The census counted two systems and a surface has one

Two drafts of twenty-two thousand ran the whole four-by-four catalogue through one question — how much horizontal crown line does this surface carry — and produced a result with the shape this collection likes best: an exact exception. Exactly two drafts carry none, and both are the plain weave, because a draft with no run of two anywhere is an alternation both ways, and there are two ways to write one down.

That essay also said, in its own list of what it did not do, that a bearing curve cares which crown is higher and a reflection does not. It did not follow the remark through.

A bearing curve sees one system. A plate coming down on a cloth meets whichever crown stands higher and meets nothing at all until it has sunk past the step between the two — which on a sheeting is eight micrometres, and which the same essay computed. Everything below that depth is one system’s crown line and none of the other’s.

The census summed both. So the number the catalogue was ordered by is a number no plate ever reads, and recounting it the way a surface is actually read changes the headline by a factor of two hundred and forty-seven.

What a plate meets, and what it does not #

The arithmetic is not in dispute and the reading of it is. A warp run of three contributes two spacings of crown line; so does a weft run of three; the census adds them and reports the total per square millimetre.

Add them and the object is a property of the cloth’s interlacement. It is a good quantity, it distinguishes drafts the float count cannot, and it is what a reflection would read, because lustre is a length times a width and light does not care which of two crowns is a few micrometres higher.

Do not add them and the object is a property of the cloth’s outside. At the top of the bearing curve there is one system and one only. The other arrives at the step, seven or eight micrometres down, and everything the bearing curve says about a light touch — which is nearly all a hand ever applies — happens above it.

Those are two different questions and the second is the one the surface account is about.

Recounted, four hundred and ninety-four drafts touch at points #

Running the catalogue again and counting each system separately gives three numbers where there was one.

Under either single-system count, 494 drafts touch at points. They are the drafts whose higher system has no run of two anywhere — an alternation in one direction only — and there are far more of those than there are alternations in both directions, which is the whole of why the number moves.

The published two remain, and they remain the exception they were said to be. A plain weave has no run of two in either system, so it carries no crown line under any of the three counts, and it is the only draft in the catalogue with that property. The finding survives; what changes is that it stops being the answer to the question the surface account was asking, because 492 other drafts also present a surface of isolated summits to a plate.

That is a large correction to a published number and it is worth being plain about where it came from. Nothing in the arithmetic was wrong. The census computed exactly what it said it computed; the essay reported it as a property of the surface; and the two are the same thing only for a cloth whose crowns are level, which no cloth’s are.

The two conventions are two different orderings #

If which system bears were a detail, the two single-system counts would be small perturbations of each other. They are not.

Only the two plain weaves carry the same crown line in both systems. Everything else in the catalogue has a warp answer and a weft answer, and they are different.

That is a stronger statement than it first sounds. The two counts are not two noisy measurements of one ranking; they are two rankings of the same objects with almost nothing in common, and a draft that is best by one is frequently near worst by the other. A three-and-one twill, the maximum under the warp count at 1.400, carries no warp-… carries almost nothing under the weft count, because its weft is on the face at one crossing in four and never twice running.

So a designer searching the catalogue for a particular surface is searching one of two lists, and which one is decided before any draft is chosen.

What decides it is a number with no draft in it #

The thing that picks the list is the step between the two crowns, and the step comes from the crimp division.

Peirce’s closure condition fixes the sum of the two systems’ wave heights and leaves their division free. This collection’s standing position is that the crimp ratio is not a measurement but a convention, and every figure here is computed at an even division because an even division for every draft makes a comparison of drafts.

The level point is at 0.836 and the convention is at 1, so the two are not far apart. A sheeting sits on the warp side of the level point by a division of about a fifth — which is to say the whole of the catalogue’s ordering rests on a quantity nobody measures being nearer one than 0.836.

Other cloths sit differently. A poplin’s level point is at 0.585 and its step at the convention is twenty micrometres, so it is further from the boundary; a muslin is at 0.828 and a duck at 0.864, both closer to it than the sheeting.

A cloth whose crimp divides unevenly enough reads the other list. The arithmetic does not say that any real cloth does; it says the question is open, it is decided by one number, and the number is a convention.

The U-shape goes with the sum #

The recount takes a second finding with it, and this one is the original essay’s own explanation of itself.

The census reported that crown line is a U-shaped function of balance: a draft with twelve of its sixteen intersections warp-up carries 1.400, one with eight carries 1.136 on average, one with four carries 1.300 — so the most balanced cloths carry the least and the most one-sided the most, in both directions. The essay gave the mechanism in one line: the two systems share the sixteen intersections, a balanced draft chops both systems’ faces into short runs so both contribute little, and a one-sided draft gives one system long runs and the other almost none, but the other was contributing almost nothing anyway.

Every word of that is about the sum.

The warp’s own crown line rises steadily with the warp-up count and never turns. 0.000 at four, 0.264 at six, 0.589 at eight, 0.964 at ten, 1.400 at twelve. There is no floor, no balanced minimum, and nothing at all of the shape the essay spent a section explaining.

So the U is not a fact about woven surfaces. It is a fact about adding two mirror-image monotone quantities, which is the shape of every sum of a rising line and a falling one and has been known to anybody who has added two lines. The original essay’s mechanism is correct and it is a mechanism for the arithmetic rather than for the cloth.

And the practical reading inverts with it. The essay concluded that the most balanced cloths have the least crown line and the most one-sided the most, in either direction — so a designer wanting little crown line should choose a balanced draft and would get the same answer whichever face went out. Under the single-system count the answer is the opposite: put the face with the fewest intersections outward. A three-and-one twill’s warp face carries 1.400 and its weft face carries nothing, and the two are the same cloth turned over.

That is a design instruction the summed census could not give and could not have given, because it had averaged away the only thing a plate can see.

Which four hundred and ninety-four they are #

The drafts that touch at points under the single-system count have a description, and it is not the one the two plain weaves have.

A draft carries no warp crown line exactly when no end is on the face at two consecutive picks — every warp run is a single crossing. That is an alternation down every column, or something weaker: a column with isolated ups and long runs of down also qualifies, because a run of down contributes nothing to the warp’s crown.

So the 494 are not a family of near-plain weaves. Sorted by balance they run 24 drafts at four warp-up intersections, 144 at five, 216 at six, 96 at seven and 14 at eight — and they stop there, because a draft with nine warp-up intersections in sixteen cannot avoid two of them landing in one column together. The twenty-four at four warp-up are the one-and-three permutations, the weft-faced single-mark weaves, which the original census already named as the maximum of its own U at the other end.

The fourteen at eight warp-up are the interesting ones. They are balanced drafts with no warp run of two anywhere, and two of them are the plain weaves. The other twelve are balanced, present a surface of isolated warp summits, and carry a full complement of weft crown line underneath — so they are cloths whose bearing surface is a lattice of points and whose second surface, eight micrometres down, is ribbed. A plain weave is points at both depths; those twelve are points and then lines, which is a surface no draft in the summed census could be distinguished as having.

Which is the honest form of the original finding #

The repair is not to withdraw the census. It is to say which of three things is being reported, and each of them is true of something.

Both systems summed is the cloth’s interlacement read as a length. It is what a reflection sees, it is what the lustre census uses, and it is convention-free — complementing a draft exchanges the two terms and leaves the sum alone at equal setts.

The higher system alone is what a bearing curve reads and what a light touch meets. It is convention-dependent, and the convention is the crimp division.

The lower system alone is what a plate meets once it has sunk past the step, which on a sheeting is eight micrometres and is a depth a hand’s pressure does not reach.

The published census is the first and the surface account wanted the second. The two plain weaves are the exception under all three, which is why the headline finding was never wrong — only differently important than it read.

And the exception is robust for a reason worth keeping #

There is one thing in the original census that no convention can move, and it is the property that made it worth a census.

A plain weave has no run of two anywhere. That is a statement about the matrix and not about heights, so it holds under the warp count, under the weft count, under the sum, at any crimp division, at any sett and for any yarn. Every other draft in the catalogue has at least one run of two somewhere, and whether it shows depends on which system is on top.

So the finding that survives is the one stated as a property of the matrix, and the finding that moved is the one stated as a property of the surface. That is a general caution worth carrying out of this account: a census whose conclusion is “exactly n objects have this property” is only as robust as the property’s own definition, and a property defined through a model inherits every convention the model has.

The plain weave is the exception because it is the only alternating matrix. It is not the exception because of anything about crowns, and the two statements were the same sentence in the original essay.

What it does to the rest of the account #

Three earlier essays read the census’s number, and the recount does not touch all three the same way.

The curve that says what a cloth touches with established that the bearing curve’s behaviour at the top is decided by one bit of the draft — whether the longest float is one crossing or more than one. That is untouched and is in fact strengthened, because it was always a statement about the bearing system alone: the bit it turns on is whether the higher system has a run of two, and the recount is exactly the census of that bit.

How much of a cloth is touching turns a depth into a pressure and finds well under one per cent of the plan in contact at any load a hand applies. Untouched, for the same reason — it reads one surface.

Two drafts of twenty-two thousand is the one that moves, and it moves in its ordering rather than in its arithmetic. Its narrowness result survives in a changed form: the summed quantity has an interquartile range that is narrow, and the single-system quantity is spread from nothing to 1.400 across the catalogue. The single-system count discriminates far better than the sum does, which is the opposite of the worry that essay ended on — that a designer choosing among twenty-two thousand drafts is choosing among surfaces that mostly differ by twenty per cent.

So the recount gives back something the original took away. Its headline is less clean — 494 rather than 2 — and the quantity underneath it is a far better instrument: it separates drafts the sum cannot, it runs monotonically with a property a designer controls, and it is the quantity a plate actually reads.

What was recounted, and how #

The catalogue is enumerated as before — every four-by-four binary matrix in which every end and every pick interlaces — and each draft’s maximal runs of face are found cyclically in each system. A run of length n contributes n − 1 spacings, in that system’s own pitch; the three counts are the warp’s total, the weft’s total and their sum, each divided by the repeat’s area. The two single-system counts are then sorted independently and the places compared. The level crimp is found by bisection on the difference of the two crown heights, sweeping the division logarithmically and solving each state through the same closure every other solved cloth here uses.

Five things are checked. Some draft carries no crown line under each of the three counts, so all three questions have exceptions. The drafts with none under the summed count are a subset of those with none under either single count, which is the containment the arithmetic demands and which a sign error would break. The two conventions order most of the catalogue differently, at more than half. The worst draft moves a quarter of the list or more. And the drafts the two conventions agree about are a minority — which is stated loosely and comes out at two.

The construction, the setts and the crimp division are inputs; the catalogue is the four-by-four enumeration.

Where the recount stops #

It is still one construction. Every crown line here is computed at one cloth’s spacings, so the warp’s and weft’s contributions carry different pitches and the two maxima differ by the sett ratio — 1.400 against 1.300, which is 28 ends over 26 exactly, as the original census established.

The step is a geometric step and a real surface has hairs on it. A light touch never reaches the crowns at all: the hair layer carries everything a hand applies, and the eight micrometres that decide which system bears are inside the depth the hairs occupy. So the whole of this argument is about a surface an instrument reaches and a finger does not.

And a crown is taken as flat where it is a run. Every float lies straight on its supports in this model, so a plateau is a plateau; a real float is pressed by the threads it crosses and bows a little between them, which rounds the ends of every ribbon and shortens none of them.

Still open: which of the two lists a real cloth reads #

The whole essay ends at a measurement that has not been made and that would settle it.

The crimp division of a real woven cloth is measurable: take a length of cloth of known dimensions, ravel it, straighten the threads and measure how much longer each system is than the cloth it came from. That is an ordinary laboratory operation, it needs no instrument not already assumed here, and it gives the division directly.

What makes it worth doing is not the division; it is the sign of the step. A sheeting’s crowns are level at a division of 0.836 and the convention is at 1, so the question is whether a real sheeting’s warp crimps more or less than 84 per cent of what its weft does. The answer is one number and it decides which of two orderings the whole four-by-four catalogue has for that cloth — and, by the same arithmetic, which system wears, which system shines in a bearing measurement, and what a thickness gauge is reading.

Nothing here predicts it, and this collection said from the beginning that nothing can.

Who found it, and when #

The four-by-four surface census is the earlier essay’s and its recount is here. Bearing curves and the reading of a surface by its highest points are surface metrology’s, and the observation that a plate meets one system is in the earlier essay’s own list of caveats — where it sat, correctly stated and unfollowed, until now.