A figure shows by its shine, not its step

A white linen damask has a pattern in it that anybody can see across a room, and the cloth is one colour throughout, woven from one warp and one weft at one sett. Something about the figure is returning light differently from the ground, and this collection has already given an answer.

The claim #

A damask’s figure is visible because its crowns run at right angles to its ground’s, not because it stands proud.

Two consequences, and the second is a correction.

The contrast reverses when the cloth is turned. Nothing about a height difference does that. A step is a step from every direction, and its shadow moves but its sign does not.

And the step this collection computed is real and is doing almost nothing. A figured cloth has a step in its surface established that a region interlacing less often is pressed less often and stands about fifty micrometres proud. That is correct arithmetic and it is not what a reader is seeing.

Why a step returns no light #

A height difference between two regions of a surface is invisible to a reflection unless something about the slopes differs, and a step does not change any slope except at its own edge.

Under a diffuse illumination — a room, an overcast sky, a light box — the two regions return the same fraction of what falls on them, because they are made of the same threads at the same angles. Under a directional light the step casts a shadow, but only at its edge and only at a grazing incidence, and the shadow is a line at the boundary rather than a difference between the two areas.

Fifty micrometres is also small against the surface’s own relief, which is 380 micrometres peak to valley on this cloth. The step is an eighth of the roughness the two regions already have, and it is being asked to be visible against it.

What is actually different #

The figure and its ground are complements. A warp-faced satin has all its crown line in the warp; its complement has all of it in the weft. The two lengths are nearly equal — 18.5 millimetres per repeat against 17.1, differing only because the two thread spacings differ — so the two regions have almost exactly the same amount of reflecting surface.

What they do not share is its direction. A thread can only mirror light arriving in the plane across itself, because that is the only plane its normals lie in. The figure’s crowns lie along the warp and need light from across the warp; the ground’s lie along the weft and need it from across the weft.

So under a directional light one of the two regions is reflecting and the other is not, and which one depends entirely on how the cloth is held.

What was counted, and how #

The two regions are built as two surfaces at the same construction, and the azimuth sweep is run on each.

At three degrees of tolerance and a tilt of nine, with the light coming from across the warp, the figure returns about twice the ground’s specular area; a quarter turn away it returns about half. The peak-to-trough modulation of each region alone is a factor of ninety-seven for the satin — the arithmetic of the previous rung — and the contrast between the regions is smaller than that because each region has a floor supplied by its own point crowns.

The reversal is the part that could not have come from a step. A contrast that changes sign when a cloth is rotated in its own plane is a statement about orientation, and a height difference has no orientation.

Why the contrast is only two to one #

Each region alone modulates by a factor of ninety-seven as the cloth is turned, and the contrast between the two regions is only two. The gap between those numbers is worth accounting for, because it is where the argument could be overstated.

Each region has a floor, and the floor is the other system’s point crowns. A warp-faced satin has its weft on the face at one intersection in eight, in isolated summits with no plateau. Those summits sit on the weft’s own cylinder, so they reflect light from across the weft — which is exactly the azimuth at which the satin’s warp plateaux have gone dark. So when the figure’s own crowns stop reflecting, the figure does not go black; it drops to whatever its minority system supplies.

The arithmetic: at the azimuth where the figure peaks it returns about 3.2 per cent of its plan and the ground about 1.6; a quarter turn away the figure returns 0.7 and the ground 1.6. The ratio is 2.0 one way and 0.47 the other, and the reason it is not 100 is the floor.

That floor is a design quantity. A construction whose minority system contributes less — a longer satin, or one whose binding points are made with a finer or a more deeply set thread — would show a deeper contrast. The classical damasks are eight-end and above for reasons that are usually given as pattern definition and float length, and this is a third reason with the same direction.

The trade already knew, and said it another way #

Every account of damask says the pattern shows “because the figure and ground reflect light differently”, and every weaving text tells a reader to view a damask from an angle rather than face on. The second instruction is the whole of this essay stated as practice.

The trade also knows the reversal, though it is described as a property of the cloth rather than of the viewing. A damask tablecloth is laid so that the pattern runs one way; turn it and the figure and ground swap from light to dark. Napkins folded from the same cloth show the effect within a single fold.

What was missing was the mechanism and the size. The mechanism is that a crown is a cylinder’s ridge and a cylinder’s normals lie in one plane. The size is a factor of two between the two regions and a factor of ninety-seven within either of them, both computable from the draft before the cloth exists.

Two essays on this site disagreed, and this one decides #

The interesting part is that the collection had already said both things, in two places, without noticing.

A damask is its own complement says in its first line that the pattern is carried by direction alone — figure and ground being one satin two ways up, with the same float length, the same interlacing count and the same threading.

A figured cloth has a step in its surface computes a fifty-micrometre step between the two regions and offers it as why the figure shows.

Both are correct arithmetic and they are not compatible as explanations. The step essay’s quantity is real and has consequences — a figured cloth’s regions do differ in thickness, in local density and in how they take a press — but a fifty-micrometre step in a surface with a 380-micrometre relief is not a visible difference under any illumination that does not cast a raking shadow, and it cannot reverse.

The shape of the mistake is worth naming, because it is a common one and it is not carelessness. A quantity was computed, it was real, it was the only quantity available, and it was assumed to be the cause. Having a number is not having a mechanism.

The same argument on a stripe #

A damask is one weave inside a region and another outside it. A stripe is the same thing arranged in bands, and the shine argument transfers with one difference that is worth naming.

A warp stripe — bands of two weaves running down the cloth — has its boundaries parallel to the warp, so the two regions’ crowns are at right angles across a boundary that runs along one of them. A weft stripe has the same regions with the boundary across them. The visibility of the two is different, and it is different at the boundary rather than in the bands: an edge parallel to a set of crowns is a sharp edge, and an edge crossing them is a ragged one, because the crowns are interrupted at whatever phase the boundary falls on.

That is a genuinely visible difference in cloth and it is usually attributed to the drafting of the stripe. It is at least partly geometric, and it predicts that a warp stripe of two weaves has cleaner edges than a weft stripe of the same two — with the caveat that a great deal else differs between the two, including how the loom makes them.

The two azimuths at which a damask disappears #

If the contrast is a function of the azimuth and the two regions are a quarter turn apart in it, then the contrast is not merely variable — it has zeros, and they can be located without computing anything.

Write C(φ) for the specular area a region returns with the light at azimuth φ. The figure returns C(φ) and the ground returns C(φ + 90°), because the ground is the figure’s draft with the systems exchanged. A cloth of one weave is symmetric about each of its thread directions, so C is symmetric about 0° and about 90° and repeats every 180°. Those three facts force

C(45°) = C(135°),

so the ratio between figure and ground is exactly one at 45° and at 135°, and nowhere else in half a turn.

A damask has two blind azimuths, at forty-five degrees to both thread systems and half a turn apart. Between them the contrast runs from two to one down through unity and out to one to two, so a cloth turned slowly through ninety degrees goes bright, flat, and inverted.

That is testable across a dinner table and it is worth stating as an instruction rather than a result: lay a damask square to the light, not diagonally. A cloth set on the table at forty-five degrees to the window is a cloth showing no pattern at all, and the fault will be attributed to the cloth.

Which makes a damask design a two-polarity design #

The reversal has a consequence for what can be drawn on a damask that no account of the mechanism seems to draw out, and it constrains the design rather than the cloth.

A pattern that reverses is a pattern with no fixed figure and no fixed ground. Whatever a designer intends as the motif will be the darker region from one direction and the lighter from the other, and a viewer walking round a table sees both. So a damask design has to work in both polarities, which rules out a whole class of design that works in one.

Anything relying on figure being lighter than ground fails. A motif that reads as an object against a background — a white bird on a dark field — becomes a dark bird on a white field, which is a different picture and often a worse one, because the eye assigns figure and ground by lightness before it assigns it by shape.

What survives is a design balanced between its two regions: motifs whose outline is legible either way up, ornament that fills its field rather than sitting in it, and areas of figure and ground in comparable proportion. That is a fair description of the classical damask repertoire — foliate scrollwork, mirrored and repeating ornament, dense allover patterning — and it has usually been explained as a matter of period taste and of what a drawloom could conveniently repeat.

It is at least also a constraint the mechanism imposes. A designer who ignores it produces a cloth that looks right on the loom and wrong on the table.

The folded napkin, which shows both at once #

The sharpest demonstration of all of this is a thing most people have seen without reading it.

Fold a damask napkin in half. The two halves are now at 180° to each other in azimuth — which by the symmetry above is not a reversal, since C repeats every half turn. Fold it into quarters, so that one half is turned ninety degrees against the other, and the two halves are at opposite polarities of the same pattern, meeting along a fold.

The motif runs continuously across the fold and swaps from light to dark as it crosses. Nothing about the cloth changes at the fold; nothing about the light changes at the fold. What changes is the direction the crowns are pointing, and it changes by exactly the quarter turn that separates a figure from its ground.

Nobody needs an instrument for that, and it is a complete refutation of the step on its own: a step fifty micrometres high cannot change sign because a cloth has been folded.

Where the model stops #

No shadowing, so the step’s own contribution is not computed at all. At a grazing incidence the step does cast a shadow and the shadow is a real signal — a raking light on a damask does show its relief as well as its shine. Nothing here can say which dominates at a given angle because only one of the two is computed.

The two regions are treated as having the same crimp, which they do not: the whole of the step essay’s argument is that they do not. Including the difference would move both crown heights slightly and would not change the orientations, which is what this essay turns on.

And there is no eye and no colour. What is computed is an area with an orientation. Whether a factor of two in specular area is visible depends on the illumination, the surround and the observer, and none of those belongs here.

What would settle it #

The account here is a computation and the earlier one was a computation, and both describe the same cloth. Deciding between them is a matter for a measurement, and the measurement is easy to specify.

Light a single-colour damask diffusely and photograph it. If the step is the cause, the pattern is visible; if the orientation is, it is not. An integrating sphere is exactly the instrument.

Then light it directionally and rotate it in its own plane. If the orientation is the cause, the contrast reverses at ninety degrees; if the step is, the contrast does not change sign at all and the shadow at the boundary merely moves.

And press the cloth flat before repeating both. Calendering removes most of the step — it flattens the region that was standing proud harder than the one that was not — while leaving the crown orientations exactly where they were. If the pattern survives a heavy calender, the step was not carrying it.

None of the three has been done here. They are stated because the claim is a claim about a mechanism and a mechanism has to be falsifiable, and because the first of them would take an afternoon.

The generalisation #

Two regions can differ in orientation without differing in amount, and an orientation difference is invisible to any measurement that integrates over direction.

That is the transferable statement, and its practical form is a warning about instruments. A diffuse-reflectance measurement of a damask returns the same number for figure and ground — correctly — and would conclude the cloth is uniform. Everything that makes the pattern visible lives in the directional part, which a diffuse measurement is built to discard.

The same trap exists wherever a structure has a grain: brushed metal, a rolled sheet, a fibre composite, a printed halftone with a screen angle. A measurement that averages over direction cannot see a pattern made of direction, and the pattern may nonetheless be the first thing anybody notices.

What a damask shares with a shot cloth #

The two constructions are the same geometry used two ways, and setting them beside each other makes the family clear.

A shot cloth puts the two orientations everywhere at once, in one uniform fabric, and separates them by turning the cloth. What changes is the whole fabric’s appearance.

A damask puts the two orientations in different places, in one uniform illumination, and separates them by position. What changes is one region against another.

Both rest on the identical fact — that a thread’s crowns reflect only across the thread — and both need the cloth to have long floats in both systems, for the same reason. A damask woven in a plain weave and its complement would be no damask at all, because a plain weave is its own complement and there would be nothing to distinguish.

That last point is a result this collection already has, and it was stated there in exactly these terms: the pattern is carried by direction alone, figure and ground being the same satin one way up and the other, agreeing on float length, interlacing count and threading and differing only in which system is on the surface. What the surface adds is the number — a right angle, a factor of two in specular area, and a reversal.

Who found it, and when #

Damask is Byzantine or earlier and the figure-and-ground reversal is part of its definition. The observation that a satin and its complement have crowns at right angles is implicit in the draft and has presumably been understood by weavers for as long as damask has been woven.

What this collection adds is the arithmetic — the two crown lines are nearly equal, the two orientations are exactly perpendicular, the contrast is about two to one and reverses — and the correction to its own earlier account, which had reached for the quantity it could compute rather than the one that was operating.

Where the ladder goes next #

To a fabric whose two faces differ for a reason with no float in it at all: a jersey has two surfaces, where the legs of the loops on one side and the heads on the other present crowns running in different directions — the same argument as this one, in a fabric with no warp and no weft.

Sideways, the finish that destroys all of it: raising moves the surface onto the hairs, after which a cloth has no crowns, no orientation and no figure.