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

Every calculation made here about a net curtain has been made on a flat piece of cloth. The open area is the fraction of a square centimetre that is hole, the veil is the rest, and the two are added with the light on each side. That is the cloth as it leaves the loom and as it lies on a table.

No curtain hangs that way. A curtain is cut wider than its window — half as wide again, twice, two and a half times, according to the heading and the taste — and the surplus is taken up in folds. The trade calls the ratio the fullness, sells heading tapes that produce a stated one, and describes it as a matter of appearance: a curtain at its window’s own width looks like a sheet, and one at two and a half times looks like a curtain.

It is not a matter of appearance. The fullness is an angle, it is the angle at which a line of sight crosses the cloth, and by the arithmetic already in hand for a view taken obliquely, the fullnesses in ordinary use are past the angle at which the cloth’s holes shut completely.

A fullness is a secant #

Take the curtain in plan — looked down on from above, with the window a straight line and the cloth a zigzag in front of it. Each flank of the zigzag is a straight run of cloth making some angle α with the window’s own plane.

A length $\ell$ of cloth lying at $\alpha$ spans $\ell\cos\alpha$ of window. The whole curtain is $fL$ of cloth across a window of span $L$, so

$$fL\cos\alpha = L, \qquad\text{and therefore}\qquad \cos\alpha = \frac{1}{f}.$$

Nothing else is in it. Not the cloth, not the heading, not the weight, not the drop. A fullness of 1.5 stands every flank at 48.2 degrees, a fullness of 2 at 60 degrees exactly, 2.5 at 66.4 and 3 at 70.5.

And a viewer on the pavement, looking straight through the window, meets the flank at that angle from its own normal. A gathered curtain is a curtain looked at obliquely, at an incidence set by a number printed on a packet of heading tape.

A hole is a channel, and a channel closes #

What an incidence does to a cloth is already settled. A hole is a channel, not an opening: the gap between four threads is a short prism one cloth-thickness deep, so a line of sight entering at an angle is offset sideways by the thickness times the tangent of that angle, and the clear part of the hole shrinks by the offset. Past

$$\alpha = \arctan\!\left(\frac{\text{gap}}{\text{thickness}}\right)$$

the offset exceeds the gap and the line of sight is blocked by the thread wall, which is to say the cloth is opaque along that direction however open it is along the normal.

Every quantity in that expression is computable from a cloth’s own construction. A voile’s clear gap is 0.287 millimetres and its thickness 0.265, so it shuts at 47.3 degrees. The fullness that reaches 47.3 degrees is its secant: 1.48.

A net curtain is hung at two or two and a half.

Not one cloth in the table reaches a fullness of one and a half before it shuts. The one that goes furthest is the voile, which is the openest and thinnest of them and the one actually used for net curtains; every heavier cloth shuts sooner, because thickness rises faster than gap as a cloth is closed up. So the general statement is not about a particular curtain: any cloth hung in the ordinary way has flanks that pass no line of sight at all.

Half of it is bought in the first eight per cent #

The fall is not gradual, and where it is steepest is the part a designer would not guess.

A voile’s contrast halves at a fullness of 1.079 — an eight per cent surplus of cloth, a flank angle of 22 degrees, and a curtain nobody would describe as gathered. Its open area halves at 1.16. By 1.3 the view is a fifth of what it was. By 1.48 the flanks are shut and the curve goes flat.

From there to two and a half times, which is where the trade’s advice actually sits, the view falls by a further quarter and no more: 4.2 per cent open at the closing fullness, 3.0 at two and a half, 2.8 at three. The last half of the cloth bought buys almost nothing, because the flanks it adds are already as shut as flanks get, and what it adds to the crests it takes away in how sharply they turn.

That inverts the way the quantity is usually discussed. Fullness is described as a dial running from thin to luxurious, and by this arithmetic it is a switch with a very short travel: the first tenth does half the work, the first half does nearly all of it, and everything past one and a half times is being bought for how the cloth hangs.

The veil does not close, which is why the privacy is more than the light #

Two things about a curtain fall as it is gathered and they do not fall together.

The open area falls because the holes shut. The veil — the light the threads themselves return and pass — does not fall at all: whatever the incidence, a thread still fills the part of the line of sight the hole has stopped filling, and it is still lit by whatever is on either side of it. Gathering a curtain takes the picture out of the view and leaves the glow.

That is exactly the mechanism a sheer hides whichever side is darker rests on, driven here by how the curtain is hung rather than by where the viewer stands. What a viewer sees of a scene is the image through the clear lines of sight divided by the image plus the veil, so a fall in open area with no fall in veil is a fall in contrast very nearly in proportion, and a curtain that has lost nine tenths of its holes has lost nine tenths of the picture while remaining exactly as bright a piece of cloth.

The two numbers separate the two claims. At two and a half times the voile’s open area is a sixteenth of its flat value and the contrast of the view is a twentieth — 1.58 per cent of the scene’s own contrast flat, 0.077 per cent hung. The cloth is not appreciably dimmer and it is twenty times harder to see through. A passer-by sees a curtain rather than a window, and the room behind it still has daylight in it.

What comes through is crest #

Past the closing fullness the flanks contribute nothing at all, and the arithmetic changes character: the whole of what a gathered curtain passes comes from the narrow bands where the cloth turns.

At a crest the cloth swings from one flank angle through the normal to the other, so the sightlines crossing it run over every incidence from the flank’s down to nothing. Those are the only places in a hung curtain where a hole is open. The crest is a small share of the span — taken here as a seventh — and it is open at about twenty per cent, giving the three per cent the whole curtain reads.

So a net curtain seen from the street by day is not an evenly hazy screen. It is a set of narrow bright lines, one at every fold, separated by cloth that is completely shut — and a figure woven into such a curtain is a figure read at one openness in its crests and at none in its flanks, and what a viewer can see of the room is confined to those lines. A curtain drawn back and re-hung moves them.

That is the honest weakness of the account and it is worth stating where it bites. Below the closing fullness the crest share hardly matters and the answer is the flank’s oblique opening, which is computed from the cloth. Above it the answer is proportional to the crest share, which is an assumption about how a pleat is shaped. Every ordinary curtain is above it. So the fall to the closing fullness is a result and the level afterwards is a parameter, and the two should not be quoted with the same confidence.

A heavier cloth shuts sooner and starts lower #

The whole argument is about gaps and thickness, so it applies to any cloth hung in folds and not only to the sheers a window carries.

The ordering across the table is not the ordering of openness. A duck is 30.4 per cent open flat and shuts at 1.15 times; a poplin is 34.0 per cent open flat and shuts at 1.12. The openest cloth is not the last to shut, because the closing angle is a ratio of gap to thickness and a duck’s coarse yarn gives it both a larger gap and a much larger thickness. A cloth’s flat open area and the fullness that shuts it are two different readings of the same construction and they do not rank together.

This is the same lesson the oblique view delivered about a cloth held up to a light: one minus the cover is the openness of a cloth with no thickness, and no cloth has none. Here the thickness decides not how much light a curtain passes but at what fullness it stops mattering how much it would have passed.

What this says about the two ways of buying privacy #

A window is made private in two ways, and the arithmetic says they are the same lever pulled at different points.

A closer cloth lowers the open area directly, and as a cloth closes its holes shut before it stops passing air — so a closer net is a dimmer room for a given privacy.

More fullness lowers the effective open area by turning the cloth, and leaves the cloth alone. The light that reaches the room through the crests is the light of a nearly-flat cloth; the light that does not reach it was stopped by the thread walls of holes it would otherwise have passed.

The second is much the cheaper of the two in light, because the shutting is a geometric effect rather than a material one — a hole that is shut to a normal line of sight is still a hole, still passing air, and still open to light arriving at the crest’s angles. A room behind a heavily gathered voile is not the room behind a cloth of three per cent open area; it is the room behind a cloth of forty-nine per cent open area, most of which is pointing the wrong way.

It is also why a curtain and a blind are different objects. A blind is flat by construction, so its privacy is entirely its cloth and it must be woven closed. A curtain gets its privacy from its hanging and can therefore be woven open — which is why net curtains are voiles rather than sheetings, and why a voile flattened against the glass is so much more transparent than the same voile hung.

The light falls twentyfold and the warmth rises by the fullness #

The same surplus of cloth is read completely differently by the three things a curtain is asked to do, and putting them side by side is the clearest statement of what gathering is.

Light: down by a factor of twenty, because a line of sight is a straight line and a shut channel stops it.

Warmth: up by the fullness, and by nothing else. A fabric is warm because of the air it holds still, and the still air in a hung curtain is the air in its cloth times however much cloth is in front of the window — so two and a half times the cloth is two and a half times the resistance, with no angle in it anywhere. A curtain’s insulation is linear in the very quantity its transparency is nearly independent of past 1.5.

Air: barely touched. A shut line of sight is not a shut channel. Air entering a hole at the flank angle turns inside the cloth and leaves through a neighbouring hole, and half of a cloth’s air goes through a tenth of its holes in any case — a flow finds the widest route and a photon does not get to choose one. So the draught through a gathered net is the draught through f thicknesses of a nearly-flat cloth, which is a fall of a factor of a few rather than of twenty.

That is an unusually clean separation of three properties a single construction is normally asked to trade against each other. A designer choosing a cloth trades them: a closer weave is warmer, darker and less permeable together. A designer choosing a fullness does not trade at all — the light is bought in the first half-turn, the warmth goes on accruing for as long as cloth is added, and the air hardly notices either. The fullness is a lever the cloth does not have, and it is set by a tape rather than by a loom.

What was computed, and how #

The flank angle is the arccosine of one over the fullness, derived from the arc length rather than assumed: a zigzag at that angle laid across a metre of window is checked to be exactly f metres of cloth, at four fullnesses. The oblique opening at that incidence is built here — the clear gap less the thickness times the tangent, in each direction, over the cell — and the closing angle is the arctangent of gap over thickness. The crest is averaged over sixty-one incidences from the flank angle down to nothing, and the curtain’s reading is the flank and crest weighted by their shares of the span. The contrast is the same veiled view every sheer result here uses, with the image the oblique opening and the veil the rest.

Four things are checked. A curtain at a fullness of one is the flat cloth, to the last bit, in both open area and angle — the check that the whole apparatus reduces. Gathering never opens a cloth and never makes it easier to see through, at every step of the sweep, which is the check on the oblique opening being the right way up. A fold is not its own mean angle, because the oblique opening is a clipped linear fall rather than a straight line, so averaging over the crest and evaluating at the crest’s mean incidence give different answers. And past the closing fullness a flank passes nothing at all, tested five per cent beyond it.

The cloths, their setts, their counts and their thicknesses come from this collection’s own table. The crest share and the light ratio are inputs.

Where the model stops #

A fold is not a corner. The plan here is a zigzag of straight flanks meeting at points, and a real curtain hangs in rounded folds whose curvature is set by the cloth’s own bending stiffness and its weight. A rounded fold spreads the crest’s band of incidences over a wider share of the span and removes the discontinuity at the flank angle, which would soften every curve here without moving the closing fullness — that is a property of the cloth and not of the fold.

The crest share is assumed. It is the one number in the account with no cloth in it, and past the closing fullness it is the only number that matters. A proper treatment would solve the fold’s shape from the cloth’s bending stiffness and the heading’s spacing, which is the same elastica problem a thread between two crossings solves, one scale up.

The folds are vertical and the sightline is horizontal. A viewer standing to one side of a window meets the two flanks of every fold at two different incidences — one closer to the normal, one further from it — so a gathered curtain looked at obliquely is more transparent on one flank of every fold than on the other. That is a real and visible effect, it is why a gathered net looks striped from an angle, and it is not computed here.

Nothing here is about the light that travels along the folds. A curtain is a set of cloth channels, and light entering a crest at a shallow angle can run down between two flanks and emerge somewhere else. Every line of sight in this account is straight.

And a curtain moves. The whole calculation is for a curtain at rest with its folds evenly spaced, and a draught, a radiator or an opened door changes every flank angle at once.

Still open: whether a heading tape’s fullness is the cloth’s fullness #

The relation between fullness and flank angle takes the whole surplus into the folds evenly. A heading tape does not: a pencil pleat gathers the cloth into tight bunches at the tape and lets it fall open below, and a pinch pleat takes its surplus in three folded returns at intervals with flat cloth between them. Both leave the fullness at the top quite different from the fullness a metre down, and the flat regions of a pinch-pleated curtain are cloth at no incidence at all — fully open, in bands, where the zigzag model has none.

So a pinch-pleated curtain should be measurably more transparent than a pencil-pleated one of the same fullness, in bands whose width is the flat run between pleats. What that does to the numbers here is a weighted average over a heading’s own geometry rather than over one angle, and which heading gives the most privacy for the least cloth is a question the arithmetic is ready for and has not been asked.

Who worked it out #

Fullness is trade practice with a long history and a precise vocabulary — tapes are sold for stated ratios and curtain makers quote them — and it is universally explained as an effect on appearance and drape. The oblique opening of a woven cloth is new here, computed for a viewer standing to one side of a window; turning it round, so that the cloth moves and the viewer stands still, is what this account does with it. The identity between fullness and the secant of the flank angle, the closing fullness of each cloth, and the finding that ordinary curtains are hung well past it were computed directly.