A crease along the twill fades as one line

A worn crease in denim is a line of white where the fold’s ridge has been rubbed through the dyed skin of the warp to the pale core under it. A crease across a twill fades in dashes laid that crease two ways on a ring-dyed 3/1 twill and found two patterns: along the warp, lines as long as a float’s crown, broken where each float dives; across the warp, a row of dots one end wide.

It ended on the one orientation it had not tried. A twill’s floats do not only run along the warp; they step, one end to the next, along the twill’s diagonal. A crease laid along that diagonal should meet a float’s crown, then the next end’s crown one pick along, then the next — and on a 3/1 twill every crown it meets would sit at the same place in its own float.

Whether the dashes join into a line was the question, and the answer is that they join completely.

The twill’s line is a direction in the cloth #

A 3/1 twill’s draft is unchanged by moving one end across and one pick along. That is what a twill is: the diagonal is not a thread, but a property of the whole draft, a direction in which the pattern repeats itself exactly.

In the cloth that direction has an angle, and the angle is set by the spacing of the threads rather than by the weave. One step along the line crosses one end spacing and one pick spacing, so the line lies at

$$\alpha = \arctan\frac{p_1}{p_2}$$

from the warp, where $p_1$ is the distance between ends and $p_2$ the distance between picks. For the light denim used throughout this work — 50 tex both ways at 24 ends and 16 picks a centimetre — that is 33.7° from the warp. Which way it leans is read off the draft: of the two diagonal steps, one leaves the draft unchanged and the other does not.

Along that line lie the crowns. Each end’s float of three picks rises to one crown, the highest point of the end over its float, and one end over, the next float’s crown is one pick further on. The crowns of every end in the cloth lie on straight lines at 33.7°, spaced 1.39 millimetres apart.

A fold lined up with the crowns meets them all #

A crease is a fold, and a fold presents its ridge to whatever rubs it. Bent round a radius $R$, a point a distance $d$ from the ridge stands lower by $d^2/2R$ than it would flat, so a flat abrader meeting the fold takes the ridge’s own crowns first and a band either side that widens as it goes deeper.

When the ridge runs along the warp, it lies along one end, and that end’s crowns come and go: three picks of float, one pick of crossing, three of float. A float fades late and hard, and the crease fades in float-long lines with a gap at every crossing. When the ridge runs across the warp it crosses every end at one point, and the crease fades in dashes one end wide.

When the ridge runs along the twill’s line, it lies along a line of crowns. Every crown under it is at the same height, because every crown is the same place in the same kind of float, and the fold lowers them all by the same amount. The abrader meets them all at once, and cuts them all alike, and the white it bares is continuous from one end’s crown to the next.

The line is made of thirty-two ends #

The white line in the opening figure looks like one thing, and it is not made of one thing. Along the twill’s line each step from one crown to the next crosses one end, and the step is $\sqrt{p_1^2 + p_2^2}$ — 0.75 millimetres for this denim. So the 24 millimetres of line drawn there are bared from thirty-two different ends, each contributing three quarters of a millimetre of white from the top of one of its floats.

No end in the cloth runs along that line. Every warp end lies along the warp, 33.7° away from it, and every one of them is crossed by the crease exactly once per repeat. The line is a property of how the ends’ floats are arranged relative to one another, which is exactly what the diagonal is not a thread found of a twill’s visible diagonal: a line the eye reads as continuous, supplied in short pieces by threads that all run another way. The crease reads the twill’s line off the cloth the same way the eye does, and bares it.

The other diagonal crosses the floats #

The opposite diagonal has the same angle and none of the property. One step along it crosses one end and goes one pick back, so consecutive crowns under it are two picks apart along the float direction rather than aligned: the ridge meets the end of one float, then the middle of the next end’s, then the start of the one after, and it crosses each float obliquely rather than riding along its crown.

So the other diagonal fades like a slightly better version of across the warp: dashes, averaging 0.93 millimetres and never longer than about a millimetre, covering 66 per cent of the crease. It is better than across the warp because each dash is an oblique slice of a float rather than a cut through one end, and worse than the warp because no float’s crown lies along it for more than a fraction of its length. The twill’s two diagonals differ by more than forty times in the length of their longest white run over 24 millimetres, and without limit over a longer crease, which is the largest difference between any two directions on the cloth.

One direction reads all white #

Sweeping the crease through every angle shows how singular the line is.

The 3/1 twill reads 100 per cent white on its own line, against 80 per cent along the warp and 66 on the opposite diagonal, 30 across the warp. The line is a spike, not a hill: two degrees either side, the share is back on the broad shoulder at 70 to 77 per cent. The 2/2 twill has the same spike at the same angle, reaching 86 per cent, and the plain weave, whose crowns are single points, has nothing there worth the name.

The shoulder itself leans. From across the warp to along it, the crease whitens more as it turns towards the floats, and it keeps whitening past the warp towards the twill’s line, then falls away past it. A crease and its mirror image across the warp do not fade alike: at 15° either side of the warp they read 79 and 73 per cent; at 30°, 79 and 67; on the line and its mirror, 100 and 66.

The line joins as the wear deepens #

The dashes do not join at once, and watching them join is how the lead’s question gets its answer in full.

At the lightest wear the twill-line crease is behind the one along the warp: 29 per cent white against 53 at twenty micrometres. Each crown under it is bared only at its very top, a short dash, and the dashes are separated by the saddle between one end’s crown and the next. The warp crease, lying along one end, starts with its whole float crown to work on.

By 25 micrometres the twill-line crease has caught up, and by 40 micrometres it is an unbroken line, 97 per cent white with runs the length of the strip. The warp crease is still 72 per cent white at that depth, and it does not reach 99 per cent until 100 micrometres, when the abrader has cut down past the crossings themselves. The line joins when the cut is deep enough to reach down the saddle between consecutive crowns, and on the twill’s line that saddle is shallow, because the next crown is only one pick away.

A sharp fold cannot join the crowns #

The saddle has a width as well as a depth, and the fold decides whether the band it presents is wide enough to span it.

A sharp fold is a narrow band: at 0.5 millimetres’ radius the abrader can reach only 0.56 millimetres either side of the ridge at this wear, and along the twill’s line the crease is 77 per cent white in dashes. At a millimetre the band is 0.79 millimetres each side and the line is 95 per cent white; at two millimetres it is complete. So the lead’s other guess was right as well: whether the dashes join depends on whether the band the fold presents is wider than the step from one crown to the next — and for this denim the step is spanned at a fold of about a millimetre and a half, which is a soft crease rather than a pressed one.

Two degrees off, a vernier #

A crease is never exactly on a line of anything, so the useful question is what happens near the line rather than on it.

Laid at a small angle Δ to the twill’s line, the ridge sits on one line of crowns and drifts off it, at a rate of $\sin\Delta$ per millimetre of crease, until it is nearer the next line than this one. While it stays near, the crowns under it whiten in an unbroken stroke; when it passes between two lines, the white breaks into short dashes; then it arrives on the next line and the stroke begins again. The crease fades as a series of long strokes separated by stretches of dashes, and the strokes’ length is set by the angle.

The strokes are 0.69 millimetres over the sine of the mismatch long: 32 millimetres at one degree, 18 at two, 9.6 at four. The constant is measured, not derived, and it is half the 1.39 millimetres between one line of crowns and the next — which reads as the crease holding a line of crowns while it is within a quarter of that spacing of it. That reading has not been tested at another sett.

It is the arithmetic of a moiré, which is a vernier and magnifies the error too: two regular structures a small angle apart beat, and the beat is long in proportion to how nearly they agree. Here the two are a line of crowns and a fold, and the beat is a white stroke. A crease within a few degrees of the twill’s line is the only crease on the cloth whose white runs are centimetres long, and the nearer it is, the longer they get.

Every twill does it, and a plain weave cannot #

The mechanism needs two things: floats long enough to have crowns worth the name, and a draft that repeats along a diagonal. Every twill has both, and the angle of its line is the same arctangent of its setts.

The warp-faced twills do it best: the 3/1 reaches a full line and the 2/1 91 per cent. The balanced 2/2 reaches 86, because half of what lies on its line of crowns is weft, which a denim leaves undyed and which bares nothing. A plain weave is unchanged along both of its diagonals, so it has two lines rather than one, and its crowns are points on both: no direction on a plain weave joins them, and it reads 30 to 36 per cent white whichever way the crease runs.

What this says about a pair of jeans #

A garment puts creases at every angle, and the model says which of them fade as lines.

The ones that do are within a degree or two of the twill’s own line. On a right-hand twill that line rises to the right on the face of the cloth, so it is a particular direction on each panel of a pair of jeans, and a crease that happens to lie along it — the diagonal fold across the thigh when a leg is bent, say — fades as a continuous white line where its neighbours at other angles fade as dashes or strokes. The same crease mirrored onto the other leg does not: the other leg is cut from the same cloth, so its crease meets the twill at the mirror angle, which on the 3/1 twill reads two thirds white in dashes a millimetre long at most.

That is a prediction a reader can check against any worn pair: of two mirror-image creases, the one leaning with the twill fades as a longer, more continuous line. What sets how different they are is how near the crease is to the line, and a crease fifteen degrees off it still reads 79 per cent white in runs up to 3.6 millimetres, where its mirror reads 71 per cent in runs of 1.4.

The model named #

The face is the plain-weave geometry this account solves for a light denim, 50 tex both ways at 24 ends and 16 picks a centimetre, with a twill’s floats drawn over it, and the warp ring-dyed to the depth one dip gives at affinity 10. The fold lowers the face by $d^2/2R$ at a distance $d$ from its ridge; the abrader is flat and takes everything above a plane a stated wear below the highest point it meets. A stretch of crease reads white if anything across its band at that stretch is bared core. The twill’s line is found from the draft, as the one diagonal step that leaves it unchanged, and its angle from the setts.

Every share is averaged over sixteen positions of the ridge across the repeat, placed at golden-ratio steps. Evenly spaced positions are a trap for a crease that lies exactly along a thread, which keeps its place in the repeat for its whole length; with sixteen positions the along-warp figure is 80 per cent rather than the 83 that six evenly spaced positions gave in the essay on crosswise creases, and the across-warp figure 30 rather than 31. The ordering is unchanged.

What was counted #

For the 3/1 twill, 2/2 twill and plain weave, 61 angles at three-degree steps from across the warp to across it again, plus 18 angles within two degrees of each diagonal; for the 3/1 twill, ten wears from 15 to 100 micrometres in four directions, ten fold radii from 0.35 to 8 millimetres in two, and nine mismatches from three quarters of a degree to twelve on both sides of the line, over 96 millimetres of crease each; and four weaves in three directions. At 0° and 90° the calculation was required to agree with the one along and across the warp, which counts on a grid aligned with the threads.

What the picture cannot show #

The fibre a real crease tears loose. Abrasion at a crease does not cut a clean plane; it lifts fibre from the bared core and the dyed skin alike, and the lifted fibre blurs every edge in these maps. Whether that blur is enough to close the gaps between a twill-line crease’s dashes at light wear, and so bring the line forward, is not computed.

A real crease is not straight. It curves, it varies in sharpness along its length, and it flattens as it is rubbed. The vernier’s strokes assume a straight crease at one angle; a crease that curves through the twill’s line fades as a line only where it passes through it.

The ring is sharp and the warp round. A ring-dyed yarn whitens all at once because its ring has an edge; the smooth profile a dip really leaves softens every onset here without moving it. And a real denim is set closer than this one, which makes its line steeper and its crowns closer together, so the joining wear and the vernier’s constant would both move.

Who worked out which part #

Fading along creases is a denim finisher’s craft, and whiskers are made deliberately by sanding and lasers along paths chosen by eye. That a twill has a line, and that its angle comes from the setts, is as old as the naming of twill direction. That a crease along the line fades as one line, that the line joins at a stated wear and fold radius, and that near it the white comes in strokes whose length goes as one over the mismatch, are computed here on one model denim and nowhere tested against cloth.

Still open: whether a crease holds its angle #

Everything here takes the crease’s angle as given. But a fold in a woven cloth is not free to lie at any angle: a crease is a fold the crimp cannot supply, and the ease with which a cloth folds depends on the direction, because the crimp that a bend spends lies along the threads. A twill has one more direction of its own, its line, and a fold along it bends every end and every pick through the same share of the bend.

Whether a twill folds more easily along its own line than across it, so that creases in a worn garment drift towards the one angle at which they fade as lines, is a question about the fold rather than about the fade. If they do, the line in this essay is not a coincidence of angle that a few creases happen to meet; it is where creases go.