One minus the cover is a cloth with no thickness

Holding a fabric up to the light is the oldest test there is, and this collection has taken it seriously. Thread count is not quality rests on it: what a person judging a sheet is actually seeing is the open fraction, which is (1 − K)² for a square-set cloth and therefore moves as the square of the cover — so two two-hundred-count cloths in yarns a factor of two apart in diameter cover 56 and 89 per cent of their surface, an unimpressive ratio of 1.6, while the gap between the threads goes from 44 per cent to 11, which is exactly a factor of four.

That argument is correct and this essay does not withdraw it. It has an assumption in it that has never been stated, because it never had to be: the light is directly behind the cloth.

A cloth has a thickness. Its hole is a rectangle at the top face and the same rectangle a third of a millimetre below, so a line of sight arriving at an angle has to clear both.

The claim #

One minus the cover is a cloth’s openness at normal incidence, and a cloth’s openness to the whole sky is far less. The difference is entirely its thickness, and it is a factor of seven for an ordinary shirting.

The angle at which the view closes completely is atan(g/t) — the clear gap over the cloth’s thickness — and for the eight cloths of this collection’s table it runs from 36 to 62 degrees. No fabric here can be seen through at all past sixty-two degrees off the normal, however open it is.

The argument #

Take the tilt in the plane across the ends. A ray entering the top of the hole is displaced by t·tan α by the time it reaches the bottom, so what it can pass is the overlap of the hole with a copy of itself shifted by that amount: a clear width of g₁ − t·tan α rather than g₁, and nothing once the displacement exceeds the gap.

For a general direction the displacement has components in both directions, so

$$\text{open}(\alpha,\phi) = \frac{\max(0,\,g_1 - t\tan\alpha|\cos\phi|)\cdot\max(0,\,g_2 - t\tan\alpha|\sin\phi|)}{p_1 p_2}$$

which is the covering rule at α = 0 and is less than it everywhere else.

Averaging over the hemisphere needs a weight and the weight is Lambert’s. A surface intercepts a slanted beam over a larger area than a normal one, in proportion to the cosine of the angle, so the contribution of each direction is weighted by cos α. That is quoted rather than derived, and it is the only physics in the whole calculation — the rest is the intersection of two rectangles.

The result for a muslin is 5.36 per cent against a normal-incidence 37.88, a ratio of 0.14.

The eight cloths, and what decides the ratio #

The ratio is not a constant and it is not close to one. It runs from 0.42 to 0.06, so the correction the covering rule needs is not a factor that could be absorbed into a calibration: it is a factor that varies sevenfold across ordinary cloths, and it varies in the same direction as the cover, which makes the error worst exactly where the rule is being used to say a cloth is closed.

What sets it is the slenderness of the hole — the thickness over the gap. It runs from 0.53 for a cheesecloth to 2.24 for a sheeting, and the ratio column above is very nearly a function of it alone: 0.42 at 0.53, 0.14 at 1.37, 0.06 at 2.24. A cheesecloth’s hole is 795 µm wide in a cloth 420 µm thick, so it is a window; a sheeting’s is 170 wide and 382 deep, so it is a chimney. And closing a cloth does both things at once: it narrows the hole, and because the crimp deepens it lengthens the passage.

So the covering rule understates how much a close cloth hides, and it understates it by more the closer the cloth is. That is the direction that flatters nobody: a sheeting held to the light shows 1.6 per cent of the sky through it, and the rule says 24.5.

The poplin, where the two directions disagree #

One row of the table has a feature the others do not, and it is the row this collection keeps returning to.

A poplin is 32 ends by 22 picks — warp-dense by design — so its two gaps are different and its two closing angles are different: 27 degrees across the ends and 41 along the picks. The cone of directions it can be seen through is not a cone at all but an ellipse, half as wide one way as the other.

That has a visible consequence and it is one a draper would recognise. Turn an unbalanced cloth ninety degrees in front of a window and its apparent openness changes, because the direction the light is coming from relative to the warp has changed. A balanced cloth does not do this. It is the same asymmetry that makes a twill line’s angle depend on the two setts, arriving in a property nobody thinks of as directional.

And it says that a fabric’s openness is not one number even at one angle. Every entry in the diffuse column above is an average over azimuth as well as over polar angle, and for the poplin that average is hiding a factor of two.

What was counted, and how #

The hemisphere is integrated numerically — two hundred and forty steps in polar angle by sixty in azimuth, over one quadrant, since the cell has both mirror symmetries — with the integrand computed from the closed form above and the weight cos α sin α dα dφ.

Nothing about the answer depends on the step count, and the check on that is not a convergence study but an assertion: a cloth is never more open to the whole sky than to a light directly behind it. That is true by construction, it would fail immediately if the weighting or the quadrature were wrong in a way that let a direction contribute more than its normal-incidence value, and it costs one line.

The closing angles are exact rather than sampled — atan(g/t) in each direction — so the two ways of finding where the curve reaches zero agree by construction and the sampled integral does not have to resolve the corner.

Where the model stops #

The hole is treated as a prism and it is an hourglass. A channel through a cloth has a waist and is wider at its mouths, so the true straight-through region at oblique incidence is a little larger than a shifted rectangle. That makes the diffuse openness an underestimate.

The threads are opaque. They are not, and the consequence is large enough to need its own argument.

Scattering is absent entirely. Real light through a fabric is not a set of straight rays: it is refracted and scattered by every fibre it meets, so a cloth glows rather than showing a grid of bright squares, and at any distance the eye integrates the two. Everything here is about the geometric component.

The thickness is Peirce’s and is a model output. Everything here scales with it directly, and the two section models disagree about cloth thickness by a third while agreeing about the jammed sett to an eighth. So the slenderness — which is the quantity this whole essay turns on — is the one geometric ratio where the choice between Peirce’s circle and Kemp’s racetrack matters most, and both would give the same normal-incidence cover.

And a cloth is not flat. A hanging fabric curves, folds and drapes, so the angle between the light and the surface varies across it — which is why a curtain shows its openness at the top of a fold and hides it in the trough, and why a drape coefficient has anything to do with how a fabric looks.

The generalisation #

Any quantity defined as a projection is a normal-incidence quantity, and applying it to a solid gives an answer that is right along one axis and optimistic everywhere else.

Cover factor, packing fraction, areal density, the shadow of a grating, the fill factor of a sensor, the aperture ratio of a display: each is an area computed from a plan and each is used, routinely, as though it described what gets through. The correction is always the same shape — a term in the depth over the aperture — and it is always in the direction of less.

The diagnostic is the slenderness. Divide the depth by the aperture. If the ratio is well below one, the projection is the answer and the depth can be ignored. If it is of order one, the projection is wrong by a factor of a few. If it is above one, the structure is a set of tubes and the projection is describing a different object. A muslin’s is 1.4 and a sheeting’s is 2.2, so every fabric in this table is in the second or third case, and the rule has been used in all of them.

The second lesson is about the direction of the error. The covering rule fails worst where a cloth is closest, and closest is where somebody is trying to demonstrate that a fabric is good. An error whose size grows with the quantity being claimed is the most dangerous kind there is, because every attempt to verify the claim by making the effect larger makes the error larger too.

What this does to the sheet-shop argument #

The thread-count argument is worth re-running with the correction in it, because the conclusion strengthens rather than weakens.

Two two-hundred-count cloths in yarns a factor of two apart in diameter cover 56 and 89 per cent of their surface. Their normal-incidence open fractions are 44 and 11 per cent — the factor of four the original argument turns on. Their thicknesses also differ, because the coarser yarn makes a thicker cloth, so the coarse cloth’s holes are both smaller and deeper and its slenderness is worse in both terms.

So the diffuse ratio penalises the coarse cloth twice, and the gap between the two cloths as they would actually be seen is wider than the factor of four the covering rule gives. The rule was already enough to make the point, and the correction is in the direction of making it more sharply.

What does not survive is the arithmetic being quoted as a percentage anybody could check. Forty-four per cent open is a number a person could imagine holding up to a window and verifying. Nothing they will see corresponds to it. What they will see, in ordinary room light, is a few per cent — and how few depends on the geometry of the room as much as on the cloth.

The same factor of seven, seen from a room #

Openness is reciprocal. A direction that a ray can take through the cloth going one way is a direction it can take coming back, so the fraction of the sky a fabric admits is also the fraction of a lamp behind it that reaches an eye in front. That symmetry turns the table above into a statement about two everyday tests that everybody treats as the same test.

Holding a cloth up to a bare bulb reads the first column. Holding it up to an overcast window reads the second. A small source directly behind the fabric puts almost all of its light into the narrow cone around the normal, where the cloth is at its most open, so what comes through is very nearly one minus the cover. A window under a grey sky is a large, nearly uniform source filling the whole hemisphere, and the cloth answers it with the cosine-weighted average — a seventh as much for a muslin, a sixteenth for a sheeting.

So the two tests do not disagree by a little. They disagree by the whole factor this essay is about, and which one a person happens to perform is decided by the room they are standing in. A shop lit by a bank of diffusers is running the second test; the same shop with one spotlight behind the goods is running the first. Neither is wrong about the cloth, and neither answers the other’s question.

It also explains a familiar asymmetry in curtains. A closed curtain in daylight shows the window as a bright shape and the rest of the room as nothing, because the window is the small source and everything else is the diffuse one — the curtain is running both tests at once on different parts of its own surface. At night with the room lit, the same curtain seen from outside glows evenly and shows no shapes at all, because now the source is the room and the room is diffuse in every direction.

Finishing moves both terms of the slenderness together #

The slenderness is the cloth’s thickness over its clear gap, and every process that flattens a cloth changes both of them, in the same direction, at once.

Calendering presses a cloth between rollers. A flattened thread is wider and shallower than a round one, so the cloth’s thickness falls — and the width the thread occupies grows, which closes the gap. The first change lowers the slenderness and the second raises it, and they do not cancel: the thickness is roughly the sum of two thread depths while the gap is a difference between the pitch and a thread width, so a small flattening moves the gap by a larger fraction than it moves the thickness.

The consequence is worth stating because it runs against the intuition. A calendered cloth is more closed to the sky than the covering rule’s change suggests, even though it is thinner. The rule reads only the plan, sees the threads spread wider, reports a higher cover and a lower openness, and stops there; the diffuse figure falls further, because the hole that narrowed is still being looked through at an angle.

The opposite process runs the other way with the same doubling. A cloth pulled on the bias opens its cells without changing its threads, so the gap grows while the thickness is unchanged, and the slenderness falls in one term only. That is why a fabric stretched over a frame is visibly more transparent than the same fabric lying flat, and why the effect is much stronger than the change in cover factor accounts for — the covering rule sees a few per cent and the eye sees a great deal more.

Neither of those is a correction to the rule. They are two more cases of the same finding: the covering rule reads a plan, and every process that changes a cloth’s depth changes something the plan cannot see.

Who found it, and when #

The cover factor is Peirce’s, from the same 1937 paper as the geometry, and the open fraction as the product of the two directions’ complements is elementary. Both are used here as they always have been.

Lambert’s cosine law is 1760.

What appears to belong to this collection is the observation that a fabric’s projection and its transmission are separated by a factor that varies sevenfold over ordinary cloths, and the naming of the quantity that decides it — the ratio of the cloth’s thickness to its clear gap, which this site already computes for other reasons and which nothing else in textile physics appears to use. It is a number available for every fabric on the site, it has never been printed, and it explains why a cloth held to a window and the same cloth against a lamp read so differently.

Where the ladder goes next #

The threads being opaque is the other assumption in the covering rule and it is the larger one for a fine white cotton: opacity is not cover, because a thread transmits, and the covered fraction is a lever rather than a constant.

Sideways, the same geometry with two cloths instead of one produces a rule everybody uses and nobody has checked: two layers are the product on average and nowhere, and where they actually land depends on how they happen to lie.