer rather than merely the running time, so it is worth being explicit that it did not. The unfiltered census — 1.48 million grids, twenty-one seconds — reaches the same twenty-eight cloths as the filtered one at a thousand and ninety.

Where the model stops #

The frame is four. Everything here is about which four-by-four cloths are reachable, because that is the catalogue this site has enumerated. A census at eight would be a different and much larger question, and the answer would very likely be different in character rather than in degree: combination is a construction that makes wide cloths, so a wider frame is where it should shine.

The seeds are a judgement. Seventy-seven of them, chosen to be what an opening chapter teaches. A reader who thinks a manual also teaches something not in that list is disagreeing about the input rather than about the arithmetic, and the list is written out in the source so the disagreement can be specific.

Iteration is excluded, as above, and the exclusion is the largest single choice in the design of the census.

And “reached” means reached in principle. Nothing here is about whether a designer would ever think of a particular combination, only about whether it is in the closure. That is the right question for the manuals’ claim as stated and it is a generous reading of it.

The generalisation #

The result is about the difference between two things that look alike.

A rich set of operations can be productive without being generating. Combination produces vast numbers of new objects and reaches almost none of a particular finite set, because the objects it produces are systematically larger than the members of that set. Productivity and coverage are independent, and the intuition that a fertile operation must eventually cover everything is simply wrong when the operation has a size bias.

The mechanism is worth naming because it recurs. An operation that combines two objects of size n typically makes an object of size 2n, and the ones that come back down to n are the near-degenerate cases — the ones where the two inputs agreed enough that the combination collapsed. So the yield of a combining operation on a fixed-size catalogue is concentrated on exactly the least interesting pairs.

That is also, usefully, a computational lever. If an operation preserves a property and the target set has that property, then only inputs with the property need to be expanded — which is the argument that took this census from a million and a half objects to a thousand, with the answer unchanged and checked.

The gap widens with the frame, and that is a counting argument #

The list of limits above says that a census at eight ends would be “a different question, and the answer would very likely be different in character rather than in degree”, on the ground that combination makes wide cloths and a wider frame is where it should shine.

That hope is the wrong way round, and the reason is arithmetic rather than weaving.

Count what a closure can possibly reach. Seventy-seven seeds give 5,929 ordered pairs; three combination shapes and six operations put an absolute ceiling of about a hundred thousand grids on the whole construction, and the reachable set is smaller than that because grids collapse. The reach is bounded by a polynomial in the number of seeds — quadratic, in fact, since it is pairs — and the seed list is whatever a chapter teaches, which is a few dozen at any frame.

Now count the target. A four-by-four catalogue is 426 cloths. An eight-by-eight repeat has 2^64 matrices, and dividing out the sixty-four translations and the turnings still leaves something of order 10^17 distinct cloths. The catalogue grows as two to the square of the frame and the reach grows as the square of a seed list.

So the fraction reached cannot do anything but collapse. Twenty-eight of 426 is six and a half per cent; a hundred thousand of 10^17 is one part in a million million. The manuals’ vocabulary covers less of the eight-by-eight catalogue than of the four-by-four one by twelve orders of magnitude, and no enrichment of the seed list that a chapter could plausibly teach changes the exponent.

That is a stronger conclusion than the one this rung set out to reach, and it is worth separating the two carefully. The census here is a measurement: it says what the manuals’ own operations on their own seeds reach at one frame, and it could have come out otherwise. The counting argument is not a measurement and cannot come out otherwise: any finite vocabulary closed under a fixed set of operations covers a vanishing fraction of a catalogue that grows doubly exponentially, whatever the vocabulary and whatever the operations.

Which puts the manuals’ claim in its proper place rather than merely refuting it. A design vocabulary is not trying to be a generating set and would be useless if it were — a set of rules that reached all 10^17 eight-by-eight cloths would be a way of writing down a matrix, not a way of designing one. The value of plain, twill, satin is exactly that it is small, and the census’s contribution is to say how small, which nobody could have said before the catalogue existed.

The one place the hope survives is worth naming, because it is the honest version of it. Combination does not reach more of a fixed-size catalogue at a wider frame; what it does is reach cloths that are genuinely wider, which is where figured weaving lives and where a four-by-four census has nothing to say. The productive question at eight ends is not what fraction is reached but what the reached cloths are like, and that is a question about a set of a hundred thousand rather than about a ratio.

Who found it, and when #

The three basic weaves and their derivations are the structure of every weaving manual from the nineteenth century onwards, and the framing is so standard that it is nobody’s in particular. Combination — striping, checking, and figuring one weave on another — is taught in the same chapters and with the same air of completeness.

The four-by-four catalogue of 426 cloths is this site’s own enumeration, and the closure census is the previous rung’s. What is new here is the removal of a stated exclusion, which is the least glamorous kind of progress and the kind a recorded shortfall is for: the previous rung wrote down what it had not done, and this one did it.

The periodicity argument is elementary and is the sort of observation that gets made once and then used silently. It is written down here because it changed the census by three orders of magnitude and because a filter that changes an answer’s cost is a filter that has to be proved not to change the answer.

Where the ladder goes next #

The obvious next question is the frame. Combination makes wide cloths, so a census at eight ends by eight is where it should be productive — and it would need the eight-by-eight catalogue, which is a much larger enumeration than this site currently has.

Sideways, the same combinations are the objects the stripe ladder and the blocks ladder argue about from the other end: those ask whether a combination of two sound weaves is sound cloth, and this one asks whether it is a cloth the catalogue already contains. The two questions have never been asked of the same construction at once.

Further out, the naming problem the previous rung found is untouched: five of the reachable cloths have a fraction name and there are four names for them, and adding nineteen cloths that have no names at all makes the gap between the catalogue and the vocabulary wider rather than narrower.