No cloth derives into more than three others

The three basic weaves do not generate the rest put the manuals’ opening sentence to the catalogue and found it false by a wide margin: starting from plain weave and every twill, and applying every derivation the manuals name, reaches nine of the 426 cloths that exist at four by four. What combining two weaves reaches admitted the one operation that account had excluded and got the reach to twenty-eight, leaving ninety-three per cent of the catalogue outside.

Both are enumerations, and an enumeration answers how much without answering why. The nine could have been nine because the operations are weak, or because the seeds are badly chosen, or because four by four is a small frame. It is none of those.

Every derivation the manuals name is one of two things: a relabelling of the grid — reversing the ends, reversing the picks, turning the draft a quarter — or a complementation, which is counterchanging. Both are invertible. A set of invertible operations closed under composition is a group, the reach from any starting point is that point’s orbit, and an orbit is bounded by the group.

So the question stops being an enumeration and becomes an arithmetic one, and the answer is much sharper than the count suggested.

The largest reach in the catalogue is four cloths #

Closing the manuals’ five derivations together with the four ways of writing one cloth down gives a group of 256 elements, and running it over the catalogue cuts the 426 cloths into 157 orbits.

Twelve cloths derive into nothing at all — their orbit is themselves, so every named operation returns the cloth it was given. Eighty-three derive into exactly one other. Sixty-two derive into three others. The catalogue’s largest reach is four cloths including the starting one.

That is a far stronger statement than the nine. The nine is a fact about a particular set of seeds; four is a ceiling on every seed there is, and it holds for the manuals’ basic weaves, for any weave a designer might prefer to start from, and for any sequence of operations however ingeniously composed.

The manuals’ claim is not merely false about its magnitude. A claim that a small set of seeds generates a large catalogue would be repaired by taking more seeds. This one cannot be repaired that way: to reach 426 cloths from orbits of at most four needs at least 107 seeds, which is not a generating set but a list.

An orbit drawn out #

The abstraction is easier to trust with an instance of it, and the instances are small enough to draw complete.

Reaching is symmetric, which is the property that makes an orbit an orbit rather than a reachable set, and it follows from the operations being invertible rather than from anything about cloth. Reversing a draft twice gives it back; counterchanging twice gives it back; turning it a quarter four times gives it back. So if A derives into B then B derives into A, and “the cloths derived from A” is the same set as “the cloths A is derived from”.

That has a consequence for how the manuals’ claim should be read. “Derived from plain weave” is not a direction of travel; it is a membership. A cloth is in plain weave’s orbit or it is not, and if it is then plain weave is equally derived from it.

One invariant puts a floor under it, with no enumeration at all #

The census establishes the orbit count. An invariant establishes a lower bound on it without counting anything, and it is worth having both because the second needs no computer.

A relabelling moves marks about and does not create or destroy them — which is the same reason a draft’s float spectrum survives a reversal, so the number of warp-up intersections survives every reversal and every quarter turn. Counterchanging exchanges it with sixteen less itself. So the unordered pair {w, 16 − w} is unchanged by every generator and therefore by every element of the group.

Two cloths with different pairs are in different orbits, necessarily, and no ingenuity closes the gap.

Five classes is a floor of five on the orbit count, from one line of reasoning. The enumeration finds 157, so the floor is far below — which says there are many finer invariants the intersection count does not see, and finding them would be the way to raise the floor without enumerating.

There is a second invariant that costs nothing to state and is sharper, and it is worth recording even though it has not been pushed through the census. A relabelling permutes a draft’s rows and columns, so it permutes the multiset of run lengths in each system; the quarter turn exchanges the two multisets; and counterchanging exchanges each system’s runs of face with its runs of back. So the unordered pair of run-length multisets, each taken up to complementation within its own system, is invariant too — and it is a far finer object than a single integer. Two cloths agreeing on it may still be in different orbits, but two disagreeing on it certainly are.

That is the shape every improvement to the floor would take: find a quantity the relabellings permute and the counterchange exchanges, and it is automatically invariant. The catalogue is full of them.

The floor is worth having anyway, because it settles the question in the form the manuals put it. A plain weave has eight warp-up intersections of sixteen and a 3/1 twill has twelve, so they are in different classes and neither derives into the other — before any enumeration, before any catalogue, and by an argument a reader can check in their head.

Which explains the nine exactly #

With the orbit structure in hand, the earlier census’s number stops being a measurement and becomes arithmetic.

The seeds were plain weave and the twills at four ends. Plain weave sits in one orbit; the 1/3, 2/2 and 3/1 twills sit in others, and the four orbits between them hold nine distinct cloths. Nine is a sum of orbit sizes, not a measure of how far derivation can travel, and it would have been the same nine had the census run for a thousand steps.

That also settles the shape of the earlier caveat. The first census noted that its restriction to four-by-four excluded derivations that make a wider repeat — reversing a twill over eight ends, for instance — and called its own count a lower bound. The orbit argument is not a lower bound: within the frame, four is exact and final. What the frame excludes is genuinely a different question, and the honest statement is that derivation inside a frame is bounded by a group and derivation that leaves a frame is not derivation in the same sense, which is the point the combination census made from the other side when it observed that a combination of two four-end weaves is eight ends wide.

Why the group is 256 and the orbits are 4 #

The two numbers look inconsistent — a group of 256 acting on 426 objects with orbits of at most four — and the reason is worth setting out, because it is where the whole argument earns its keep.

The group acts on drafts, and the catalogue is of cloths. A cloth is a draft up to the four ways of writing the same cloth down: started one end along, started one pick along, turned half round, turned over. Those four generate a subgroup of considerable size, and every element of it acts trivially on the catalogue, because a cloth is already the equivalence class.

So the group that actually moves cloths is the quotient, and the quotient is small. The 256 is mostly relabellings that are not derivations at all — they are ways of writing down the cloth one already has — and what remains once they are divided out is a handful of operations that do something.

That is the sharpest version of the finding. Most of what a manual calls a derivation is a change of description, and the operations that change the cloth are few enough that their orbits are tiny. Reversing a draft in the warp and reversing it in the weft and turning it half round are three names for a family of relabellings that a weaver has already been told are the same cloth in the section on how to read point paper.

The twelve cloths that derive into nothing #

Twelve orbits hold a single cloth, and they are worth naming because a cloth that is its own whole orbit is a cloth every named operation returns unchanged.

That is a strong symmetry. It means the cloth is its own reverse in the warp, its own reverse in the weft, its own quarter turn and its own counterchange — or that each of those happens to give a draft the catalogue already calls the same cloth. A cloth fixed by counterchange is a cloth identical to its own negative, which needs exactly eight warp-up intersections and an arrangement that the complement reproduces, and the plain weave is the obvious member.

So the twelve are the most symmetrical objects in the catalogue by this particular measure, and they are the ones about which the manuals’ claim is most nearly vacuous: a derivation applied to one of them produces the cloth it started from. A weaver told that every cloth is derived from the basic weaves, working with one of these, can derive for as long as they like and will not leave the draft in front of them.

The relation to the seventeen plane groups a draft can have is close and is not identity, and the difference is instructive. That census reads a draft’s symmetry as a two-coloured plane pattern and finds which group it belongs to. This one asks which of a specific list of operations fix it. A draft can be highly symmetric in the plane-group sense and sit in an orbit of four, because the plane group’s operations include translations the catalogue has already quotiented out, and it can have a small plane group and be fixed here, because counterchange is the operation that matters and the plane-group census treats colour separately.

The two censuses are over the same objects with different groups, and neither implies the other. Running them together — the orbit of a cloth under derivation, against its plane group — is a table the two enumerations would support and nobody has built.

What the manuals actually offer, restated #

Reading the five operations back with the group in hand separates them into two kinds, and only one kind was ever a derivation.

Reversed, reversed in the weft, reversed both ways, turned a quarter. These are relabellings of the grid. They do not change a cloth’s float lengths, its interlacing count, its layer count or its integrity, and in a catalogue that has already quotiented out the four ways of writing one cloth down, most of them do nothing whatever. They are instructions for reading point paper, not for making cloth.

Counterchanged. This one changes the cloth. It exchanges the two systems’ faces, it sends a warp-faced weave to a weft-faced one, and it is the reason eighty-three orbits have two members rather than one. It is also the operation a damask uses to make its whole pattern — the one place in this collection where a derivation is doing real design work.

So the manuals’ vocabulary of five is a vocabulary of one and a half. That is the finding at its most compact, and it is why the reach is four: a group generated by one useful operation and a handful of relabellings has orbits the size of the useful operation’s own order, which for an involution is two, doubled where a relabelling happens to bite.

The generalisation, which is about invertibility #

The transferable statement is short and it is not about weaving.

A set of invertible operations cannot generate more than it can undo. If every operation in a vocabulary has an inverse in the same vocabulary, the vocabulary partitions its domain into orbits, no orbit is larger than the group, and no amount of composition gets outside one. Generation in that setting is not a process but a classification, and asking how far it reaches is asking for an orbit size rather than for a path length.

The way to reach further is not to try harder with the same operations; it is to add a non-invertible one. Adding threads, doubling a system, combining two weaves into a wider frame — those are not invertible, they leave the frame, and they are exactly the operations the earlier censuses had to admit before the reach moved at all.

So the manuals are right that there are a few basic weaves and wrong about what follows from that, and the two halves come apart cleanly: the basics are a small set of well-chosen starting points, and the operations offered alongside them cannot travel. A textbook offering the same five operations as “five ways to write a cloth down differently” would be making the true claim in the same number of words.

What was computed, and how #

The five derivations the manuals name are applied as maps on the sixteen bits of a four-by-four draft, together with the four operations that write one cloth down another way. The group they generate is closed as a set of functions — two words are the same element exactly when they agree on a spanning set of masks — and the closure runs to completion rather than to a cap. Each cloth’s orbit is then the set of canonical cloths its images reduce to, over every element of the group, and the orbits are collected. The invariant is the warp-up intersection count taken up to complementation, and every orbit is checked to lie inside one class rather than assumed to.

Five things are checked. The derivations generate a finite group of modest order, which is the statement that the closure terminated rather than being truncated. No orbit is larger than the group, which is the group-action bound checked rather than cited. The catalogue falls into many orbits, over twenty, which is the finding. Every orbit lies inside one invariant class, which would fail if the invariant were not one. And the class count is at most the orbit count, which is the floor stated as an inequality.

The catalogue and the operations are built here; the group theory is arithmetic.

Where the argument stops #

It is a statement about one frame. Every operation here keeps a four-by-four draft four by four, and the manuals’ vocabulary includes operations that do not — doubling a thread, combining two weaves, backing a cloth. Those are not in the group and the bound says nothing about them.

The four-way equivalence is a convention adopted here. Whether a cloth started one end along is “the same cloth” is a decision, it is the decision the whole catalogue rests on, and a different convention would give a different group and different orbits. The orbit sizes would change; the argument that reaching is bounded by a group would not.

And the floor is a weak one. Five classes against 157 orbits means the intersection count is a coarse invariant, and a sharper one — the float spectrum, the plane group, the layer count — would raise the floor considerably. None of them has been run through this argument, and doing so would replace the enumeration entirely.

Still open: what the finest invariant is #

The enumeration gives 157 orbits and one invariant gives a floor of five, and between them sits a question with a definite answer: is there a set of invariants that separates the orbits exactly?

Such a set would be a complete invariant for derivation — a short list of numbers computable from a draft, agreeing exactly when two drafts derive into each other. With it, the question “is this cloth derived from that one” becomes a comparison of a few integers rather than a search, and the catalogue’s orbit structure becomes something a weaver could apply by inspection rather than something a computer reports.

Candidates that are known to be invariant under part of the group are already to hand — the longest float, the interlacing count, the plane group, the layer count — and it has never asked which of them survive complementation, which is the one generator that is not a relabelling. That is a small computation over a catalogue already enumerated, and it would turn a census into a criterion.

Who found it, and when #

The claim that the basic weaves generate the rest is in every weaving manual and is older than any of them. The enumeration that refuted it is in two accounts below. Reading the derivations as a group, and finding that the largest orbit in the catalogue holds four cloths, was done here.