Turn the cloth and the shine changes hands

Shot silk — changeant, chatoyant, iridescent taffeta — is a cloth with one colour in the warp and another in the weft that appears to be one colour from one angle and the other from another. It is usually explained by saying that the eye sees more of one system at one angle, which is true, vague, and does not say why the transition is so sharp or why the effect needs no particular fibre.

The claim #

A thread can only mirror light arriving in the plane across itself, because that is the only plane its normals lie in — so the two systems of a woven cloth reflect a quarter turn apart, exactly, whatever the tolerance.

Two consequences.

The effect needs no dye that changes and no interference. It is available in any cloth whose two systems differ in anything a reflection can carry — colour, lustre, fibre — and it is entirely geometric.

And the separation is exact rather than approximate. The two maxima are ninety degrees apart at every tolerance the arithmetic will admit, and the assertion is written against that rather than against any particular area.

The argument, in one sentence #

Take a straight thread lying along the y axis. Its surface is a cylinder, so its normal at any point is (sin φ, 0, cos φ) — the y component is exactly zero.

At the standard gloss geometry a patch reflects into the eye when its normal is within the tolerance of the half-vector, and the half-vector points in the direction the plane of incidence bisects. A normal with no y component can only be near a half-vector with no y component, which means the plane of incidence must contain the x axis, which means the light must arrive from across the thread.

That is all of it. The warp’s crowns lie along the warp, so they need light from across the warp; the weft’s lie along the weft and need light from across it; and the two directions are a quarter turn apart because the two systems are.

What was counted, and how #

The half-vector is tilted off the vertical by a stated angle and swept round the azimuth in twenty-five steps, and at each step the fraction of the plan within the tolerance of it is counted — separately for the two systems, using the same sampled height field throughout.

The tilt has to exceed the tolerance or the check passes for the wrong reason. At zero tilt the half-vector is straight up and has no azimuth at all; at a tilt smaller than the tolerance, the cone contains the vertical at every azimuth and both systems reflect everywhere. So the sweep is run at a tilt three times the tolerance, and the assertion is checked at four tolerances from two degrees to six.

On a balanced two-and-two twill in sheeting the warp peaks at 0° and the weft at between 82.5° and 90°, depending on the sampling; the separation is between 82 and 90 degrees at every tolerance. Each maximum is a real area of the order of a per cent of the plan rather than a sampling accident, which is asserted separately — because a check that a maximum exists somewhere is satisfied by noise.

The absolute areas move with the sampling and the ratios do not, so the ratios are what is quoted below. A grid at forty samples per thread spacing resolves a strip a few micrometres wide only approximately, and every figure in this essay says what grid it used.

The one number that had to be checked #

An assertion of the form “the two maxima are a quarter turn apart” is exactly the shape this collection has got wrong five times: an assertion written against the numbers in front of it rather than against the claim. So it was written the other way round, and the way it was written matters.

It asserts the separation, not the areas. The separation must lie between sixty and a hundred and twenty degrees at every tolerance, which is a statement about the geometry; the areas at the two peaks may be anything at all, and on an unbalanced cloth they are very different.

It additionally asserts that each system reflects a real area somewhere in the turn. Without that clause, a warp-faced twill whose weft contributes nothing but noise passes — the sweep finds a maximum, because a maximum of noise is still a maximum, and it lands wherever the sampling happens to be luckiest. That is how the check failed the first time it was written, on a three-and-one twill whose weft has almost no visible plateau, and the fix was to run it on a balanced cloth and to require both peaks to exceed a tenth of a per cent.

And the tilt is tied to the tolerance. A tolerance wider than the tilt admits every azimuth at once and the separation collapses — correctly, and it is the one way the check could have passed for a reason that had nothing to do with the claim.

Why the trough is so deep #

The sharpness is the part the usual explanation cannot supply, and it comes from the same place as the sharpness of the lobe.

A warp crown’s along-thread tilt is zero over a plateau — exactly zero, not small. So as the plane of incidence rotates away from across the warp, the half-vector acquires a component the warp’s normals cannot match at all, and the warp’s contribution does not decline gradually: it goes to whatever the transitions provide and stays there.

The transitions are what fills the trough, and they fill it very little. A turning thread’s normals do have an along-thread component, so a crown near its turn can mirror light from an oblique azimuth — but the arc within the tolerance is a few micrometres long against a plateau of hundreds, so the residual is a per cent or two of the peak.

The numbers say how much. Sweeping one system’s specular area round the azimuth and taking its peak against its trough gives a ratio of 2 for a plain weave, 25 for a two-and-two twill and 97 for an eight-end satin. That is why shot silk works and why it works best on a floated cloth: the plain weave, which is all transition and no plateau, barely modulates at all, while a long float switches almost completely.

And a point crown is azimuth-selective too, which is the part that surprised the arithmetic. A crown with no plateau still sits on a cylinder, so its transverse normals are confined to the plane across its own thread even though its along-thread normals are not. That is why the plain weave’s ratio is two rather than one: the selectivity is a property of a thread being a thread, and the float decides only how strong it is.

The modulation against the float, and what it is a figure of merit for #

The three ratios above — 2 for plain weave, 25 for a two-and-two twill, 97 for an eight-end satin — sit on a pattern worth stating, because the pattern decides the construction.

Divide each by its own system’s float length. The twill’s float is two threads and gives 12.5 per thread; the satin’s is seven and gives 13.9. Above a float of two the peak-to-trough ratio is very nearly proportional to the float, at something like thirteen or fourteen per thread of it, which is what a plateau against a fixed transition length would give. Two points is thin evidence for a proportionality and it is offered as a reading of the sweep rather than as a law.

Plain weave is nowhere near that line: proportionality would put it at thirteen and it returns two. It has no plateau at all, so its ratio is the bare cylinder selectivity — the floor every cloth inherits from a thread being round, which the previous section identifies and which is exactly what is left when the float is taken away.

That gives the design rule its shape. A shot cloth must switch both ways, so what matters is not the larger of the two systems’ ratios but the smaller — and the smaller is what a balanced construction maximises. An eight-end satin gives one system a float of seven and the other a float of one: 97 against 2, and the cloth turns from a bright warp to nothing rather than from one colour to the other. A two-and-two twill gives both systems two, so the figure of merit is 25 rather than 2.

Under any float limit the best shot cloth puts the whole allowance into both systems equally, which is the opposite of what a lustre specification asks for, and is why the constructions on the two ladders diverge at this point rather than agreeing.

What a shot cloth actually needs #

The arithmetic gives a specification, and it is not the one usually given.

Both systems must float. A system with no plateau contributes nothing to its own peak, so a cloth with all its float in one system — a warp-faced satin, say — shines from the warp at one azimuth and from nothing at the other. Shot cloths are woven plain or in balanced twills for exactly this reason, and taffeta, the classical shot construction, is a plain weave with a heavy rib.

The two systems must differ in something. Colour is the usual choice and is not the only one, and a colour order is a partition of the warp: two systems of the same colour but different lustre give a cloth that changes in brightness rather than in hue, which is what a cloqué or a two-lustre satin does.

And the cloth must be smooth enough to keep its plateaux flat. A raised or brushed cloth has fibre on top of it pointing in every direction, and the effect disappears entirely — which is why no shot fabric is ever napped.

Where this sits relative to the other angle effects #

This collection has two other essays about a cloth whose appearance depends on how it is looked at, and all three have different mechanisms. Setting them beside each other is the useful part.

Watered silk is a beat between two grids. Fold a ribbed cloth on itself, press, and a figure appears at a scale neither ply has — the difference of two wavevectors, enormous because the angle is tiny. It depends on the cloth having been folded and pressed and it survives being turned.

Colour and weave is a pattern in the plane. Thread a colour order into a draft and a figure appears that is nowhere in the interlacement, at the scale of the repeat, and it is there from every angle.

And the shot effect is a change in which system is reflecting. It is at no scale at all — the cloth is uniform — and it exists only in the specular direction. Take a shot cloth out of the specular and it is the average of its two colours from every angle at once.

The three are separable by a test. Rotate the cloth in its own plane: the beat moves, the colour-and-weave pattern rotates with the cloth, and the shot effect switches. Move the light out of the mirror direction: the beat and the pattern stay, and the shot effect vanishes.

What it does to a cloth of one colour #

The effect is described as a colour effect because that is the fabric it is sold in, and it is present in every woven cloth whether or not the two systems are differently dyed.

A plain white sheeting turned under a lamp changes in brightness by the same ratio as a shot cloth changes in hue, and by the same mechanism: at one azimuth the warp’s crowns are reflecting and at the other the weft’s, and if the two systems have different counts, different setts or different twists then they are returning different amounts.

That is the origin of the directional sheen every fabric has, and it is why a length of cloth laid out with half of it turned end for end shows a visible join — a defect of the same family as a fault that shows because it is periodic. The join is not a fault in the cloth and not a difference between two pieces; it is one piece with its two halves presenting their crowns to the light in two different orientations.

The trade calls that shading and treats it as a defect of laying-up. The arithmetic says it is unavoidable in any cloth with an anisotropic surface, which is every woven cloth, and that its size is the ratio between the two systems’ specular areas — computable, before the cloth is cut, from the draft and the two counts.

Where the model stops #

There is no colour anywhere in this. What is computed is which system’s crowns are oriented to return a source into an eye. What that does to the appearance of a cloth of two colours needs an eye, a light and a colorimetry, and none of them belongs here.

No shadowing, again. At the grazing angles a shot cloth is actually admired at, a crown of one system hides part of the other, and the balance between the two moves with the elevation as well as the azimuth. Nothing here computes it.

The sweep is at one tilt. A real viewing has a source of some elevation and an eye at another, and the half-vector’s tilt is set by their difference; the sweep here holds the tilt and turns the azimuth, which is one line through a two-dimensional space.

And the sampling is coarse for narrow tolerances. At one degree the reflecting strip is three micrometres wide and a grid at forty samples per thread spacing cannot resolve it, so the sweep is run at two degrees and above and the assertion says so.

Why a garment is cut with the grain #

The shading result has a consequence that every cutting room knows as a rule and nobody derives.

A marker is laid so that every piece of a garment runs the same way up the roll. The reason given is that the cloth has a nap or a pattern direction; on a plain undyed cloth with neither, the rule still holds, and cutters still enforce it.

The arithmetic says why. Two pieces of one cloth cut a half-turn apart present their two systems to the light in opposite orientations, so a sleeve set into a body the wrong way up reflects differently from it at every angle at which either is reflecting at all. The difference is the gap between the two peaks of the sweep, which on a balanced cloth is small and on an unbalanced one is a factor of two.

That also predicts which fabrics are worst. A cloth whose two systems have very different crown lines — a warp-faced twill, a sateen, anything one-sided — has the largest asymmetry, and is exactly the sort of cloth on which a reversed panel is most obvious. A balanced plain weave is the most forgiving, and is the cloth on which the rule is least strictly kept.

The generalisation #

A surface made of parallel cylinders is a one-dimensional mirror, and a surface made of two families of cylinders at right angles is two of them, superposed and independent.

That is the transferable statement. Anything with a grain — brushed metal, a corduroy, a bundle of fibres, a bird’s feather — reflects into a fan perpendicular to the grain and can only be lit into the eye from across it. Two grains at right angles give two such fans, and rotating the object switches between them at ninety degrees with a deep trough in between.

The corollary is that any anisotropic gloss carries the direction of the structure that made it. A reflection is a measurement of a normal, a normal is perpendicular to a surface, and a surface made of lines has normals confined to a plane. Nothing about the material enters at any point.

Who found it, and when #

Shot silk is at least as old as silk weaving and its trade explanations are ancient and various. That it is a geometric effect rather than an optical one is not a new claim — every account that mentions the warp and weft being seen at different angles is saying so — but the accounts stop there.

What is added here is the exactness: that a straight thread’s normals have no component along the thread, so the separation is a quarter turn rather than approximately one, and the depth of the trough is the ratio of an arc to a plateau. Both are consequences of the same height field that decides the cloth’s contact, and neither needed a new model.

Where the ladder goes next #

Out of the weave and into the finish, where the surface is destroyed rather than exploited: raising moves the surface onto the hairs, and a napped cloth has no crowns, no plateaux and no anisotropy at all.

Sideways, the same two systems seen not at two angles but at two positions is what makes a figure visible in a single colour — and it turns out that a figure shows by its shine rather than by its step, which corrects a claim this collection had carried a long time.