The width of a bolt is worth what its faults leave it

A fault map is worth most where the grade is worst cut a bolt as a line. Faults sat at points along its length, panels were laid end to end, and a cutter with the map could break the marker wherever a fault fell and start the next panel just past it. That freedom recovered twelve panels in fifty-five on a moderately faulty bolt, and a sweep from one end was provably the best a marker could do.

It closed on the question it could not answer. A bolt has a width too, and a cutter can move panels across it. What does the second dimension add? The essay made a prediction without computing it: across the width a bolt is only a few panels wide, so there is little room to move, and the second dimension should add much less than the first.

That prediction is right for two kinds of fault and wrong for the third, and the third is the one a missing end makes.

Three shapes of fault #

Faults in a woven cloth come in three shapes, and the shape is set by which thread went wrong.

A point is local: a slub, a knot, a small hole, a spot. It spoils a panel only if it lies inside it, in both directions.

A weft bar runs the whole width at one place along the piece. A mispick, a stop mark where the loom was restarted, a pick of the wrong yarn — a mispick is one row in the wrong place, and a row runs from selvedge to selvedge.

A warp streak runs the whole length at one place across the width. A missing end, a tight end, an end of the wrong yarn — every warp fault is a line down the piece, because an end is on the beam for the whole of it. What a missing end does to the weave is the same all the way down.

Each shape takes away one of the cutter’s freedoms entirely. A weft bar cannot be dodged by moving across, because it is there at every position across; a warp streak cannot be dodged by moving along, because it is there at every position along. Points can be dodged either way.

Four markers, one of them a ceiling #

A bolt 1.5 metres wide takes three panels of 450 millimetres across it with 150 to spare. Four markers are compared.

Rigid tiles: three fixed lanes, panels end to end from the start, any panel with a fault in it thrown away. The cutter knows nothing.

Fixed lanes, cut around: the same three lanes, each swept along the piece and broken wherever a fault falls — the whole of the first essay’s freedom, used lane by lane.

Lanes placed too: the width’s freedom added in two ways and the better one kept. Either each lane is placed across the width wherever suits it, rigid along its length, without overlapping its neighbours; or each panel may shift across within its lane’s share of the spare 150 millimetres. Both are real markers.

No marker can beat this: each lane’s panels allowed to use the whole spare width, with the lanes permitted to overlap one another. That is not a marker at all, because two panels could claim the same cloth — which is exactly why it is useful. Every real marker yields at most this many panels, so the true best marker, whatever it is, lies between the third and the fourth.

That bracket is what lets the question be answered without solving the packing problem. The first essay’s sweep was provably optimal along one dimension; in two dimensions nothing so simple is, and nothing here claims to have found the best marker. What is found is a ceiling on what the width could possibly add.

Against weft bars the width is worth nothing, exactly #

Put only weft bars on the bolt, at 0.35 a metre, and the four markers yield, averaged over eight bolts:

120.4 rigid, 139.9 cut around along the piece, 139.9 with lanes placed, and at most 139.9 by any marker at all.

The ceiling equals the along-only marker to the panel on every one of the eight bolts, and it has to: a fault that runs the whole width spoils every position across at its place along, so there is nothing a sideways move can do. The second dimension adds exactly nothing, and the first adds 19.5 panels.

This is the case the first essay’s one-dimensional answer was secretly about. A bolt whose faults are all weft faults is a line, and treating it as one loses nothing.

Against points the width is worth a sixth to two fifths of the length #

Scatter point faults at 0.6 a square metre — about forty-five on the bolt — and the length’s freedom recovers most of what can be recovered:

128.3 rigid, 145.5 cut around along the piece, 148.4 with lanes placed, and at most 152.3 by any marker.

The length buys 17.2 panels. The width buys at least 2.9 — a real marker gets them — and at most 6.8, because no marker can beat the ceiling. So the second dimension is worth between a sixth and two fifths of the first against points, and the prediction was right: across the width there is little room to move, and it buys correspondingly little.

The reason the width is weak is visible in the strip. Moving a lane sideways helps only when a fault lies within the spare width of a lane’s edge, so that a step of at most 150 millimetres takes it outside; a fault in the middle of a 450-millimetre lane cannot be dodged sideways by any amount the bolt allows. Moving along helps with every fault, because the bolt is as long along as it is.

Against a missing end the length is worth nothing #

Now put only warp streaks on the bolt, at a rate that gives two on each, and the answer turns over:

82.5 rigid, 82.5 cut around along the piece, 89.4 with lanes placed, and at most 89.4 by any marker.

Cutting around along the piece recovers nothing — not a panel on any of the eight bolts — because a streak is there at every position along, and a lane it passes through is spoilt from end to end however the lane is broken. The width recovers all 6.9 panels there are, and here the bracket closes: the best real marker equals the ceiling, so 6.9 is not a range but the answer.

So the first essay’s prediction fails for the fault that the whole account of grading began with. A missing end is the fault a length freedom cannot touch and a width freedom can, and the width’s value against it is set entirely by whether the streaks leave room for a lane between them.

Mixed faults make the width worth as much as the length #

A real bolt carries all three shapes at once, and the mixture is where the prediction does worst.

Take points at 0.3 a square metre, weft bars at 0.15 a metre and one warp streak. Averaged over eight bolts:

87.9 rigid, 101.6 cut around along the piece, 115.0 with lanes placed, and at most 116.5 by any marker.

The length buys 13.7 panels and the width buys between 13.4 and 14.9 — as much as the length, on a bolt only three panels wide. One streak is enough to do it, because a streak spoils a whole lane of fifty-five panels and a sideways step of a few centimetres can save the lot, while a point or a bar spoils one panel and a lengthwise step saves one.

That asymmetry is the whole finding in one line: a fault’s cost is the length of cloth it spoils in the direction it runs, and a freedom is worth the faults it can dodge. A streak spoils a lane’s whole length and is dodged across; a bar spoils one panel’s length across all lanes and is dodged along.

The slack decides how much the width can buy #

The width’s freedom is the spare cloth left when the panels are laid across — 150 millimetres for three 450-millimetre panels on a 1.5-metre bolt. It is a property of the panel size, and it moves.

At 300 millimetres five panels fill the width exactly and at 500 three do; at both, the width’s freedom is zero and buys exactly nothing. At 400 millimetres three panels leave 300 spare — the most in the sweep — and the width buys between 5.8 and 13.1 panels against the length’s 15.9. So the second dimension’s weakness against points is not a law about bolts; it is a statement about how much cloth the panel size leaves unused across them.

That turns a planning decision into a trade. A marker planner choosing a panel width to fill the bolt exactly is minimising the scrap along the selvedges, and in doing so is also giving up the only freedom that can dodge a warp streak. A bolt cut to fill its width is a bolt whose missing ends cannot be avoided.

Why the panels cannot simply be turned #

There is an obvious third freedom and it is worth saying why no marker here uses it. A panel turned through a right angle would meet a warp streak as a bar and a weft bar as a streak, and a cutter free to turn panels could choose, fault by fault, which freedom to spend.

A garment panel cannot be turned, because it has a grain. The bias cut and the selvedge is the long version: a panel cut off its intended grain drapes, stretches and shrinks as a different cloth, and a marker that turned pieces to dodge faults would be trading a visible fault for an invisible one that appears at the first wash. A patterned cloth adds a second reason, because a skew steps at every seam and a turned panel matches nothing.

So the grain fixes which of the bolt’s two directions each fault shape takes away, and the essay’s asymmetry is not an artefact of rigid panels. It is what a woven cloth’s two thread systems do to a cutter.

A dodged streak is worth dodging even where nobody would see it #

It might be objected that a missing end is often invisible — that a cutter can leave it in a panel and nobody will notice — and then the whole value of the width’s freedom disappears.

Sometimes that is right. Which weave hides a fault found that a weave’s float structure decides how visible a broken end is, and that the answer is not the one the trade gives. But the cloth a missing end leaves is weaker along that line whether or not it shows, since the crossings it should have made are not there, and a panel with a lengthwise line of missing interlacings is a seam-slippage and tearing hazard in exactly the direction a garment is loaded. A cutting room that grades by eye will keep some of those panels; the arithmetic here counts what it would keep if it did not.

What the map is for, restated #

A grade charges by the length and a cutter pays by the panel found that the standard four-point grade says nothing about yield, and recommended recording positions instead. The first marker essay found that a position along the piece is worth twelve panels in fifty-five. This one adds the position across.

A map that records only where along the piece a fault is — which is what an inspection frame’s length counter gives for free — throws away the one coordinate a warp fault needs. A warp streak recorded as “at 23 metres” is recorded where it starts, and it runs to the end; recorded as “at 920 millimetres across” it is the whole of what the cutter needs to place the lanes. For the other two shapes the along-coordinate carries almost everything.

So the map’s two coordinates have different owners. The along-coordinate serves the weft faults and the points, and it is the one the trade already records. The across-coordinate serves the warp faults, and a warp fault is the most expensive fault a bolt carries, because it spoils cloth for the whole length of the piece.

The model named #

The bolt is 50 metres by 1.5 metres, and its faults are placed at random from a fixed seed, so every figure reads the same bolts: points uniformly in the rectangle, weft bars uniformly along it, warp streaks uniformly across it, each at its own stated rate. Panels are 450 by 900 millimetres unless the slack sweep says otherwise, and a panel is spoilt by any fault inside it.

Each lane is swept along the piece and a panel started at the earliest position where some position across its band leaves it clear — a greedy that is optimal within one lane for the same reason the first essay’s was, since panels in one lane never meet across the width. Lanes are placed across by a short dynamic programme over the three lanes on a five-millimetre grid, subject only to not overlapping. The ceiling gives each lane the whole spare width and lets the lanes overlap.

Required bolt by bolt, not shown: rigid ≤ along only ≤ lanes placed ≤ ceiling, on every bolt of every case. And against weft bars alone, the ceiling must equal the along-only marker exactly — the check that the width’s freedom is really being given the whole spare cloth and still finds nothing to do.

What was counted #

Four cases — points only, weft bars only, warp streaks only, and all three mixed — on eight seeded bolts each, cut four ways: thirty-two bolts and a hundred and twenty-eight markers, with every panel of every lane placed. Then seven panel widths from 300 to 700 millimetres on eight bolts each for point faults, another two hundred and twenty-four markers.

Where the model stops #

The best marker is bracketed, not found. For points the bracket is 2.9 to 6.8 panels wide, and the true answer is somewhere in it. Closing it is the packing problem the first essay declined, and the argument here does not need it: the ceiling alone is enough to say the width buys less than the length against points, and the closed bracket for streaks says exactly what it buys there.

All panels are one size. A real marker nests pieces of many sizes and shapes, which makes both freedoms more valuable, because a small piece can use a gap a large one cannot. What survives is the structure: a fault’s shape removes a freedom entirely, whatever the pieces are.

Faults are sharp. A real streak wanders across a few ends and a real bar fades at its edges. Neither changes which freedom can dodge it.

And the rates are illustrative. The eight bolts in each case were chosen for a comparable loss to a blind marker, not measured from a mill. The ordering of the three shapes does not depend on the rates; the size of each gain does.

Who worked it out #

Marker planning is an old practical art and a modern software problem, and cutting around faults is standard in cutting rooms that have a fault map. That warp faults run the length and weft faults the width is the first thing an inspector learns.

What this essay adds is the separation. The value of each coordinate of a fault map is fixed by the shape of the faults, and it can be bounded without solving the marker by putting a real marker below and an impossible one above. The first essay’s one-dimensional sweep turns out to be the exact answer for weft faults and nearly the answer for points, and no answer at all for the fault the grading account started from.

Still open: whether a streak’s cost changes the specification #

The account of grading has found three times that a point system charges by the length of a fault and a cutter pays by the panel. A warp streak charged as a fault the length of the piece costs the grade its full penalty for every metre; to a cutter with the across-coordinate it costs a lane at worst and nothing at best, depending on where it sits relative to the spare width.

That suggests a specification that charges a warp fault by its position across the bolt — nothing within the spare cloth of a selvedge, a whole lane elsewhere — rather than by its length. Whether such a charge would rank bolts the way cutting rooms actually value them is a comparison between this arithmetic and a mill’s own cutting records, and it has not been made.