Textile Structure
how cloth is put together
Series

Opacity — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · pattern
  2. Two layers of one cloth, against how they happen to lie. Two identical muslins laid over one another and slid across each other by one thread spacing. In register the pair passes 37.9 per cent — as much as one cloth, because every hole is over a hole — and it falls linearly to 12.5 per cent before rising again at the next thread. The rule everybody uses is that two layers pass the product of their open areas, which is 14.3 per cent. That number is the average of this curve over all offsets, exactly — an identity, not a fit — and it is the answer at two points on it and nowhere else. Nothing about a real pair of layers is at its average, and the openness varying from place to place across a folded cloth is what a moiré is.

    Two layers are the product on average and nowhere

    Everybody knows what two layers of a cloth pass: the product of their open areas. That figure is exactly right — it is the mean of the true answer over every way the two layers can lie — and it is the answer at two registrations out of a continuum. In register a doubled cloth is as open as a single one; half a thread out it can be shut completely. The variation across a folded curtain is what a moiré is.

    part 2 · pattern
  3. 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.

    part 3 · pattern
  4. 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.

    part 4 · pattern
  5. 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.

    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.

    part 5 · pattern
  6. A white thread 70% open figure behind a net 60% open, from both sides. A lozenge figure in the inner curtain of a pair, drawn as the street sees it and as the room sees it through a plain net 60% open, with the street 30 : 1 as bright as the room. From the street the figure's contrast against its ground is −7.0% where a single curtain would have given −24.6%; from the room it is +6.9% where a single curtain would have given +5.4%. Each panel is shaded relative to its own brighter region, both at one gain, so the two steps are to scale against each other, and the contrast is stretched 9 times to be visible at all — the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the light between the two curtains, which is 19.80 times the room's illuminance from the street and 0.57 from the room, nor the thread optics, which are assumed.

    A net in front gives the figure to the room

    The commonest double curtain is a plain net outside a patterned one, and the plain net does not merely dim the pattern. It weakens the figure from the street by a factor of three and a half and strengthens it from the room by a quarter, so a pattern four and a half times stronger outside than in becomes one the room sees slightly better. A figure designed to be invisible from indoors reappears at nearly three per cent, and the day-and-night exchange a single curtain obeyed exactly stops holding at all.

    part 6 · pattern
  7. A voile hung at 2.5 times its window, seen in plan. A curtain of voile gathered to 2.5 times the width of its window, drawn in plan with the window above it and a line of sight crossing a flank. A length of cloth spans its own length times the cosine of its flank angle, so a fullness of 2.5 stands every flank at 66.4 degrees and a line of sight normal to the window meets the cloth at that incidence. This cloth's holes close completely at 47.3 degrees, which is a fullness of 1.48, so at 2.5 times every flank passes no line of sight at all and the whole of what comes through arrives at the crests. Flat the cloth is 49.3% open and hung it is 3.0%. What the plan cannot show is the cloth's own drape, which rounds every fold drawn here as a corner.

    A curtain is gathered so that it is seen edge-on

    A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

    part 7 · pattern
  8. What a passer-by is looking at, for rooms of four reflectances. The street's view of a window behind a sheer white thread 49% open at 30 : 1, split into the part that is an image of the room, arriving through the holes, and the part that is the cloth's own threads lit by the street. a room reflecting 8% contributes 0.4% of the view; a room reflecting 20% contributes 1.0% of the view; a room reflecting 35% contributes 1.8% of the view; a room reflecting 60% contributes 3.1% of the view. The veil is identical in every row because the cloth and the street are: only the furniture changes. What the chart cannot show is where in each room that reflectance sits, and a dark room with a lit lamp in it is not its own average.

    What a sheer hides is decided by the furniture

    Every result these essays have produced assumed the room and the street reflect the same three tenths of the light on them. That was never a fact about cloth. A room reflecting a twentieth is hidden two hundred and thirty-eight times better than the street it faces and one reflecting four fifths only fifteen times, with the same curtain at the same window — and the day-and-night exchange they rested on, exact for equal scenes, is out by more than half the glow for an ordinary pair of unequal ones.

    part 8 · pattern
  9. What a sheer multiplies a street reading's errors by. The factor by which an error in reading the light from a window is multiplied on its way into the room's reflectance, against the street-to-room light ratio, for three sheers and a room reflecting three tenths. It is the reading over the room's own image, the reciprocal of the share of the view that is room. white voile: 67.3 at 30 : 1, 3.6 at equal light, 1.6 at night; grey voile: 28.7 at 30 : 1, 2.1 at equal light, 1.3 at night; black net: 2.1 at 30 : 1, 1.1 at equal light, 1.0 at night. What the chart cannot show is the thread optics, which are assumed values here and not measured ones.

    A sheer's privacy is the error it multiplies

    Everything a passer-by sees of a room through a sheer is the room's image plus a veil the street lights, and the veil can be known without going in. So one subtraction and one division ought to read the room's reflectance from the pavement. They do not, and the reason is the curtain's whole purpose: every error in the veil arrives in the answer multiplied by the reading over the image — sixty-four through a white voile by day, three and a half at equal light. The privacy a sheer gives and the precision it denies are one number.

    part 9 · pattern
  10. What a pane of glass does to a sheer's multiplier. The factor by which an error in reading a window is multiplied on its way into the room's reflectance, against the angle a passer-by looks at the glass from, at 30 : 1, for three sheers behind a pane of window glass. The dashed level beside each is the same cloth with no glass. white voile: 64.0 bare, 59.8 behind glass straight on, 65.7 at 60°, 92.7 at 75°; grey voile: 27.3 bare, 28.7 behind glass straight on, 34.7 at 60°, 61.7 at 75°; black net: 2.0 bare, 5.2 behind glass straight on, 8.8 at 60°, 25.4 at 75°. The white voile is easier to read through glass up to 56.7°, because the pane dims the street on its threads by more than it mirrors; the black net is harder at every angle. The pane mirrors a street reflecting 0.3; a sky in the mirror would be brighter, and nothing here models one.

    Glass hides a black net and not a white voile

    A window pane in front of a curtain mirrors the street, and a mirror is a veil with no thread in it. It also dims the street's light on the cloth. For a white voile the two nearly cancel, and straight on the dimming wins, so the voile is slightly easier to read through glass than without it. For a black net the mirror is most of what a passer-by sees: its multiplier goes from 2 to 5 head-on and to 25 from along the pavement. The glass makes the net private, not the voile. The angle at which it stops helping does not depend on the time of day, and a polarising filter at Brewster's angle takes the mirror out altogether.

    part 10 · pattern
  11. What an upper window's glass does to a white voile. How many times harder a room is to read through a white voile behind glass than through the bare cloth, against how far the passer-by stands out from the façade, for a window centred 1.5, 4.5 and 7.5 m up in a street twelve metres wide between ten-metre terraces, under an overcast sky, at 30 : 1. Below one the glass helps. ground floor: 0.95 from 2 m, 0.95 from 5 m, 0.95 from 9 m; first floor: 2.34 from 2 m, 1.55 from 5 m, 0.94 from 9 m; second floor: 4.92 from 2 m, 1.86 from 5 m, 1.49 from 9 m. The first floor's curve drops by a step at 6.3 m, where its pane stops mirroring the sky and starts mirroring the terrace opposite. What the chart cannot show is a clear sky, which is brighter again near the horizon.

    An upper pane mirrors a sky no curtain can afford

    A pane in front of a sheer helps a reading only while what it mirrors is dim: for a white voile, below about half the brightness of a white wall lit like the window. A street-level pane mirrors the terrace across the road and stays under that. An upper pane, seen from the pavement below, mirrors the sky, and an overcast horizon — the darkest sky there is — is already twice the limit. Whether a first-floor pane shows sky depends on how far out from the façade the passer-by stands and on nothing else, so the same window is harder to see into from close up than from across the street.

    part 11 · pattern
  12. What tilting a window pane does to a curtain behind it. How many times harder a room is to read through a white voile behind glass than through the bare cloth, against the tilt of the pane — leaned forward to the left, tilted back to the right — for windows centred 1.5, 4.5 and 7.5 m up, seen from 3 m out in a street twelve metres wide between ten-metre terraces, under an overcast sky. Each curve jumps where its pane starts to mirror the sky: the ground floor tilted back past 18.6°, the first floor leaned forward less than 9.7°, the second floor leaned forward less than 25.6°. Upright the three read 0.95, 1.82, 2.89; leaned forward 26°, 0.97, 0.94, 0.93. What the chart cannot show is a canopy or blind over a shop window, which a pane leaned forward would mirror instead of the street.

    A pane leaned forward loses the sky and finds no darker mirror

    A pane tilted by some angle sends a passer-by's line of sight off at twice that angle more, and the passer-by's place along the street still drops out. Tilted back about twenty degrees, a street-level pane mirrors sky from everywhere in the street and nearly doubles what a white voile hides. Leaned forward ten degrees, a first-floor pane stops mirroring sky from three metres out, and the voile behind it reads better through glass than bare. Leaning further finds nothing darker: under an overcast sky the road is a brighter mirror than the terrace opposite, by a seventh.

    part 12 · pattern

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