What a sheer hides is decided by the furniture

Four accounts of this account have priced a net curtain. A sheer hides whichever side is darker and nothing about the cloth decides which; a figure in one is a negative from one side and trades sides exactly between day and night; a plain net hung in front of a figured one weakens the pattern outside and strengthens it in.

Every one of those carries a number that has never been examined. The room and the street were both taken to reflect three tenths of the light falling on them. It is a reasonable figure for an average outdoor scene and an average interior, and it sits in every expression from the covering rule’s own amendment onward, it was stated as an assumption each time, and it has been the same number in every expression.

It is also doing more work than any of them said. One of the two scenes decides, on its own, the whole of what a passer-by can see through a curtain; the two together decide whether the day-and-night exchange this account treats as an identity holds at all. Neither is a property of cloth, and a curtain maker controls neither.

The veil is the same for every room #

A passer-by looking at a lit window through a sheer receives two things. Through the holes, an image of the room: the open area times whatever the room’s surfaces return, under the room’s own light. From the threads, a veil: the covered fraction times what the thread throws back at the street’s light and passes forward from the room’s.

Write them for a white voile 49 per cent open at thirty to one, in units of the room’s illuminance:

$$\text{image} = 0.49\,\rho_{\text{room}} \qquad \text{veil} = 0.51 \times (0.60 \times 30 + 0.15) = 9.256$$

The second expression has no room in it. The threads are lit overwhelmingly by the street — the street’s contribution is 9.18 of the 9.256, and the room’s 0.077 — so the veil a passer-by sees is decided by the cloth and by the daylight, and is identical in a dark room and a white one.

Everything that varies is the first expression, and it varies over a factor of seventeen between a room reflecting a twentieth and one reflecting four fifths.

So a sheer’s privacy is not a property of the sheer. It is the ratio of one number the curtain fixes to another the occupant fixes, and the occupant’s number is the one with the range in it.

What the street looks like does not enter #

The symmetry a reader would expect here is not there, and its absence is the cleanest result in the essay.

A passer-by’s view of the room depends on the room’s reflectance, the cloth, and the ratio of the two illuminances. It does not depend on the street’s own reflectance at all — nothing a viewer standing on the pavement can see of a room is affected by whether the houses opposite are white stucco or dark brick.

That looks wrong for a moment and is not, and it is the same asymmetry a curtain hung in folds shows from the other direction — one side’s geometry, one side’s light. The street’s reflectance decides what the room’s viewer sees looking out, because that is the scene arriving through the holes from the other direction. Looking in, the street is not a scene at all: it is a light source, and its brightness has already been counted, once, in the ratio.

The consequence is that the two directions have different sensitivities. Darkening the street opposite by a factor of six — a bright terrace against a row of soot-black warehouses — changes what the room sees out from 63 per cent of the street’s own contrast to 22, and changes what the street sees in by nothing whatever.

A dark room is hidden fifteen times better than a pale one #

Putting the two together gives the quantity a curtain is bought for: how much better the room is hidden than the street is.

Between a dark room and a white one the advantage falls by a factor of fifteen, and no curtain has been changed. The line is a straight consequence of the image being linear in the room’s reflectance while the veil is constant, so it is very nearly a reciprocal — halving the furniture’s reflectance doubles the privacy.

The practical readings are short and they are not the ones curtain advice gives.

A white-walled room needs a closer net than a dark one for the same privacy, and the difference is not marginal. A lamp beside the window is worse than the same lamp across the room, because what matters is the reflectance of what the holes look at. And repainting is a privacy decision, which is an odd thing for arithmetic about cloth to have concluded and follows directly from the veil being street-lit.

It also explains a familiar observation that is usually put down to the curtain. A room looks perfectly private in the morning and exposed in the afternoon when the sun has come round on to a pale wall inside it: the ratio has moved a little and the room’s effective reflectance a great deal, and the arithmetic says the second is the larger lever.

Two unequal scenes break the day-and-night exchange #

The assumption is load-bearing in a second place, and this one is structural rather than a matter of degree.

A figured sheer’s reciprocity — that a region glows from the room at a ratio R exactly R times what it glows from the street at 1/R, so whatever the street sees by day the room sees by night — was established exactly, to the last bit of a double, over five tones, three opennesses and six ratios. It is exact, and it is exact because the two scenes were given the same reflectance.

With unequal scenes it departs, and the departure has a closed form: the room’s glow at R less R times the street’s glow at 1/R is

$$\text{open area} \times R \times (\rho_{\text{street}} - \rho_{\text{room}})$$

— nothing else, at every openness, tone and ratio tried. The open area is there because the term is entirely about the holes: the threads are lit by illuminances, which the exchange treats correctly, and only the scenes carry a reflectance that fails to cancel.

More than half, which is the point. This is not a correction to be noted and neglected; for an ordinary asymmetric window — a dark sitting room facing a sunlit pale street — the reciprocity’s prediction and the truth differ by the larger part of the answer. The identity is a special case and the special case is the one every earlier account computed.

What survives is the qualitative half. The brighter side is still the hidden one, the figure still trades polarity between day and night, and every null still exists. What does not survive is the exactness — the night view is no longer the day view handed over unchanged, and a figure designed to vanish from one side by day does not vanish from the other by night unless the two scenes happen to match.

There is a room bright enough to defeat any sheer #

The last question the assumption was hiding is whether a sheer can fail outright, and the useful way to ask it is backwards: not how bright the street must be, but how reflective the room would have to be for a stated share of what a passer-by sees to be the room rather than the cloth.

Since the image is linear in the room’s reflectance and the veil does not contain it, that is one division. For a share s,

$$\rho_{\text{room}} = \frac{\text{veil}\times s}{(1 - s)\,\text{open area}}$$

and the answer is a reflectance, which can be compared against what rooms are actually made of. A reflectance above one is a room brighter than a perfect white, which is to say the sheer cannot be defeated by furniture at all.

A white sheer cannot be defeated by a room in full daylight. At 49 per cent open it needs a room of 18.9; even at 85 per cent open it needs 4.5. The white thread’s veil, lit at thirty to one, is simply too bright for any furniture to compete with, which is why a white voile is the standard net curtain — and why a hairy sheer being closer and brighter than its construction says moves the threshold in the direction that helps and why nobody has ever been advised to choose one for a dark room.

A black net is another matter. Its threads return four hundredths of what falls on them, so its veil is a fortieth of the white one’s and a real room can outshine it: at 70 per cent open it is half room at a reflectance of 0.52 and a quarter room at 0.17. That is the arithmetic behind the reputation of dark sheers, which are chosen for the view out — a black sheer of the same openness shows twelve times more of the room by day than a white one — and it now has a threshold attached rather than a ratio.

At dusk, everything is defeated #

The ratio was thirty to one throughout, which is a bright day. It does not stay there, and the veil is the thing that falls with it.

At four to one a white voile 49 per cent open needs a room of 2.65 to be half the street’s view — still impossible, but by a factor of under three rather than nineteen. A black net of the same openness needs 0.19, which is a room with a pale carpet.

So the margin a white sheer enjoys is a daylight margin, and it closes through the afternoon rather than at nightfall. The moment a room becomes visible is not the moment its lamps go on; it is the moment the light outside falls far enough that the cloth’s own veil stops outshining the furniture, and that is decided by the furniture’s reflectance and the cloth’s tone together.

What moves in the accounts below, and one thing that stops existing #

The assumption is in every result this account has produced, so the honest thing is to say which of them move and by how much. They divide into three kinds.

The identities survive. That a hole’s glow and a thread’s glow balance somewhere; that the brighter side is the hidden one; that a figure’s contrast is a difference over a sum; that the null is one division — none of those contains a scene reflectance in a place where it fails to cancel, and all of them hold for any pair of scenes.

The levels move, and by a good deal. Openwork in a white net vanishes from the street at one to four and from the room at four to one, which is the account’s most-quoted pair of numbers. Those are $(\rho_{\text{room}} - \tau)/\rho$ and $\rho/(\rho_{\text{street}} - \tau)$, so they are functions of the furniture and the view. With a room reflecting a fifth and a street two fifths the pair becomes one to twelve and 2.4 to one — a band that has both shifted and narrowed, so a curtain whose figure keeps one polarity from both sides over a wide range of light in an average room keeps it over a much narrower range in a dark one. The same is true of a pair of curtains, whose nulls move with the front net as well.

The matched figure moves with them. A black region hidden from the room at thirty to one sits at 76.2 per cent open with equal scenes; at a fifth and two fifths it sits at 69.0, at an eighth and nearly a half at 66.7, and in a pale room reflecting a half facing a dark street at 91.4. That is a swing of twenty-five percentage points of open area in a design specified to the second decimal place, and the cause is entirely in the room and the view.

And one result stops existing. Openwork’s null from the street needs a hole’s glow to reach a thread’s, and a hole’s glow is the room’s reflectance while a thread’s floor is what it passes forward. In a room darker than a white thread’s forward scattering — below fifteen hundredths — there is no light ratio at all at which openwork vanishes from the street, because even with the street pitch black the thread is brighter than the room behind it. The expression refuses rather than returning a negative ratio, which is the right behaviour and is a stronger statement than a moved number: in a dark enough room, a white openwork figure is a dark figure from the street at every hour of the day and night, and the polarity the figured-sheer account traded between the sides never trades.

The same limit has a mirror in the other design. A black figure hidden from the room needs an open area, and in a pale room facing a dark street the arithmetic asks for one above one — no figure of black thread can be hidden from a white room that faces a dark street, at any openness, because the room’s own light on the near face of the threads already outshines what a hole could show.

What was computed, and how #

Each view is the image over the image plus the veil, exactly as this account’s earlier accounts compute it: the image is the open area times the far scene’s reflectance times its illuminance, and the veil is the covered fraction times what the thread returns at the near illuminance plus what it passes at the far one. Every luminance is in units of the room’s illuminance. The room’s reflectance is the swept variable and the street’s is stated; the thread optics are the three assumed sets this account has carried throughout, interpolated through the grey for intermediate tones. The room that defeats a sheer is one division, because the image is linear in the reflectance and the veil does not contain it.

Four things are checked. With equal scenes the reciprocity is exact, at three opennesses, three tones and three ratios, to the last bit of a double — which is the check that the earlier accounts were entitled to it. With unequal ones the departure is exactly the open area times the ratio times the scenes’ difference, at the same nine combinations, which is the closed form rather than an observation that something moved. A darker room is hidden better at every step of a sweep from a twentieth to four fifths. And no room of a possible reflectance shows half through a white voile in full daylight, while one does through an open black net — a pair of results in opposite directions, so that a sign error in the division would fail one of them.

The room and street reflectances, the light ratios and the thread optics are all inputs. What is computed is what a given set of them implies.

Where the model stops #

A room is not one reflectance. It has a window wall, a dark sofa, a white ceiling and a lamp, and what a hole looks at is whichever of those lies along its own line of sight. The single figure used here is an average over a scene that a viewer does not average over: a pale wall directly behind a seated person is the reflectance that matters for whether the person is seen, and it is not the room’s mean.

Nothing here is a claim about recognition. A figure’s own boundary is a few threads wide and this arithmetic is an average over many. Four per cent of a view being room rather than cloth is a number about light and not about whether a shape is identifiable, and a moving figure is found at contrasts a still one is not.

The thread optics are assumed and they carry the veil. Every threshold in this essay is a ratio of a room’s reflectance to a thread’s, so a white thread returning half rather than six tenths of its light moves every one of them. The three sets used here bracket real sheers and measure none.

The scenes are matt and evenly lit. A wet road, a car windscreen or a low sun on glass opposite is a specular source, and a specular source in the street is not a reflectance at all.

And the light ratio is one number for a whole window. Sunlight falls on part of a room and not the rest, and a window with a patch of sun on the floor inside it has two ratios at once.

Still open: whether a room’s reflectance can be read from outside #

Everything here treats the room’s reflectance as an input, and the arithmetic offers a way to measure it that needs no access to the room.

The street’s view of a window is the room’s image plus a veil that the cloth and the daylight fix. Both of the latter can be obtained without entering: the cloth’s openness and tone from a sample, the illuminance from a meter held at the glass. So a photograph of the window, calibrated against the cloth beside it, would give the image — the same move two layers at an unknown registration force, where the mean is knowable and the instance is not — and one division would give the reflectance of whatever the holes are looking at.

Whether that reading is stable enough to be worth anything is the question. The image is a few per cent of the total in daylight, so the division is a small difference between two larger numbers, and the thread optics enter it linearly. It would be a poor way to measure a room and it may be an adequate way to detect a change in one, since the cloth and the light can be held fixed between two exposures and only the furniture moves. Neither has been tried here.

Who worked it out #

Photometry of this kind is standard and the two expressions are elementary. Their application to a net curtain is new here, and the assumption examined here — that a room and a street reflect the same fraction — was stated honestly in every essay that used it and never tested. The closed form for the reciprocity’s departure, the independence of the street’s own reflectance, and the room expressed as a reflectance rather than as a light level were computed directly.