A repeat has to fit the width

The rung below is about a repeat as a mathematical object: the matrix tiled over the plane, the lattice of translations that leaves it alone, and the unit that is smaller than the rectangle anybody drew. Every statement in it is about an infinite pattern.

A warp is 1,800 ends wide and has an edge at each side. Somewhere between them, a repeat is cut through.

The arithmetic, which is a division with a remainder #

A warp of E ends carries a selvedge at each side: a band of ends threaded to their own firmer weave, because the edge of a cloth takes the beat-up and the temple and a plain-weave selvedge does not fray. Twenty-four ends a side is ordinary.

What is left is the body, E − 2s, and the design has to fit into it. A repeat of r ends gives ⌊body/r⌋ whole repeats and body mod r ends left over.

For a 1,800-end warp with 24-end selvedges the body is 1,752, and a 48-end repeat gives 36 whole repeats and 24 ends over — exactly half a repeat, unaccounted for.

That leftover has to go somewhere and there are only three places: at one selvedge, split between both, or absorbed by changing the repeat. The trade takes the second, so the pattern is centred and each selvedge carries twelve ends of a broken repeat.

Fitting is a divisor question, and divisors are rare #

The obvious response is to choose a repeat that fits, and the arithmetic says how much choice that leaves.

The repeats that divide 1,752 exactly are its divisors. Between eight and 240 ends — the whole range a figured design might use — there are six of them: 8, 12, 24, 73, 146 and 219. Six out of 233 candidates, which is 2.6 per cent.

Other warps do a little better and none does well. A 1,200-end warp has a body of 1,152 = 2⁷ × 3², which is unusually rich in divisors and gives fifteen; a 2,400-end warp gives seventeen; a 3,600-end warp twelve. Two to eight per cent, everywhere, and the row-to-row variation is about how many small prime factors the body width happens to have rather than about anything a weaver controls.

And the ones that do fit are not the ones a designer wants. The six available on the 1,752-end body are 8, 12 and 24 — far too small for a figure — and then 73, 146 and 219, which are odd multiples of a prime and correspond to no shaft count, no satin order and no design scale anybody would choose.

So a repeat that fits is not a choice so much as a coincidence, and the design that takes it is being designed by the warp.

Which is why the remainder is normal #

Sweep every repeat from eight to 240 across the 1,752-end body and the remainder averages 0.478 of a repeat — which is what it must, since a repeat picked without regard to the width leaves a remainder uniform on 0 to r − 1.

So the normal case is not “a small remainder”; it is half a repeat, and the two selvedges each carry a quarter of one.

That is a large piece of broken pattern and it is visible in every figured cloth anybody has ever seen. Look at the edge of a damask tablecloth, a jacquard upholstery or a printed-effect woven furnishing and the design does not run out neatly: it is cut, somewhere in the middle of a motif, and the cut is in the same place on both sides. It is the same cut a figured cloth’s blocks would show if the design’s own region boundaries fell there instead.

The trade’s word for handling it is centring, and centring is a decision about where to put a fault rather than a way of avoiding one.

Which fraction reads as a border and which as a mistake #

The remainder is a fraction of a repeat, and the fraction is what a reader sees rather than the count of ends.

A remainder of two ends out of fifty is four per cent of a repeat — a sliver, invisible at any normal viewing distance, and a designer would call it a good fit. A remainder of twenty-four out of forty-eight is half a repeat, which cuts a motif through its middle and is unmistakable.

Between them is a judgement and it is the designer’s, not the arithmetic’s. What can be said is where the arithmetic puts a design: the remainder is uniform on the repeat, so a quarter of repeats leave under a quarter of a repeat over and a quarter leave over three quarters.

There is one deliberate use of the second case and it is worth naming, because it inverts the whole problem. A border is a design that does not fit on purpose: the body carries a repeating pattern, the two edges carry something else, and the something else is exactly as wide as the remainder would have been. A designer who has a border to place does not need the repeat to fit at all; a designer who does not has a broken motif to place.

How many repeats across is a design decision hiding in the same sum #

The quotient is usually read as an intermediate step and it is the number that decides what the cloth looks like.

At 24 ends per centimetre a body of 1,752 ends is 73 cm wide. A 48-end repeat is 2 cm across, so the design shows 36 times between the selvedges; a 240-end repeat is 10 cm and shows seven times.

Those are two completely different cloths. Thirty-six repeats across reads as a texture — the eye does not resolve one motif, it sees a field — and seven reads as a pattern, with individual motifs that can be looked at. Somewhere between them the design changes from something the cloth is to something the cloth carries.

And the sett is in it, which means the same draft on the same warp gives different design scales in different cloths. A 48-end repeat at 24 ends per centimetre is 2 cm and at 40 ends per centimetre is 1.2 cm, so a designer moving a draft from a coarse cloth to a fine one has shrunk the motif by nearly half without touching it.

That is the reason a repeat is quoted in ends and a design is discussed in centimetres, and the reason the two are so easy to confuse. The draft owns the first, the cloth owns the second, and the sett is the exchange rate.

The reed is a third grid and it has to agree too #

The body has been divided by the repeat, and there is another division of the same ends that the designer does not do.

The warp is drawn through a reed several ends to a dent, so the ends are grouped — and that grouping beats against the weave repeat with the same divisor arithmetic. A designer who has fitted a repeat to the body has fitted one of the two grids and left the other where it was.

The practical consequence is small and sharp. A repeat that is a whole number of dents keeps the same relation to the reed at every repeat across the cloth; one that is not shifts by a fraction of a dent each time and comes back into step after lcm(repeat, ends-per-dent) ends. Since the number of ends per dent is a small number and the repeat is a large one, the two nearly always share a factor — so the shift is usually a short cycle rather than an incommensurate wander.

But it is a third constraint on the same integer, and it is the one nobody writes down. The repeat has to suit the design, divide the body if it can, and sit well against the denting — and only the first of the three is what the repeat is chosen for.

The selvedge is not free either #

The arithmetic above treats the selvedge as a fixed band, and it is not quite.

A selvedge is threaded to its own weave — usually a plain or a rib, because those grip the weft and resist the temple — so its ends do not weave the design and are not available to it. Twenty-four ends a side out of 1,800 is 2.7 per cent of the warp, which is small; on a narrow ribbon warp of 200 ends it is a quarter.

And the selvedge’s ends come off the same beam at the same rate as the body’s, so they have to take up at nearly the same rate. A plain selvedge beside a satin ground is exactly the crimp mismatch two layers of a double cloth have, at a much smaller scale — which is why a selvedge is often given its own small beam on a wide loom, and why on a loom without one the selvedge is set more openly to compensate.

So the body width is not simply the warp minus a constant. It is the warp minus whatever the selvedge construction needs, which depends on the ground weave, and a designer who changes the ground has changed the body and therefore the fit.

The turn, which halves the problem and moves it #

There is one construction that changes the arithmetic rather than absorbing its result, and it is what a tablecloth is.

A turned design is mirrored at a centre line: the right half is the left half reflected, so the pattern runs out from the middle of the cloth to both selvedges. It is threaded as a point draw, it costs no extra shafts — the harness does not grow with a reversal — and it changes what has to fit.

An untied repeat has to fit the whole body. A turned design has to fit half of it, because the two halves are the same and only one of them is designed. That sounds like more room and is less: a half-repeat that divides half the body is a whole repeat that divides the body, so a design turned between two ends can use only the even divisors of the body, which are a subset of what an untied repeat can use. On the 1,752-end body it keeps 8, 12, 24 and 146 and loses 73 and 219, and over every even body from 1,000 to 4,000 ends it fits about two thirds as many. Only a turn on a single end changes the set, to the even divisors of one end less, and that needs an odd body — where it roughly doubles the choice, because an odd body admits no even repeat untied at all. The point tie’s own arithmetic works through both.

And the break moves from the selvedges to the centre. A turned design whose half-repeat does not divide the half-body leaves its remainder in the middle of the cloth, where the two mirrored halves meet, and there it is a seam in the pattern rather than a cut motif at the edge.

Which of those is worse is the design’s own question and the answer is usually the centre, because a cloth is looked at in the middle and handled at the edge. That is why a turned design is normally arranged so the centre is a deliberate motif — the reflection axis is drawn as a feature, and the remainder is pushed back to the selvedges by adding a plain band.

So the turn does not solve the fitting problem; it gives a designer a choice of where to have it, which is one more decision of the same kind as centring. Every option here is a decision about placement and none is a way of making the remainder go away, because a finite width divided by a repeat has a remainder and no construction changes that.

Which is the second place this collection’s infinite cloth runs out #

The finite-width problem has appeared once before on this site and it appeared as a surprise, which is worth noticing because it is the same surprise.

A tube, a double-width cloth and two separate cloths are the same draft to the last square, and what separates them is how many free selvedges the finished piece has — four, two or none. A repeat has no selvedges in it, so the criterion this collection is built on reports two layers for all three and is right every time and useless every time.

That is exactly the shape of this rung. A repeat has no edges, so nothing computed from a repeat can say what happens at one; and what happens at one is a broken motif, a firmer weave, a different take-up and a decision about centring, none of which is in the matrix.

So the finite cloth is where two quite different things have now gone missing, and both were invisible from inside the notation rather than merely hard. The pattern is a general one for a site that counts matrices: what the matrix cannot say is a list, and “it has edges” belongs on it twice.

What is different here is that the missing thing is arithmetic rather than topology. The layer count needed a different kind of object — a finite fabric with a boundary — before it could tell a tube from two cloths. The fitting problem needs only one more integer, the warp’s width, and everything follows from a division. It was missing because nobody put the integer in, not because the notation could not hold it.

What was counted, and how #

The fit is integer arithmetic and is exact. Whole repeats, remainder, and the split — floor and ceiling of half the remainder, which puts the odd end on one side and is what centring does.

The divisor counts are enumerated rather than derived from a factorisation. Every candidate repeat in the range is tried against the body and the ones that leave no remainder are collected, which is slower than factorising and is checkable by hand at any single point.

The claim that fitting is rare is asserted over four warp widths, not at one, because “how many divisors does this number have” is exactly the sort of claim that is a fact about the number somebody chose. And it is asserted in both directions: fewer than a fifth of repeats fit, and at least one does — a warp with no fitting repeat at all in the range would be a different and more interesting problem.

The mean remainder is computed by sweeping rather than quoted as “half a repeat” from the uniformity argument, because the sweep is over a finite range of repeats against one body and the uniformity is asymptotic. It comes out at 0.478, which is close enough to a half to say so and not close enough to assume.

Where the model stops #

The turned case was argued before it was computed, and the argument was wrong. It said the halved set was usually the richer one. Enumerated, a turn between two ends fits only the even divisors of the body, never more than an untied repeat on any even body, and only a turn on a single end of an odd body gives a designer more.

The selvedge width is taken as given. How many ends a selvedge needs is a question about the temple, the beat-up and the weft’s insertion, and it has an answer this collection does not have.

Nothing here is about the length. A design also repeats down the piece, and the piece has two ends — so the same arithmetic applies in the other direction, with the difference that a piece’s length is cut rather than warped and can therefore be chosen to fit. That asymmetry between the two directions is real and is the reason the width is the hard one.

And “reads as a border” is a judgement. Where the boundary lies between an invisible sliver and a broken motif depends on the design’s own scale, on the viewing distance and on whether the cloth is seen with both edges at once. No threshold is proposed and none should be read into the fractions above.

Who found it, and when #

Fitting a repeat to a width is the first thing a jacquard designer does and it is in every design manual, usually as a worked example rather than as a rule: here is a warp of so many ends, here is a repeat, here is the number of repeats and what is left.

What the manuals do not say is how rare an exact fit is. The reason is probably that they are teaching the calculation rather than surveying it, and a designer doing one calculation has no occasion to notice that the answer comes out untidy nearly always.

The divisor framing and the two to eight per cent are this collection’s, and the useful part is the inversion: the question is not how to make a repeat fit but where to put the part that does not. That is a design decision with three answers — centre it, put it all at one selvedge, or design a border — and only the third makes it stop being a compromise.

Where the ladder goes next #

The repeat has now been fitted to a warp, which is a count of threads fixed at warping and not negotiable. The other direction is not like that at all: a piece’s length is cut, so a repeat there has to fit a panel rather than the cloth — and every panel of a patterned cloth pays its own allowance, which multiplies where the width’s remainder is paid once.