The three basic weaves do not generate the rest

Open any weaving manual at the first chapter and the same sentence is there. There are three basic weaves — plain, twill and satin — and everything else is derived from them: doubled, reversed, broken, pointed, rearranged, combined, counterchanged.

It is the opening claim of the whole subject, and it is stated with the confidence of a completeness result. This site has the complete catalogue of the smallest interesting repeat, four ends by four picks, in which 22,874 drafts interlace everywhere and 426 of them are distinct cloths. So the claim can be tested rather than repeated.

Take the basic weaves. Apply every derivation the manuals name. Close the set under everything that leaves a cloth the same cloth. Count what is reachable.

Nine. Nine of the 426.

The operations, and which of them do anything #

The manuals’ operations divide into two kinds and telling them apart by eye is exactly what nobody can do.

Some leave the cloth the same cloth. Sliding the writing one end or one pick along produces a different drawing of the same fabric; so does turning the paper half round, and so does turning the fabric over, which reverses the ends and complements every square at once. This site’s count of 426 already divides all of those out — that is what makes it a count of cloths rather than of drawings.

Some make a different cloth. Reversing the threading, reversing the treadling, turning the draft a quarter turn so warp becomes weft, and counterchanging so that every warp-up becomes weft-up. Those are the operations a manual means by derivation.

They have to be in one census because the derivation question is about the two together: what the manuals claim is that starting from a few weaves and applying the second kind, everything is reached — and the count has to be taken modulo the first kind or it counts drawings rather than cloths.

How the census was run, generously #

A derivation can widen a repeat. Reversing a four-end twill takes eight ends to write, and a cloth written on eight is not in a four-by-four catalogue — so running the closure only at four would miss anything that widens and then happens to repeat inside its own writing, and would make the manuals look worse than they are.

So the census is run in two frames. At four, the seeds are plain weave, every twill a repeat of four admits, every satin four ends admits, the hopsacks, and the pointed, reversed, broken and diamond constructions at widths two and four. At eight, the seeds are the same list computed at eight — plain, all sixty-four twills a repeat of eight admits, both eight-end satins, both hopsacks, and the four reversal constructions at widths four and eight. In each frame the closure is taken under all six operations, and anything that reduces to a four-by-four cloth is collected.

The two are then unioned, which can only ever make the manuals look better.

Frame four reaches nine cloths from fifteen seeds and 196 derived grids. Frame eight reaches five, from seventy-seven seeds and 962 grids — all five of them already in the nine. The union is nine of 426, which is 2.1 per cent.

What is missed is ordinary cloth #

The obvious objection is that the 417 are junk — separable drafts, long floats, things no weaver would use. They are not.

Of the nine reached, all nine are single cloths, with floats of one, two and three. Of the 417 missed, 411 are single cloths and six are separable. The float distributions overlap completely: the reached set runs from one to three and the missed set from two to three, and the only reason the missed set does not reach one is that plain weave is the unique four-by-four cloth with no float at all, and it is one of the nine.

So the missed set is 411 perfectly ordinary sound cloths with exactly the same range of float lengths as the reached ones. The derivations do not reach a distinguished part of the catalogue and leave the rest. They reach 2.1 per cent of it and the rest is indistinguishable.

What the claim probably means #

It is worth being fair to the manuals, because the sentence is not stupid and it is not quite a mathematical claim.

What a first chapter is doing is teaching a vocabulary and a set of moves. Its claim is closer to every weave met in a mill can be understood as a modification of one of these three than to the closure of this set under these operations is the whole catalogue. The first is a claim about what is woven, and it is very nearly true — the weaves in commercial use are overwhelmingly twills, satins, hopsacks and their reversals, and the catalogue’s other 417 are cloths nobody has ever put on a loom.

But that is a different statement with a different content, and the difference matters in one direction: it says the derivations describe the corner of the catalogue that tradition selected, not the catalogue. Which raises the question the count makes unavoidable — is that corner selected because the rest is unusable, or because the rest was never looked at?

This site has a partial answer already. Of the 22,874 four-by-four drafts, about a dozen have names. Of the 426 cloths, five have a fraction name. The naming, the derivations and commercial use all pick out the same tiny neighbourhood, and none of the measures this site takes off a matrix — float, interlacings, balance, shafts, layers — separates that neighbourhood from the rest.

What was counted, and how #

The catalogue is computed twice by different routes and the two are required to agree. This site’s census builds the symmetry group by closure and counts orbits by Burnside’s lemma; the machinery here reduces every draft to the least mask reachable from it under the same equivalences and counts the distinct results. Neither is checking the other’s arithmetic — they are checking that “the same cloth” means the same thing in both places, which every number in this essay rests on.

The closure itself is a breadth-first search over grids in a stated frame, with the operations applied as permutations of cells. Everything reached is checked to be in the catalogue before it is counted, because an operation that produces a draft in which some thread never interlaces would otherwise be compared against a set it is not in.

Two assertions guard the finding and both can fail. The closure must not reach the whole catalogue, which is what makes the essay’s claim a claim; and it must reach more than the weave it started from, which is what makes the operations non-trivial. If a future change to the operation list made the first fail, the machinery would refuse to return rather than quietly agreeing with the manuals.

And the comparison between the two sets asserts three things: that most of the sound cloth is outside the derivations, that the two sets reach the same longest float, and that the missed set is overwhelmingly sound rather than separable. Each is a way the finding could have been an artefact.

Why the closure is so small #

Nine out of 426 is a very small number and it is worth asking what makes it small, because the answer is not that the operations are weak.

The operations are a group acting on the catalogue, and a group’s orbits are as large as its own size allows. Six operations, closed under composition, generate a group of modest order; an orbit under a group of order g has at most g members. So the reachable set from any one seed is bounded by the group’s size before any question about weaving arises, and the whole closure is bounded by the number of seeds times that.

That is the arithmetic reason and it is decisive. Fifteen seeds and a group of a few dozen elements cannot reach four hundred cloths whatever the operations are, and the only way to reach the catalogue is with a construction that is not a fixed finite set of moves.

The manuals’ operations are also unusually redundant. The census found four of nine producing a cloth already reached, and the reason is that most of the manuals’ moves are symmetries in disguise: reversing a treadling and turning the paper are the same operation on some weaves, and turning a draft a quarter turn is a symmetry of many of the seeds outright. So the effective group is smaller than the list of names suggests, and a list of nine operations is doing the work of rather fewer.

And the seeds are drawn from a single construction. Plain, twill, satin and hopsack are all shift-rule weaves or products of them — every column a rotation of one column, which is 1.2 per cent of the catalogue on its own. Applying symmetry operations to a set of shift-rule weaves reaches shift-rule weaves and a small halo around them, because the operations are themselves rearrangements rather than constructions.

So the two per cent is not a failure of the derivations to be applied thoroughly. It is what a finite group acting on a small, highly structured seed set can reach, and the answer would have been small for any comparable list of moves on any comparable list of starting points. What the count adds is the size — and the size is what turns “not everything” into a number worth writing down.

It also says what would have made the closure larger, which is not a longer operation list. Adding seeds does: every seed brings its own orbit, so a manual that started from thirty weaves rather than fifteen would reach roughly twice as many cloths and still be nowhere near four hundred. The only route that scales is a construction whose output is not an orbit at all — which is what a combination is, and why combination is the one omission from this census that could change its order of magnitude.

What a generating set would have to look like #

If the manuals’ operations reach two per cent, the obvious question is what would reach the rest.

The answer is uninteresting and that is itself informative. The four-by-four catalogue is generated trivially by “write down any binary matrix in which every thread interlaces”, which is not a construction anybody could teach and is not what a first chapter is for. Between that and the nine there is no natural family that anybody has proposed, because nobody has had a reason to look for one: a weaver needs a vocabulary of cloths that behave in known ways, not a generating set for a combinatorial catalogue.

So the honest reading of the two per cent is not that the manuals are wrong about weaving. It is that they are not making a mathematical claim at all, and the sentence that opens every first chapter is a statement about a tradition’s vocabulary wearing the grammar of a completeness result. Measuring it is worth doing precisely because the grammar is misleading and the measurement is cheap.

Where the model stops #

This is one repeat size and it is the smallest interesting one. At six by six or eight by eight the catalogue is very much larger and the derivations reach more absolutely; whether they reach a larger or smaller fraction is not computed here, and the honest guess is smaller, because the catalogue grows faster than any fixed set of operations can.

The operation list is the manuals’ and it is finite. “Rearranged” is a real derivation and at four ends the only rearrangements available are the cyclic ones and the reversals, which are in the list; at eight ends a satin-order rearrangement exists and would reach more. That is a genuine limitation and it means the nine is a lower bound rather than an exact answer to “what can be constructed”.

Combination is excluded, and it is the largest omission. Manuals also derive by combining — a stripe, a check, a figure — and this site has three essays on what those produce. A combination of two weaves is a new cloth and can certainly be outside the nine. What it cannot be is a small correction: a stripe of two four-end weaves is at least eight ends wide and is therefore not a four-by-four cloth at all.

And nothing here says the 417 are useful. They are sound cloth with ordinary floats. Whether any of them has a property worth having is a question about handle, appearance and cost that no count answers.

The generalisation #

The shape of this is a generating claim that has never been checked against a complete enumeration, and the reason it survives is that until there is a catalogue there is nothing to check it against.

That is a common position and it is not a disgrace. A field with no complete enumeration of its objects has no way to distinguish “these operations generate everything” from “everything anybody has seen was made this way”, and the two are indistinguishable from inside. What produces the difference is a census, and a census needs the objects to be finite and decidable — which weaves at a fixed repeat happen to be and most things are not.

The second half of the generalisation is the more uncomfortable one. When the constructive vocabulary of a field reaches two per cent of its objects, the vocabulary is not a description of the space. It is a description of a path through the space that somebody took, and the fact that everything anybody has made lies on that path is evidence about the makers rather than about the space.

Who found it, and when #

The three basic weaves are as old as writing about weaving, and the derivations are the ordinary content of a nineteenth-century textile-design course. Every one of them is correct: reversing a twill does produce a herringbone, doubling plain weave does produce a hopsack, counterchanging does produce the reverse face.

The complete four-by-four catalogue is recent, and this site’s own version of it is some years old. The 426 comes from quotienting 22,874 drafts by the symmetries that leave a cloth unchanged, and the count is checked two ways.

Running the derivations against the catalogue appears not to have been done, and the reason is probably that the two belong to different literatures: the derivations are in design manuals and the enumeration is in combinatorics, and neither has much reason to look at the other. What makes the comparison worth making is that it is cheap once both exist, and that the answer is not 60 per cent or 90 per cent — it is 2.1, which is a different kind of answer.

Where the ladder goes next #

The next thing to ask about the same catalogue is not what generates it but what can be said about it: four notations, what each can express, and whether expressing a cloth identifies it. A fraction name reaches five of the 426 and does not identify the five; point paper reaches all of them and does; a four-part draft reaches what a loom of stated size can weave; and a float specification, which is what a fabric is actually bought against, sorts the whole catalogue into three classes.

Sideways, the catalogue itself is every cloth at four by four and how many cloths there are; the family the derivations enumerate best is the twills; and the reason one of the three basic weaves is missing at this size is there is no six-end satin, which rules out four for the same reason.