A cloth's derivation class is its census of small patches

No cloth derives into more than three others. Every derivation a weaving manual names is a relabelling of the grid or a counterchange, both invertible, so together they generate a group — 256 operations at four by four — and the 426 cloths fall into 157 orbits of at most four. Whether one cloth derives from another is whether they share an orbit, and the answer was found the way an orbit is always found: by applying all 256 operations and looking.

That account ended on the question a weaver would actually ask. Is there a short list of numbers, readable off a draft, that agree exactly when two cloths derive into each other? Such a list would be a complete invariant. It would turn “is this cloth derived from that one” from a search into a comparison, and it would say what derivation preserves rather than what it does.

The answer is yes, and it is not the list anybody would write first. None of the measures the essays on derivation have quoted is enough, singly or together. What is enough is a census.

The orbits being told apart #

The object every measure below is tested against is the orbit census itself, and its shape matters for what “telling apart” means.

An orbit of one is a cloth every derivation returns unchanged. An orbit of two is, almost always, a cloth and its counterchange. An orbit of four adds a reversal or a quarter turn that the cloth does not already have as a symmetry. So an invariant has two jobs at once: it must not change under any of these operations, which is what makes it an invariant, and it must change between any two orbits, which is what makes it complete. Most measures do the first easily and the second badly.

What a derivation cannot change #

An invariant has to survive every generator of the group: reversing the ends, reversing the picks, turning the draft a quarter, counterchanging, and starting the repeat somewhere else. A measure that survives all five survives every sequence of them.

The number of warp-up marks is moved by a counterchange to sixteen less itself, so it survives as an unordered pair. The integrity count survives, because a counterchange reverses every above-and-below relation and leaves which threads are connected alone. The plane group survives, because every derivation is itself a symmetry of the grid and commutes with the cloth’s own. The interlacing count survives, and so does the multiset of changes of face per end and per pick, taken as an unordered pair.

Two need care. Float lengths as usually quoted — the longest warp float and the longest weft float — do not survive: a counterchange turns runs of face into runs of back, which the usual count does not see. Counting runs of face and of back in both systems, and taking the four lists up to the exchanges a counterchange and a quarter turn make, does survive. And the number of distinct ends and distinct picks — what a loom must thread and lift — survives as an unordered pair, since a quarter turn exchanges them.

Every one of the seven was checked on every cloth of every orbit, and every one is constant on all 157.

Seven measures tell 120 orbits apart #

A measure separates two orbits if it takes different values on them. The more classes it cuts the orbits into, the closer it comes to a complete invariant.

Singly, the familiar measures are coarse. The integrity count splits the orbits into two classes, which is right — almost all cloths at four by four hang together — and useless for identification. The mark count gives five, the plane group twelve. The run lengths are by far the finest single measure, at sixty classes, which is the invariant the earlier account proposed and did not push through the census.

All seven together give 120 classes. That leaves 37 orbits unaccounted for: fifty-nine orbits sitting in twenty-two classes where every familiar measure agrees and the cloths still do not derive into each other.

Only three of the seven carry anything the others do not #

Adding the measures one at a time, most informative first, shows how little the list of seven really holds.

The runs of face and back alone give sixty classes. Adding the plane group gives a hundred. Adding the count of distinct ends and picks gives 120 — and the other four add nothing at all. The mark count, the interlacing count, the integrity count and the crossings per thread add nothing on this catalogue once the three are known: the runs already fix how many marks there are and how many changes of face the repeat holds in all, and the integrity count is one for nearly every cloth.

The confusions left are all among the larger orbits. Every one of the twelve cloths that derive into nothing is separated by the familiar measures; the fifty-nine orbits the seven cannot tell apart are 24 orbits of two and 35 of four, sitting in classes of two, three, four and five. A cloth with no derivative is distinctive enough to be recognised by counts; a cloth with three derivatives can have a stranger elsewhere in the catalogue that matches it on everything but arrangement.

Two cloths the familiar measures cannot tell apart #

The smallest confused class holds two orbits, and its two representative cloths are worth drawing, because they look like near neighbours and they are not relatives.

The two drafts agree on every measure above. Both carry seven marks, sixteen interlacings, one layer, no symmetry beyond translation, identical runs of face and back, two changes of face on every thread in both systems, and three distinct ends with four distinct picks. A specification written in any of those terms would admit both, and no weaver’s derivation turns one into the other.

What differs is where the runs sit relative to one another. Each draft has the same floats; they are arranged against their neighbours differently. That is exactly the kind of fact no count over whole threads can see, because each count is taken along one thread at a time.

A census of two-by-two patches is closer and still not enough #

The natural next measure looks across threads. Slide a two-by-two window over every position of the repeat, record the pattern of marks it sees, and count how often each pattern occurs; take the counts up to the eight turnings and reflections of the square and a counterchange, since a derivation can apply any of them.

That census is invariant — a derivation moves the windows about and turns their contents, and the census taken up to those turnings does not notice — and it sees arrangement, because a window straddles two ends and two picks at once.

It separates 127 orbits alone, more than all seven familiar measures together. With the seven added it separates 153. Three classes remain confused, holding seven orbits.

Two-by-two patches still miss a relation three threads apart. The pair above has the same local pairs of face and back in the same numbers; the way those pairs chain across a third thread differs. A window of two cannot see the third.

A census of three-by-three patches separates every orbit #

Widen the window to three by three and the census separates all 157 orbits. Two four-by-four cloths derive into each other exactly when they contain the same three-by-three patches, in the same numbers, up to turning the square over and counterchanging.

That is a complete invariant of derivation on the catalogue, and it answers the question as it was put. “Is cloth A derived from cloth B” no longer needs the group: count the nine-intersection windows in each draft, put each count in its least form under the square’s symmetries and a counterchange, and compare two lists.

It also says something about what derivation is. A derivation preserves a cloth’s local neighbourhoods and nothing coarser has to be preserved for two cloths to be relatives. The manuals present derivation as an operation on the whole draft — reverse it, turn it, counterchange it — and the census says the same relation can be read without ever handling the whole draft: from the small pieces of it a three-thread window sees.

A census is what a specification could carry #

The practical content of a complete invariant is a statement about specifications. A lifting plan says nothing without a threading, and a specification that names a cloth by measures rather than by its draft — “a four-end weave, seven marks, sixteen interlacings, one layer” — admits every cloth those measures fit. On this catalogue the seven familiar measures admit, in the worst case, cloths from five different derivation families. A buyer asking for “that weave or anything derived from it” and writing the request in those terms has asked for up to five unrelated cloths.

A census of three-by-three patches is an unusual thing to put in a specification, and it need not be put there literally. What it establishes is that a specification can be complete without being a picture: a list of local counts, each checkable by looking at a small piece of cloth, pins a four-by-four weave down to its derivation family exactly. The runs, the plane group and the distinct ends and picks — three measures a designer already reads off a draft — get within 37 orbits of that, and the census closes the rest.

That also bears on the float, which decides so much else. The float spectrum is the finest single familiar measure here, and it is still a measure along one thread at a time. Every confusion it leaves is a confusion about how floats on neighbouring threads sit against each other — which is exactly what a firmness, a lustre or a slippage depends on, and what a count along threads cannot say.

Why three, and not a smaller or a larger window #

The window size that works is not arbitrary, and the reason is visible in the frame.

A four-by-four draft on a torus has sixteen windows of every size. A one-by-one window sees only the mark count. A two-by-two window sees four of the sixteen intersections at a time and can confuse cloths whose pairs match and whose chains of pairs do not. A three-by-three window sees nine intersections — more than half the repeat — so any two of its sixteen overlapping views share at least four intersections and usually six, and the census’s constraints on how the views fit together are tight enough, on this catalogue, that two cloths with the same census always turn out to be one cloth turned, reflected or counterchanged. That is an observed fact about 157 orbits rather than a proof, and it is the kind of fact a larger repeat could break.

A four-by-four window would trivially work, since it sees the whole repeat, and it would be no simplification of the search at all. Three is the smallest window that is complete here, and it is the answer to “what is the finest invariant” in the only sense that matters for a draft: the coarsest local view that still loses nothing.

Whether three remains enough at larger repeats is not established. At six by six a three-by-three window sees a quarter of the repeat rather than more than half, and the argument above suggests the complete window grows with the repeat.

What reading a cloth from its fragments resembles #

The census has a close relative in a field with nothing to do with weaving. A genome is sequenced by breaking it into short overlapping reads and asking which k-mers — strings of k letters — occur and how often. Whether the k-mer counts determine the sequence depends on k against the sequence’s repeated stretches: short windows confuse sequences that share their short pieces in different arrangements, and a long enough window pins the sequence down.

A draft is a two-dimensional string on a torus, a patch is a two-dimensional k-mer, and the pattern is identical: a window too small to span a draft’s repeated local structure confuses cloths; one large enough reads the cloth back. The difference is that a draft’s reading is wanted only up to the symmetries of weaving, so the census is taken up to them, and at four by four three is enough.

What the census says about the basic weaves #

The three basic weaves do not generate the rest, and the census gives that finding a second form. A cloth derives from plain weave or from a twill exactly when its three-by-three patches match the seed’s in kind and number. Plain weave’s patches are two checkerboards and nothing else; a 2/2 twill’s are the four placings of one diagonal band a three-intersection window can take. Any cloth containing a single patch of any other kind is outside both orbits, and at four by four nearly every cloth contains one.

That turns the manuals’ opening sentence into something checkable on a swatch. To see whether a cloth is “derived from” a basic weave in the manuals’ own sense, it is enough to look at small neighbourhoods of it: if any nine-intersection patch is one that plain weave or the twill never contains, the answer is no, and no reversal, turn or counterchange will change it.

What combining two weaves reaches is a different operation — setting two weaves side by side in blocks — and it is not in the group, so the census makes no claim about it. It is worth noticing why: combining introduces patches along the boundary between the two weaves that neither weave contains, which is precisely the kind of change a derivation can never make. The census says derivation preserves neighbourhoods; combination manufactures new ones, and that is why it reaches further.

Where the census’s promise stops #

Four by four only. Every count is over the 426 cloths of the smallest interesting repeat. The claim that a three-by-three census is complete is a fact checked on 157 orbits, not a theorem about every repeat.

The group is the manuals’ group. Derivations that change the repeat — doubling a weave to eight ends, reversing a twill over a width it does not fill — are outside it, and the census says nothing about them.

A census is a list, not a number. A complete invariant here is a table of up to sixteen window counts. It is a comparison a machine makes easily and a person makes slowly, and whether a shorter invariant — a handful of integers — can also be complete is not answered: the census proves completeness is reachable with local information, not that it needs this much of it.

Still open: whether the complete window grows with the repeat #

At four by four the complete window is three, one less than the repeat. The obvious guesses at larger repeats are that it stays at one less than the repeat, which would make the census barely better than the draft itself, or that it stays near three, which would make derivation a genuinely local relation at every size. The catalogue of six-by-six cloths is too large to take whole, but the question does not need the whole catalogue: a random sample of six-by-six drafts, each paired with a random non-derivation partner chosen to share its familiar measures, would show whether a three-by-three census still tells the pairs apart, and at what window it first always does.

Who found it, and when #

The manuals’ derivations are as old as weaving instruction, plane groups and orbit counting are standard mathematics, and k-mer counting is standard in sequence analysis. The orbit census of the four-by-four catalogue was made in the account below. Testing which familiar measures are invariants of derivation, finding that all seven together separate 120 of 157 orbits, and that a census of three-by-three patches separates all of them while a census of two-by-two patches does not, was done here.