No cloth shines under a sky

Every specular figure on this account is drawn at a tolerance of two degrees, and the number has been carried from the first essay without being examined. It is a stand-in for the source: a mirror returns a source’s image, so a patch of cloth counts as shining when its normal lies within the source’s own half-angle of the specular direction.

Two degrees is a lamp. The sun’s disc is a quarter of a degree. An overcast sky is the whole hemisphere.

Sweeping that one number turns out to say something the account has not said, and it is not a correction — it is a quantity every result so far has been holding fixed without noticing that it had been chosen.

Two different powers, and the reason is in a factorisation #

Lustre is a length times a width is this account’s second essay and it is where the argument comes from. The specular area factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply.

Both factors depend on the tolerance, and they depend on it differently.

The width is flat + b·sin ε. For a round section the flat run is nothing, so the width is proportional to the angle.

The length is the crown line plus one arc of D·ε at each transition. A float reflects into a line: a run of L picks over presents L − 1 thread spacings of plateau between its two turns, and a plateau’s normals are all vertical, so a plateau shines however narrow the source is. A transition does not — it sweeps through the weave angle, and only a strip of ε either side of vertical is within tolerance.

So the two weaves behave differently in the limit:

$$\text{a float: } \text{area} \propto \varepsilon \qquad\qquad \text{plain weave: } \text{area} \propto \varepsilon^2$$

because a plain weave has no run longer than one and therefore no crown line at all. Its whole specular area is arcs, and an arc shrinks with the source exactly as the width does.

That is why a plain weave’s line on the log frame has twice the slope of the others. It is not a material difference, a finish or a yarn; it is the difference between having a plateau and not.

Which makes the contrast inverse in the source #

Dividing one by the other gives the result the account was missing:

The contrast halves every time the source doubles, and the product of the two is constant to within a few per cent over the narrow half of that range.

It is not exactly constant, and saying how it fails is better than quoting a range of validity. Writing the contrast times the angle as a straight line,

$$\text{contrast} \times \varepsilon = A + B\,\varepsilon,$$

the intercept A = 44.23 is the float’s crown line divided by the plain weave’s transitions, and the slope B = 0.2500 is the ratio of the two weaves’ own transition counts — an eight-end satin turns twice in eight ends and a plain weave twice in two, which is exactly a quarter.

Both coefficients are counts off the draft, and neither has a yarn or a finish in it. The inverse law is in error by a tenth once the source is 17.7 degrees wide, which is a large window seen from close to.

And it dies at sixty degrees #

The closed form is a narrow-strip approximation and cannot be taken to the wide end, which is exactly where the interesting question is. Counting the reflecting elements directly off the surface field instead answers it.

At a source of sixty degrees’ half-width every weave reflects 0.6563 of its face, to four figures, and the four curves are identical beyond it. At forty-five the contrast is already 1.12 — within what a casual look would call the same.

The reason is straightforward once the geometry is in hand. At sixty degrees the tolerance takes in every normal a thread’s curved section presents up to the point where it is hidden by its neighbour, so the whole visible surface is inside the acceptance. A weave cannot differ from another weave in how much of its face reflects when all of every face reflects.

An overcast sky is a hemisphere, which is ninety degrees of half-width and well past the collapse. So the result is not an asymptotic curiosity:

A satin and a plain weave of the same yarn are indistinguishable in lustre out of doors on a grey day. Not nearly indistinguishable — identical, to the fourth figure of this model.

A plain weave is the only one with no plateau at all #

The mechanism rests on a claim it is worth checking rather than assuming, because everything above follows from it: that plain weave’s crown line is exactly zero.

A crown line is the plateau a float presents — a run of L picks over gives L − 1 thread spacings of flat between its two turns. Plain weave’s every run is of length one, so L − 1 is nought, every run is a turn, and the sum is zero rather than small.

That is not an approximation and it is not a property of this cloth or this yarn. It is the definition of plain weave, and it makes plain weave a singular point of the whole family rather than one end of a range: every other weave in the four-by-four catalogue has at least one run of two somewhere, so every other weave has a crown line and therefore a linear term, and plain weave alone has only the quadratic one.

The consequence is that the contrast against plain weave is the largest contrast a given source can produce. A 2/2 twill against a plain weave is 14.7 at two degrees and an eight-end satin is 22.4; the two floated weaves against each other are 1.5. Almost all of the lustre difference in a fabric range is the difference between having a float and not, and the choice among floats is a refinement of a tenth.

Which explains a set of practices that look like superstition #

Several habits of the trade fall out of one number, and none of them has ever been given this reason.

Silk is shown under a spotlight. A merchant’s shop and a fabric library light their samples with small sources, and the effect is a contrast of 44 rather than 3. The cloth is the same cloth; the room is doing the work.

A gloss meter’s reading is meaningless without its geometry, and gloss standards specify it exactly — the acceptance apertures are a few degrees, and a difference of a few degrees between two instruments is a difference of tens of per cent in the ratio they report. That is not an instrument tolerance; it is the quantity being measured changing.

And a shot silk is a fabric for indoors. Turn the cloth and the shine changes hands is this account’s fourth essay: a warp crown’s normals lie in the plane across the warp, so a warp float can only mirror light arriving from that direction, and turning the cloth hands the shine to the weft. That effect needs a source narrow enough to have a direction. Under a sky there is no direction and the cloth does not change colour when it is turned.

The same argument runs the other way for a nap. A hair layer veils a highlight found that fibre ends standing above the crowns return light with no direction in it, which flattens the swing as a cloth is turned. A wide source does the same thing from the other side — the cloth’s directionality is destroyed by the illumination rather than by the surface — and the two are indistinguishable to a viewer.

The finish moves one factor and the source moves both #

Setting this essay beside the third one puts the account’s two levers in their places, and they are not the same kind of lever.

A calender acts on the width alone. It flattens the section, the plateau on top of the thread widens, and the specular area multiplies by twenty-four with none of the gain in the length. Because the width factor is common to both weaves, a calender scales every cloth on the page by the same number and leaves the contrast untouched.

A weave acts on the length alone. The crown line spans a factor of sixty across the four-by-four catalogue, and that is the whole of what a draft contributes.

A source acts on both, and it acts on them with different powers. The width goes as the angle and the length goes as a constant plus the angle, which is why the source is the only one of the three that changes a ratio. That is a stronger kind of lever than either of the others, and it is the one nobody controls, because it belongs to the room rather than to the cloth.

So the account’s three levers rank cleanly by what they can do. A finish changes how much a cloth shines. A weave changes how much one cloth shines against another. A source decides whether the second question has an answer at all.

What a designer should take from it #

The practical form is a question about where a cloth will be seen, and it has a threshold rather than a preference.

Below a few degrees the float length is worth choosing. The contrast is 22 at two degrees, and moving from a five-end satin to an eight-end is worth 24 per cent more specular area — a visible difference, in a shop or under a lamp.

Above about thirty degrees it is not. The contrast is 1.7 and falling, and the difference between a satin and a twill is under a tenth. Every other property a float decides — abrasion, firmness, how closely the cloth can be set — is unaffected, so the argument is not that the float stops mattering; it is that the reason for choosing it stops being lustre.

And a finish cannot rescue it. A calender buys the width by a factor of twenty-four, and the width is one of the two factors — so a calender multiplies both weaves’ areas by the same thing and leaves the contrast exactly where it was. Pressing a cloth makes it shinier and does not make it more distinguishable from the cloth beside it, at any source size.

The one place the collapse is a design tool #

A result that says a distinction disappears reads as a loss, and there is one construction in this account for which it is the mechanism.

A figure shows by its shine rather than by its step is the account’s own finding about damask: a single-colour figured cloth is read because its figure and its ground reflect differently, not because one stands above the other. Everything on this page applies to that reading, and it applies to both halves of it at once.

A damask under a narrow source is at its most legible, because the figure and the ground are two weaves and their contrast is inverse in the source. Under a sky it is invisible — not faint, but gone, by the same sixty-degree collapse.

That is a much stronger statement than the usual one about damask needing good light, and it is testable in an afternoon: photograph a white damask napkin under a bare bulb and under an overcast sky, and the figure should be plain in the first and absent in the second. Nothing about the cloth changes between the two photographs.

And it says what a damask designer is really choosing. A shading’s tone steps are exact and its lustre steps are not: a graded damask varies both the fraction of warp on the face and the float length, and the first is a tone that survives any illumination while the second is a lustre that does not. So a shaded damask read under a wide source keeps its tones and loses its lustre grading entirely — which is half of the design, and the half that was the harder to compute.

What was counted, and how #

Two methods, each used where it is valid, and both run over the band where they overlap. closed form is this account’s own factorisation and is a narrow-strip form: it treats the tolerance as a strip either side of the specular direction, which is right below a few degrees and wrong at thirty. sampled count counts the field element by element and is right at any angle and useless below four degrees, because a sampled field has no element within a quarter of a degree of anything.

The crossover between them is where the argument would be weakest, so the sweep is deliberately run past the closed form’s range rather than stopped at it, and the drift is required to be monotone — which is the shape the arcs’ growing contribution must produce.

The law is required below four degrees to within three per cent and the departure above it is required to rise, rather than a single tolerance being chosen to make both true.

The two coefficients are required to be positive rather than fitted and quoted: the intercept is a crown line over a transition count and the slope is a ratio of transition counts, and both are structural.

And the mechanism is required directly: plain weave is required to have a crown line of exactly zero, and every floated weave a crown line greater than zero. That is the sentence the whole argument rests on, and it is checked rather than reasoned about.

What this cannot say #

The source is treated as a disc of uniform brightness. A real sky is brighter near the horizon and a real window is a rectangle, so the effective half-width is a shape rather than an angle, and a rectangular source gives an anisotropic acceptance that would interact with the lobe’s own anisotropy. The collapse at sixty degrees would survive that; the numbers in the middle of the range would move.

Nothing here is about brightness. A narrow source is also a bright source per unit solid angle, so a satin under the sun is both more contrasty and more dazzling than one under a sky, and the eye’s response to the second is not linear. Everything above is a ratio of areas, which is the right quantity for a comparison between cloths and the wrong one for a prediction about what is seen.

And the diffuse return is not modelled at all. Every fibre scatters as well as reflects, and the diffuse part has no direction in it — so the contrast a viewer sees is the specular ratio diluted by whatever the diffuse background contributes, which is the mechanism the hair layer was found to work through. The collapse computed here is the specular term reaching equality; the visible collapse happens earlier, and by how much needs the diffuse term this account does not have.

Who found it, and when #

That gloss depends on the geometry of measurement is the foundation of every gloss standard, and specular gloss has been defined by an illuminating angle and an acceptance aperture since the 1930s. That a glossy surface needs a small source to look glossy is common knowledge among photographers and is the reason a softbox exists.

What is not anywhere is the exponent. The statement that a plain weave’s specular area goes as the square of the source’s angular size while a floated weave’s goes as the first power is a statement about cloth, and it needs the factorisation of the specular area into a crown line and a section width — which is this account’s own second essay and is not in the optics literature because the optics literature does not have a crown line.

The slope being a ratio of transition counts is the part worth carrying. It says the departure from the inverse law is decided by the draft and by nothing else — not by the yarn, the finish, the sett or the fibre — and it is the kind of result that only appears when a quantity that has been held fixed for five essays is finally swept.

Still open: what a real room does #

Everything above is one source of one width, and a room has several of different widths at once, plus the walls.

The quantity that would settle the practical question is the effective width of a real illumination, which is not an average of the sources but a weighting of them: each source contributes its own specular term and the contrast is the ratio of the sums. A room with a small lamp and a large window therefore has a contrast between the two single-source answers, weighted by brightness rather than by area — so a weak spotlight in a bright room buys less than its own number suggests, and by exactly how much is a sum this arithmetic can do as soon as anybody measures a room.

The measurement is a photograph of a mirror. A flat mirror in the place the cloth will be, photographed from where the viewer will stand, shows every source at its true angular size and brightness, and the sum over that image is the effective width. That is an afternoon with a camera and it would turn every ratio on this page into a number for a particular room.