Textile Structure
how cloth is put together
Pattern and colour

A figured sheer is a negative from one side

A net curtain with a pattern in it carries two patterns, one for each side, and by day they are opposites. A more open figure in a white voile is a dark figure a quarter below its ground from the street and a light one five per cent above it from the room, and after dark the two views trade places exactly. The figure vanishes from the street at one light ratio and from the room at another. And a figure can be made that the room cannot see at all while the street sees it nearly black — along one line of openness and thread tone, and never from both sides at once.
17 min read 7 figures A mechanism, not a materialStructure before fibreMeasured, not claimed

Worth reading first: A sheer hides whichever side is darker · Opacity is not cover · A figure shows by its shine, not its step.

A sheer hides whichever side is darker, and the reason is not in the cloth. What comes through its holes is an image of the far side; what comes off its threads is a veil lit from both sides; and the balance between them is set by how much brighter one side is than the other. That account treated every sheer as one construction from edge to edge.

Few net curtains are. A lace has a dense figure on an open net; a woven sheer carries openwork stripes and spots; a devoré has regions burnt thin. Each is a sheer whose open area or thread tone changes from place to place, and the question that account left was whether a pattern could be designed to read as a figure from the street and as plain haze from inside.

It can, on one line, and the reason it can says something larger first. A figure in a sheer is not a fixed picture that the two sides see more or less clearly. Each side sees a figure of its own, made of a difference in glow, and by day the two figures are negatives of each other.

What a region of a sheer glows with

Every patch of a sheer sends light towards a viewer by two routes, and the account of the one-way curtain already has both.

Through its holes it passes an image of the far side: the fraction of the patch that is hole, times how much light the scene behind reflects, times the light falling on that scene. From its threads it sends a veil: the fraction that is thread, times how much of the viewer’s own side’s light the thread throws back, plus how much of the far side’s light it passes forward by scattering.

Add the two and a patch has a glow, and a figure is nothing more than two regions whose glows differ. How strongly it shows is the difference between them over their sum — a contrast that is negative when the figure is darker than its ground and positive when it is lighter.

The quantities are the assumed ones every sheer here carries, because what a thread passes by scattering is the quantity opacity is not cover showed cannot be read off its construction. A white thread throws back six tenths of the light falling on it and passes fifteen hundredths forward; a mid-grey one a quarter and eight hundredths; a black one four and two hundredths. The room and the street each reflect three tenths of their light. The ground throughout is a white voile at its own construction, 49 per cent open.

By day, with the street thirty times brighter than the room, that ground glows 9.35 times the room’s illuminance from the street and 7.02 times from the room. Most of the first is street light thrown back by its threads; most of the second is street light arriving through its holes. The two sides are looking at almost entirely different light.

Openwork is a dark figure from the street

Put a figure into the voile that is the same white thread and 70 per cent open instead of 49. Nothing else about it is different.

From the street, a more open region has less thread in it to throw back the daylight, and what it shows through its extra holes is a dim room. It is darker than the ground by 24.6 per cent. From the room, the same extra holes are windows on to the bright street, and the thread that was removed was only returning the room’s own weak light. It is lighter than the ground by 5.4 per cent.

A white thread 70% open figure in a white thread 49% open ground at 30 : 1, from both sides. A lozenge figure in a sheer, drawn as the street sees it and as the room sees it, with the street 30 : 1 as bright as the room. The ground is white thread 49% open and the figure white thread 70% open. Each region glows with the scene behind it through its holes and with its threads lit from both sides; from the street the figure's contrast against the ground is −24.6% and from the room +5.4%, a negative number being a figure darker than its ground. Each panel is shaded relative to its own brighter region, as an eye adapted to that view would see it, and both panels share one gain so that the two steps are to scale against each other. The contrast is stretched 3 times to be visible at all, so the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the absolute glow, which from the street is 9.35 and from the room 7.02 times the room's illuminance for the ground, nor the thread optics, which are assumed values.
Fig. 1 A lozenge figure of white thread 70% open in a white voile ground 49% open, with the street thirty times brighter than the room, as each side sees it. From the street the figure is 24.6% darker than its ground; from the room it is 5.4% lighter. Each panel is shaded against its own brighter region, the way an eye adapted to that view would see it.

The same cloth, at the same moment, is a dark lozenge to the passer-by and a pale one to the occupant. Nothing reversed in the curtain. What reversed is which of a region’s two contents is the brighter, and from the street the thread is brighter than the hole while from the room the hole is brighter than the thread.

The street’s figure is also the stronger of the two by a factor of four and a half. From the room the ground is already mostly window, and adding more window to it changes it proportionately little.

A denser figure reverses the other way

A lace is the opposite construction: an open net ground with the figure worked more closely. Take the figure 30 per cent open in the same 49 per cent ground.

From the street the denser figure throws back more daylight than the net around it and is 15.6 per cent lighter. From the room it blocks more of the bright window and is 5.7 per cent darker.

A white thread 30% open figure in a white thread 49% open ground at 30 : 1, from both sides. A lozenge figure in a sheer, drawn as the street sees it and as the room sees it, with the street 30 : 1 as bright as the room. The ground is white thread 49% open and the figure white thread 30% open. Each region glows with the scene behind it through its holes and with its threads lit from both sides; from the street the figure's contrast against the ground is +15.6% and from the room −5.7%, a negative number being a figure darker than its ground. Each panel is shaded relative to its own brighter region, as an eye adapted to that view would see it, and both panels share one gain so that the two steps are to scale against each other. The contrast is stretched 4 times to be visible at all, so the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the absolute glow, which from the street is 9.35 and from the room 7.02 times the room's illuminance for the ground, nor the thread optics, which are assumed values.
Fig. 2 The same ground with a denser figure, white thread 30% open, at the same thirty to one. From the street the figure is 15.6% lighter than its net; from the room it is 5.7% darker, a figure standing against the window rather than glowing on it.

That is the look a white lace curtain has from each side on a bright day — a white pattern on a grey net from the pavement, and a grey pattern against a bright net from indoors — and the arithmetic produces it with no property of lace in it other than that its figure is closer than its ground.

So there are two kinds of figure and each is a negative from one side. An open figure is dark outside and light inside; a close one is light outside and dark inside.

The street sees by day what the room sees by night

Turn the lamps on and draw the street down to a thirtieth of the room’s light, and the two views trade places — not approximately but exactly.

The reason is in the two glows. From the street a patch shows the room through its holes and threads lit mostly by the street; from the room it shows the street through its holes and threads lit mostly by the room. Swap which side is brighter and each view becomes the other view scaled by a constant, and a contrast does not notice a constant. With the two scenes equally reflective, the glow from the room at any ratio is exactly that ratio times the glow from the street at its reciprocal.

So the openwork figure at one thirtieth is 5.4 per cent lighter from the street and 24.6 per cent darker from the room: the day’s two figures, handed to the opposite sides. The lace does the same. Whatever a figured sheer shows the street in daylight, it shows the room after dark, figure for figure.

At equal light the two views are the same view, as they were for the room behind a plain sheer, and the openwork figure is 9.7 per cent darker than its ground from both sides.

Each side has a light ratio at which the figure disappears

Between thirty to one and one to thirty the openwork figure has to pass from one negative to the other, and on each side it does so by vanishing.

From the street, a hole’s worth of the room glows three tenths and a thread’s worth glows six tenths of the street’s light plus fifteen hundredths of the room’s. The two balance when the street is a quarter as bright as the room, and at that ratio an openwork figure shows the street nothing at all. From the room, a hole’s worth of street glows three tenths of the street’s light and a thread’s worth six tenths of the room’s plus fifteen hundredths of the street’s, and they balance when the street is four times as bright.

A figured sheer's contrast from both sides against the light ratio. The contrast of a lozenge figure against its ground in a sheer, seen from the street and from the room, as the street goes from a hundredth as bright as the room to three hundred times as bright. The ground is white thread 49% open and the figure white thread 70% open. A positive contrast is a lighter figure and a negative one darker. It vanishes from the street at 1 : 4 and from the room at 4 : 1. At equal light both sides see −9.7%. What the curves cannot show is the smallest contrast an eye reads as a figure, which depends on its size and on the light.
Fig. 3 The openwork figure’s contrast against its ground from both sides, as the street runs from a hundredth as bright as the room to three hundred times as bright. From the street it passes through nothing at one to four, from the room at four to one; the curves cross at equal light, where both sides see the figure 9.7% darker, and each is the other reflected about that line.

Neither null has anything to do with how open the figure is. Openwork of any size vanishes from the street at one to four and from the room at four to one, because the balance is between a hole’s glow and a thread’s glow, and the figure only decides how much of each it trades. Between the two ratios a white net’s openwork is darker from both sides; outside them it is a negative from one.

That band is a room lit about as brightly as the street — an overcast evening, a lit room at dusk. On a bright day and at night the figure is always the wrong way round from somewhere.

A black net’s band runs the other way

The nulls are set by the thread, so they move with its tone, and not only in size.

For a mid-grey thread the band nearly closes: the figure vanishes from the street at one to 1.14 and from the room at 1.14 to one. For a black thread it opens again and turns over. A black thread’s veil is so faint that from the street its holes outshine its threads until the street is seven times the room’s light, and from the room they outshine them until the street falls to a seventh. So black openwork is lighter than its ground from both sides between one to seven and seven to one, and a negative only outside that.

The light ratios at which openwork vanishes from each side of a sheer. For a figure that differs from its ground only in open area, the ratio of street to room light at which it shows nothing from the street and the ratio at which it shows nothing from the room, for white, mid-grey and black threads: white, 1 : 4 and 4 : 1; mid-grey, 1 : 1.14 and 1.14 : 1; black, 7 : 1 and 1 : 7. Between its two ratios a figure keeps one polarity from both sides; outside them it is lighter from one side and darker from the other. What the chart cannot show is the thread optics' uncertainty, which moves every ratio.
Fig. 4 The light ratios at which openwork vanishes from the street and from the room, for white, mid-grey and black thread, on one axis of street light over room light. White: one to four and four to one; mid-grey: one to 1.14 and 1.14 to one; black: seven to one and one to seven. Between its two nulls a figure keeps one polarity from both sides.

The turning point is the one the veil essay found at equal light: whether a thread, lit from both sides, glows brighter or dimmer than the scene it stands in front of. A white thread returning and passing three quarters of the light it gets outshines a scene reflecting three tenths, so its openwork reads as holes into something dimmer. A black thread returning six hundredths reads the other way. The mid-grey, at a third, is almost exactly the scene’s own brightness, which is why its openwork nearly disappears at equal light from both sides at once.

A figure the room cannot see

Now the design question can be answered.

A region’s glow from either side is a straight line in its open area — so much hole, so much thread — so for any thread tone there is exactly one open area at which the region glows from a given side precisely as the ground does. At thirty to one, from the room, a white region matches only at the ground’s own 49 per cent. A region three quarters of the way to white matches at 61.4 per cent open, a mid-grey one at 68.9, and a black one at 76.4.

From the street the matching regions go the other way. To glow as the voile does under daylight a thread has to throw back nearly as much of it, so only threads more than 58 per cent of the way to white can match at all, and they must be closer than the ground: at three quarters of the way, 28 per cent open.

Where a figure is invisible from each side of a sheer, at 30 : 1. Every region a sheer could have, by open area and thread tone, marked where it glows exactly as a ground of white thread 49% open does, with the street 30 : 1 as bright as the room. The line of regions invisible from the room runs from the ground to black thread 76.4% open; the line invisible from the street leaves the ground towards closer, still pale cloth and meets the edge at tone 0.59. The two lines share only the ground itself, so no other region is invisible from both sides at this ratio. What the plot cannot show is how visible a region just off a line is, which the contrast curves give.
Fig. 5 Every region by open area and thread tone that glows exactly as the white voile ground does, from the room and from the street, at thirty to one. The room’s line runs from the ground to black thread 76.4% open; the street’s runs from the ground towards closer, paler cloth and meets the edge 58% of the way to white. The corner on the room’s line at mid-grey is where the tone interpolation changes between its two segments.

The two lines meet only at the ground. So at any light ratio but one, no figure can be invisible from both sides of a sheer; the one exception is equal light, where the two sides see the same thing and the two lines are one. Any figure the room cannot see, the street can.

Nearly black from the street, and nothing from the room

Take the end of the room’s line: a black-thread figure 76.4 per cent open in the white voile, at thirty to one.

A black thread 76% open figure in a white thread 49% open ground at 30 : 1, from both sides. A lozenge figure in a sheer, drawn as the street sees it and as the room sees it, with the street 30 : 1 as bright as the room. The ground is white thread 49% open and the figure black thread 76% open. Each region glows with the scene behind it through its holes and with its threads lit from both sides; from the street the figure's contrast against the ground is −89.5% and from the room 0.0%, a negative number being a figure darker than its ground. Each panel is shaded relative to its own brighter region, as an eye adapted to that view would see it, and both panels share one gain so that the two steps are to scale against each other. The contrast is stretched 1 times to be visible at all, so the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the absolute glow, which from the street is 9.35 and from the room 7.02 times the room's illuminance for the ground, nor the thread optics, which are assumed values.
Fig. 6 A black-thread figure 76.4% open in the white voile ground, the open area at which it glows exactly as the ground does from the room at thirty to one. From the room there is no figure at all; from the street it is 89.5% darker than its ground, a lozenge very nearly black against a glowing net.

From the room the curtain is an even haze. From the street the figure is 89.5 per cent darker than its ground, which is as close to black on white as a sheer can come. It is the pattern the question asked for, and it is not subtle: it is the strongest figure in this essay, hidden completely from one side.

It has a much older relative in paper. A watermark is a region of a sheet made thinner or thicker than the rest, and it shows plainly when the sheet is held against a window and faintly or not at all when it lies on a desk, because by transmitted light what matters is how much the region passes and by reflected light how much it returns. The matched figure is a watermark turned round: tuned so that the view through it, which is the room’s view of a bright window, carries nothing, while the view of what it returns, which is the street’s, carries everything.

How long the match holds

A match made at one ratio is exact at that ratio only, and a day does not keep one. What matters is how quickly the figure comes back from the room as the light moves.

A figured sheer's contrast from both sides against the light ratio. The contrast of a lozenge figure against its ground in a sheer, seen from the street and from the room, as the street goes from a hundredth as bright as the room to three hundred times as bright. The ground is white thread 49% open and the figure black thread 76% open, the open area at which it glows exactly as the ground does from the room at 30 : 1. A positive contrast is a lighter figure and a negative one darker. It vanishes from the street at 1 : 30 and from the room at 30 : 1. At equal light both sides see −36.9%. What the curves cannot show is the smallest contrast an eye reads as a figure, which depends on its size and on the light.
Fig. 7 The black figure’s contrast from both sides against the light ratio. From the room it is nothing at thirty to one and within two per cent of nothing from fifteen to one up to several hundred; it reaches four per cent at ten to one and sixteen at three to one, and at equal light both sides see it 36.9% darker. From the street it is nearly black by day and vanishes at one to thirty.

Slowly, on the bright side. From the room the figure stays within two per cent of its ground from fifteen to one up to about 450 to one, which is most of a day’s range at a window; it reaches 4.0 per cent at ten to one and 15.7 at three to one. The match is forgiving by day for the reason the room’s figures were weak to begin with: the room is looking mainly at the street through the holes, and a small mismatch in a faint veil is lost against it.

It is not forgiving towards evening. At equal light the figure is 36.9 per cent darker from both sides, and after dark reciprocity finishes the job. A figure hidden from the room by day is hidden from the street by night, and the room then sees it 89.5 per cent dark — the passer-by’s daytime view, handed indoors with the lamps.

So a sheer can keep a figure from one side, and the side it keeps it from is the darker one. That is the same rule the plain sheer obeyed about the room behind it, arriving at the pattern in the cloth: what the brighter side hides is the dimmer side’s view.

A damask trades figure and ground by turning, a sheer by lighting

The pattern field has met a reversal of figure and ground before, and it came from somewhere else entirely.

A damask’s figure shows by its shine, not by its step: its figure and ground are the same satin used two ways, one warp-faced and one weft-faced, and because a warp crown’s normals lie across the warp, turning the cloth a quarter turn changes which of the two shines. A damask napkin turned on the table swaps its figure and ground.

A figured sheer swaps them with no direction anywhere in it. Its threads are treated as matt, and nothing about the reversal depends on where the light comes from or which way the cloth is turned — only on which side of it is brighter. The damask’s reversal is geometric and the sheer’s is photometric, and they share only the fact that a figure made of one material in two arrangements has no fixed sign.

What a figure in a sheer is made of

The arithmetic needs two numbers for each region, its open area and its thread tone, and says nothing about how a loom or a finisher produces them.

Openwork on shafts comes cheapest from grouping ends and floating picks, which is a mock leno and is exactly a region more open than its ground. A denser figure comes from more picks or heavier yarn in the figure. A darker region needs a coloured thread in it, laid in as a figuring weft or printed, and a devoré changes openness and fibre together. A figure’s edge brings questions of its own that a figure’s boundary raises in any weave, and the model takes its regions to be large beside the threads.

What is new here is only that a figure of this kind is a scale of glows, not a scale of greys. A weave is a halftone screen with a stated number of tones, and a shading fixes the tone of each step in the draft. A sheer’s figure has two tone scales, one per side, and the light chooses which is shown to whom.

One glow, two sides, one division

Each region’s glow from the street is its open area times the room’s reflectance, plus its thread fraction times the thread’s reflectance at the street’s light and its forward scattering at the room’s, all in units of the room’s light; from the room the roles of the two lights and two scenes exchange. A figure’s contrast is the difference of figure and ground over their sum. Matching a region to its ground from one side is one division, because the glow is linear in the open area. Openwork’s nulls are the ratios at which a hole’s glow equals a thread’s. Tones between black, grey and white interpolate the two thread numbers linearly through the grey.

The reciprocity — glow from the room at a ratio equal to that ratio times glow from the street at its reciprocal, and so equal figure contrasts — was confirmed exactly for five tones, three open areas and six ratios. The openwork’s polarity at thirty to one, its nulls at one to four and four to one, its single polarity at three ratios between them, and the vanishing at each null were confirmed directly; so were the black figure’s match at 76.4 per cent, its invisibility from the room and near-blackness from the street, and that at thirty to one no region of five other tones is invisible from both sides.

What the glow leaves out

The thread optics are assumed. The white, grey and black threads are the three sets of numbers the one-way curtain used, and they bracket real sheers without measuring any. Every null, every match and every percentage moves with them; the reciprocity, the reversal by day and the single shared point of the two lines hold for all of them.

The two scenes are even and equally reflective. A real room has a dark sofa and a white wall, and a real street has trees and a pale house opposite. A figure matched against a street reflecting three tenths is matched against that alone, and in front of a brighter patch of the view it reappears. The reciprocity needs the two scene reflectances equal as well; unequal, the night view is the day view scaled by their ratio, and no longer exactly.

Nothing here says what an eye can see. Each panel is shaded against its own brighter region, which is roughly what adaptation does and is not a model of vision; whether two per cent across a lozenge of lace reads as a figure depends on its size, the light and the viewer.

The holes are clean. Fibre ends standing in a hole block a line of sight as a thread does and are lit as a thread is, so a hairy sheer’s regions are all a little closer and a little brighter in their veil than their construction says, and every match moves towards more open cloth to make up for it.

The regions are large and matt. The model takes each region to be many threads across, so that its open area is a fraction and its glow an average; at a boundary a few threads wide, and in a shiny polyester sheer that throws light back in a direction, neither is true.

Still open: a figure behind a plain sheer

The commonest double curtain is a plain net in front of a patterned one, and every result here is for a single layer. The back layer’s figure reaches the street only through the front layer’s holes, and it arrives against the front layer’s own veil, which is lit by the street as brightly as ever. By the arithmetic of the plain curtain that should dilute the street’s figure by roughly the front layer’s image share at the street’s ratio, which by day is a few per cent — but the back layer’s glow is itself lit partly through the front layer, and two layers pass the product of their open areas only on average. Whether a figure in the back curtain can be hidden from the street by the plain one in front, and whether the room still sees it, needs both layers’ glows followed through each other, and has not been done here.

Where this goes next

Seeing through a white voile, from the street and from the room. How much of a scene's contrast survives a white voile, whose open area is 49%, looked through from the street into the room and from the room out to the street, against how much brighter the street is than the room. The scene comes through the clear lines of sight and nothing else; everywhere else the viewer sees thread lit from the viewer's own side, which returns light with no image in it. On a bright day, a hundred times brighter outside, 0.5% of the room's contrast reaches the street and 65% of the street's reaches the room. At equal light both are 28% — the curves cross at a ratio of one whatever the cloth — and with the lamps on after dark the room is the side on show, at 63%. What the plot cannot show is the threads' own optics: their reflectance and transmittance are assumed values for a white sheer rather than measurements, and only the crossing point is independent of them. A sheer hides whichever side is darker A net curtain hides a room by day and shows it at night, and the cloth has nothing to do with which. What a viewer sees through a sheer is an image through its clear lines of sight against a veil of lit thread, and the only thing that decides their balance is how much brighter one side is than the other. At equal light the two views are identical for every cloth there is. And a black sheer of the same openness shows twelve times more of the room by day than a white one. A damask's figure and ground trade places when the cloth is turned. A satin 8 figure on a sateen 8 ground in sheeting — one cloth, one set of threads, one sett, and the ground is the figure's own complement. Their total specular areas are within a few per cent of one another, so neither is intrinsically the brighter. What differs is the direction: the figure's crowns run with the warp and the ground's with the weft. So the contrast between them is 2.0-to-one with the light coming from 8° and 0.47-to-one from 90° — it reverses, exactly, a quarter turn apart. That is what makes a damask visible in one colour, and it is not the step in its surface: the step is fifty micrometres and returns no light at all under a diffuse illumination, while this contrast is a factor of 2.0 and is present whenever there is a direction in the light. A figure shows by its shine, not its step A damask is one cloth in one colour and its pattern is plainly visible. This collection attributed that to the step in its surface — fifty micrometres of relief, computed from the interlacing rates. The step is real and returns almost no light. What makes the figure visible is that its crowns run at right angles to the ground's, so the two trade places when the cloth is turned. Figure and ground on 8 ends. A damask is a satin and its own complement. The figure is warp-face and the ground is weft-face, they carry the same longest float and need the same shafts on the same threading, and the whole of the pattern is carried by which system is on top. A damask is its own complement The pattern is carried by direction alone. Figure and ground are the same satin, one warp-face and one weft-face, with the same longest float, the same shaft count and the same threading — so the cloth's most famous effect costs it no structural difference whatever between the two areas.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Clear openingFigure and groundOpacityOpen areaScatteringTransmittanceVeil