A lifting plan says nothing without a threading

The rung below measured the first of the three notations for something larger than a weave. This is the second, and it differs from the first in a way that has to be said before anything can be counted: it is not a notation, it is a pair of them.

A lifting plan says which shafts rise on each pick. On its own that is a statement about a machine and not about a cloth — the same plan on two different threadings produces two different drawdowns, and there is no draft it names.

The pair is a factorisation #

A harness factors a draft: end j is up on pick i exactly when shaft threading[j] is up on pick i. So the drawdown is a product of two things — a map from ends to shafts, and a map from picks and shafts to lifts — and the notation is that product written out.

That is a factorisation and not an encoding, which is what makes its behaviour different from the profile’s. A profile draft is a substitution: two weaves go in, one drawdown comes out, and the operation is deterministic in one direction. A factorisation has the two properties factorisations always have:

• its image is constrained — not every draft factors at a given size;
• and it is not unique — a draft that factors at all factors many ways.

Both are measurable and both say something about what a weaver is actually doing.

The image is the shaft census #

The constraint is the one this collection has had for a field. Two ends can share a shaft exactly when their columns are identical, so a draft factors on S shafts precisely when it has at most S distinct columns — and that is the whole of it.

Running the four-by-four sweep by distinct columns gives the harness’s image at every size:

So a two-shaft loom weaves 98 of the 22,874 four-by-four drafts, a three-shaft loom 5,282, and a four-shaft loom all of them — which it must, because four ends cannot have more than four distinct columns.

The image is a nested chain, one image per shaft count, and it exhausts the sweep at the repeat’s own size. That is a completely different shape from the profile’s single small set.

The redundancy, which is the weaver’s freedom #

The second property is the one nobody counts and it turns out to be useful.

A draft with c distinct columns, woven on S shafts, can be threaded in S × (S − 1) × … × (S − c + 1) ways — the number of injective assignments of its column classes to shafts — and the lifting plan follows from the threading. So a four-column draft on four shafts has twenty-four threadings and on eight shafts has 1,680.

Every one of them produces exactly the same cloth. The notation is therefore redundant by that factor, and a designer choosing among them is choosing nothing about the fabric.

Except that they are not choosing nothing about the loom. Where the heddles go is the collection’s own account of the difference: the threadings differ in how many heddles hang on each shaft, and a shaft with three times its neighbour’s load is three times as heavy to lift and opens its shed against three times the friction.

So the notation’s redundancy is exactly the weaver’s freedom to balance the harness, and the two are the same number. That is a satisfying place for a redundancy to come from — most notations’ redundancies are waste, and this one is a design variable with a name.

Neither image contains the other #

The ladder’s own question was which of the notations can express a cloth the others cannot, and the cross-tabulation answers it.

The profile draft reaches 306 drafts. The harness at three shafts reaches 5,282. The second number is seventeen times the first and the second set does not contain the first:

192 of the profile’s drafts need all four shafts, so no three-shaft loom weaves them and the harness’s three-shaft image misses them entirely. And 64 of the 98 two-shaft drafts are outside the profile’s image, so a two-shaft loom weaves cloths a block design cannot describe.

The two images are therefore incomparable: each notation expresses cloths the other cannot, and neither is a restriction of the other in any sense.

That is not the answer the question invites. Notations for one subject usually form a hierarchy — the rung below found four notations for a single weave and they nested cleanly, with point paper on top — and these two do not, because they are restricting different things. A profile restricts the design’s structure and a harness restricts the cloth’s columns, and those are unrelated properties of a matrix.

Which is why a designer needs both #

The practical form is short and it explains a division of labour the trade has always had.

A designer works in a profile and a weaver works in a harness, and the two documents are not translations of each other. A profile draft that produces a drawdown needing more shafts than the loom carries is a design that cannot be woven, and the profile does not say so — the shaft count is not in the notation, because the notation’s cells are blocks and shafts are counted over columns.

So the check that a design is weavable is a third computation, done on the drawdown, using neither notation’s own vocabulary. That is what the harness’s own budget is for, and it is why the drawdown exists at all — the same gap a figured cloth’s cost is measured across: it is the common object the two notations both project onto, and the only place a question involving both can be asked.

A drawdown is not a notation anybody works in. It is the meeting point of two that people do, and it is generated rather than written — which is exactly how this collection’s own figures produce it.

The third notation, which is not a notation either #

The ladder named three and the jacquard card is the last, and it is worth a paragraph because it is the degenerate case that makes the other two legible.

A card is one pick’s lifts, verbatim, one hole per hook. There is no threading to factor against, because a jacquard gives every end its own hook — so the “threading” is the identity and the card is the drawdown, a row at a time.

Its image is therefore everything whose repeat fits the hooks, which for a machine of six hundred is every draft this collection could ever enumerate. And its redundancy is one: there is exactly one card set per design.

So the three sit at three points of one axis:

• a profile is small, ambiguous about where each weave starts, and shaped like a design;
• a threading and lifting plan is medium, redundant by a computable factor, and shaped like a loom;
• a card set is complete, unique, and shaped like nothing at all.

And the economy runs backwards along the same axis. The profile is a few dozen cells, the harness pair is a few hundred, the card set is one hole per end per pick — which is why the machine that needs no notation is the machine that costs the most to instruct.

What the pair costs to store, which is where the dobby lives #

The three notations sit on an axis of expressiveness and they sit on another of size, and the second one is what a machine actually pays.

A threading is one shaft number per end of the repeat. At E ends and S shafts that is E log₂S bits — 8 bits for a four-end repeat on four shafts, and 3,600 for an 1,800-end repeat on four.

A lifting plan is one bit per shaft per pick. At P picks that is S·P bits — sixteen for a four-by-four on four shafts, and it does not grow with the width at all.

That asymmetry is the whole economics of the shaft loom. The threading grows with the cloth’s width and the lifting plan does not, so a repeat run out across a wide warp costs threading and nothing else — which is why the harness does not grow and why a four-shaft loom weaves a repeat of any width.

And it says exactly what a dobby stores. A dobby holds the lifting plan and not the threading, because the threading is in the heddles and stays there between designs; so the machine’s memory is S·P bits and a design change is a change of chain rather than a rethreading.

So the pair’s two halves live in two places with two costs, and the notation’s shape follows the machine’s. A profile draft has no such split because it is not a factorisation, and a jacquard card has none because there is nothing to factor.

The one thing a lifting plan can say that nothing else can #

Everything above has treated the pair as a way of naming a draft, and there is a question it answers that the draft cannot.

A lifting plan says what the loom does, pick by pick, in order. A drawdown is a matrix and a matrix has no order in it — the picks are rows and any permutation of them is another matrix that the collection’s own measures would treat as a different draft with the same everything.

So a lifting plan carries time, and the two things that depend on it are not visible in a draft at all.

The shed’s sequence. Two consecutive picks that lift the same shafts require no shed change and two that lift complementary sets require every shaft to move. So a lifting plan has a cost per pick which depends on its neighbours, and a plan reordered to reduce shaft movement weaves faster and strains the warp less — on a cloth the draft says is identical.

And the beat-up’s history. A pick is beaten against the cloth already formed, so what a pick meets depends on the picks before it, and a rearranged lifting plan makes a cloth with the same matrix and a different fell geometry. A mispick is one row in the wrong place is the collection’s own account of a fault of exactly this kind — a defect that is invisible in the repeat and visible in the cloth.

So the pair’s image is smaller than the draft’s and its content is larger, which is the reverse of the usual trade and is why a lifting plan is what a machine is given rather than a drawdown. A drawdown is what the cloth is; a lifting plan is what to do.

What was counted, and how #

The shaft census is the site’s own, unchanged: every draft in the four-by-four sweep, sorted by the number of distinct columns. What is new here is reading it as a notation’s image rather than as a harness requirement, which is the same numbers under a different question.

The profile’s image is the rung below’s enumeration, also unchanged, and the cross-tabulation intersects the two sets rather than recomputing either.

The finding is asserted in both directions, which is the whole of what “incomparable” means: the profile must reach at least one draft in the largest shaft class, and it must miss at least one draft in the smallest. A one-way assertion would be satisfied by one image containing the other.

And the redundancy is arithmetic rather than an enumeration. S!/(S − c)! is the count of injective maps, and it is stated rather than counted because the counting would be a check on a factorial.

Why the drafts a two-shaft loom weaves are so few #

The 98 is a striking number and it is worth a moment, because it says something about the whole chain of images.

A four-end draft on two shafts has at most two distinct columns, so its four columns take one of two values — which is 2⁴ = 16 assignments — and each of the two values is one of the 2⁴ = 16 possible columns. That is 16 × 16 × 16 before the constraints, and the constraints cut it to 98.

The constraints are the interlacing condition, which this collection’s sweep applies: every end and every pick must reach both faces. A two-shaft draft with both columns equal is a cloth of one shaft and is not cloth at all; a draft whose picks do not all interlace is out; and what survives is 98.

So the smallest image is small for a structural reason and not because two shafts are a hard limit — a two-shaft loom weaves 0.43 per cent of the drafts of its own repeat, and it weaves plain weave, every rib, every hopsack and every one of the doubled family, which is a great deal of the world’s cloth.

That is the same observation the profile’s rung makes about its own narrowness, arriving from the machine’s side. A tiny image can cover most of what is made, and the ratio of image to sweep says nothing about a notation’s or a machine’s usefulness on its own.

The chain says it three times over. Two shafts: 0.43 per cent, and most of the world’s cloth. Three shafts: 23 per cent, and almost nothing anybody weaves — a three-shaft loom is a curiosity. Four shafts: everything, and the twills. The share and the usefulness are not related at all, which is the general lesson of measuring images and is worth having before the numbers are quoted anywhere.

Where the model stops #

Four by four is a very small repeat, and the harness’s image exhausts it. At four ends a four-shaft loom weaves everything, which is not true at any larger size — an eight-end repeat has up to eight distinct columns and an eight-shaft loom is an ordinary one. So the chain of images is much longer at a real repeat and the top of it is not the whole sweep.

The threading is taken as a straight assignment of ends to shafts. A real threading can be pointed, broken or skipped, and those are the same map read differently rather than a different kind of object — but the count of threadings above assumes any injective assignment is available, and a mill threading by hand has preferences the arithmetic does not.

And nothing here is about the tie-up. A treadle loom’s notation has a third part, the tie-up, which factors the lifting plan again into treadles and their shaft connections. That is a further restriction and a further redundancy, and it is a different machine’s notation rather than a different notation for this one.

Who found it, and when #

The factorisation of a draft into threading and lifting is as old as the shaft loom and is how every weaver has written a draft for centuries. That the shaft count is the number of distinct columns is elementary and is in every manual, usually as a rule for reading a threading off a drawdown.

Reading the shaft census as a notation’s image is this collection’s, and so is the cross-tabulation against the profile’s. The finding — that the two are incomparable — is the sort of thing that could only turn up by measuring both, because the intuition points the other way: a profile produces a drawdown and a drawdown is woven on a harness, so it feels as though the profile must be inside the harness’s reach.

It is not, and the reason is stated in one line: a block design can need every shaft the repeat has. A two-by-two block of two weaves reaches drafts with four distinct columns, and a designer who has never counted them has no reason to expect it.

Where the ladder goes next #

Three notations have now been measured and all three are notations for a repeat — a pattern that tiles the plane. The document a mill actually works from is none of them: it is a specification, and a specification names a cloth by its yarn, its setts and its weave together, which is a fourth kind of object with a fourth kind of image and a much less obvious failure mode.