A crease across a twill fades in dashes

A float fades late and hard, a crossing early and soft wore a ring-dyed face flat and found that a twill’s floats, whose crowns are lines, bare their undyed cores along their length once the cut passes the ring. It ended on the obvious objection: a garment is not worn flat. A knee bends, a hip folds, the cloth at the back of the knee concertinas, and wherever the cloth is folded the ridge of the fold stands proud and is rubbed first. The fade follows the crease, and a crease has a direction.

On a warp-faced twill the floats have a direction too. So the same crease can run along the floats or across them, and the question the flat face could not ask is whether the two look different. They do, and not in how much white they make. They differ in its shape: along the floats a crease bares a line, across them it bares a row of dots.

A fold is a flat abrader on a curved face #

Fold a cloth round a radius $R$ and its outer face is a cylinder. The line along the top of the fold, the ridge, is the highest; a point a distance $d$ to either side of it stands lower by

$$\Delta z = \frac{d^2}{2R}$$

to within the small-angle approximation, which at the distances that matter — a millimetre or so, against radii of one to several — is accurate to a few per cent. Whatever rubs a folded garment meets the ridge first and then a band either side that widens as the rubbing goes deeper, exactly as a crease is a fold the crimp cannot supply placed the fold’s strain on its outer face.

So a crease is the flat face of the essay before with one change: the height of every point is lowered by $d^2/2R$ before the abrader’s plane is applied. Everything else is unchanged. The face is the site’s height field for a light denim, fifty tex both ways at twenty-four ends and sixteen picks a centimetre; the ring is the one a twenty-second dip leaves at an affinity of ten, fourteen per cent of the warp’s radius; and the question at each point of the plan is whether the plane has cut into the warp’s undyed core.

The crease is read the way an eye reads it: as a line. A stretch of the crease counts as white if anywhere across the band at that stretch the core is bared. The count is the share of the crease’s length that is white and the lengths of the white runs along it, averaged over six positions of the ridge in the repeat, because a real crease falls wherever it falls.

Along the warp: float-long lines #

Put the ridge along the warp. On a 3/1 twill the warp is on the face three picks in four, in floats three picks long, so the ridge runs along whichever ends lie under it and meets their floats end to end.

At each float the ridge’s crown is the float’s crown line, the same line lustre is a length times a width measured, and a flat cut through a line bares a strip along it. So along the warp, the crease bares the float’s core over the whole float’s crown, then stops where the end dives under a pick, then starts again on the next float. At fifty micrometres of wear into the ridge, 83 per cent of the crease’s length is white, in runs averaging 1.7 millimetres — very nearly a float’s crown, three picks less the turns.

Deeper, the gaps close. The dips between floats are the next thing the cut reaches, and once they are cut through the runs join: by a hundred micrometres the crease reads as one unbroken white line along the whole strip. A satin, whose floats are longer and whose dips are further apart, joins its runs sooner.

Across the warp: one-end dashes #

Turn the crease a right angle, so the ridge runs along the weft. Now the ridge crosses every end in turn, and at each one it meets the end at a single point of its float: a place on the crown line, but only a point of it.

A cut through a crown line at a single point, across the line, bares a patch as wide as the part of the end’s section below the cut and as long as the band the fold lets the abrader reach. Across the warp, the crease reads as a row of short dashes, one per end it crosses, each about as long as an end is wide. At fifty micrometres, 31 per cent of the crease’s length is white, in dashes of 0.15 millimetres, and the longest never exceeds 0.21 — the width of one warp end’s bared core. The dashes do not join at any depth drawn, because between every two ends there is a gap, or a weft, that has no core to bare.

The across-crease also whitens more slowly. Its share of white climbs from 15 per cent at thirty micrometres to 49 at a hundred and ten, where the along-crease has gone from 65 to a hundred. What an eye sees is a line along the floats and a dotted line across them, and the dotted one stays dotted.

The same core, arranged two ways #

The difference is not that one orientation bares more.

Per millimetre of crease, the twill bares much the same core either way: at fifty micrometres of wear, 0.138 square millimetres along the warp against 0.125 across it, and the curves stay within a quarter of each other at every depth. The abrader takes what the fold presents, and the fold presents the same band of cloth whichever way it runs. The weave decides only how that white is arranged along the crease: gathered into float-long lines when the crease follows the floats, scattered into end-wide dots when it crosses them.

The plain weave, drawn with the twill in both curves, bares about a quarter as much core per millimetre, because its crowns are points and a fold lowers a point’s neighbours as fast as a line’s. It also bares it in short pieces whichever way the crease runs, as the next figure shows.

A plain weave does not care which way #

On a plain weave both creases fade in dashes. Along the warp the runs average 0.24 millimetres, across it 0.12: a crown of a plain weave is a point on its end, as long as a pick is wide and as wide as an end, so whichever way a crease runs, it meets a row of points. The runs differ by the ratio of the pick spacing to the end spacing and by nothing else. The white covers 32 per cent of the along-crease at fifty micrometres and 24 of the across one, and neither ever joins into a line at the depths drawn.

That completes the picture the flat face began. A float fades late and hard: a line crown takes more wear to reach and then bares more along its length. A crease adds direction to that: it samples a line crown along its length if it runs along the floats, and across its width if it does not.

Whiskers are dotted and train tracks are lines #

The two orientations are the two families of fade that denim wearers name.

Whiskers are the fan of short white creases at the front of the hips, and honeycombs the lattice of them behind the knees. Both are creases across the leg, where the cloth folds as the hip and knee bend, and on jeans the warp runs down the leg. So whiskers and honeycombs are creases across the warp, and the arithmetic says they should fade as rows of end-wide dashes, not as solid lines — which is what a whisker is under a lens: a row of short white marks, one on each float the fold crosses, softened into a line only by the fibre torn loose over them.

Train tracks are the pair of white lines that form along the outer seams of a worn pair of selvedge jeans, where the seam allowance underneath stands the cloth proud in a ridge running down the leg. That ridge runs along the warp, and the arithmetic says it should fade as continuous lines broken only at each float’s end, joining into unbroken lines with further wear — which is what train tracks are, and why they are called tracks rather than dots. A seam stands proud and wears first found why the seam is where the wear goes; this says why it shows as a line there and as dots at the hip.

Knees tear across the leg #

The two orientations differ in strength as well as in look, and the difference is the same arrangement read the other way.

A cloth loses its strength before its mass found that abrasion weakens a thread at its thinnest place, so what matters for strength is how many threads are cut and where, not how much fibre has gone. A crease across the warp cuts into every end it crosses, once each, at the same line: the dashes are a row of wounds, one per end, lined up across the cloth. The warp-way strength of the cloth at that line is the strength of every end at its thinnest place, and all of them are thinned there together. A crease along the warp cuts into the one or two ends under its ridge, along their length: it wounds few ends, deeply, and leaves their neighbours whole.

So the across-crease is a line of weakness across the leg and the along-crease is not. That is where denim fails. Knees blow out across the leg, in a horizontal tear along the crease the knee makes, because every warp end has been abraded at that line and they fail together; the seams that fade in train tracks rarely split lengthwise, because a longitudinal crease wounds a handful of ends and the weft holds across them. The dashes that make a whisker look dotted are the same row of wounds that makes a knee tear straight.

It also says why the across-crease forms in the first place on cotton. Which fibres crease, and why there are two answers put cotton near the bottom of the recovery ranking: a cotton fold is kept, and a kept fold is rubbed in the same place wear after wear. On a warp-faced cloth, where balance and the unbalanced cloth found three quarters of the face to be warp, every such crease is a crease through the ends — which is why the fade and the tear both belong to the warp.

The sharper the fold, the narrower the band #

A sharper fold presents a narrower band to the abrader, since $d^2/2R$ grows faster off the ridge. Along the warp that barely matters: the band is narrow but it still runs the whole length of each float under the ridge, and 69 per cent of the crease is white at a half-millimetre radius, rising to all of it at eight. Across the warp the share rises from 26 to 38 per cent over the same range, because a wider band lets the dashes lengthen a little across each end.

So a sharp crease across the warp is the most dotted fade a twill can make, and a gentle crease along it the most solid. A finisher placing whiskers with a laser or a template chooses both the direction and the radius, and with them the texture of the fade, independently of how much colour is removed.

What the model assumes #

The fold is a cylinder of fixed radius and the abrader a plane, which is the simplest way to put a crease on the flat face and the right one for a fold held while it is rubbed. A real fold flattens under the rubbing, and a real rub is not planar: a knee inside a leg rubs the fold from within as well as from without. Both would widen the band.

The face is the plain-weave geometry with the floats drawn over it, more open than a real denim, exactly as in the flat essay; the counts are of the covered face. The ring is sharp at the half-strength depth. The crease is read as a line, a stretch white if any point across the band is bared, which is how an eye resolves a crease at arm’s length; read at the scale of the dashes themselves, the across-crease is a row of separate marks, which is the point.

The crease is required to whiten along a 3/1 twill’s floats in runs more than five times as long as across them, and over more of its length; to whiten a plain weave in runs shorter than half a millimetre whichever way it runs; to bare much the same core per millimetre along and across the twill at a hundred micrometres of wear; and to refuse a crease that runs neither way.

What the figures cannot show #

A crease on the bias. Real creases are rarely exactly along or across the warp: a whisker fans out from the fly at a range of angles, and a crease at an angle to the floats would sample each float’s crown over a length that shortens as the angle opens. The two orientations here are the extremes, and a crease at forty-five degrees on a twill whose diagonal runs at some angle of its own would lie between them in a way that depends on whether it runs with the twill line or against it.

And the torn fibre. Abrasion breaks fibres as well as cutting them, and the broken ends stand out of the surface in a pale fuzz over every bared core. It is the fuzz that turns a row of dashes into what reads as a soft white line at a distance; the dashes are underneath it, and they are what a lens finds.

Who worked out which part #

Whiskers, honeycombs and train tracks are the denim trade’s names, and their positions — at the hips, behind the knees, along the outseams — are the folds and seams of a pair of jeans. The flat face and the ring are the dyeing essays’. The fold as a lowering of $d^2/2R$ off the ridge is the ordinary geometry of a bent surface.

What is added here is the crease’s direction: that along a twill’s floats a crease bares float-long lines and across them end-wide dashes, with much the same core either way; that a plain weave fades in dashes whichever way its crease runs; and that this is why whiskers are dotted and train tracks are lines.

Still open: a crease on the twill’s own diagonal #

A twill’s floats are not only along the warp; they step, one end to the next, along the twill line at an angle set by the setts. A crease running along that diagonal meets a float’s crown, then the next end’s crown shifted by one pick, then the next, and on a 3/1 twill each crown it meets is at the same place in its float, so the white would come in dashes arranged along a line — and whether they join depends on whether the step from one end’s crown to the next is shorter than the band the fold presents.

That is the one orientation in which the twill’s own structure and the crease could line up, and the diagonal is not a thread found that the twill line is a property of the whole draft rather than of any one end. Whether a crease along it fades as a line made of many ends — a line no single float supplies — is a computation on the same face with the ridge turned to the twill’s angle.