A crepe cannot be structureless

Every other weave on this site is designed to have a structure. A twill has a line; a satin has an unbroken surface; a honeycomb has a gradient. A crepe is designed to have none of those — a cloth whose surface reads as a fine irregular grain with no direction in it at all, so that the eye finds nothing to follow.

That is an unusual specification because it is stated negatively, and negative specifications are the ones worth making exact. This one turns out to have a bound.

What “no line” means, exactly #

A line in a cloth is a direction along which the surface repeats sooner than it has to. Slide a 2/2 twill one pick down and one end along and it lands exactly on itself: every warp-up sits on a warp-up. That is what makes the diagonal visible — the eye is a correlator and it has found a shift at which the pattern agrees with itself perfectly.

So write warp-up as plus one and weft-up as minus one, and define the correlation at an offset as agreements minus disagreements when the draft is slid that far over itself. At no offset at all the answer is the whole repeat. At the twill’s own step it is also the whole repeat, and that is the line. A draft with no line is a draft whose correlations away from the origin are all small.

Two facts about that quantity fall out before any design work is done, and both are exact.

The total is fixed #

Sum the correlation over every offset, including the origin, and the answer is the square of the draft’s own balance — the number of warp-ups minus the number of weft-ups, squared. For a balanced draft that is zero.

The origin contributes the whole repeat, sixty-four on an eight-end draft. So the correlations away from the origin must sum to exactly minus sixty-four, whatever the draft is, however it was designed and whoever designed it.

Structure can be moved about. It cannot be removed, and the amount to be moved is decided by the repeat before anybody picks up a pencil. A crepe is not a draft with no correlation; it is a draft that has spread its correlation thin.

The satin method cannot work on an even repeat #

Every weaving manual gives the same construction for a crepe: take a satin base, which already has its interlacings as far apart as a repeat allows, and superimpose a second one on it.

On an even repeat that construction cannot work, and the reason is two lines of arithmetic.

A satin of move m is invariant under sliding one pick down and m ends along — that is what a satin is. So it is invariant under sliding u picks and um ends for every u. Take u to be half the repeat: um is nm/2, and m is coprime to n, so on an even repeat m is odd, so nm/2 is congruent to n/2 modulo n.

Every satin on an even number of ends is therefore invariant under the half-repeat diagonal shift, whatever its move and whatever its shift. And so is any superposition of them, because a translation that fixes each part fixes the union.

On eight ends the moves are three and five, and both satins correlate perfectly with themselves at four picks down and four ends along. Any crepe built by superimposing them has a perfect line at half a repeat diagonally, however many satins are used and however they are shifted. It is not a design failure; it is arithmetic, and the site’s generator asserts it for every available move rather than quoting it.

Which explains something the manuals do not: crepe repeats in the trade are odd, or the crepe was made by another method.

The other method, and the search it admits #

The method that does admit a search is rearrangement.

Take a short binary word — the base column, up for so many picks and down for so many — and give every end in the repeat the same word rotated by its own amount. A twill is the case where the rotations run 0, 1, 2, 3…; a satin is the case where they run in a constant step. A crepe is the case where they run in no pattern at all, and which orders count as no pattern is exactly what a search settles.

The space is small enough to walk. Balanced words of eight with runs of two or less, taken up to rotation, are five; rotations of the ends up to a common shift are 5,040; drafts that are one cloth with no float longer than three are 14,610. Every one of them is scored on its worst correlation at any offset.

The floor is one-dimensional and has nothing textile in it #

The search reaches 32 out of 64 and stops there, and the reason is not that it looked in the wrong place.

Consider the offset of some picks and no ends. Sliding a draft straight down leaves every end lying against itself, so the correlation is the sum over the ends of each column’s own correlation at that shift — and every end carries the same word. So the vertical correlation is the end count times a number the base column decides, and no rearrangement whatever can move it.

That turns the first design decision into a question about binary words. The correlations of a balanced word of eight sum to zero and each is even, and the origin holds eight, so the seven others sum to minus eight. The best that can be done is to spread that as evenly as the arithmetic allows, and the best available is a worst value of four.

Four times eight is thirty-two, which is what the search found.

What was counted, and how #

Three enumerations, each run rather than recalled.

The base columns. All 256 words of eight, filtered to the balanced ones, reduced to the lexicographically least of their rotations, and filtered again to runs of two or less. Five survive. Each one’s periodic correlation is computed and checked against two identities: it must equal the word length at no shift, and the whole set must sum to the square of the word’s balance.

The rearrangements. All 5,040 orders with the first end fixed, against each base, giving 25,200 drafts, of which 14,610 are one cloth with floats of three or less. Every one is scored by the full correlation grid — sixty-four offsets, sixty-four products each — and the grid’s own two identities are asserted on every single draft rather than on a sample.

The steady cases. Any order whose rotations advance by a constant step is a twill or a satin, and every one of those must score the whole repeat at one offset. That is asserted, and so is the complementary claim that the best crepe found does not — which is the pair that makes the search’s answer mean something rather than merely being a number.

The best draft is balanced, thirty-two warp-ups in sixty-four, with a longest float of three and 0.72 interlacings per intersection: firmer than a twill, looser than plain, and closer to plain than to anything else. That last is not incidental. A crepe is a firm cloth, because the only way to have no long float in any direction is to interlace often, and the trade’s association of crepe with a lively, dense handle follows from the same constraint that makes the surface quiet.

Where a crepe sits among the weaves #

Set the four surfaces side by side by the one number this essay is about, and the ordering is not the one the trade’s vocabulary suggests.

Plain weave, the twill and the satin all correlate perfectly with themselves somewhere. That is not a criticism of any of them — plain weave is bought for firmness, the twill for its line, the satin for its unbroken surface — but it does mean that the crepe is the only one of the four whose defining property is a number rather than a description, and it is the only one that can be searched for.

It also explains why crepes are hard to design and easy to recognise. The other three are constructions: state the rule and the draft follows. A crepe is the output of a filter over a space, and the float limit and the connectivity criterion are two of its constraints while the correlation is the objective. Nothing about the three constructions prepares a designer for that, which is why the pattern books hand over examples instead of methods.

The mean is worth a line of its own. Two drafts can share a worst correlation and differ enormously in how much correlation is left everywhere else, and the best crepe’s mean absolute correlation away from the origin is 6.1 out of 64 where the steady case’s is 64. That second number is what makes the surface read as a grain rather than as a pattern, and it is the one a designer is actually optimising even when they think they are avoiding a line.

The floor falls as the repeat grows, and by how much #

The vertical argument that sets the floor is one-dimensional, and running it at a general repeat says exactly what a larger crepe buys.

The floor is the end count times the base column’s own worst correlation: eight ends times four, on an eight-end draft, giving thirty-two. The repeat is the square of the end count, sixty-four. So the floor as a fraction of the repeat is the word’s worst correlation divided by the end count, and the numerator is the part that barely moves.

A word’s off-origin correlations sum to minus its own length, spread over one fewer offsets than that, so the best achievable worst value grows very slowly with length — it is bounded below by a little over one and is in practice a small even number for any length anybody would use. The denominator, meanwhile, grows in proportion. So the relative floor falls roughly as one over the repeat: doubling the repeat roughly halves it.

That is the arithmetic behind a rule the trade states without one. Crepes are woven at sixteen and twenty-four ends rather than at eight, and the reason usually given is that a larger repeat gives a designer more room. It does, and the room is quantified: a sixteen-end crepe whose base word reaches a worst correlation of two would have a floor of a fifth of its repeat against the eight-end draft’s half.

It also says where the effort belongs. The floor is set entirely by the base column, and the rearrangement search cannot touch it — so a designer choosing a base word is making the decision that bounds the answer, and every subsequent choice is about approaching a bound already fixed. Choosing the word well at a long repeat is worth more than any amount of rearranging at a short one.

Only the absolute mean is free #

The identity fixes a total, and that has a consequence for how two crepes may be compared which is worth stating because it rules out the obvious comparison.

The off-origin correlations sum to minus the repeat, over sixty-three offsets. So the mean signed correlation is minus sixty-four over sixty-three — about minus one — for every balanced draft on eight ends, the best crepe and the worst alike. Averaging the signed values distinguishes nothing, because the identity has already decided the answer before the draft is drawn.

What is free is the absolute mean, and it is free precisely because the signed sum is fixed: a draft can reach its total with sixty-three small negatives or with one enormous positive and one enormous negative pair, and those are entirely different cloths. The best crepe’s mean absolute correlation is 6.1 out of 64 and the steady case’s is 64, and both average to minus one.

That is the clearest statement of what a crepe designer is doing. They are not reducing a total, which is not theirs to reduce; they are minimising the spread of a set whose sum they cannot touch. Every crepe carries the same amount of structure and they differ in how finely it is divided, which is a description of a grain.

Where the model stops #

Correlation is not the eye. The quantity measured here is a shift-and-compare over the interlacement, and what a reader sees is light off a surface of round threads at some sett with some twist and some lustre. Two drafts with identical correlation grids can look quite different if their yarns do, and the colour order threaded into them can produce a pattern the interlacement never hints at.

The float limit is a stated one. Three was chosen because it is the trade’s working limit for a firm crepe; changing it changes the field and the floor does not move, because the floor is set by the base column and the base columns are filtered on runs rather than on floats.

The family is a construction, not the space. Rearranging one base column is one of the trade’s two methods and the whole space of eight-by-eight drafts is 2⁶⁴. Nothing here says a better crepe does not exist outside the family; what it says is that inside the family the answer is exactly the bound the base column sets, which is a stronger statement than a search usually gets to make.

And the identity is about one repeat. A real crepe is woven at a repeat of sixteen or twenty-four, and the arithmetic scales: the off-origin correlations sum to minus the repeat, so a larger repeat has more room to spread the same relative amount of structure. That is the reason crepes are woven large and it is the direction the next rung goes.

Who found it, and when #

The correlation identity is the Wiener–Khinchin relation in its most elementary form, and it is a hundred years old in signal processing and older in optics. Nothing in this essay needed a transform: the sum of the correlations over all offsets is the square of the sum of the entries, which is one line of rearrangement.

The bound on binary sequences is a harder and much better-studied problem. Asking for a sequence whose periodic correlations are all zero away from the origin is asking for a perfect binary array, and they are known to be rare — length four is the only length anybody has for the ±1 case, and whether any longer one exists is an open question with a great deal of computation behind it. A crepe designer on eight ends is standing at the near end of that question without knowing it.

The weaving side is nineteenth-century pattern-book knowledge, given as recipes. Watson’s Textile Design and Colour lists the methods — satin bases, superimposition, rearranged twills, and “irregular” drafts placed by hand — and gives no way to compare two crepes. The comparison is what a correlation grid supplies, and it says that the recipes are not equally good: the satin method on an even repeat is provably the worst of them, and it is the one given first.

Where the ladder goes next #

Both rungs so far are about what a draft does to a surface. The next is about what a draft does to the space between threads: a mock leno makes holes with no thread crossing another, by grouping ends that nothing separates — and the holes it makes are as stable as friction, where a real leno’s are as stable as topology.