How many shafts a draft needs

Everything counted on this site so far has been a property of the cloth. Float length, interlacings, cover, layers, plane group — take a piece of fabric, look hard enough, and every one of them is present in the object. The number this essay is about is not in the object at all. It is in the loom, and it decides which fabrics exist.

A shaft is a frame carrying a set of heddles, and every warp end threaded through those heddles rises and falls together. A loom has some number of them: four on an ordinary countermarch, sixteen or twenty-four on a dobby, and none at all on a jacquard, which is a different machine for a reason this ladder gets to later. The number is fixed when the loom is bought.

So a weaver looking at a draft asks a question no measurement of the cloth answers: can this be threaded on the loom in the room?

The answer is one pass over the matrix #

Two warp ends can share a shaft exactly when they do the same thing on every pick — when their columns in the draft are identical. If two ends differ anywhere, they must rise at different moments and cannot be tied to the same frame; if they never differ, no cloth on earth can tell whether they were separate.

So the shaft count is the number of distinct columns in the matrix. That is the whole of it. It is one pass, it needs no search, and it has been available since the draft became a matrix — which is to say it has been computable on this site since its first essay about weaves and was not computed.

The same argument in the other direction gives a second number that is easy to conflate with the first and is not the same. Two picks that lift the same set of shafts are the same lift, and a dobby chain stores lifts rather than picks — so the number of distinct rows is what the pattern chain costs. A repeat of forty-eight picks that only ever calls for four distinct lifts is four lags of chain, not forty-eight.

The threading and the lifting plan together are a factorisation of the draft: end j is up on pick i exactly when shaft threading[j] is up on pick i. And a factorisation is worth precisely as much as the check that it multiplies back out. It is easy to get wrong and the error is silent — number the shafts by first use in the wrong direction, or take distinct rows where distinct columns were meant, and the count comes out entirely plausible while the cloth comes out different. So every figure in this ladder weaves its drawdown from the threading and then asserts it against the draft, and a disagreement stops the build.

What the census says #

The four-by-four sweep has been the backbone of this site since the foundation: every draft on four ends and four picks in which each end and each pick interlaces at least once, which is 22,874 of the 65,536 possible matrices. It has been asked how often a draft falls apart, which plane groups it realises, how colour orders collapse it. It has never been asked what a loom would charge for it.

Three things fall out, and the third was not expected.

Nothing needs one shaft. A single shaft means every column is identical, so every pick is either over the whole warp or under it — which is a pick that never interlaces, and the census filter removed those before counting. One shaft is not a cheap weave; it is not a weave.

Two shafts is a very small world. Ninety-eight drafts out of nearly twenty-three thousand, which is four in a thousand. Plain weave is one of them and so is every ribbed variant of it, and the fact that so much of the world’s cloth comes out of that four-in-a-thousand corner says more about looms than about aesthetics.

Every draft that falls apart needs the full harness. The 144 separable drafts — the ones that look like fabric and are two fabrics — are distributed 0, 0, 144 across the shaft counts. Not one of them weaves on two or three shafts.

That last one deserves care, because it is a measurement and it is easy to over-read. It does not say that economy of harness prevents the failure; it says that at this repeat size the failure never occurs in the cheap corner. The reason is not mysterious once looked at — a draft on three shafts has two ends behaving identically, and duplicating an end tends to tie the fabric more firmly rather than less — but that is an explanation offered after the fact, and this site’s habit is to say so.

Why ninety-eight, exactly #

The two-shaft count comes out of the enumeration, and it is one of the cases where the enumeration can be checked against an argument rather than merely trusted. The argument is short and it explains the shape of the answer as well as its size.

A two-shaft draft assigns each of the four ends to one of two frames, with neither frame empty and the first end on the first frame by convention. That is 2³ − 1 = seven threadings, and the census finds seven, each carrying the same number of drafts.

Now fix a threading and ask what the two frames may do. Each frame’s behaviour over the four picks is a four-bit column, call them A and B. Two conditions apply. Every end must interlace, so neither A nor B may be all-up or all-down: fourteen choices apiece. Every pick must interlace too, and a pick sees only two distinct values — so if A and B ever agree at a pick, that whole pick is uniform and the pick does not interlace.

Therefore A and B must disagree at every pick, which is to say B is the complement of A, and it is not a free choice at all. Fourteen values of A, each fixing B, gives fourteen drafts per threading and 7 × 14 = ninety-eight, which is what the sweep counts.

The consequence is more interesting than the arithmetic. A two-shaft cloth is not merely cheap to thread; it is forced into a shape. The two groups of ends do exactly opposite things at every moment, which is plain weave with the single ends replaced by blocks — every rib, every basket, every warp cord. There is no other kind of two-shaft cloth, at this repeat or any other, and that is a proof rather than a count. It also explains why no two-shaft draft in the census has a float of four: a constant column is exactly what the complement condition forbids.

And the same argument gives five thousand one hundred and eighty-four #

The two-shaft count was worth deriving because it explained the shape of its own answer. The three-shaft count is fifty-three times larger and comes out of the same argument, and following it through says why the jump is so violent.

Start with the threadings. Four ends into three non-empty groups, with the groups numbered by first use, is the Stirling number S(4,3) = 6: one pair of ends shares a frame and the other two are alone, and there are six ways to choose the pair.

Now fix a threading and count what the three frames may do. Each frame is a four-bit column A, B, C. Three conditions apply and only the first is the same as before.

No frame may be constant, or its ends never interlace.

No pick may be uniform. A pick now sees three values rather than two, so the condition is that A, B and C are not all equal at that pick — which forbids two of the eight patterns per pick and leaves six.

And the three frames must be distinct, or the draft needs fewer shafts than three and belongs in another row of the census.

Counting them in that order: six patterns per pick over four picks is 1,296. Removing the triples with a constant frame, by inclusion and exclusion, leaves 906 — the subtraction is 486, the pairs add back 102, and the all-constant case removes six. Then the triples with two frames equal: if A and B coincide, the uniform-pick condition forces C to be their complement exactly, so there are fourteen such triples per pair and the three cases cannot overlap. That is 42 more, leaving 864.

Six threadings at 864 apiece is 5,184, which is what the sweep counts.

The structural reading is the interesting half, and it is the answer to a question the census raised and left standing: why is three shafts fifty-three times as large a world as two?

Because at two shafts the second column is forced and at three it is not. With two frames, every pick must have them disagreeing, which pins B to the complement of A and leaves fourteen drafts per threading. With three frames, a pick needs only some disagreement, which is a far weaker condition — six of eight patterns rather than one of two — and the counts diverge accordingly.

That is also the reason a three-shaft draft is never one of the separable ones, which the census reports and does not explain. A three-shaft draft has two ends behaving identically, so the digraph has a duplicated vertex, and a duplicate can only add edges to a component rather than split one. The census result is a consequence of the counting rather than a coincidence of it — though the argument here is offered after the measurement rather than before, which is worth saying plainly.

Why the number is not a property of the fabric #

This is worth stating plainly because it is the one thing about the shaft count that is genuinely different from everything else this site counts.

Take two drafts that need four shafts and forty. Wear them, wash them, hold them to the light, measure their floats and their cover and their drape. Nothing distinguishes a cheaply-threaded cloth from an expensively-threaded one as cloth. The number lives in the machine, and the machine is not present in the finished object.

That has a consequence which runs the other way and is much less obvious. A property invisible in the fabric has shaped almost every fabric that exists, because the drafts anyone ever wove are the drafts somebody’s loom could hold. The catalogue of historical weaves is not a sample of the space of weaves; it is a sample of the space of weaves filtered by an economic constraint that leaves no trace in the product.

That is a strong claim and this essay does not prove it. What the census establishes is the shape of the filter — how much of the space each budget reaches — and the shape is severe enough to make the claim worth taking seriously. The direct evidence would be a survey of surviving cloth against shaft count, which is a historian’s work and not a matrix’s.

The thing a straight draw cannot do #

Every draft above is a straight draw: the ends are threaded 1, 2, 3, 4, 1, 2, 3, 4 and the repeat of the cloth is the repeat of the threading. A straight draw spends one shaft per end of the repeat, so on a straight draw the two numbers are equal and the loom’s shaft count is the widest repeat it can weave.

Almost every book says exactly that, and it is true of a straight draw and false in general. The next rung of this ladder is entirely about how false it is: a reversed twill on ninety-six ends weaves on four shafts, and the repeat outgrows the harness without bound.

What was counted, and how #

The threading is built by walking the columns of the matrix left to right, keying each on the string of its entries, and numbering a column the first time it is seen. That numbering is arbitrary — any permutation of the shafts weaves the same cloth — but it is the numbering a weaver would use, which is to say the one that puts the first end on the first frame.

The lifting plan is then read off: shaft s is up on pick i exactly when the ends assigned to it are up on pick i, and because those ends are identical by construction there is no ambiguity about which to ask.

The drawdown is then woven rather than copied. Every cell of it is produced by looking up the shaft of its end and asking whether that shaft is up, and the result is compared cell by cell against the draft the threading was taken from. The comparison is not decoration: the factorisation is the claim, and a factorisation nobody multiplies back out is a claim nobody has checked.

Over the census, each of the 22,874 drafts is threaded in turn and the shaft counts accumulated. The total is asserted back against the census size, and the range is asserted to lie between one and four, because a draft on four ends cannot need five shafts and a count of zero would mean the walk never ran.

Where this stops being about the matrix #

Three limits, and the first two are the kind that get quietly forgotten.

Shafts are not the only thing a loom is short of. A harness has a maximum number of heddles per shaft, so a draft needing four shafts across a very wide warp may still be unweavable because one of those four carries too many ends. Nothing in the matrix knows how wide the cloth is.

The lifting plan has its own budget. A treadle loom is limited by how many treadles a weaver has feet for, and a dobby by the length of its chain. A draft with four shafts and forty distinct lifts is cheap in one currency and not in the other, and quoting the shaft count alone is the same error as quoting a thread count without a weave.

Nothing here says the threading is unique. The shaft count is minimal and the partition of ends into shafts is unique, but their numbering is not, and neither is the physical order in which the frames sit in the loom. Two threadings that differ by a permutation weave identical cloth and behave differently at the loom, because a shed opens more cleanly when the frames it lifts are not all at the back. That is a real consideration in a weaving shed and there is nothing about it in the matrix.

Who found it, and when #

Nobody found this, which is part of why it took this collection so long to compute. The identity of the shaft count with the number of distinct columns is not a theorem anybody published; it is what the loom is, and it has been obvious to every weaver since the harness was invented, in the way that the arrangement of one’s own kitchen is obvious.

What is new is asking it of a census rather than of a draft. The counting-out of a design space by the machine required to make it is a modern habit — it is the same move as asking how many chess positions are reachable, or which knots can be tied with a given number of crossings — and it needs the enumeration to exist first.

An earlier essay here of this site recorded the gap explicitly, in a section headed what is recorded as not done: “Nothing computes how many shafts a draft needs. It is the number of distinct columns in the matrix, one pass, and it is the strongest predictor of which of the twenty-two thousand drafts anybody ever wove.” That note is now three sentences of code and a census column, and the prediction in its last clause is the part still owed evidence.

Where the ladder goes next #

Three directions, and the ladder takes them in order.

The threading is where the economy is, and a harness that does not grow with the repeat is the whole reason a dobby is worth owning. The lifting plan is the other budget, and what a dobby actually stores is not what most descriptions of one suggest. And past both of them is the machine that gives up on shafts entirely, where every end is its own shaft and the constraint moves somewhere else altogether.