A float reflects into a line

The lustre of a fabric is usually filed under fibre, and silk gets the credit. This collection established early on that it is a property of the weave — a cylinder reflects a line rather than a point, so the highlight a cloth returns is exactly as long as its longest float. That essay made the argument with a drawing. The surface makes it with a computation, and the computation says something the drawing could not.

The claim #

A specular reflection is a statement about a surface normal and nothing else, the distribution of normals over a repeat is computable from the draft, and it is not round.

Three things follow.

A straight thread has no along-thread spread at all. Its normals lie in the plane across it. So it scatters into a fan in that plane, which an eye at a distance sees as a bright line running along the thread.

A turning thread does have along-thread spread, and the amount is the weave angle. Everything that dulls a cloth dulls it by putting tilt into the along direction, and everything that brightens a cloth takes tilt out.

And a plain weave’s lobe is round. With no float anywhere, every crown is a point, curved equally along and across, and there is no preferred direction. The anisotropy of the reflected lobe comes out at exactly one — which is the control that makes the satin’s 2.48 mean something.

The geometry, stated #

Reflection is taken at the standard gloss geometry: a source and a view at equal and opposite angles in one plane, so the half-vector is vertical and a patch of surface reflects into the eye exactly when its own normal points within some tolerance of straight up.

The tolerance is stated and never fitted. How near counts as reflecting is a choice, it changes every absolute number, and it changes no ordering — so every result here is quoted at a stated tolerance and checked across the whole range from half a degree to ten.

What is computed is an area fraction of a surface whose normal lies in a stated direction. There is no eye anywhere in it, no spectrum, no illuminant and no unit of brightness; the moment a question needs a receptor it has left this collection.

The plain weave, which is the control #

An enumeration is only worth having if something could have come out the other way, and the plain weave is what says this one could.

Its crowns are points: doubly curved, by the thread’s own radius across and by the crimp arc along, and the two radii are not equal but they are of the same order. So its normals sweep in both directions and the lobe it returns has no preferred axis.

The measured anisotropy of a plain weave’s lobe is 1.00. That is not a fitted agreement or a near-miss; it is what a surface with no straight portion anywhere has to give, and it comes out of the same sampler that gives the satin 2.48 without anything being changed between the two runs.

What was counted, and how #

The surface is the height field built from the draft and the Peirce solution, sampled on a grid over one repeat. At every sample the two slopes are known exactly — the transverse slope from the thread’s own section, the along slope from the centre line — so the normal is exact and no differencing is involved.

Each normal is then mirrored: for a beam arriving from straight above, the returned direction is 2(n·l)n − l, which for a vertical source is decided by the normal alone. The returned directions are binned by their two deflection angles and the marginals give the two spreads.

The along-thread spread has an analytic form and it is the weave angle. Over a plateau the along-slope is exactly zero; over a transition it runs from zero at the crown to the weave angle at the steepest part of the straight run and back. So a thread’s along-thread tilt never exceeds θ, and the whole lobe is contained within 2θ in that direction — about seventy-four degrees for an ordinary sheeting, which is a wide fan and is entirely produced by crimp.

The two spreads, and why one of them is the crimp #

Setting the two directions side by side makes the mechanism explicit.

Across the thread, the spread is the whole half-turn of a cylinder, limited only by whether a returned ray is going up or down. Nothing about the weave changes it, nothing about the sett changes it, and no fabric finish changes it except one that stops the section being a cylinder.

Along the thread, the spread is twice the weave angle and nothing else. Over a plateau the surface is horizontal and the along-slope is exactly zero; over a transition it rises to θ and comes back. A cloth whose threads never turn would have no along-thread spread at all and would be a mirror in that direction.

So the anisotropy of the lobe is a competition between a fixed quantity and a variable one, and the variable one is the fraction of the surface that is plateau. A satin, most of whose face is plateau, has most of its returned light in a narrow band. A plain weave, none of whose face is plateau, has all of its returned light spread by the crimp in both directions.

That is why the anisotropy is a clean discriminator and the total is not. Two cloths can return the same total specular area from quite different arrangements; only a cloth with straight portions can return it into a narrow band.

Where the highlight lies on the cloth #

The consequence a reader can see is where the bright part of a cloth actually is, and it is not distributed the way an intuition about “the shiny weave” suggests.

At two degrees of tolerance the highlight of that satin occupies 1.4 per cent of the plan and the highlight of a plain weave 0.06 per cent. The whole difference between a lustrous cloth and a matt one is a factor of twenty on an area of about one per cent, and ninety-nine per cent of both fabrics is doing something else with the light entirely.

That is worth stating plainly because it disposes of a common way of talking. A satin is not shiny all over. It is matt over almost all of itself with a set of very bright lines in it, and the eye integrates.

What this adds to the earlier account #

Why satin shines established the length of the highlight and got it right: the bright line is as long as the float. What it could not say is the other factor.

A highlight is a length times a width, and the two come from opposite ends of the fabric. The length is the crown line, which the draft supplies. The width is what the thread’s section offers inside the tolerance, which the yarn and the finish supply. Neither knows anything about the other, and they multiply.

That decomposition is the whole content of the next rung, and it is why a calendered plain weave and an as-woven satin are not two routes to the same place: one buys width and the other buys length.

A weave with no lustre at all, and why it does not exist #

The arithmetic invites an obvious question: what would a completely matt weave be?

It would be a cloth whose surface has no patch anywhere with a normal near the vertical — which is impossible, because a thread that comes to the top of the cloth has to be horizontal at the moment it is highest. Every crown, of every kind, on every fabric, has a point where its normal is exactly vertical.

So the minimum specular area is not zero. It is the area within the tolerance of that point, summed over all the crowns there are, and it is the plain weave’s 0.06 per cent. Nothing in the four-by-four catalogue is duller, and the reason is a matter of counting: a plain weave has the most crowns of any draft — every intersection is one — and each contributes a small ellipse of nearly-flat surface.

The consequence is a floor and it is not far below what an ordinary cloth returns. To get below it a fabric has to stop having a woven surface: it has to be raised, so that what is on top is fibre ends pointing in every direction, or it has to be made of a fibre that scatters rather than reflects. Both are what a matt fabric actually is, and neither is a weave.

Where the model stops #

There is no shadowing and no interreflection. At a grazing enough angle a crown hides the valley behind it, and a real cloth bounces light between its own threads several times before it escapes. Neither is computed, so every fraction here is exact at normal incidence and an upper bound at a grazing one.

The reflection is treated as specular from a smooth surface. A yarn’s surface is neither smooth nor a single interface: it is fibres, with air between them, and what leaves it is part surface reflection and part light that has entered, scattered among the fibres and come back out. Only the first is geometric. The second is a matter for a subject this collection does not do.

And the hairs are in the way here too. The outermost fibres of a spun yarn stand off it, and a hair lying across a crown scatters into every direction at once. A hairy cloth’s measured lustre is therefore below its geometric lustre by an amount that is a spinning property, which is one of the reasons singeing changes a fabric’s appearance out of all proportion to the 0.2 per cent of its mass it removes.

The tolerance is a stand-in for a real instrument. A gloss meter has an aperture and a source of finite size, and the two together define an acceptance that is not a simple cone. The tolerance here is a cone because a cone is the honest simplification, and every ordering is checked across the whole range of it.

What a gloss measurement is measuring #

A gloss meter shines a beam at a stated angle and collects what comes back at the mirror angle through a stated aperture — a measurement with an instrument in it, like every other one on this site. Read against the above, three things about the number it returns become predictable.

It depends on which way the specimen is turned, for any cloth with floats, and by a factor of about two at a tight aperture. Gloss standards for textiles say to measure in a stated direction relative to the warp for exactly this reason, and the reason is usually given as “because fabrics are anisotropic”, which is true and unhelpful.

It falls as the aperture is opened, in the sense that the contrast between weaves falls: a wide aperture collects the crimp-spread light along with the specular and the ordering compresses. A tight aperture is measuring the plateau and a wide one is measuring the whole surface.

And it is dominated by about one per cent of the specimen. A gloss reading on cloth is a measurement of the crown lines and of nothing else, which is why it is so sensitive to pressing, brushing, singeing and handling — all of which act on exactly that one per cent — and so insensitive to things that change the bulk of the fabric.

The contrast between weaves is the reciprocal of the aperture #

The essay notes that a tight aperture measures the plateau and a wide one measures the whole surface, and the difference has an exponent rather than being a matter of degree. It falls out of the same two-directions argument the lobe does.

A plateau’s specular area goes as the tolerance. Along the thread the surface is exactly horizontal, so every point of the plateau’s length is within tolerance however tight it is; across the thread, the strip whose normal lies within ε of vertical is sin ε of the width. One direction is unrestricted and the other scales, so the area goes as ε.

A crown’s specular area goes as its square. A crown is curved in both directions, so both extents scale, and the area goes as ε².

So the ratio between a long-floated weave and a plain one goes as 1/ε, without limit as the aperture closes.

The essay’s own numbers check it. At two degrees the eight-end satin returns 1.4 per cent of its plan and the plain weave 0.06 — a ratio of twenty-three. Tripling the tolerance to six degrees should triple the satin’s area and multiply the plain weave’s by nine, taking the ratio to about eight — and the satin’s highlight figure at six degrees is drawn at just that.

Three things follow, and all three are about instruments rather than fabrics.

A gloss meter’s aperture sets how much weave contrast it reports. Two laboratories with different acceptance angles will rank a set of fabrics in the same order and disagree about the spread by the ratio of their apertures. That is a systematic disagreement with a known form, and it is the kind that gets attributed to specimen preparation.

The tightest practical aperture is the most discriminating instrument, which is the opposite of the usual instinct about signal. A wide aperture collects more light and less information, because the extra light it collects is the crimp-spread return that every weave has in common.

And a measurement at a stated aperture can be converted to another. The scaling is ε for a floated weave and ε² for a crowned one, so a reading at one aperture predicts the reading at another once the draft is known — which turns two incomparable numbers into one, and needs nothing but the matrix.

The caution is that both scalings are small-angle statements. Past about ten degrees the sin ε stops being ε, the crown’s two curvatures stop being comparable, and shadowing begins to matter — so the reciprocal law holds over the range gloss meters actually use and not beyond it.

The generalisation #

The dimensionality of a highlight is the dimensionality of the flat part of the surface it comes from.

A point that is flat gives a point highlight. A line that is flat gives a line. A plane that is flat gives a plane, which is a mirror. That is the whole rule, it is why a scratched metal looks the way it does and why a brushed finish has a grain, and cloth is a case of it with the unusual feature that the flat parts are specified — a matrix decides where they are and how long.

The corollary worth carrying is that anisotropic gloss is evidence of anisotropic geometry, always, and never of anything about the material. A fabric that shines differently in two directions has straight portions running in one of them.

That last point is worth one more sentence, because it explains a piece of trade practice that looks like superstition. A fabric wanted matt is raised, brushed or given a fibre with a delustrant in it — never given a different weave. The weave lever exists and is worth a factor of twenty in the bright direction; in the dull direction it is worth almost nothing, because the floor is set by the crowns every draft must have. Lustre is a property a weave can add and cannot take away.

Who found it, and when #

The observation that a cylinder reflects into a fan is elementary optics and belongs to nobody in particular. Its application to fibres was made by everyone who ever looked at a skein of silk.

What this collection contributes is the computation: the distribution of normals over a repeat, derived from the same height field that decides the cloth’s contact, so that the lobe of a weave can be produced before the cloth is woven and the plain weave’s roundness is a result rather than an assumption.

Where the ladder goes next #

To the arithmetic: lustre is a length times a width makes the decomposition exact, censuses it over the whole four-by-four catalogue, and finds a factor of sixty between the brightest draft and the dullest.

Then to the two levers separately. A calender buys the width and multiplies a highlight twenty-fold without touching the draft; and turning the cloth changes which system shines, which is shot silk and needs no dye that changes.