Only a float can be raised

A raising machine is a drum covered in wire teeth, or in the older version teasel heads, running against the cloth. The teeth pick at the surface and pull fibre ends out of the yarns, so that a layer of loose fibre stands up off the fabric. That layer is the nap, and it is what makes a flannel warm, a moleskin soft and a blanket a blanket.

The teeth need something to catch. A tooth passing over a thread that is bound down at both ends of every span it has cannot lift anything; there is no slack anywhere and no unsupported length to hook under. What it needs is a float — a run of thread lying on the surface, crossing two or more of the other system without going under any of them.

Which means the whole question is a question about the draft, and it is decidable.

The plain weave cannot be raised #

Start with the case that decides the trade’s whole vocabulary.

A plain weave alternates at every intersection: every end goes over one pick and under the next, all the way across, and every pick does the same between the ends. The longest float anywhere in it is one, in both systems.

A float of one is a thread crossing a single thread of the other system and going straight back under. There is no unsupported span. A raising tooth passing over it meets a surface bound at every point, and it lifts nothing.

So a plain weave cannot be raised at any setting of any machine, and this is not a matter of degree. It is not that plain weave raises poorly; it is that the operation has no purchase.

This is why every napped fabric in the trade is a twill or a satin. Flannel is a raised twill. Moleskin is a raised sateen. Melton is a raised and milled twill. Blanket cloth is a twill or a broken twill. The vocabulary looks like a set of conventions and is actually a complete list of what is available.

The census answer #

This site enumerates every draft on four ends and four picks in which each thread interlaces at least once — 22,874 of them — and has asked that census several questions already: how often a draft falls into two cloths, which plane groups occur, how many shafts each needs.

Here is another one, and it has a sharp answer. Asking each draft whether it has a float of at least two on either face:

Exactly two drafts of the 22,874 have no float to catch anywhere, and they are the plain weave and its complement — which are the same cloth with the two faces exchanged. Nothing else in the whole census is unraisable.

That is a stronger statement than the trade’s rule and it says the same thing. Flannel is a raised twill not because twills are traditional but because plain weave is the only thing that cannot be, and everything else in the design space is available.

The zero at the bottom is a definition #

The bottom row of that table is zero and it would be easy to read as a surprise. It is not; it is the census’s own entry condition, restated.

A float of four in a four-pick repeat is a thread that never goes under anything. Such a thread does not interlace, and a draft containing one is excluded from the census by construction — the entry condition is that every end and every pick interlaces at least once.

So the zero is not the answer to how many drafts have very long floats; it is the answer to how many non-interlacing drafts are in a census of interlacing drafts. raisingCensus asserts it as exactly zero for that reason, so that a reader meeting the row understands it as a tautology rather than as a finding.

Asserting a tautology is worth doing when the alternative is that somebody reads it as data. It is also a live check: if the enumeration ever admitted a non-interlacing draft, this row would go non-zero and say so.

The binary answer is not the useful one #

Ninety-nine point nine nine per cent raisable is not a useful number, and it is worth saying why the census question is nevertheless worth asking.

The binary question — is there any catchable float — is nearly always yes. What varies enormously, and what decides whether a fabric raises well, is how much of the face is catchable. That is the fraction raisable() computes: the proportion of the surface lying in floats long enough to be caught.

The twills and satins reach 100 per cent because every one of their floats is at least two, so the entire face is available. What separates them in practice is float length rather than coverage: a longer float gives the tooth more to lift, produces a deeper nap, and takes more fibre out of the cloth’s structure per pass.

So the design variable is float length, which is the same variable this site has been following since the second essay, pointing in the same direction it always does.

The face matters and the first version got it wrong #

raisable() takes a face, because a cloth is raised on one side, and this is where the first implementation of the census was wrong in a way worth recording.

It asked only about weft floats. That is defensible for a weft-faced fabric and wrong for a warp-faced one, and it counted the 3/1 twills as unraisable — which is precisely the construction most flannels are made from. A warp-faced twill has its long floats on the warp, and a raising machine set against that face catches them exactly as it would catch weft floats on the other side.

The corrected census asks both faces and counts a draft raisable if either has a catchable float. That is the right question for can this fabric be napped at all, and the per-face figures remain available for how much nap does each side give.

The general shape is one this site has met before: a question asked about one system when it should have been asked about both, giving an answer that is plausible, self-consistent and backwards for exactly the case that matters most.

What was counted, and how #

raisingCensus walks all 22,874 drafts, builds each as a weave, and runs raisable() on both faces at each threshold. The float runs come from floats(), which measures cyclically — rotating so a run does not straddle the repeat boundary — and that is the same function the figure’s marks are drawn from, so the picture and the count cannot disagree.

Three assertions run. A machine needing a longer float can raise no more cloths than one needing less, which is monotonicity and would fail on an off-by-one in the run measurement. Exactly two drafts are unraisable at a threshold of two, which is the finding, asserted so that a change to the census or to the float walk announces itself. And zero at four, which is the definition above.

Why the answer is a property of the matrix at all #

It is worth pausing on what kind of result this is, because it is unusual in a field otherwise full of models with parameters.

Every other essay here computes something that depends on a friction coefficient, a packing factor, a loom tension or a measured constant. This one depends on none: the question is there a run of two or more is answered by reading the draft, exactly, with nothing to be uncertain about.

That puts it with the site’s oldest results — the integrity criterion, the satin theorem, the plane groups — rather than with the rest of this field. A finishing question turned out to have a matrix answer, which is not the usual direction of travel, and it happened because the raising machine’s requirement is topological rather than mechanical: it needs an unsupported span to exist, not a particular force.

A deep nap is bought in shafts #

The census’s own boundary — that a four-pick repeat holds a float of at most three — is stated above as a limitation. Read the other way it is a design rule, and it prices the one variable a napped cloth is actually sold on.

A wire needing a float of k needs a weave with a float of at least k, and a float of k needs a repeat of at least k + 1, because a thread that floats over its whole repeat never interlaces. So the shallowest construction that will feed a given wire is decided before any yarn is chosen:

repeat ≥ nap depth + 1.

And a repeat costs shafts. A regular satin of order n needs exactly n shafts and there is no cheaper threading for it; a twill of repeat n needs n on a straight draw. So the harness a napped cloth requires is

shafts ≥ float + 1,

which turns a surface property a customer can feel into a machine requirement a mill has to own.

Read against the trade’s own napped cloths it lands where it should.

Every one of them is at or just above the floor, and the ordering of nap depth across the four is the ordering of their shaft counts. The deepest naps in the trade are woven on the largest harnesses, and the reason is not that a heavy cloth needs a complicated draft — it needs exactly one thing from the draft, which is a long float, and a long float is what a shaft count buys.

Two consequences the binary census cannot reach.

The raisable fraction rises with the repeat and the answer stops being interesting. At a repeat of four, 22,872 of 22,874 drafts feed a wire needing a float of two; at a repeat of eight the same threshold excludes only the plain weave again, and the useful question has moved to which drafts feed a wire needing five or six. That census has not been run here and its shape is predictable: as the threshold rises the survivors thin out towards the satins, because a satin is the arrangement that concentrates its whole repeat into one run.

And a deep nap and a fine cloth pull against each other. A long float is set more closely than a short one, so a deeply napped construction is also a densely set one — which is more yarn per square metre before any fibre has been raised out of it. That is part of why napped goods are heavy, and it is a structural reason rather than a matter of the fibre standing up: the cloth underneath the nap was already the heavier cloth.

Where the model stops #

A float is necessary and it is not sufficient. Whether a fabric raises well depends on the fibre — a short-stapled, loosely twisted, woollen-spun yarn gives up fibre ends readily and a long-stapled, hard-twisted, combed cotton does not. Two cloths with the same draft and different yarns raise entirely differently, and none of that is in the matrix.

The threshold is a machine setting, not a fact about cloth. How long a float a tooth needs depends on the wire’s angle, the drum’s speed relative to the cloth, and the tension. The census reports the answer at a stated threshold rather than asserting one.

Four by four is a small repeat. The census is the site’s standard enumeration and it is not the whole design space; a twelve-end satin has floats no four-by-four draft can hold. What the census establishes is the shape of the answer, and the shape does not change with the repeat.

And raising is not the only way to a nap. Sueding uses abrasive rollers rather than wires and works on the fibre ends already at the surface, so it works on cloths a wire cannot raise — including, to a limited extent, plain weaves. That is a different operation with a different mechanism and the float argument does not apply to it.

Raising is directional, and so is the cloth after it #

One property of the operation does not follow from the float map and belongs here because it decides how a raised fabric is used.

The wires meet the cloth at an angle and travel in one direction relative to it, so the fibres they lift are laid over in that direction. A raised cloth therefore has a nap direction — run a hand along it one way and it is smooth, the other way and it resists — and it reflects light differently along and against the nap.

The consequence is a rule every tailor knows: all pieces of a garment cut from a napped cloth must be laid the same way up. A panel cut the wrong way round is visibly a different shade of the same colour, in the same piece of cloth, and no amount of pressing repairs it.

That costs fabric. Laying a pattern one-way is markedly less efficient than laying it both ways, and the loss is real money on a long marker. So a napped fabric is more expensive to make up than its price per metre suggests, before anything else is considered.

The direction also decides wear. Fibres lying over are protected; fibres standing up are abraded first. A cloth brushed with the nap wears differently from one brushed against it, which is why brushing instructions on a coat are not fussiness.

What the census cannot be asked #

The four-by-four census is this site’s standard enumeration and it has a boundary that this question runs into sooner than most.

A four-pick repeat can hold a float of at most three — a float of four is a thread that never interlaces. So every question about long floats is a question the census cannot answer, and the raising question is exactly such a question: a deep nap wants a float of five or eight, which needs a repeat of at least six or nine.

What the census establishes is the shape of the answer at the small end: almost everything is raisable, one construction is not, and the threshold sorts the design space monotonically. Those three facts do not change with the repeat.

What it cannot establish is the interesting end of the distribution — how the available nap depth is distributed across, say, all twelve-end satins — and that would need a larger enumeration than this site currently runs. It is recorded here as a question with a known method rather than as a limitation without one.

There is also a reading of the census result that is about the design space rather than about raising. The plain weave is the only draft on four ends and four picks with no float anywhere, which makes it the unique extremal member of the census in a property nobody had previously asked about — and it is extremal in the same direction it is extremal in interlacing count, firmness and crimp. That one draft keeps appearing at the end of every ordering this site computes is not a coincidence: it is the maximally interlaced cloth, and almost every quantity here is a count of interlacings in disguise.

Who found it, and when #

The observation is as old as napped cloth, which is to say older than any written account of it. Teasels were used for raising in Roman times and the fuller’s teasel was a cultivated crop in Europe for centuries; the wire card replaced it in the nineteenth century and gave more control and a harsher action.

That plain weave cannot be raised has never needed discovering — it is immediately obvious to anyone who tries — and what is not obvious, and appears to be unrecorded, is that plain weave is the only thing that cannot. The census makes it a complete statement rather than an observation about the fabrics anybody happened to try.

That is the general value of enumerating rather than sampling, and it is this site’s founding habit. A trade rule says use a twill. A census says use anything except one draft and its complement, which is the same advice with the boundary drawn.

Where the ladder goes next #

Raising takes fibre out of the yarns and stands it on the surface. Fibre standing on the surface is fibre that is no longer in the load path, so the next rung computes what a nap costs — and finds that the trade-off runs along the same axis as everything else the float decides.