A hair layer veils a highlight

Turn the cloth and the shine changes hands is the most striking figure in this site’s shine ladder. A straight thread’s normals lie in the plane across it and have no component along it, so a warp float can only mirror light arriving from across the warp; turn the cloth a quarter and the weft takes over. Sweeping one system’s specular area round the azimuth gives a peak-to-trough of two for a plain weave, twenty-five for a twill and a large factor for an eight-end satin.

That is shot silk, arrived at with no dye that changes and no interference in it, from a binary matrix and a section. And it has a condition attached that the essay could not see: it needs a fibre with no ends.

The cloth #

Shot fabrics — taffeta, shot silk, two-tone linings — are made of filament. That is a fact of the trade so consistent that nobody records it as a fact; it is simply what those fabrics are. The explanations offered are about lustre and about fineness, and both are reasonable and neither predicts that a spun yarn cannot do it.

The hair layer predicts exactly that, and the arithmetic has one moving part.

The claim #

A hair layer does two things to a directional reflection and only one of them matters. It blocks, which takes the same factor off the peak and the trough and changes no contrast at all. And it returns light of its own, which is added to both — and because a hair population points every way at once, its return has no azimuth in it. Adding a constant to both ends of a ratio of a hundred destroys the ratio.

The consequence is a prediction the trade has been obeying for centuries without stating: a shot effect needs a fibre with no staple length, and this site’s own table of fibres says which those are.

Blocking is not the mechanism, which is the whole point #

It is tempting to say that the hairs get in the way, and that is the wrong explanation in an instructive manner.

A specular ray goes down through the layer and comes back, so it is attenuated by the square of the layer’s transmission. On an ordinary spun cotton that factor is about 0.85; on a raised cloth it is a few thousandths. Either way it multiplies the peak and the trough by the same number, and a ratio is unchanged by multiplying both of its terms.

So blocking makes a cloth duller and does not make it less shot. A veil in front of a mirror dims the reflection and does not spoil the geometry; only something that adds light spoils the geometry.

What the hairs add, and why it has no direction #

A hair is a cylinder. This site has already computed what a cylinder does to a beam: a round thread reflects into a fan, and the strip of its projected width whose normal lies within a tolerance of the mirror direction is sinε of it. That construction is the shine ladder’s own and it is applied here to a much smaller cylinder.

So the hairs’ specular return is the fraction of the plan they block, times sinε — a small number, of order three parts in a thousand at a four-degree tolerance on an ordinary cloth.

And it has no azimuth. A thread in a cloth runs one of two ways; a hair points wherever it happened to escape. Averaged over a population there is no preferred direction, so the hairs’ return is the same whichever way the cloth is turned.

Adding a constant to a ratio #

Now put the two together. The measured contrast is

(peak · through + hair) / (trough · through + hair)

The trough of an eight-end satin is very nearly zero — that is what makes the effect striking — so the hairs’ small addition is large compared with the trough and negligible compared with the peak. The denominator moves by a lot and the numerator by almost nothing.

On an ordinary spun cotton the contrast falls from what the bare surface gives to about five to one. Singeing, which truncates the population and removes nearly all of its length, recovers most of the way back. Raising, which multiplies it, takes the contrast to one — the cloth looks identical from every direction, which is what a flannel does.

That is the whole mechanism and it is three lines of arithmetic on top of that sweep. What it needs from this ladder is only the coverage, which is a population’s total length times a fibre’s diameter.

Why the fibre with no ends is the one every shot fabric is made of #

This site’s STAPLE table was written for an argument about twist. It records that a filament has no staple length at all, and it lists which fibres those are: silk, polyester, nylon, glass, carbon, aramid. A filament yarn has two fibre ends in the whole package.

So a filament yarn has no hair population, and therefore no isotropic return, and therefore the full bare contrast.

Every shot fabric in the historical record is silk, and every modern one is polyester or nylon. Nobody has ever made a shot cotton, and the reason is not that cotton is dull — a mercerised cotton satin is very lustrous, and this site’s own arithmetic says the lustre is a crown line times a section’s width and has nothing to do with the fibre. A shot cotton fails not on brightness but on contrast, and the failure is a fibre-end count.

That is a prediction the model makes and the trade has been obeying without stating, which is the most satisfying shape a result in this collection can have.

Why singeing is on every lustre finishing route #

Singeing listed five things that move when a cloth loses under one per cent of its mass, and lustre was one of them. The arithmetic is now available.

The gain in brightness is real and modest: the transmission factor goes from about 0.85 to nearly 1, so a singed cloth returns about a sixth more specular light. The gain in contrast is much larger, because it is the isotropic term that has been removed rather than the blocking one.

So singeing does two things to shine and the larger one is the one nobody quantifies. A finisher describing a singed cloth as “brighter” is describing the smaller effect; what has actually happened is that the cloth’s shine has become directional, which is what the eye reads as lustre in the first place. A surface that returns the same amount of light in all directions looks matte however much light it returns.

The exchange a designer is actually making #

Put the two lustre levers side by side and the account is unexpectedly clean.

The draft buys length. Lustre is a crown line times a section’s width, and the draft owns the length: an eight-end satin carries far more horizontal crown line per square millimetre than a plain weave, and the whole four-by-four catalogue spans a large factor in it.

The calender buys width. Pressing the threads flat replaces a fan of normals with a plane of them, and the shine ladder showed that the gain lands entirely in the width factor and none of it in the length.

And the flame buys direction. Singeing changes neither factor — it does not touch the crowns and it does not flatten a section — and it removes the undirectional term standing in front of both.

Three operations, three different parts of one expression, and only the third of them is about the fibre rather than the cloth. A finishing route that wants maximum lustre does all three, in that order, and the order is not arbitrary: singeing before calendering, because a calender presses the hairs into the surface where a flame can no longer reach them.

What was counted, and how #

The sweep is re-run rather than quoted: the azimuthal peak and trough come from this site’s own azimuthSweep on an eight-end satin in sheeting at the essay’s stated tolerance, so a change to the shine machinery would move this essay’s numbers rather than leaving them stranded.

Four assertions. That the bare contrast is large, so there is something to destroy. That hairs reduce it, and reduce it more the more there are. That the ordering runs filament, singed, spun, raised — four states in a strict sequence, which is a stronger statement than any pairwise comparison. And that a raised cloth’s contrast is under one and a half to one, which is the claim that the effect is gone rather than merely reduced.

Nothing in this essay is fitted to a measurement of a shot fabric. The coverage comes from the hair population, the sinε from the shine ladder’s own cylinder construction, and the peak and trough from a census.

The one shot fabric that is not filament, and why it works #

There is an apparent counter-example and it is worth chasing because it confirms the mechanism rather than denting it.

Worsted mohair and some hard-spun worsteds do show a directional two-tone effect, and they are staple yarns. What they have in common is a very long staple, a very hard twist and a severe finishing route — cropping, singeing and pressing — and every one of those acts on the same term.

A long staple means fewer ends. The population’s density is 2n/L times the shell share, so doubling the staple halves the ends per millimetre before anything else happens. Mohair at a hundred and fifty millimetres against cotton at twenty-eight is a factor of five.

A hard twist lowers the escape fraction, because the binding propagates further back up the spinning triangle.

And cropping and singeing truncate what is left. Cropping shears the surface mechanically, which is a truncation at a greater height than a flame’s, and the two together leave very little reach.

So the apparent counter-example is a staple yarn that has been driven as far toward the filament limit as a staple yarn can go, by three independent routes, and it recovers part of the contrast. The mechanism predicts a partial effect and that is what such cloths show — a shot mohair is not a shot silk, and nobody who has seen both would confuse them.

The colour version of the same floor #

Everything above is about a specular contrast, and the commonest shot fabric in the world does not work that way. Shot taffeta is a plain weave, its two-tone effect comes from a warp and a weft of different colours, and a plain weave’s specular contrast is two to one — nothing like enough to be the mechanism.

The same additive floor destroys it, and following that through says why taffeta is filament too.

The colour effect is a ratio of areas seen. Look along the warp and the crowns of the warp threads face the eye; the weft is foreshortened and partly hidden behind them, so the warp’s colour dominates. Turn the cloth a quarter and the weft dominates. The two views differ in the fraction of the visible surface belonging to each system, and that fraction swings with the azimuth for the same reason the specular area does — a thread’s crown presents itself across its own direction and not along it.

Now add a hair layer. Hairs escape from both systems, so what they return is a mixture of the two colours, in whatever proportion the two yarns’ hairiness happens to give — and, as above, with no azimuth in it whatever. It is a constant of the mixed colour added to both views.

So the arithmetic is the arithmetic of this essay with a colour where the specular fraction was, and the conclusion is the same: a multiplicative veil would dim the cloth and leave the two-tone alone; an additive mixture washes it out. A shot taffeta in spun cotton would read as a single muddy colour that barely changes with the angle, which is a fair description of what a two-colour spun poplin looks like.

Two things follow that are worth stating separately from the specular case.

The colour effect does not need a float. It needs each system to dominate one view, which a plain weave achieves through the crown geometry alone — so taffeta can be the most tightly interlaced weave there is and still be shot, where the specular effect needs the long floats of a satin. That is why the two shot constructions in the trade are so different from each other: a shot taffeta is plain-woven and a shot satin is not, and they are exploiting two different contrasts that a hair layer spoils by the same route.

And it explains the one thing plain weave is best at here. The specular contrast of a plain weave is only two to one, so a hair layer barely touches it — an additive floor destroys a ratio of a hundred and leaves a ratio of two nearly intact. The weaves with the most striking effect are the ones most easily spoiled, exactly because the trough is what the hairs are being compared against. A satin’s shot effect is spectacular and fragile; a taffeta’s colour effect is modest and, in filament, robust.

That inverse relation is the sharpest form of the whole mechanism. Bare contrast and tolerance of a hair layer are reciprocal, and a designer choosing between the two shot constructions is choosing where on that reciprocal to stand.

Where the model stops #

There is no shadowing and no interreflection, which the shine ladder already recorded: every specular fraction is exact at normal incidence and an upper bound at a grazing one. A shot cloth is admired at a grazing angle, so the bare contrast here is an upper bound and the veiled one is less affected — which narrows the gap the essay is about.

The hairs’ return is treated as a specular fan and nothing else. A real fibre also scatters diffusely from its interior, and for a dyed fibre that return is coloured. Both would add to the isotropic term and strengthen the conclusion.

Albedo is absent. How much light a fibre actually returns depends on its refractive index, its surface and its dye, none of which is in this site’s subject or its machinery. What is computed is the geometric fraction of the hair population lying within the tolerance, and every contrast here is a ratio of geometric fractions.

And the tolerance is a parameter. The hairs’ return goes as sinε and the cloth’s specular area goes as ε in one factor and a constant in the other, so the ratio between them moves with the tolerance. The orderings survive its whole range; the numbers do not.

The generalisation #

A multiplicative disturbance leaves a ratio alone and an additive one destroys it.

The transferable shape is worth carrying beyond cloth. Any measurement whose value is a contrast — a signal against a background, a peak against a trough, a figure against a ground — is immune to anything that attenuates both and is at the mercy of anything that adds a constant to both. Attenuation is usually the thing people worry about, because it is visible; a small additive floor is usually invisible and is what actually sets the limit.

This collection has met the same shape once before and did not name it. A damask’s figure and its ground differ by a contrast that reverses a quarter turn apart, and a fifty-micrometre step in a surface with hundreds of micrometres of relief returns no light at all under diffuse illumination — which is an additive floor swamping a signal, exactly as here.

Who found it, and when #

Shot fabrics are ancient and their restriction to filament is universal practice. The optical explanation usually given is that filaments are smooth and parallel, which is true and is about brightness rather than about direction.

The azimuthal sweep is this site’s own. What is added here is that the effect is destroyed by an additive isotropic term rather than by attenuation, that the term is a hair population’s own specular return, and that the fibres with no such term are the ones the table already lists as having no staple length.

Where the ladder goes next #

The shine ladder has now been round the outside of the cloth twice — once for the crowns and once for what stands above them — and the second pass reverses part of the first. What the same layer does when something touches it rather than looks at it is a light touch never reaching the crowns; what it does when a drop lands on it is the hairs deciding the sign of the wetting.

And the same isotropic material, seen as ink rather than as light, is a printed edge feathering — where the hairs carry colour off the printed area into the unprinted one, and the reach rather than the coverage is what decides how far.