How sharply a weave lets a cloth fold

A fold across the warp needs the warp on the outside of it to be longer than the warp on the inside. A woven cloth’s way of finding a length is to move crimp, and the amount of crimp a cloth has is a number this collection computes for any construction.

That number is an average. It is the extra length a warp end carries over the whole repeat divided by the cloth the repeat covers, and using it for a fold assumes the crimp is available where the fold is. It is not. A warp end’s crimp is made where it turns from over to under, and between turns it lies straight — a float presses on nothing and carries no crimp either; a fold that falls in the middle of a float finds a piece of warp with nothing to give.

For a plain weave the distinction is empty, because every end turns at every pick. For a satin it is the whole story.

The claim #

How much crimp a fold finds is a property of the matrix, and for weaves with long floats it is zero for a large fraction of the warp.

Two numbers say it and they say different things.

The mean is how many turns a fold window contains, averaged over the ends of a repeat. It is the interlacing rate times the window’s width, and it is what an average-crimp argument would have predicted.

The zero fraction is the share of end-and-position pairs where the window contains no turn at all. It is not visible in the mean, it is not derivable from the float length alone, and it is the quantity that decides how much of a fold is handed straight to the fibres.

Counting it #

A warp end turns at pick i when it is over at i and under at i+1, or the reverse — reading the repeat cyclically, because a repeat tiles. Slide a window of W picks across the repeat and count the turns inside it, for every end and every position.

For a three-pick fold across the standard catalogue, the means fall as the interlacing rate does: a plain weave gives three turns per end, a 2/2 twill and a 3/1 twill give 1.5, a five-end satin 1.2, a 4/4 twill and an eight-end satin 0.75.

The zero fractions do not follow the same order. A plain weave, a 2/2 twill, a 3/1 twill and a 2/2 basket have none at all. A five-end satin has 20 per cent, a 4/4 twill 25, and an eight-end satin 50.

Why the float does not order it #

A 4/4 twill and a five-end satin have the same longest warp float — four picks — and different zero fractions, 25 per cent against 20. An eight-end satin and a 4/4 twill have the same interlacing rate — 0.75 turns per end per three-pick window — and the satin has twice the zeros.

So neither the float nor the rate orders the census. What decides it is how the turns are spaced.

A satin’s warp end goes under for one pick and over for the rest, so its two turns are adjacent and it has one long clear run. A 4/4 twill’s end goes under for four and over for four, so its two turns are four picks apart and it has two short runs. Same number of turns, same longest float, different arithmetic: a clear run of L picks contains L − W + 1 windows that miss every turn, and two runs of three contribute two while one run of six contributes four.

That closed form is computed a second way and asserted against the sliding count, which is a real check because the two share no arithmetic — one slides a window across a matrix and the other sums over the gaps between turns.

The flat column, and where it comes from #

There is a surprise in the census and it is in the column nobody looks at.

Every twill and every regular satin has the same mean at every fold position across the repeat. The fold does not care where it lands. Only the basket varies — from one turn to two, a factor of two, depending on which pick the fold falls between.

The reason is a symmetry this collection identified while sorting satins. A weave fixed by a one-end step, with some accompanying shift of picks, has an orbit of n rather than n², so moving the fold along by one pick is the same operation as moving it across by one end — and averaging over the ends erases it. A basket has no such step, and its two positions are genuinely different.

That correspondence is exact across the catalogue and is asserted as such: a weave’s fold-average is flat at every position precisely when a one-end step fixes it. Nothing about the fold argument suggested that a classification of satins would turn up in it.

And what the flat average hides is the whole of the effect. An eight-end satin’s mean is flat at 0.75 turns per end, and half its ends have none at any given fold. So a satin creases unevenly from end to end rather than from place to place, which is what the broken, glassy look of a crease in a satin actually is: a line of ends that were straight across the fold and took its whole length difference in their fibres, alternating with ends that had somewhere to put it.

How the window width changes the answer #

A fold window of W picks is a fold whose bend is spread over W pick spacings. A sharper fold is a narrower window.

At two picks the answers spread out: a plain weave and a 2/2 twill still have no zeros, a 3/1 twill acquires 25 per cent, a five-end satin 40, a 4/4 twill 50 and an eight-end satin 62.5. At four picks nearly everything is clear — only the eight-end satin retains any zeros at all, at 37.5 per cent, because it is the only weave in the catalogue with a clear run longer than four.

That gives the practical statement. A weave has no zeros at any window at least as wide as its longest float, which is asserted rather than observed, and follows immediately from the definition of a clear run. So a cloth’s vulnerability to a fold is set by a comparison of two lengths: the number of picks the fold’s bend is spread over, and the weave’s longest float.

A plain weave is immune at every width. A satin is exposed at any fold sharper than its float, which for an eight-end satin means anything spread over fewer than seven picks — which is nearly every fold there is.

What a designer can do with it #

The census turns into three pieces of advice, and the second is the one that is not obvious.

A fold that must not mark wants a short float. That is the direct reading and it is what the trade already does: a cloth intended to be folded, packed, pleated or sat on is rarely a satin, and where a satin is used for such a thing it is usually backed or bonded so that the fold radius is set by something other than the cloth.

A fold that must mark evenly wants a one-end step. A pleat, a permanent crease, a fold that is part of the design rather than a defect: what makes it look deliberate is that it is the same all the way along, and that is exactly the flat-average property. Every twill and every regular satin has it; a basket does not, and a basket-woven pleat is a pleat whose appearance depends on which pick it was folded at, which is not a thing a maker can control to one pick.

And the exposure is a comparison of two lengths rather than a property of the weave. A weave has no zeros at any fold spread over at least as many picks as its longest float, so a satin folded gently is as well behaved as a plain weave. What makes a satin crease badly is not the satin, it is a fold sharper than the float — which is nearly every fold, which is why the qualification is usually dropped.

The third of these also says where the intervention is. Making the fold gentler is available to a maker in a way that changing the weave is not: an interlining, a seam allowance turned twice, a radius at a hem. Each of those raises the number of picks the bend is spread over, and each moves a satin down the census toward the plain weave’s answer.

How many weaves fold evenly, counted #

The flat-average property is worth more than a note about baskets, because it is decidable from the weave’s own construction and the weaves that have it can be counted.

A weave is fixed by a one-end step exactly when it is generated by rotating a single column word down the picks — every twill and every regular satin, and nothing else. So the weaves that fold evenly are the orbits of a binary word of length n under rotation by one, and counting orbits under a cyclic group is a standard exercise.

At a repeat of n picks the number of such orbits is

(1 ÷ n) · Σ over d dividing n of φ(d)·2^(n/d),

and two of them are the constant words — all warp up and all warp down — which are not weaves at all, because a thread that never turns does not interlace.

Four on a four-pick repeat, which are the plain weave, the two three-and-ones and the two-and-two twill counted once. Thirty-four on eight, which is where the ordinary satins live. And the count grows very nearly as 2ⁿ/n, so at any useful repeat the flat-folding weaves are a vanishing fraction of the drafts available — a four-by-four census holds 426 cloths and four of them fold evenly.

Two things follow that the fold census alone does not say.

A pleat wants one of these and there are not many. The property that makes a pleat look deliberate — the same appearance all the way along, whatever pick it happened to be folded at — is exactly the flat average, and the designer’s choice is therefore from a list of four at a repeat of four and thirty-four at a repeat of eight. Every basket, every hopsack, every block construction and every stripe is outside it.

And the list is closed under the operations a manual teaches. Reversing, turning over and counterchanging all carry a rotation orbit to a rotation orbit, so a designer who starts inside the list and applies the chapter’s own derivations stays inside it. That is a rare piece of good news about the manuals’ vocabulary, and it is the reverse of the finding this collection’s derivation census reports: the operations are poor at reaching the catalogue and excellent at preserving a property once it is had.

The caution is that flatness and exposure are independent. A weave can fold perfectly evenly and have half its ends with nothing to give, which is precisely the eight-end satin: it is on the list of thirty-four and it has the worst zero fraction in the catalogue. Evenness is about position along the cloth and exposure is about which end, and the census measures both because neither predicts the other.

What was counted, and how #

Every number here is counted from the binary matrix and nothing else. The turns are cyclic, because a repeat tiles and a turn between the last pick and the first is a real turn.

The mean over positions is asserted to equal the window width times the weave’s own interlacing rate, which is a check on the sliding count against a quantity this collection computes independently.

The zero fraction is computed twice — once by sliding a window and once by summing L − W + 1 over the clear runs — and the two are asserted equal.

And the two ends of the range are asserted rather than described: a weave whose longest float does not exceed the window has no zeros, and one whose float does exceed it has some. That pair of assertions is what makes the census a statement about floats rather than a table.

Where the model stops #

Crimp is treated as being at the turns and nowhere else. A real thread’s curvature is distributed over a finite arc around each crossing rather than concentrated at a point, so an end whose nearest turn is just outside the window still has a little to give. The correction shrinks the zero fraction and cannot remove it, because a seven-pick float is very much longer than the arc at either end of it.

The fold is treated as being across the warp only. Every count here is of warp turns against a fold whose axis lies along the weft. A fold the other way is the transposed question with the same machinery, and for a weave that is not its own transpose the two answers differ — which is the shape of the finding this collection’s figured-cloth ladder reported about rectangular blocks.

Nothing here computes a strain. The census says how much crimp is available at a fold and not what happens when it runs out, which needs the fibre-level arithmetic from the crease ladder. Joining the two properly would need the crimp available per end to be converted into a local length, which needs the section geometry at the fold and is a rung this collection has not built.

And the catalogue is seven weaves. It spans the range from a plain weave to an eight-end satin and it is not a census over all weaves of a given repeat, which is a thing this collection has done elsewhere and could do here.

The generalisation #

An average is a statement about a whole and a fold is a statement about a place, and substituting one for the other is a mistake with a definite sign. The average always overstates what is available locally, because the places where the resource is concentrated are not the places the demand falls.

The same substitution error recurs wherever a periodic structure meets a localised demand: the reinforcement in a slab and the crack that opens between two bars, the stiffeners on a panel and the impact between them, the sampling of a signal and the event that lands between samples. In each case the design quantity is an average per unit length and the failure is a question about a gap.

The narrower lesson is about what a matrix is for. A property that is stated as a rate per repeat can usually be re-asked as a distribution over positions in the repeat, and the second question is often the interesting one and is usually free to answer once the first has been. Nothing in this rung needed a new measurement or a new model; it needed the same matrix, read with a window on it.

Who found it, and when #

That a satin creases and marks badly is trade knowledge as old as satin, and it is usually attributed to the long floats being unsupported and to the smooth surface showing every mark. Both are true and neither is this.

The interlacing rate and the float length are standard fabric-structure quantities. That regularity in a satin is a translational symmetry is this collection’s own, from the census that enumerated satins as arrangements rather than as recipes.

Reading the crimp available at a fold as a windowed count over the matrix, and finding that it is flat in position exactly for the weaves that symmetry fixes, appears to be new here. It is the kind of question that only becomes askable once a fold has been established as a competition between crimp and fibre strain, which is the rung beside this one.

Where the ladder goes next #

The census says how much crimp a fold finds and not what happens when there is none, and that is the fibre’s question: some fibres break at the fold and others survive and fail to return.

Sideways, the same matrix decides the cloth-level fold radius through the thickness and the budget, which is where the crimp route gives out entirely. And the clear runs counted here are the same objects the float ladder counts for abrasion and for lustre, met from a third direction.