A net in front gives the figure to the room

A figure in a sheer is a pattern in its glow, and by day the street and the room see opposite figures: a more open region is a quarter darker than its ground from the pavement and a twentieth lighter from the armchair. All of that was worked out for a single layer of cloth.

Almost nobody hangs one. The ordinary window carries a plain net against the glass and a second curtain — figured, laced, devoré, or merely heavier — behind it, and the pattern a passer-by is being shown reaches them only through the plain net’s holes, arriving against the plain net’s own lit threads. The question the single-layer account left was whether the net in front hides the figure behind it.

It does, from the street, by rather more than the single-layer arithmetic predicted. And it does something the single-layer arithmetic could not have predicted at all, because with one curtain there is nothing for it to happen to: the room’s view of the figure gets better. The pattern that was four and a half times stronger outside than in is, behind an ordinary net, very slightly stronger inside.

What a hole passes and what a thread passes are not the same quantity #

The single-curtain expression collapsed two things that have to be kept apart once there is a second layer, and the collapse is invisible while there is only one.

A hole shows a scene: a viewer looking through it sees the room or the street, whose luminance is a reflectance times whatever light is falling on it. A thread is lit by an illuminance: it throws back a share of the light on its near face and passes forward a share of the light on its far face, and neither of those carries a picture of anything. With one curtain the scene and the light both belong to the same side of it, so the two terms can be written in one expression and nothing is lost.

Put a second curtain between them and they part company. The space between two curtains has a light level of its own — brighter than the room and dimmer than the street — and it is that light, not the street’s, that the inner curtain’s threads are lit by from outside. Following the light through the pair takes two lines for each direction of view.

Looking in from the street, with the plain net in front and the figured cloth behind it:

$$B = o_b\,\rho_{\text{room}} + (1 - o_b)\bigl(\rho_b R\,T_f + \tau_b\bigr)$$

$$F = o_f B + (1 - o_f)\bigl(\rho_f R + \tau_f T_b\bigr)$$

where o is an open area, ρ and τ a thread’s reflectance and forward scattering, T a whole layer’s transmittance — its open area plus what its threads pass — and R the street’s light over the room’s. Looking out from the room the same two lines run the other way. Setting either curtain’s open area to one gives the other curtain’s single-layer glow exactly, which is the check the expression has to pass and which the first version of it did not.

The front net’s image share is its open area, not its transmittance #

The first line of the expression carries the result the single-layer account had guessed at, and corrects it.

The guess was that the street’s figure would be diluted by roughly the front layer’s image share. That is right in kind. What it got wrong is which quantity the share is made of. A thread passes light forward and carries no picture with it — that is the whole of what τ means in this collection, the transmittance the covering rule leaves out — so the part of the net that can show anything behind it is its holes and nothing else.

So the image of the inner curtain that reaches the street is $o_f B$, with the net’s open area in it and not its transmittance. On a net sixty per cent open with white thread the two differ by a tenth: 0.60 against 0.66. The difference is small and it is the difference between a statement that is exactly true and one that is nearly true, and the second kind has caught this collection before.

The net’s own glow moves with the figure, and cancels part of it #

The second term is the one no estimate would have reached, and it runs the wrong way.

A net’s threads are lit from the street in front and from the gap behind. The gap’s brightness from the room’s side is the inner curtain’s own transmittance — so a more open region of the inner curtain lets more room light into the gap, which lights the back of the net’s threads a little more, which makes the net glow a little more exactly where the figure is. The figure is a darker region from the street; the net’s answering glow is brighter there; the two subtract.

Written out, the front net’s gain on any difference behind it is

$$o_f + \frac{(1 - o_f)\,\tau_f\,(1 - \tau_b)}{\rho_{\text{room}} - \rho_b R\,T_f - \tau_b}$$

and by day the bracket in the denominator is negative, because the inner curtain’s threads lit at twenty to one outshine the room behind its holes. So the second term is negative and the gain is below the net’s open area rather than equal to it. At a net sixty per cent open the figure that survives is 28.7 per cent of the bare figure where the image share alone gives 34.3 — the estimate overstates by a fifth.

That is a small correction with a general shape. An intervening veil that is lit by what it is hiding does not merely swamp a signal; it partially cancels it, and the cancellation is invisible to any account that treats the veil as a constant added to the view.

Closing the net improves the room’s view #

The room’s half of the arithmetic is not the street’s with the labels changed, because the figure is in the near curtain from the room and in the far one from the street. A viewer indoors sees the inner curtain directly; what the net in front does is change what that curtain is seen against.

And what it is seen against is the street. A sheer hides whichever side is darker precisely because the bright side’s light swamps the dim side’s image, and the room’s weak view of its own figure was weak for that reason: the ground is already mostly window, so adding window to it changes it very little. A net in front takes street light out of the window faster than it takes figure out of the figure, so the room’s contrast rises.

The numbers run one way from end to end. With no net the room sees the openwork figure 5.4 per cent lighter than its ground. Behind a net ninety per cent open, 5.7; eighty, 6.1; sixty, 6.9; forty, 8.0; twenty, 9.6. Over the same range the street’s view falls from 24.6 per cent to 0.7.

So the pair has a crossing that neither curtain has alone. At a net about sixty per cent open the two sides see the figure about equally strongly — 7.0 per cent from the street and 6.9 from the room — and below that the room sees it better than the street does. A single figured sheer cannot do that at any openness or any light ratio by day: its street figure is the stronger one for the whole of daylight.

A pair does not obey the curtain’s reciprocity #

The single-curtain account rested on an exact identity. A region glows from the room at a light ratio R exactly R times what it glows from the street at 1/R, so whatever a figured sheer shows the street in daylight it shows the room after dark, figure for figure, and at equal light the two sides see the same figure.

Neither survives a second layer, and the reason is that the pair is not symmetric: the figure is in one of the two curtains and not the other.

At one to thirty — lamps on, street dark — the openwork figure behind the sixty-per-cent net is 7.1 per cent lighter from the street and 9.5 per cent darker from the room. The day’s figures were 7.0 per cent darker from the street and 6.9 per cent lighter from the room. The polarities still trade, and the sizes no longer match. The curtains have exchanged which is nearer the bright side, and since only one of them carries a pattern, the exchange is not the identity it was.

At equal light the break is plainer still. One curtain shows both sides the same figure at a ratio of one, always, for any cloth. The pair shows the street 1.95 per cent and the room 3.63 — different figures from the two sides of a curtain lit identically from both. The asymmetry is in the stack, not in the light, and it is there at every ratio including the one where a single curtain has none.

The practical form is that a double curtain cannot be designed by working out one layer and applying a discount. The discount is different on the two sides, it is different by day and by night, and it does not vanish when the light does.

The pair has a null, and it is nearly where the simple rule puts it #

If a figure can be hidden from the street behind a net, there is an open area at which it is hidden exactly, and the single-layer account suggests where to look: the net delivers a reduced light ratio $R\,T_f$ to everything behind it, so the figure should vanish where a bare curtain’s figure would vanish at that reduced ratio.

The street’s view of the pair is linear in the inner curtain’s open area — once through the image the net’s holes carry, once through the inner curtain’s transmittance, which sets the gap’s brightness and therefore how hard the net’s threads glow — so the null is one division, exactly as the bare curtain’s was.

The simple rule is good for an open net and poor for a close one, which is what the second term says it should be: the term the rule omits carries a factor of one minus the net’s open area, so it disappears as the net opens and grows as the net closes. Behind a net ninety per cent open the two agree to a twentieth of a percentage point; behind one twenty per cent open they are four points apart, which for a designer setting an openness is a real difference and for anybody reasoning about the mechanism is the whole of it.

And the room sees the hidden figure at every point on both curves — 18.7 per cent dark behind a net thirty per cent open, 15.6 behind a sixty, 14.2 behind a ninety. A pair of curtains can carry a figure invisible outside and plain inside, which is the one design the single layer could not produce: its own one-sided figures were hidden from the room and showed the street.

The figure designed to be invisible indoors reappears #

The single-layer account’s strongest result was a figure of black thread 76.4 per cent open in a white voile, matched so that the room sees nothing at all while the street sees a lozenge 89.5 per cent darker than its ground — a watermark turned round, invisible from one side and nearly black from the other.

Hang a plain net in front of it and both halves of that go.

The street’s near-black lozenge becomes a fifth of itself, which is expected. What matters is the other half: the figure the room could not see becomes visible, faintly but definitely, at 2.8 per cent. The match was a single division against a stated light ratio, and the net in front changes that ratio for everything behind it — so a figure tuned to vanish at thirty to one is looking at twenty to one and no longer vanishes.

That is a general caution about any design of this kind. A match is a point, not a region, and every element hung in front of a matched cloth moves the point. It is the same fragility a shaded weave’s tone steps have, where a scale designed at one setting is a scale at that setting only. The single-layer account already noted that the match held only within a band of light ratios; what this adds is that a second curtain is itself a change of light ratio, so the design is undone by furnishing rather than by weather.

A denser figure behind the net #

Everything above is openwork — a figure more open than its ground, which is what a mock leno or a devoré produces. A lace is the other construction: an open net ground with the figure worked closer, and its contrast reverses on both sides relative to openwork.

Both results hold with their signs turned over, which is the check that they are about the pair’s arithmetic and not about openwork. The net’s gain is a gain on a difference, and it does not ask what sign the difference has; the room’s improvement comes from street light being removed from the room’s view, which does not ask either.

What was computed, and how #

Each curtain is a region with an open area and a thread tone, the tone interpolating the collection’s three assumed sets of thread optics — white, mid-grey, black — linearly through the grey. Every luminance is in units of the room’s illuminance. The gap between the curtains carries an illuminance from each side, $R\,T_f$ from the street and $T_b$ from the room, and each curtain’s glow towards a viewer is its open area times whatever scene or glow lies beyond it, plus its thread fraction times its reflectance at the illuminance on the near face and its forward scattering at the illuminance on the far face. The light is followed once through, and a figure’s contrast is the difference of the figure’s and ground’s glows over their sum, as in the single-curtain account.

Four things are checked. Either curtain opened to nothing is the other one alone, from both sides, to the last bit of a double, for three inner curtains at four light ratios — which is the reduction that makes the expression trustworthy and is what the first version of it failed, by lighting a thread with a scene rather than with an illuminance. The front net’s glow follows the inner curtain’s transmittance and not its pattern, so it changes by under two per cent between a figured and a plain inner curtain while the figure changes the view by twenty. The gain on a difference behind the net is the closed form above, checked against the finite difference to nine places and checked to lie below the net’s open area. And the pair’s null shows the street exactly nothing at the open area one division gives, while the room sees it by more than two per cent.

The net’s openness, the ground’s, the figure’s and the light ratios are the model’s inputs. The thread optics are the assumed values every sheer result here carries.

What the two lines leave out #

There is no light bounced between the curtains. A net’s threads throw back six tenths of what falls on them, and some of that lands on the inner curtain and comes forward again. One pass is the whole of what is computed; a full account would sum a series whose ratio is the two curtains’ reflectances times their overlap, which for a white net against a white voile is not negligible and would raise every glow a little. Whether it raises the figure and the ground by the same proportion — which would leave every contrast here unchanged — is exactly the question, and it has not been answered.

The two curtains are taken as independent screens with nothing between them. In fact they hang a few centimetres apart, so a viewer at an angle sees a hole in one against a thread in the other, and their two grids beat: a net over a voile makes a moiré through a harmonic, and two sheers make a moiré that walks with the viewer. This account uses each curtain’s open area as a number rather than as a grid, which is the same averaging that makes two layers the product of their open areas on average and at no particular registration. Everything here is therefore a statement about the mean over registrations, and the variation across a window is the moiré.

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 net is effectively closer and its veil brighter than its construction says. Every openness here is a construction rather than a measurement, and what a cloth is open to at an angle is smaller again.

The regions are large and matt. Each curtain is many threads across at every point, so its open area is a fraction rather than a pattern; at a figure’s own boundary that is not true, and a shiny synthetic net returns light in a direction rather than diffusely.

And nothing here is a claim about seeing. Seven per cent across a lozenge may read as a figure or as nothing depending on its size, the light and the viewer, and the panels are shaded against their own brighter region because that is roughly what adaptation does and not because it is a model of vision.

Still open: what a lining does that a net does not #

The pair here is two sheers. The other ordinary arrangement is a sheer against the glass and an opaque or near-opaque curtain behind it, drawn at night and open by day — and when it is drawn the inner layer is not a figured screen but a wall. Every expression above survives that as a limit, with the inner curtain’s open area at nothing and its transmittance at whatever the lining passes, and the interesting case is the half-drawn one: a window that is sheer over part of its width and lined over the rest, seen from the street as two regions of one curtain.

By the arithmetic here those two regions differ enormously — the lined part shows the street only the net’s own veil and the sheer part shows it the room — so a half-drawn pair should read from outside as a figure with a contrast far beyond anything a woven figure reaches. Whether it does, and whether the boundary between them is the sharp edge the region model implies or is spread by the light bouncing in the gap, needs the interreflection this account leaves out, and has not been worked here.

Who worked it out #

Hanging a net against the glass with a second curtain behind it is older than any of the arithmetic and is what most windows in most streets carry. The single-layer account of a figured sheer is the one below, and the estimate it left — that a front layer would dilute the street’s figure by about its image share — is the guess this account was written to check. The two-layer transport, the correction from transmittance to open area, the cancelling second term, the room’s gain and the break of the reciprocity were computed directly.