A profile draft is a notation whose alphabet is weaves

The rung below measured four ways of writing a weave down — point paper, a fraction name, a four-part draft, a float specification — and asked of each what it can express. The answer was that only point paper can say everything, and that the others fail in four different ways.

It ended by naming what it left open: the notations that are not about a single repeat. A profile draft that carries a block design, a lifting plan a dobby stores, a card a jacquard reads. Each of those is a notation for something larger than a weave, and the interesting question is which of them can express a cloth the others cannot.

This is the first of the three, and its image is small enough to count.

What a profile draft is #

A profile draft is a small grid. Each cell is a block, each block carries one of two weaves — a figure weave or a ground weave — and the whole thing stands for a drawdown b times as large in each direction, where b is the block’s size in threads.

It is a notation whose alphabet is weaves rather than intersections, and it is how every figured cloth is actually designed. A damask designer draws a shape on a grid of blocks; the weaves are chosen once, for the whole design; and the drawdown is produced from the two by substitution.

So the notation is a substitution, and its image is exactly the set of drafts obtainable by substituting two weaves into a block pattern.

The image, enumerated #

That set is finite and small enough to walk. At a four-by-four repeat with two-by-two blocks:

• there are sixteen two-by-two matrices, so 256 pairs of weaves;
• there are four blocks, so sixteen assignments;
• 4,096 combinations, and the distinct drafts among them are the notation’s image.

The result is 1,696 of the 65,536 four-by-four matrices, of which 306 interlace — 1.34 per cent of the 22,874 the four-by-four census is over.

One and a third per cent. The notation that every figured cloth is designed in can express one draft in seventy-five of its own size.

Which is a fact about designs and not a defect #

That number sounds like an indictment and it is not, and saying why is the point of measuring it.

The 306 are the ones anybody weaves. A figured cloth is a figure weave in one region and a ground weave in another; that is what “figured” means, and it is what the notation was built to say. The 22,568 drafts outside the image are not designs a profile draft fails to reach — they are drafts with no block structure at all, which no designer would draw and no design would produce.

So the notation’s narrowness is a fit rather than a limitation. It is narrow in exactly the direction the design space is narrow, which is what a good notation does.

The comparison with the rung below is instructive here. A fraction name is narrow in the wrong direction: it reaches only the twills, so it fails on satins and everything else and its failures are the interesting cloths. A profile draft’s failures are drafts nobody wants, which is a completely different kind of narrowness and cannot be seen without counting both.

The block size is the notation’s resolution #

The image shrinks as the block grows and the control makes that visible.

A one-by-one block reaches everything. With blocks of a single intersection the two “weaves” are a filled square and an empty one, the profile is the draft, and the image is all 65,536 matrices. That is the degenerate case and it is the one that says the enumeration is right.

A two-by-two block reaches 1.34 per cent. Coarsen the alphabet and the notation says less.

And a block equal to the repeat reaches two drafts — the two weaves themselves — because there is one block and it takes one value or the other.

So the block size is the notation’s resolution, and it trades against expressiveness exactly as a resolution does. That is the same trade a woven outline makes in the cloth: a block is the smallest thing a design can place, and the finer the block the more the design can say and the more it costs.

What the notation cannot say, which is where the census lives #

The profile is a substitution and a substitution has a boundary problem, which is the whole subject of the figured-cloth field.

A profile draft says nothing about what happens at a block boundary. It says this block is figure and that one is ground; it does not say what the intersections at the join do, because the join is not a cell of the notation at all. The drawdown has a boundary and the profile does not.

That is not a small omission. The whole apparatus of the figure census — the coarse cases, the loose threads, the genuine two-dimensional separation that only the connectivity computation can see — lives at those boundaries, and every one of them is invisible in the notation a designer works in.

So the profile draft is a notation whose image is small and whose failures are all at the edges of its own cells. A designer reading a profile back cannot tell whether the cloth hangs together, and neither can the notation.

Where the image’s boundary actually falls #

Knowing that 306 of 22,874 are reachable is one thing; knowing which is more useful, and the reachable set has a description.

A draft is in the image exactly when it is block-constant in the right sense: cut it into two-by-two blocks, and every block must be one of at most two distinct two-by-two matrices. That is the substitution read backwards, and it is a test anybody can apply to a draft in a few seconds.

So the image is the set of drafts with at most two distinct blocks, and the count follows from that rather than from the enumeration — which is why the enumeration is worth running as a check on the description rather than as the primary result.

Two consequences fall out and neither is obvious from the count.

Every draft in the image has at most two distinct blocks and therefore at most two distinct block-columns, so its shaft count is bounded by twice the shafts either weave needs. A block-structured cloth is cheap on the harness for a structural reason, which is what a figure costs the loom approached from the notation rather than from the design.

And a draft with three distinct blocks is outside the two-weave image and inside the three-weave one. So the notation’s image grows in steps as the alphabet does, and the steps are large: three weaves at two-by-two blocks reaches a great deal more than two, because the constraint being relaxed is the strongest one there is.

The three larger notations, and what separates them #

The rung below’s open question was which of the three can express a cloth the others cannot, and the profile’s image is the first term.

A profile draft is a substitution and its image is the block-structured drafts. Bounded, small, and exactly the design space.

A lifting plan is a factorisation. It says which shafts rise on each pick, and with a threading it produces the drawdown — so its image is the drafts expressible on the shafts available, which is a different bound entirely and is the harness’s own.

A jacquard card is neither. It is one pick’s lifts, verbatim, so it is not a compression at all: its image is every draft whose repeat fits the hooks, which is everything.

So the three are a strict hierarchy in expressiveness and the reverse in economy, and the middle one is the only one that is a genuine trade. The profile is small and cheap, the card is complete and expensive, and the lifting plan sits between them at a boundary the loom sets.

What the notation buys, which is not compression #

It is tempting to describe the profile draft as a compression — a small grid standing for a large one — and the arithmetic says that is the wrong word.

A profile of B by B blocks with b-thread blocks stands for a drawdown of Bb by Bb. The notation carries B² bits plus the two weaves; the drawdown carries B²b² bits. So the ratio is b², and at four-thread blocks that is sixteen to one.

But a compression is lossless and this is not. The profile cannot express most drafts of its size at all, which is what the 1.34 per cent measures. It is not a smaller encoding of the same information; it is an encoding of less information, chosen so that what is left is what a design has.

The distinction matters because it says what the notation is for. A compression is for storage and a restricted notation is for thinking. A designer working in blocks is not saving paper; they are working in a space where every point is a design, so that nothing they draw is nonsense.

That is a property the drawdown does not have: nearly every four-by-four matrix is a draft nobody would want, so a designer working on point paper is working in a space that is 98.7 per cent noise. The profile’s narrowness is what makes it usable, and it is the same argument a fraction name fails: a fraction name is narrow too, and its space is not the space of designs.

And a third thing the notation cannot hold #

The boundary is the obvious omission and there is a second one that is easier to miss, because it does not look like an omission at all.

A profile draft records which weave goes in each block and not where each weave starts. Each of the two weaves has a starting point — a satin’s move can begin at any of its n cosets, a twill’s diagonal at any end — and the profile has no cell for either.

That matters because the relative origin decides more than it looks like it should: two cloths built from the same profile and the same two weaves, differing only in where one of them starts, are different cloths with different boundaries and different failure behaviour.

So the notation is ambiguous as well as narrow. Its image is 306 drafts and a single profile-and-weave-pair does not name one of them: it names a family, indexed by the two weaves’ own starting points, and the designer’s own habit of “starting both weaves at the corner” is a convention filling a gap the notation left.

That is a different kind of gap from the boundary one. The boundary is outside the notation’s cells; the starting point is a parameter the notation forgot. A notation can be extended to carry a starting point — two small numbers would do it — and it cannot be extended to carry a boundary without ceasing to be a substitution.

What was counted, and how #

The image is enumerated and not derived. Every pair of weaves and every assignment of blocks is built, the resulting drafts are reduced to a bitmask, and the distinct masks are counted — 4,096 combinations at the two-by-two block, which is small enough to do exhaustively and large enough that a closed form would be worth doubting.

The interlacing subset is taken against the site’s own sweep, so the denominator is the same 22,874 every other census here is over and the two numbers are comparable.

The control is asserted rather than mentioned. A one-by-one block must reach exactly 2^(n²) drafts — all of them — because there the profile is the draft. An enumeration that got that wrong would be wrong everywhere and would look plausible.

And the enumeration refuses what it cannot do. The cost is (2^b²)² × 2^(n/b)², which at a four-by-four block is 8.6 × 10⁹, so the guard computes it and refuses rather than attempting it. A block equal to the repeat is refused for a different reason: it is a degenerate case with one block in it.

What a designer loses by working in it #

The notation is the design space, so its gaps are the designer’s blind spots, and it is worth listing them as such rather than as properties of a notation.

A designer working in blocks cannot see whether the cloth hangs together. The connectivity failure the figured-cloth census finds — a genuine two-dimensional separation with no long float and no loose thread — is invisible in the profile and invisible in the drawdown to the eye. It has to be computed, and it is computed from the thing the designer did not draw.

A designer working in blocks cannot see the phase. Two designs that are the same profile are not the same cloth, and which of the family arrives depends on how the drawdown was generated rather than on any decision.

And a designer working in blocks cannot see the resolution failure. A block smaller than either weave’s repeat produces a drawdown in which neither weave completes, and the result is neither weave — which is a fault with a threshold the notation does not carry, because the notation does not know how large its blocks are in threads.

That last one is the sharpest, because it is a unit problem. A profile draft’s cells are dimensionless: they are blocks, and how many threads a block holds is not in the notation. So the same profile at a four-thread block and at a two-thread block are the same drawing and are a sound cloth and a failure.

Which makes the block size the one number a profile draft must be accompanied by and does not contain, and it is exactly the number a woven outline’s staircase is measured in.

Where the model stops #

Two weaves, and a design can have more. A profile draft with three or four weaves — a figure, a ground and a binding — is ordinary in practice, and its image is correspondingly larger. What is measured here is the two-weave case, which is the commonest and is the one the block notation’s own convention assumes.

The blocks are square and equal. A real profile has blocks of different sizes, laid out to suit the design, and a rectangular block is not half a rule — the census’s own finding is that unequal blocks behave differently at their boundaries. The image of a variable-block profile is not computed.

And four by four is a very small repeat for a figured cloth. A real damask is on eight-end satins with blocks of four or eight, so the repeat is thirty-two or sixty-four ends and the enumeration is out of reach. The 1.34 per cent is a measurement at the size that can be measured, and whether the share falls or rises with the repeat is not known — the combinatorics push both ways.

Who found it, and when #

The profile draft is old and universal: every design manual teaches it, every jacquard designer uses it, and its relation to the drawdown is explained as a substitution in all of them.

Measuring its image is this collection’s, and it follows the rung below’s method exactly: take a notation, ask what it can express, and count. What is different here is the interpretation. A narrow image was a defect for the four single-weave notations, because their failures were cloths anybody might want; it is a fit for the profile, because its failures are drafts nobody would draw.

That distinction is the useful part, and it needs both numbers to see. A notation’s image is worth measuring; the number on its own says nothing about whether the notation is good.

Where the ladder goes next #

The lifting plan is the second of the three and it is the one with a real trade in it: it says which shafts rise on each pick, its image is bounded by the harness, and unlike the profile it is what the loom actually stores. What it can and cannot express is a question about the machine rather than about the design.