A crepe is flat in its draft and not in its surface

A crepe weave is designed to have no line in it anywhere. This collection settled how well that can be done: the correlations of a draft with itself sum to a number fixed by the repeat alone, so structure can be spread and never removed, and a search over 14,610 rearrangements finds drafts that reach the floor exactly.

The search reaches the floor. What it does not do is choose.

The claim #

The criterion that selects a crepe weave has thousands of winners, a second criterion separates them, and the weave this collection draws is not on the frontier of both.

Three statements, and the third is a self-correction.

The correlation floor is easy. On the base in question, 4,416 of 5,040 rearrangements reach it. The floor is set by a one-dimensional fact about the base column, and once the rearrangement is disorderly at all it is reached.

The surface is a different functional and it discriminates. Crown line is a sum over face runs of (length − 1); autocorrelation is a sum over offsets of a product of columns. Neither determines the other. Scoring the same 5,040 candidates by how evenly they distribute their crown line spreads them from zero to 0.376 millimetres of standard deviation.

And sixteen candidates are at the floor of both. They exist, they are free, and the crepe this collection has been drawing since its fancy-weaves phase is not one of them.

Why the two functionals differ #

A crepe wants a surface with no structure at any scale — no twill line, no satin sparkle and none of the beat a second grid makes — and there are two quite different ways for a surface to have structure in it.

A correlation says whether the draft repeats itself at some offset. If shifting the matrix by one end and two picks reproduces much of it, the eye finds a line at that offset — a twill line, a satin sparkle, a diagonal. The correlation search is looking for the offset at which a draft most resembles itself and pushing that resemblance down.

A crown line says how much horizontal thread each end and each pick presents. If one end carries three plateaux and its neighbour carries one, the two reflect and wear and feel differently, and the cloth has a structure that has nothing to do with any offset — it is a difference between threads rather than a repeat at a displacement.

A draft can be flat in the first sense and lumpy in the second. A rearrangement can scatter its correlations perfectly while giving one end a great deal more float than another, because the correlation is a sum over the whole repeat and does not care where the float sits.

What was counted, and how #

The candidate set is the same one the correlation search uses, restricted to the rearrangements of a single base — which is the restriction that makes the comparison meaningful.

A base fixes every end’s own column of lifts, so it fixes the warp’s crown line exactly; only the weft’s varies with the rearrangement. Comparing across bases lets the search escape to the plain weave, which has no crown line at all and is trivially the most even surface there is — and is the one weave a crepe is designed not to be.

Within the base, each of the 5,040 orders is built, its crown line per end and per pick is computed, and the score is the standard deviation of the ends plus the standard deviation of the picks. Then each is re-scored by the correlation measure and the two are crossed.

The result: 4,416 at the correlation floor of 32; 28 with a spread of exactly zero; 16 in both sets. The drawn crepe has a spread of 0.236 millimetres, which is the median of the whole set and the thirty-eighth percentile of it.

Why zero is reachable at all #

A spread of exactly zero means every end carries the same crown line and every pick does too, and that sounds like it should force regularity — which is precisely what a crepe must avoid.

It does not. Regularity is sufficient and not necessary. An orderly rearrangement — every end shifted by the same step, which is a twill or a satin — gives every thread an identical column and therefore an identical crown line, and it is the worst possible draft by the correlation measure. But so do twenty-seven other rearrangements, and sixteen of those are disorderly enough to sit at the correlation floor.

That is the whole of why the improvement is free. The two criteria are not opposed, they are merely independent, and independence means the frontier of both is not empty.

The surface of the crepe as drawn #

Setting the numbers out makes the size of what is being left on the table clear, and it is not large.

The drawn crepe’s ends carry between nothing and two spacings of crown line each, and its picks likewise; the standard deviation of the two together is 0.236 millimetres. A perfectly even member of the same tie has zero. The total crown line is very nearly the same in both — the base fixes the warp’s contribution exactly and the weft’s varies by a few per cent — so the difference is entirely in how it is distributed.

What that would be worth to a reader is not computed here and is probably small. A cloth whose ends differ in crown line by a couple of tenths of a millimetre per square millimetre has ends that reflect and wear slightly differently, at the scale of one thread — which is 350 micrometres, at the edge of what an eye resolves at arm’s length.

So the improvement is free rather than important. It is worth recording for the same reason a green gate is worth keeping green: not because the defect is costly, but because a criterion that was silently doing nothing is now doing something, and the next crepe designed on this site will be chosen on both.

What this says about the earlier result #

The crepe essay’s arithmetic is untouched. The correlation identity is exact, the floor is exact, the search reaches it, and the finding that structure can be spread and never removed stands.

What has to be added is that reaching the floor is not a strong condition. An optimisation that is satisfied by 88 per cent of its candidates is not choosing between them, and the essay presented its answer as the crepe rather than as one of thousands. That is a real shortfall in how the result was stated and it took a second criterion to expose it.

The general shape is worth naming because it recurs. A search that reports a winner without reporting the size of the winning set is reporting less than it looks. If the set is a singleton the answer is a discovery; if it is most of the search space the answer is an arbitrary pick from a large plateau, and the interesting question becomes what else could be asked of the survivors.

The improvement, which is not made here #

Sixteen rearrangements of this base reach the correlation floor and have a perfectly even crown line. Switching to one of them would cost nothing: the same base, the same float limit, the same number of shafts, the same correlation floor, and a surface whose crown line is identical on every end and every pick.

It is not done here, and the reason is that the crepe this collection draws is in an existing essay and its figures, and changing the draft changes those figures and the assertions written against them. That is a piece of work with its own risk and it deserves to be done deliberately rather than as a side effect.

What is done here is the measurement, the assertion that records it, and the list. The check refuses if the tie is not wide, if none of the tied candidates is even, or if the drawn one turns out to be among them — so if any of those three facts changes, the build says so.

What a third and a fourth criterion would say #

Once a search is known to tie, the question becomes what else can be asked of the survivors, and this collection has at least two more functionals sitting ready.

The float limit, which is already applied: no run longer than three, because a long float is a line in the cloth however flat its correlations are. That constraint is in the search and it removes candidates before the tie is formed.

The relief, which is not. A crepe’s finished surface is dominated by the out-of-plane buckling that develops in finishing, where regions wanting different contractions are held at one reed pitch. That depends on the gradient of float length across the repeat, which is a third functional again, and a crepe with an even crown line could still have a strong relief gradient.

And the raisability, which is a fourth. Every run of two is a float a teasel could catch, so a crepe’s crown line is also its nap potential — and a crepe is a cloth that is never raised, so a designer might want that minimised rather than evened.

The point is not that any of these should be added. It is that the survivors of a wide tie are a resource, and a search that reports one winner throws that resource away silently. Reporting the size of the tie costs one integer.

How many criteria a base can carry #

The tie is 4,416 wide and two criteria reduce it to sixteen. That is a ratio worth reading as a budget, because it says how much more can be asked of this base before the survivors run out.

The first criterion cuts 5,040 to 4,416 — a factor of 1.14, which is barely a criterion at all. The second cuts 4,416 to 16, a factor of 276. A third of the second’s strength would leave 0.06 candidates, which is to say none.

So the base is exhausted at two criteria, and the exhaustion is not marginal: there is no room for a third demand of any consequence, and a designer who adds one will find the frontier empty and will not know whether that means the demands conflict or merely that the search space was too small.

That distinction is worth having in advance, because the two look identical from inside a search. An empty frontier means either that the criteria are incompatible or that the candidate set is too small, and the size of the tie at each stage is what tells them apart. A set that goes 5,040 → 4,416 → 16 → 0 has run out of candidates; one that goes 5,040 → 40 → 0 has criteria that genuinely fight.

Which points at the remedy and prices it. The candidate set grows factorially with the repeat and the criteria cut multiplicatively, so one more end of repeat multiplies the candidates by the repeat size and buys room for roughly one more criterion of the same strength. Going from eight to ten ends takes 5,040 rearrangements to 362,880 — a factor of seventy-two — which is enough for a third demand at the second’s strength and not enough for a fourth.

So the rule a designer wants is short. Count the tie at each stage. If the last cut left single figures, the next criterion needs a larger repeat rather than a better search. That is the same statement the essay’s own generalisation makes about reporting the size of a winning set, sharpened into a stopping rule: the tie is not merely where the next criterion lives, it is the budget the next criterion has to be paid out of.

And it explains why crepes are woven on the repeats they are. A crepe is customarily eight or sixteen ends, and eight is exactly the size at which one demand about correlations and one about the surface can both be met with a handful of drafts left over. Below eight there is no room for either and above sixteen the search is no longer enumerable by hand — which is a fair account of why the trade settled where it did, arrived at from a counting argument rather than from anything about cloth.

Where the model stops #

Crown line is not the only surface functional and probably not the best one for a crepe. A crepe’s aim is that no region of the cloth reads differently from any other, which is a statement about the two-dimensional distribution rather than about per-thread totals. A better measure would be the spatial spectrum of the height field, which this collection has the machinery to compute and has not.

The comparison is within one base. Across bases the surface measure is degenerate — it prefers the plain weave — so a joint optimisation would need the correlation constraint applied first and the surface second, which is exactly what the frontier above is.

And there is no eye in any of it. Whether a cloth “reads as a crepe” is a perceptual question about spatial frequency and contrast, and neither functional here is that. Both are statements about a matrix that a reader might reasonably expect to correlate with the appearance, and neither is a model of it.

The relief is a third quantity again, and a crepe’s real surface in finished cloth is dominated by the buckling that develops when regions of different float want different contractions. Nothing here joins the two.

The generalisation #

When an optimisation ties, the tie is where the next criterion lives.

That is the transferable statement and it is a method rather than a result. A search that reports a winner should report how many winners there are; a large tie is not a failure of the search but an invitation, and the second criterion applied to the tied set costs nothing because every member already satisfies the first.

The corollary is about how design constraints stack. Constraints that are independent can usually be satisfied together, and the way to find out is to enumerate the frontier rather than to argue about whether they conflict. Here two functionals of one small matrix looked as though they might be opposed — regularity helps one and destroys the other — and the enumeration found sixteen drafts that do both.

How wide the tie is, and why that is the surprising number #

Four thousand four hundred and sixteen of five thousand and forty is 88 per cent, and the size of that fraction deserves a sentence of its own.

The correlation floor for a base is set by the base’s own column: at an offset of a whole number of picks and no ends, every end contributes its own column’s autocorrelation at that offset, and no rearrangement can move it. So the floor is a one-dimensional property of the base word, and the rearrangement’s only job is not to make anything worse than that floor.

Making something worse requires the ends to line up — several of them sharing a displacement, so that their correlations add at one offset. A rearrangement picked at random does not do that, because a random permutation of eight rotations has very few coincidences in it. Disorder is the default and order is the special case.

That is why the floor is easy and why the search’s answer is not a discovery about the draft. The real content of the crepe result was always the identity — that the correlations sum to a constant, so structure moves and never leaves — and the search was demonstrating that the bound is attained rather than finding a rare object.

Who found it, and when #

The correlation identity and the crepe search are this collection’s own, and were built for the essay that asked how structureless a weave can be.

The tie was there from the day the search was written and nothing looked at it: the search reported its best and moved on, as searches do. What found it was a second functional arriving from an unrelated direction — a surface argument built for contact and reflection — and being applied to the same candidate list.

That is the argument for building new instruments even when the old ones are green. The correlation search was not wrong and its answer was not wrong; it was simply less determinate than it appeared, and only a different question could show that.

Where the ladder goes next #

To the surface a reader meets most often: a seam stands proud and wears first, which is a step in a bearing curve rather than a weave in a matrix, and which takes the whole of a garment’s rubbing on a couple of per cent of its area.