An unbroken line is not a clean one

No weave draws an unbroken line one thread wide. A single light end in a dark cloth is on the face only where it is up, so the line it draws is dashed; two neighbours of the same colour fill each other’s gaps in a plain weave and three in a 2/2 twill, and the fewest that leave no gap is a property of the weave, found by a search over every run of neighbours round the repeat.

That census answered one question about a narrow line and raised another in its own drawing. The three-end line in the 2/2 twill has no gap in it and it is plainly not a straight line: the lit cells step down and across with the twill, so what a reader sees is a band whose two edges move in and out with a period of four picks.

Unbroken and clean are different properties, and only the first had been counted. This is the second, and it has an exact answer with a one-line proof.

The boundary is where the outermost lit thread is #

A line of k adjacent light threads in a dark ground is not k threads wide at every crossing. At each pick some of the k are up and the rest are covered by the ground’s weft, and what a reader sees as the line’s left boundary is the position of the leftmost thread that happens to be up there.

So the boundary is a sequence rather than a position, one entry per crossing, and three quantities come out of the same table.

The swing is how far the boundary moves, in thread widths: the range of the outermost lit thread over the repeat, taken on the worse of the two sides. An edge that never moves has a swing of nought.

The wander is the range of the mean position of the lit threads, which is the quantity an eye reads at a distance. A line whose edges fray about a fixed middle looks soft; one whose middle moves looks wavy, and they are different defects.

The period is how many crossings pass before the lit set repeats. It decides whether the swing is a texture or a shape, because an eye that cannot separate two neighbouring picks averages them and one that can does not.

No line in any weave has a straight edge #

The first result is a theorem and it holds at every width, not only at the narrowest.

Take the thread at a line’s boundary. In a cloth that hangs together every thread goes under somewhere — that is the integrity criterion reading a thread that never interlaces as a thread held by nothing — so there is a crossing at which the boundary thread is covered by the ground. At that crossing the line’s edge is not where that thread is; it is at the first thread inward that is up.

So the boundary moves by at least one thread width, in every weave there is, at every width of line. Widening the band cannot help, because the thread at the new boundary has the same obligation.

That is the same shape of argument as the one about gaps, and it lands one level out. A line one thread wide is broken because its only thread goes under; a line of any width has a moving edge because its outermost thread goes under. The interlacing condition produces both.

The swing is the narrowest unbroken width less one #

The second result is the surprising one, and the census finds it with no exceptions at all.

The swing is exactly the width less one, in every draft, in both directions. It is not a correlation and it is not approximate: the width classes and the swing classes are the same partition of the catalogue, to the last draft.

The reason is the minimality, and it takes one sentence. If k is the fewest adjacent threads that leave no gap, then k − 1 of them do leave one — which means there is a crossing at which some run of k − 1 adjacent threads is wholly under the ground. Put the band there. At that crossing the band’s only lit thread is the one at the far end of it, so the boundary has retreated k − 1 threads, and it cannot retreat further without the line having a gap, which by construction it does not. The very fact that makes a line as narrow as it can be is what makes its edge as ragged as it can be.

The corollary is worth stating separately because it is what a designer would notice first. At its narrowest unbroken width a line narrows to a single thread somewhere in every repeat. A three-end line in a 2/2 twill is three threads wide, two threads wide, and one thread wide, in a rhythm four picks long; it is never an even band of three.

Widening the stripe does not clean it #

The obvious repair is to make the line wider, and the census says it does nothing.

Past the narrowest unbroken width the swing neither grows nor shrinks. It cannot grow, because a gapless line’s retreat is bounded by the run of covered threads at its edge and that run is fixed; and it cannot shrink, because the boundary thread still has to go under. The edge is a property of the two or three threads at the boundary and not of how many lie between them.

That is an unusual shape for a design lever to have. Most of what a designer can do to a stripe is continuous: wider is heavier, closer is firmer, more picks is darker. This one is a step and then a plateau — the gaps close at k and the edge stops changing at k, and everything from there to a stripe a centimetre wide is the same edge.

The cleanest edge belongs to the narrowest line #

Putting the named weaves side by side gives the table a designer would want, and it does not contain the trade-off the question implied.

The weave that draws the narrowest line draws the cleanest edge, because they are the same number. A plain weave needs two ends and swings one; a 2/2 twill needs three and swings two; the twenty-four permutation drafts need four and swing three. There is nothing to trade: the search for a narrow line and the search for a clean one return the same ranking, and the earlier account’s census is this one’s census with one subtracted.

That is a genuinely convenient result and it is worth being suspicious of, so it is worth saying exactly what is not implied. The two censuses agree about the amplitude of the edge’s movement. They say nothing about its period, and the period is where the weaves separate.

A fast swing is a texture and a slow one is a shape #

A plain weave’s line swings one thread every two picks. A 2/2 twill’s swings two threads every four. An eight-end satin’s swings one thread every eight.

At twenty-eight picks to the centimetre those are 0.7, 1.4 and 2.9 millimetres of period. An eye resolves roughly a tenth of a millimetre at reading distance and nearly a millimetre across a room, so in the hand all three are separable and at three metres none of them is — every one of these lines reads as smooth from across a room, with its own mean width and its own mean lightness.

Between those two distances they behave differently, and the ordering is not the swing’s. A plain weave’s edge alternates faster than any of them and by the least, so it is the first to average out. A satin’s edge swings the same one thread and holds each position for eight picks, so at a metre a satin pinstripe still has visible steps while a plain-weave one does not. The smallest swing and the most visible one are the same cloth, which is the thing the amplitude census cannot see and is the reason the period is measured beside it.

That also settles the question the earlier account left about the 2/2 twill. Its edge steps down and across with the twill line, two threads every four picks, so the steps run at the twill’s own angle — and a stepped edge running diagonally is exactly what reads as a rope rather than as a fuzzy line. The twill’s line looks like a cord because its edge is a diagonal, not because its edge is large.

The weft bar is the same census turned over, and it is worse #

The previous account found that a stripe and a bar of the same thickness need different numbers of threads in more than a third of the catalogue. The same asymmetry runs through the edges, and the satin is the extreme case.

An eight-end satin draws its narrowest unbroken warp line in two ends with an edge that swings one, and its narrowest unbroken weft bar in eight picks with an edge that swings seven. The same cloth, at the same moment, draws the cleanest line in this table one way and the worst one the other.

A designer drawing a check in that satin with equal thread counts in both systems is drawing a fine crisp stripe crossed by a broad haze, and nothing on point paper says so: a column and a row of the same colour look exactly alike there, which is the same blindness the width census found and which the edge makes worse rather than better.

What a grouped thread does that a single one cannot #

There is one construction in this collection that escapes the theorem’s premise, and it is the one the earlier account nominated.

A cord is a stripe with no colour in it: warp rib, weft rib and hopsack are plain weave with its threads doubled, so the threads move in groups rather than singly. A group of two ends that rise and fall together has a boundary that is the group’s own edge, and the group goes under as a unit — so the edge retreats by two threads at once rather than by one, and the swing in thread widths is worse.

But the relevant width is not the thread’s. A group is one thread for cover and two for bending, and for a line’s purposes a group of two is one thread of twice the diameter: the swing is one group, and a boundary that steps by one unit is a boundary that steps by one unit whatever the unit is made of. A cord’s edge swings the same one step and each step is twice as large, which is a worse edge in millimetres and the same edge in units of the thing the eye is counting.

So grouping does not answer the question the earlier account hoped it would. What it changes is the period, which doubles with the group, and the period was the thing that separated the weaves in the first place. A doubled plain weave swings one group every two picks where the single swings one end every two — the same rhythm at twice the scale, which is to say the same figure enlarged.

What the edge costs a check, and what it costs a shaded figure #

Two constructions in this collection turn on a boundary being where it says it is, and the swing prices both.

A check. A tartan’s crossing squares are mixtures, and the whole of a check being two stripes and a tartan one is that the same colour order runs in both systems. The boundary between two bands of a tartan is therefore two boundaries at right angles, and each swings by its own system’s number: in a 2/2 twill, two ends and two picks. At twenty-eight ends and twenty-six picks to the centimetre that is 0.71 millimetres one way and 0.77 the other, on bands that in a fine sett are four or six millimetres wide. A tenth of every band’s width is boundary that is not where the sett says it is — and since the swing at a band’s two edges is the same weave at the same phase, the two edges move together rather than independently, so the band’s width is steadier than either of its edges.

That last is worth separating out, because it is the good news in the result. The centroid wanders as far as the edge swings, so the band moves; the two boundaries move in step, so the band does not breathe. A reader looking at a tartan sees bands of a constant width lying slightly askew rather than bands of a wobbling width, and those are quite different defects to look at.

A shaded figure. A weave is a halftone screen with a stated number of greys, and a shading moves between tones by changing the weave from step to step. Every step boundary is a line boundary, so every one of them swings — and by a number that changes from step to step, because the steps are different weaves with different narrowest unbroken widths. A tone ramp’s steps are not equally ragged, which is a defect a designer choosing a shading series cannot see on point paper and which this measurement names: the step from a plain-weave region to a 2/2 twill region has a boundary swinging one thread on one side and two on the other.

How the edges were measured #

For each draft, each direction and each width, the band is placed at the start the width census found — the run of neighbours that leaves no gap — and the lit threads are listed at every crossing of the repeat. The leftmost and rightmost lit positions give the two boundaries, their mean gives the centroid, and the ranges of those over the repeat give the swing and the wander. The period is the smallest divisor of the repeat for which every crossing’s lit set equals the set that many crossings on, found by trying the divisors in order.

Five things are checked. No draft in either direction draws a line with a straight edge, at the narrowest unbroken width or at a whole repeat — which is the theorem, checked rather than argued. Every draft’s swing is exactly its width less one, tested as the width classes and the swing classes being the same partition rather than as a correlation, so a single draft out of place would fail it. Widening a 2/2 twill’s line past its narrowest unbroken width leaves the swing where it was, at every width to a whole repeat. A plain weave’s edge swings less and faster than a 2/2 twill’s, which is the amplitude and the period tested together so that an implementation which confused them would fail. And a plain weave’s two-end line is lit by exactly one of its two ends at every pick, to the last bit, which is the check that the lit-set table is being read correctly at the one case where the answer is known by hand.

Where the model stops #

A thread is a rectangle. A lit cell is drawn as a full cell of the grid and a float is drawn as lying flat, so a boundary is a step function. A real float spreads a little sideways over the threads it crosses, and a real thread beside it is pressed under, so the edge in cloth is rounded at every step and the swing measured with a glass would be a little under the swing counted here.

The ground is one colour and one weave. A line between two different weaves — a satin stripe on a plain ground — has a boundary whose two sides interlace differently, and which of the two the eye takes as the edge is not a question this measurement asks.

The period is a count of crossings and not a length. Turning it into a distance needs the sett, and the two directions have different setts, so the same period reads at different sizes in the warp and the weft of one cloth.

And nothing here is a claim about seeing. Where a stepped edge stops reading as steps depends on the contrast, the light, the distance and the viewer; the figures of a tenth of a millimetre at reading distance and a millimetre across a room are rules of thumb carried from the earlier account and are not measurements of anybody’s vision.

Still open: whether a boundary can be dented straight #

The theorem forbids a straight edge in a weave. It does not forbid a straight edge in a cloth, because a cloth has one more freedom than its matrix: the spacing of its ends, which the reed sets and which need not be even.

A stripe’s outermost end retreats at the crossings where it is covered. If that end were set closer to its neighbour than the general sett — a denser dent at the stripe’s boundary — the retreat would be shorter in millimetres while the matrix stayed exactly as it is, and a boundary end set at half the spacing would halve the visible swing. The reed is not the sett and a denting that varies across the width is ordinary practice at a selvedge, so the mechanism exists and costs nothing but a denting plan.

What it would cost the cloth is the question. Crowding two ends at a stripe’s edge changes the cover there, which changes the shade of the boundary, and a denting that shares a factor with the repeat shows as a stripe — so the repair may produce a second line beside the first. Whether a boundary can be made to look straight by denting, and at what cost in shade, has not been worked out here.

Who found it, and when #

That a woven line has a stepped edge is visible to anyone with a glass and a piece of striped cloth, and the practice of choosing a weave for a fine stripe is old. Nothing in the literature reached here counts the swing, and nothing connects it to the narrowest unbroken width — which is unsurprising, since the width itself was counted here for the first time an earlier essay. The identity between the two, the census over the catalogue and the plateau past the narrowest width were computed directly.