Textile Structure
how cloth is put together
Pattern and colour

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.
17 min read 6 figures A mechanism, not a materialStructure before fibreMeasured, not claimed

Worth reading first: A cloth is more opaque than it is closed · Opacity is not cover · One minus the cover is a cloth with no thickness.

Everybody who has lived behind a net curtain knows two facts about it. In the daytime it hides the room from the street and leaves the street perfectly visible from the room. After dark, with the lamps on, it does the reverse, and the room is on show to anybody passing.

That is usually put down to the curtain, as though a sheer were a one-way material that stops working at night. It is not. Opacity is not cover established that a sheer lets through more light than its open area and left the extra — whatever comes through the threads — as a quantity nothing here can compute. The one-way curtain turns out not to need it. What decides whether a room can be seen is not how much light the cloth passes but how the light it passes is divided between an image and a glare, and the division is set by the two rooms rather than by the cloth.

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.
Fig. 1 How much of a scene’s contrast survives a white voile, looked through from the street into the room and from the room out, against how much brighter the street is. By day the room keeps under two per cent of its contrast and the street nearly two thirds; at equal light the two views are identical; with the lamps on after dark they have changed places.

A cloth passes two different things

Light through a sheer arrives at a viewer by two routes, and only one of them carries a picture.

A ray that passes a hole in a straight line arrives from exactly where it started, so the rays that do it form an image of whatever is behind the cloth, dimmed by the fraction of the view that is hole. A ray that goes into a thread is scattered — reflected from fibre to fibre, refracted, sent out in some other direction — and when it emerges it no longer says where it came from. It adds brightness to the thread and nothing to the image.

So a sheer has two transmittances and they answer different questions. How bright does the cloth look against a window is the total, holes plus scattering, and that is the quantity the opacity essays keep finding larger than the cover suggests. How much of what is behind it can be seen is the image alone, and the image is a question of geometry: a count of the straight lines that miss everything.

A line of sight goes round a spun yarn

The first thing to settle is whether any of those straight lines pass through the threads, since a thread is a bundle of fibres with air between them.

A straight line crossing a yarn misses every fibre only if it threads the gaps all the way through. With the fibres placed at random at a packing of six parts in ten, a line meets them at a rate set by the packing and their diameter, and the chance of meeting none falls exponentially with the length of yarn it crosses.

How much of a yarn a line of sight gets through. The fraction of a round yarn's width through which a straight line passes without touching a fibre, against the number of fibres in its cross-section, at a packing of 0.6 with the fibres placed at random. The upper curve is the average across the yarn's width; the lower is the line straight through its middle. A monofilament passes about half its width, because a single cylinder is clear either side of itself. By the 71 fibres of a voile's spun yarn the middle is shut to 2.5e-4 and the width as a whole passes 1.5%, 94% of it through the outer fifth of the yarn — which is to say round its edge rather than through it. What the curve cannot show is where a real yarn's edge is: the fibres there are loose, hairy and not at the packing the rest of the yarn has, which moves the few per cent but not the shut middle.
Fig. 2 The fraction of a round yarn’s width a straight line of sight passes without touching a fibre, against the number of fibres in its cross-section. A single monofilament passes about half its width, clear either side of itself. A voile’s 12 tex cotton yarn, seventy-one fibres, passes one and a half per cent — ninety-four per cent of that round its outer edge — and straight through its middle the chance is a few parts in ten thousand.

The numbers split cleanly. A 12 tex cotton yarn of the kind a voile is woven from holds about seventy-one fibres in its cross-section — the count how many fibres make a thread takes from the tex and the fibre’s own fineness. Straight through its middle a line of sight gets past with a probability of two and a half parts in ten thousand. Averaged across its whole width, one and a half per cent of the lines get through, and ninety-four per cent of those pass through the outer fifth of the yarn — round it, not through it. A muslin’s 20 tex yarn, at a hundred and eighteen fibres, passes under one per cent, ninety-eight per cent of it at the edge.

That edge is not a sharp boundary. It is the loose, hairy margin a yarn’s diameter is ill-defined across, so what the calculation really says is that the image passes through the open area measured to the yarn’s outer contour, to within a per cent or so of the covered area. The middle of a spun yarn is shut, and the quantity nothing here could compute — how much light the thread passes by scattering — has no bearing on what can be seen through it.

The veil

Where a viewer’s line of sight does not pass a hole, it lands on a thread, and a thread is not black. It is lit from both sides, and it sends some of that light towards the viewer.

From the viewer’s own side, it throws back a share of whatever light falls on it there — a white cotton thread a large share, a black one very little. From the far side, it passes forward a smaller share by scattering. Neither carries any image. Both add to the brightness of the part of the view that is thread, and that brightness lies over the image the way fog lies over a landscape.

It is the same kind of light that a hair layer returns over a satin’s highlight, and it does the same thing to a contrast: it adds a constant to both the bright part and the dark part of whatever is behind it, so the difference between them is unchanged while their ratio collapses.

The image survives in proportion to its share of what reaches the eye. A scene whose light comes through the holes is seen against a veil from the threads, and the contrast that remains is the image divided by the image plus the veil. That is the whole model, and it has three ingredients: the open area, the thread’s two optical shares, and the light on each side.

Only a ratio of the two lights is left

Write the image as the open area times the scene’s reflectance times the far side’s light, and the veil as the covered area times the thread’s reflectance times the near side’s light, plus the covered area times its forward scattering times the far side’s light. Divide both by the far side’s light.

What is left depends on the far side’s light only through the ratio of the near side’s light to the far side’s. The absolute brightness has cancelled. A curtain looked through at noon and at dusk behaves identically if the two rooms keep the same proportion, and the thing a lamp or the sun does to its privacy is to move that one ratio.

That is why the curtain changes sides at night. Looking in, the viewer is in the street and the veil is lit by the street; looking out, the veil is lit by the room. The two views are the same function of reciprocal ratios. By day the street is tens to hundreds of times brighter than a room, so looking in the veil is huge and looking out it is small. After dark the ratio is inverted and so is everything that depends on it.

By day the room is the hidden side

For a white voile half open, a thread that throws back six tenths of the light falling on it, and a street thirty times brighter than the room, the numbers are stark. From the street, 1.6 per cent of the room’s contrast survives. From the room, 63 per cent of the street’s does. At a hundred times brighter, which is a sunny day against an ordinary lamp-lit room, it is half a per cent against sixty-five.

A person looking in sees a bright, even glow with the room’s furnishings reduced to a fiftieth of their contrast — below what an eye reads as anything but the faintest shape. A person looking out sees the street through a light haze.

Neither of those numbers is a property of the voile. Change the ratio and both move; change the cloth and the curve moves but its shape does not.

At equal light, every cloth is equally two-way

The crossing in the figure is at a ratio of exactly one, and it is worth saying why that point is special rather than a coincidence of the numbers chosen.

When the two sides are equally lit, a viewer on either side sees threads lit identically, a scene lit identically and the same holes. The two views are then the same view, whatever the cloth is made of and however open it is — a white voile, a black mesh, a heavy lace — and at equal light no sheer is one-way at all. For this voile both views keep twenty-eight per cent of their scene’s contrast; for a black one of the same construction, eighty-three.

The crossing is the one result in the model that needs none of its assumed numbers. The thread’s reflectance and forward scattering are estimates for a white sheer, a grey one and a black one, and they are stated as estimates in every figure. The crossing is a symmetry, and it holds for all of them.

After dark the curtain works the wrong way

Turn the lamps on and draw the street down to the light of the street lamps, and the ratio falls to a thirtieth or less. Looking in, 63 per cent of the room’s contrast now survives; looking out, 1.6. The occupant sees a glowing curtain and the passer-by sees the room.

The trade’s answer is well known and it follows from the arithmetic rather than from the cloth: a second, closed curtain or a blind drawn at dusk. No sheer can be chosen that keeps the room private both ways round, because at any ratio above one the room is hidden and at any ratio below one it is shown — and the ratio is what a lamp changes.

Seeing through a black voile, from the street and from the room. How much of a scene's contrast survives a black 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, 6.8% of the room's contrast reaches the street and 93% of the street's reaches the room. At equal light both are 83% — 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 93%. What the plot cannot show is the threads' own optics: their reflectance and transmittance are assumed values for a black sheer rather than measurements, and only the crossing point is independent of them.
Fig. 3 The same two views through a black voile of the same open area, forty-nine per cent. On a day a hundred times brighter outside, 6.8 per cent of the room’s contrast reaches the street and 93 per cent of the street’s reaches the room; the curves still cross at equal light, now at 83 per cent against the white voile’s 28, because a black thread returns almost no veil to either side.

The trade’s white net is the right colour

The model’s most practical result is about the colour of the thread rather than its openness, and it runs against the obvious expectation that a darker curtain is a more private one.

How much of a room a sheer shows, against its open area, at 30 times the light outside. The share of a room's contrast visible from the street through a sheer, against the sheer's open area, with the street 30 times brighter than the room, for white, mid-grey and black threads. At a voile's 49% open a white sheer passes 1.6% of the room's contrast, a grey one 3.7% and a black one 19%. Opening the cloth raises every curve, but the tone of the thread moves them further than any openness a sheer would be woven at: the veil a lit white thread throws back is what hides the room, and a black thread throws back almost none. What the plot cannot show is how visible a given contrast is, which depends on how large and how bright the thing behind the curtain is and on the eye.
Fig. 4 The share of a room’s contrast visible from the street, against the sheer’s open area, with the street thirty times brighter, for white, grey and black threads. At a voile’s openness a white sheer passes 1.6 per cent, a grey one 3.7 and a black one 19. A white sheer seventy per cent open still hides the room better than a black one thirty per cent open.

By day a black sheer of the same construction shows twelve times as much of the room as a white one — nineteen per cent of its contrast against one and a half. What hides the room was never the thread’s obstruction, which is the same in both; it is the daylight the thread throws back towards the street, and a black thread throws back almost none of it.

The same arithmetic says openness is the weaker lever. A white sheer opened from thirty per cent to seventy lets the room’s visible contrast rise from 0.7 per cent to 3.7. A black sheer at only thirty per cent open already shows 9.5. The tone of the thread moves the result further than any openness a sheer is woven at, which is why privacy nets are white and why nobody sells a black one for the purpose.

Why a dark screen is easier to see out of

The same figure read from the other side explains a choice made in exactly the opposite direction.

A white, a grey and a black voile, from both sides, at 30 times the light outside. Three voiles of the same construction and 49% open area, differing only in the tone of their threads, looked through from the street and from the room with the street 30 times brighter. The white one passes 1.6% of the room's contrast to the street and 63% of the street's to the room; the black one passes 19% and 93%. A black sheer is better to look out through and far worse at hiding the room, and it is worse for the same reason it is better: it throws back almost none of the daylight that would otherwise veil the view in both directions. What the bars cannot show is heat and fading, which a dark sheer in a sunny window buys along with its view.
Fig. 5 Three voiles of the same construction and forty-nine per cent open area, white, grey and black, looked through from both sides with the street thirty times brighter. The white one passes 1.6 per cent of the room’s contrast and 63 per cent of the street’s; the black one 19 and 93. The black one is better to look out through and far worse at hiding the room, for one reason.

Looking out through a black mesh the veil is dim as well, so the view is clearer: 93 per cent of the street’s contrast against 63 through white. That is why insect screens and the solar screens sold for keeping a view are dark. They are chosen to be looked through, and what makes them good to look through is exactly what makes them poor to hide behind — the two cannot be separated, because both are the thread’s reflectance and the thread has only one.

From the pavement the holes close first

A passer-by rarely looks at a window square-on. They look in at an angle, from a pavement below and to one side, and the angle works against them twice.

A cloth has a thickness, so a line of sight arriving obliquely has to clear a hole at the front of the cloth and the same hole one thickness behind, and the clear area narrows as the angle grows. For a voile the clear lines across the ends close entirely at forty-seven degrees. But every part of the view that is not hole is thread, and the thread stays lit however steeply it is seen.

Looking into a sheer from the pavement, at 30 times the light outside. The share of a room's contrast visible from the street through a white sheer, against how far off square the viewer stands, for cheesecloth, voile, muslin. Off the normal a line of sight has to clear a hole at the front of the cloth and the same hole one thickness behind, so the clear lines narrow and close — the cheesecloth at 62°, the voile at 47°, the muslin at 36° — while every thread in the way stays lit from the street. Square-on the three pass 3.0%, 1.6%, 1.0%; at thirty degrees 1.3%, 0.5%, 0.1%. What the plot cannot show is that a real viewer's angle and a real window's light change together as a person walks past, so the curve is swept rather than sat on.
Fig. 6 The share of a room’s contrast visible from the street through three white sheers, against how far off square the viewer stands, with the street thirty times brighter. A voile passes 1.6 per cent square-on, 0.8 at twenty degrees and 0.2 at forty, and none at all past forty-seven; a muslin closes before forty, and a cheesecloth’s larger holes keep a trace past sixty.

Square-on a voile shows 1.6 per cent of the room’s contrast; from twenty degrees, half that; from forty, an eighth. The image is narrowing while the veil holds, so the contrast falls faster than the open area does. A curtain is therefore more private to the pavement than to a house across the road, and a hole that is a channel rather than an opening closes sooner still.

The hairs are on the privacy side

The opacity essays found one more obstruction the covering rule leaves out: the fibre ends standing in the holes, which block a line of sight exactly as a thread does and cost nothing in air.

In this model a hair in a hole does two things, and both work against the passer-by. It removes a clear line of sight, so the image falls, and it is itself lit, so it joins the veil. Taking the hair layer’s useful obstruction on a shirting-weight cloth — about four points of cover, from that essay — off a voile’s clear area moves the room’s visible contrast from 1.58 per cent to 1.33, a sixth less.

So singeing a sheer makes it slightly less private and raising one makes it more, and neither changes the direction of the effect: a singed voile is still one-way by day and the wrong way round at night.

A second sheer halves the image and barely touches the veil

The commonest way to make a net curtain more private is to hang two, and the arithmetic says which of the two views the second layer actually buys.

The image now has to pass both layers’ clear lines, and two layers pass the product of their open areas on average over the ways they can lie — right as an average and at no registration in particular. For a pair of voiles that multiplies the image by a further half. The veil a viewer sees is almost entirely the layer on their own side, which is lit as it always was.

So from the street a second voile roughly halves what survives of the room, because it halves the image and leaves the street-lit veil in front of it unchanged. From the room the view out loses a similar share of its image, but against a veil lit only by the room, so the view out stays clear while the view in fades. A double sheer is a more one-way curtain, not merely a thicker one, and at night it is a more one-way curtain in the wrong direction.

It also adds something neither layer has on its own. Two sheers a few centimetres apart beat by perspective alone, and the bands that walk across a double net curtain as a person passes are the image and the veil of two layers interfering, at a scale set by the gap between them.

The same arithmetic as a one-way mirror

The police interview room’s mirror is the familiar version of this effect in glass, and it obeys the same rule for the same reason.

A partly silvered pane passes a share of the light and reflects a share, and the reflected share carries an image of the viewer’s own room — a veil that happens to be a picture. It is one-way only when the observed room is brightly lit and the observers’ room is dark; turn the observers’ lights on and the suspects see them through their own reflection. Nothing about the glass is one-way, and nothing about a sheer is either. Both are a transmitted image competing against a returned veil, and in both the side with the more light is the side that cannot be seen into.

The difference is in what is returned. A mirror returns an image of the near room, and a sheer returns an even glow. That makes a sheer the more forgiving of the two — a glow hides detail without showing the viewer a picture of themselves — and it is why a net curtain in a lit window at night reads as a lit room behind haze rather than as a mirror.

How the two quantities were found

The line of sight is a chord average. For a yarn of a given fibre count at a packing of six tenths, its width in fibre diameters follows from the packing; a line crossing it at a given offset crosses a chord of known length; fibres placed at random meet that line at a rate of four times the packing over π per fibre diameter, so it passes with the exponential of minus that rate times the chord; and the average over four thousand offsets across the width is the fraction passed. The fibre counts come from the yarns’ own tex and a cotton fibre’s fineness, rather than from a round number.

The contrast is one expression, image over image plus veil, with the open area taken from each cloth’s own construction and, for an oblique view, from the overlap of a hole’s front and back faces. What is checked about it is structural rather than numerical: that the two views are equal at a ratio of one for three openings, three thread tones and two scene reflectances; that by day the room is the less visible side and by night the more; that a darker thread shows more of the room at every openness; and that looking into a voile the contrast falls at every step of angle.

Where the model stops

The thread’s optics are assumed. A white thread returning six tenths of its light and passing fifteen hundredths, a grey one a quarter and eight hundredths, a black one four and two hundredths: these bracket real sheers and are measurements of nothing. Every number quoted above except the crossing moves with them, and the comparisons — white against black, day against night, square-on against oblique — are what survives.

The threads are treated as matt and the lighting as even. A shiny polyester sheer throws back light in a direction rather than evenly, so its veil depends on where the sun is; a window lit by a low sun is not evenly lit at all. Both change the veil’s size and neither changes which side it favours.

Nothing here says what an eye can see. A contrast of one and a half per cent is below what reads as detail in a room and above what reads as nothing in a bright patch; where the threshold falls depends on size, brightness and the viewer, and no model of vision is used.

And the holes are treated as geometric. A voile’s hole is about three tenths of a millimetre, and light passing an opening that small spreads by diffraction through an angle of about two thousandths of a radian — which at three metres blurs a point by several millimetres. That softens the image further for a distant viewer and is not included.

Still open: what a patterned sheer hides

Every sheer here is one construction throughout. A lace, a devoré or a net with a woven figure has its open area and its thread tone varying across the cloth, and by the arithmetic above each region hides a room to a different degree — so a figured sheer is a pattern in how much of the room shows through, drawn in daylight and erased at equal light. Whether a design can be made that reads as a figure from the street and as a plain haze from inside is a question about the two views at once, and it has not been worked out here.

Where this goes next

How open a batiste is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this batiste it is 40.1 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 39.0° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 6.42 per cent open — 6.3 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none. Opacity is not cover The covering rule counts a thread as a bar that stops everything, and a single fine cotton thread held to a window plainly does not. What a cloth transmits is the open area plus whatever comes through the threads, so the covered fraction is a lever rather than a barrier — and the lever is longest exactly where the rule says the cloth is most closed. Nothing here computes a thread's transmittance, and saying why is the useful half. A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument. A cloth is more opaque than it is closed Opacity is not cover — this collection established that already and left the discrepancy attributed to the thickness of the threads. Part of it is not in the threads at all. A hair standing in a hole blocks light exactly as well as a thread does and costs nothing in air. How open a muslin is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this muslin it is 37.9 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 36.1° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 5.36 per cent open — 7.1 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none. One minus the cover is a cloth with no thickness The covering rule says a cloth's openness is one minus its cover factor, and this collection derived it and has used it ever since. It is the answer for a light directly behind the cloth. Move the light and a line of sight has to clear the hole at the top of the fabric and the same hole one thickness below, so the openness falls, and it reaches nothing at thirty-six degrees. Averaged over the whole sky a muslin is a seventh as open as the rule says.

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 openingCover factorOpacityOpen areaScatteringTransmittanceVeil