A ring-dyed yarn whitens all at once

A dip dyes a depth, not a share followed the dye into a yarn and found it stopping at a depth set by the dip’s length and the fibre’s affinity for the dye, much the same in micrometres whatever the yarn’s thickness. A coarse denim warp, dipped for twenty seconds at a time, ends with a blue skin a fibre or two deep round a core that never saw the dye.

What that ring does when the yarn is worn is a question of pure geometry, and the answer has a shape that no gradual process would produce. For the first stretch of wear nothing changes. Then the yarn whitens with a vertical tangent. The threshold is exactly the ring’s depth, the steepness is a square root, and the whole of denim’s reputation for fading in sharp-edged patterns follows from those two facts.

An abrader cuts a chord #

Model the wear the simplest way that is still exact: a flat surface rubbing across the top of a round yarn and taking off everything above some level. Floats and abrasion put the rubbing where a denim’s warp floats stand highest, and a seam stands proud and wears first found the same selection at a garment’s scale: whatever sticks up is what is rubbed. At the scale of one yarn, what sticks up is the yarn’s crown, and a flat abrader cuts the crown off level.

Measure everything in units of the yarn’s radius. After a cut of depth $w$, the yarn has a flat top where the crown was. The flat is the chord of the circle at height $1 - w$ above the axis, and it is $2\sqrt{2w - w^2}$ wide. Everything above it has gone: a circular segment whose area, as a share of the whole section, is

$$m(w) = \frac{\arccos(1-w) - (1-w)\sqrt{2w - w^2}}{\pi}.$$

That is the share of the yarn’s fibre the abrader has taken, because each fibre above the chord has been cut through.

The undyed core is a smaller circle, of radius $1 - \delta$ for a ring of depth $\delta$, with the same centre. The chord meets it only when the cut is deeper than the ring, $w > \delta$, and from then on the flat top has a white stretch in its middle,

$$2\sqrt{(1-\delta)^2 - (1-w)^2}$$

wide, with blue on either side of it where the cut runs through the ring.

A yarn seen from above is still two radii wide #

What an eye sees of a worn yarn from above is its projected width, and that stays two radii wide while the abrader has only taken the crown: the untouched flanks still stand at the sides. The share of the width that is white is the core’s chord over two,

$$s(w) = \sqrt{(1-\delta)^2 - (1-w)^2} \quad\text{for } w > \delta, \qquad s = 0 \text{ otherwise.}$$

Each ring’s curve is flat at zero up to its own depth and then leaves the axis vertically. Just past the threshold, with $w = \delta + e$,

$$s \approx \sqrt{2(1-\delta)\,e},$$

so the white share goes as the square root of how far the cut has gone past the ring. For a ring of a fifth, one per cent of the radius past the threshold whitens 12.6 per cent of the width; two per cent whitens 17.8; five per cent, 27.8. The first hundredth of the radius does more than the next four hundredths put together.

The yarn dyed through is the dashed line along the bottom. It thins as it is worn and never shows a white core, because it has none. The whole shape, flat and then a cliff, is a property of the ring.

A smooth profile makes a valley with the same width #

The ring in the geometry above has a sharp edge: fully dyed outside the core’s radius, not dyed at all inside it. A real dip does not leave a sharp edge. Its profile falls smoothly with depth, from full strength at the surface to half at the ring’s nominal edge and on toward nothing, and the wear it uncovers should be expected to fade in rather than switch on.

It does fade in, a little. With the profile a twenty-second dip leaves at an affinity of ten, whose half-strength depth is eleven per cent of the radius, the middle of the cut face carries 59 per cent of the surface’s dye at a cut of nine per cent, half at eleven, 40 at fourteen, 27 at eighteen and under a tenth at twenty-eight. The white arrives as a valley rather than a step, and the valley deepens over about a tenth of the radius’s further wear.

But the valley’s width at half strength is exactly the sharp ring’s chord. A profile that falls steadily inward is at half strength on one circle and only one, and the cut crosses that circle where the sharp ring’s core would have been. So the square-root law survives the smooth profile as the law for the width of the half-strength patch, and what the smoothness adds is a soft rim to the white rather than a different growth.

The fibres the cut face shows #

The geometry above treats the yarn as a solid circle. Drawn with its fibres, the same cut reads more directly.

Every fibre the cut face presents is one the dye never reached. The dyed fibres are a band one or two fibres wide round the outside; the cut has taken the top of that band away and exposed a row of fibres from the interior, pale, with a blue fibre or two at each end of the row where the cut runs out through the ring. The yarn has lost twelve per cent of its fibre by this point. From above, the middle of it is white.

The picture also makes plain why a ring-dyed yarn does not pale evenly as it wears. An evenly dyed yarn loses its colour as its fibres are cut and roughened: the surface becomes hairy and scatters more light, and the shade lightens — abrasion takes the hairs first, and the hairs are what a pale bloom on a worn cloth is. A ring-dyed yarn does that too, over the ring, and then, at the threshold, starts showing fibres of a different colour altogether.

How much yarn a ring can hide #

The threshold is a depth, and depth is not what matters to the yarn. What matters is how much of the yarn has gone.

Read against the mass, the threshold becomes the circular segment of depth $\delta$. A ring a tenth of the radius deep first shows white after 1.9 per cent of the yarn has been worn away. A ring of a fifth, after 5.2 per cent. A ring of a third, after 11.0. The rest of each curve rises just as steeply as before, because just past the threshold the mass taken grows in proportion to the extra depth while the white grows as its square root.

The two milestones spread apart as the ring deepens. A ring of a tenth goes from first white to half white between 1.9 and 7.3 per cent of the yarn; a ring of a fifth between 5.2 and 13.0; a ring of a third between 11.0 and 22.9. A thin ring gives the alarm early and completes it quickly; a thick one waits and then takes longer. A ring deeper than half the radius never shows half its width white, since its core is narrower than half the yarn; such a yarn is worn in half before it shows its middle.

The denim a dip makes, cut #

For the seventy-four tex denim warp of the dyeing essay, dipped twenty seconds at an affinity of ten, the ring is 11.3 per cent of the radius. Its first white appears when 2.2 per cent of the yarn has been worn away, and half its width is white by eight per cent.

That puts a number on what a denim wearer sees, on a face that balance and the unbalanced cloth found to be three quarters warp and so three quarters ring. A pair of jeans does not lose a twentieth of its warp at the knees in the first months of wear, but the crowns of the floats at a crease, which floats and abrasion identified as the points that are rubbed, lose a few per cent of their section quickly. Those crowns cross the threshold while the rest of the cloth is nowhere near it, and they cross it steeply. The fade appears at the places of highest wear with an edge, and the rest of the cloth stays blue — the two-colour reading colour and weave as two colours gave the twill line, now written by wear instead of by the draft.

The contrast of that edge comes from the square root. Two neighbouring crowns that have been worn by slightly different amounts, one just short of the ring’s depth and one just past it, differ in white share not slightly but by an eighth or more of their width. A small, continuous difference in wear is turned into a large, abrupt difference in colour, and that is what makes a denim’s fading patterns look drawn rather than worn: whiskers at the hip, a honeycomb behind the knee, a frame round a pocket.

The alarm is set by the dyer #

The threshold is the ring’s depth and the dyer chooses the ring’s depth. A dip dyes a depth found three handles on it: the dip’s length, which deepens the ring as its square root; the affinity, which thins it as its square root and is set by the bath’s alkalinity; and the yarn’s packing, which a harder twist raises.

A thin ring is a denim that fades early and hard. A dyer who wants jeans that fade quickly into high-contrast patterns rings the warp thinly: short dips, many of them to build the shade, at the bath chemistry that gives the dye the highest affinity. A dyer who wants a denim that keeps its colour dyes deeper, or dyes through. The first is a fashion product and the second a workwear one, and the arithmetic of the choice is this essay’s two curves.

That is also why eight short dips are preferred to one long one. The dyeing essay found that more dips deepen the shade without moving the ring. So a denim can be dyed very dark and still fade at a thin threshold, the combination that a single long dip, which would have deepened the ring as it darkened the shade, could not give.

Wear shows up in the colour before the strength goes #

The surprising connection is with a result about strength. A cloth loses its strength before its mass found that abrasion takes material from the same place on every thread it rubs, and a thread breaks at its thinnest place, so a few per cent of the mass lost from the crowns can be a much larger share of the strength. Worn cloth fails long before it looks thin.

A ring-dyed yarn looks worn at about the moment it starts losing strength. Its white appears after a few per cent of the yarn has gone from exactly the places where strength is lost: the crowns, the thinnest sections of every rubbed thread. It works like the wear bars moulded into a tyre’s tread, which sit at a chosen depth and show when the tread reaches it. The ring is an indicator at a chosen depth of a yarn, set by the dyer rather than the tyre-maker, and it reads out on precisely the threads that are weakening.

An evenly dyed cloth has no such indicator. It pales as it is abraded, gradually, and its colour says little about where it has lost strength. A ring-dyed denim says so plainly, in a way the wearer can see, which is presumably part of why its fading is valued rather than hidden.

What the model assumes #

The yarn is a circle and the abrader a plane. A yarn in a woven cloth is flattened at its crossings, as a flattened yarn’s cover computed; its section is closer to a racetrack, and a racetrack’s flat top would expose the core along a wider front at once. That would sharpen the threshold rather than soften it. A real abrader is not a plane either: a stone in a washing machine or the inside of a trouser leg rubbing a knee has its own curvature, which would cut a curved face and whiten a narrower strip at first.

The ring is concentric. Diffusion from a uniform bath leaves the ring the same depth all round; a yarn dyed in a rope, bunched with others, may be dyed less deeply where it touched its neighbours, and a yarn whose ring is thinner on one side whitens earlier on that side.

The white is the undyed core, not a colour. How white the exposed fibre looks against the blue beside it, and at what share of white a patch reads as faded to an eye, is a question about scattering and vision that the arithmetic leaves alone. The curves say how much of the width shows fibres that carry no dye; they do not say what shade the yarn becomes.

The cut is required to show no white at any wear short of the ring’s depth for rings of a tenth, a fifth and seven twentieths; the white just past the threshold is required to approach $\sqrt{2(1-\delta)e}$; a yarn dyed through is required to show none at any wear; and a ring of a fifth is required to reach its core after 5.2 per cent of the yarn, to four figures.

What the pictures cannot show #

Fibre migration puts dyed fibre in the core and undyed fibre in the ring, fibre by fibre. The yarn was dyed after spinning, so the ring is geometric: every fibre is blue over the stretches where it lay near the surface when it went into the bath, and pale over the stretches where it had dived toward the axis. That leaves the ring’s geometry exactly as drawn. What it changes is what happens once a cut fibre is loose: a straight fibre cannot share the load found migrating fibres anchored along their whole length, and a cut one pulls partly out of the yarn as it is flexed, dragging blue stretches out over white ones. The model has no fibre ends and so no hairiness, and the hairiness a worn yarn grows is part of how a real fade looks.

And none of this is a cloth. A yarn in a twill is not uniformly exposed: the float’s crown is a line along its length, a crossing is a point, and where the abrader first reaches depth $\delta$ on a woven face is decided by the weave’s height field. That is where the threshold goes next.

Who worked out which part #

The geometry is a circle cut by a line, and nothing in it needs a citation. The ring is from diffusion, set out in the dyeing essay from Crank’s cylinder. That indigo denim fades because its warp is ring-dyed and its core is pale is the trade’s own understanding, stated in every account of denim; what the trade has not usually written down is the shape of the threshold and its steepness, which is what makes the fading patterns sharp.

What is added here is the square-root onset and its consequence for contrast; the threshold read as a share of the yarn worn away, which turns a ring’s depth into a quantity of loss; and the observation that a ring-dyed yarn is a wear indicator set by the dyer, reading out on the threads that are losing strength.

Still open: where on a twill the threshold is first crossed #

A denim’s face is a height field: warp floats three picks long standing above the weft, crowns running along each float, points where the float turns under. A flat abrader pressed on it reaches the floats’ crown lines first and the crossings later, and a cloth loses its strength before its mass already computes the bearing curve that says how much of the face is within any depth of the top.

Shifting that bearing curve by the ring’s depth would give the share of a denim’s face that is white against the depth an abrader has taken off the cloth, weave by weave. A three-one twill and a two-one twill of the same yarn should whiten differently, and by an amount that depends on how their float crowns stand above their crossings. That is one subtraction applied to a curve the surface essays already draw, and the one thing it needs that this essay does not supply is a denim construction the plain-weave geometry will accept at the setts a denim is actually woven to.