Two layers are the product on average and nowhere

Fold a piece of muslin in half and hold it to the light. The obvious arithmetic is that two layers pass the product of their open areas — 0.379 times 0.379, which is 14.3 per cent — and the obvious arithmetic is used everywhere, for curtain linings, for down-proof cambrics, for the number of plies a filter needs.

Now slide one layer over the other by two tenths of a millimetre and look again. Nothing has changed about either cloth, and the pair has gone from 37.9 per cent open to 12.5.

Fourteen point three per cent is the average of that. Exactly the average — not approximately, not to a good approximation, but precisely, and the exactness is the finding rather than an accident. It is also the answer at two offsets out of a continuum, and nothing about a real pair of layers puts it at either of them.

The claim #

The open fraction of two stacked cloths is a piecewise-linear function of their registration whose mean over all offsets is exactly the product of their open areas, whose maximum is the open area of one cloth, and whose minimum is zero for any cloth that is more closed than open.

The familiar rule is therefore not wrong — it is a statement about an expectation, being used as a statement about an object.

The argument #

Take one direction at a time. Two identical slit trains of period p and clear width g, offset by δ. The overlap of a slit with the slits of the other train is

$$f(\delta) = \frac{\max(0,\,g-\delta') + \max(0,\,g-(p-\delta'))}{p}, \qquad \delta' = \delta \bmod p$$

which is a triangular wave: g/p at δ = 0, falling linearly, flat at zero if the cloth is more closed than open, and rising again to g/p at δ = p.

Integrate it over one period. The two triangles each contribute g²/2, so the mean is g²/p² — which is (g/p)², the square of one train’s open fraction.

The two directions are independent, so the two-dimensional open fraction is the product of the two one-dimensional ones and its mean over both offsets is the product of the two means. That is the product of the two cloths’ open areas, and it is exact.

The exactness is worth a sentence. It is not an approximation that happens to be good, and it does not depend on the cloth being open or closed, balanced or not, or on the two layers being the same. Two different cloths, at unrelated setts, still average to the product of their open areas. It is the same fact that makes an expectation multiply for independent events, arriving as a statement about geometry.

When two layers can be shut completely #

The minimum of the triangular wave is zero whenever 2g ≤ p — that is, whenever the cloth’s clear gap is less than half its spacing, which is the same as a cover factor of a half or more.

So there are two regimes and the boundary is a cover of 0.5:

An open cloth cannot be shut by doubling it. A muslin’s cover is 0.40, so 2g > p in both directions and there is no offset at which the layers block each other completely. The worst a pair can do is 12.5 per cent in the plot above, against a single layer’s 37.9.

A close cloth can be shut completely, at the right registration. A sheeting’s warp cover is 0.55, so half a spacing out its two warps between them block every straight path, and the pair passes nothing at all through that direction.

That is the whole of why a lining works and a doubled scrim does not, and it is a threshold rather than a gradual thing: it turns on at a cover of a half and nowhere else.

What a moiré is, in this language #

A real pair of layers is not at one offset. If the two setts are even slightly different — because one cloth is stretched, or came off a different loom, or has been pulled off-grain — the offset drifts steadily across the cloth, so the openness runs through the whole of the curve above, over and over, at a spatial period that is the beat between the two setts.

That is a moiré, and it is the same beat this collection has already computed for watered silk and for the reed’s own mark. What is new is what the beat is a beat in: not a pattern of colour or of relief, but of transmitted light, running from as open as one cloth to as closed as the pair can get.

It also explains the everyday observation that a moiré in a folded net curtain is far more visible against a window than in reflected light. The contrast in transmission is the full range of the curve; the contrast in reflection is a much smaller thing.

What it means for a specification #

Three places where the difference between the mean and the object matters, and they pull in different directions.

A curtain lining is specified on the pair. A lining is bonded, sewn or hung at whatever registration it lands at, and the product rule is what the specification is written from. If the two cloths are at similar setts the registration will drift across the panel and the average over the panel really is the product — so here the rule is right, for the reason it is usually assumed to be wrong: the nuisance parameter genuinely is being averaged over, by the cloth itself.

A quilted or laminated pair is not. Two plies stitched or bonded together hold their registration, and a pair that was assembled in register is as open as a single layer for its whole life. That is a factor of 2.6 against the specification, in the direction of the assembly being worse than promised, and no test on either cloth alone would show it.

A doubled filter is worse than either of these, because what matters there is the largest hole through the pair, which is a maximum over offsets rather than a mean. Two layers in register have exactly the largest hole of one layer; out of register they have less. So a stacked filter’s rating is set by its worst-registered patch, and a large sheet has one.

What was counted, and how #

The mean is computed numerically over a grid of offsets and asserted against the product to a tolerance that scales with the grid, rather than being asserted equal — because the integral is exact and the sum over a grid is not, and asserting an exact equality between them would be asserting the grid.

Two further checks are made and each catches a different error:

• In register the pair is exactly one cloth. That is the value at δ = 0 and it must equal a single layer’s open area to the last digit. A version of the overlap function that reserved the wrong half of the period would fail this and would still average correctly.
• The pair can be shut only when the cloth is more closed than open, which is asserted as the condition 2g ≤ p rather than by looking for a zero in a sampled curve.

The tolerance on the mean is four over the square of the sample count, which for two hundred samples per direction is one part in ten thousand, and the agreement is at that level.

The two directions do not fail together #

The threshold at a cover of a half is stated per direction, and a cloth has two — which means there are three regimes rather than two, and the middle one is the interesting one.

Both covers below a half and no registration shuts the pair in either direction, so the doubled cloth always passes something. A voile is here.

Both covers above a half and there are registrations that shut it completely, because either direction alone suffices.

One above and one below, which is where an unbalanced cloth sits, and the pair can be shut by its close direction and never by its open one. A poplin at 32 × 22 is exactly that: warp-dense enough to block across the ends at the right offset, open enough along the picks that no offset blocks there.

The consequence is that an unbalanced cloth doubled has a preferred direction of failure, and it is the same direction the cloth is already dense in. Two poplins laid warp-to-warp can be shut; the same two laid warp-to-weft cannot, because the close direction of one is now facing the open direction of the other and no straight path is ever blocked by two dense systems at once.

That is a practical instruction with no measurement in it. Two plies of an unbalanced cloth should be laid crossed if the pair is meant to pass light evenly, and aligned if it is meant to block it — and the trade’s habit of laying a lining on the grain is the second choice made for a different reason.

It also sharpens the moiré argument. A drifting registration in an unbalanced pair runs through the whole curve in one direction and through a much shallower one in the other, so the beat is strongly directional: the bands run one way and are far more visible than the ones running the other. Anybody who has seen a net curtain folded knows the pattern has a grain, and this is where the grain comes from.

The three regimes also settle what a doubled cloth is worth as a specification, and the answer is that only one of them is specifiable at all. In the first the pair’s openness is bounded away from zero whatever happens, so a product-rule figure is at least the right order. In the third the pair’s openness in one direction runs to nothing at some registrations and not at others, so a single figure is describing a quantity that varies across the panel by everything it has. A specification written from the product rule is safe for an open cloth and meaningless for a close one, which is the reverse of where anybody would expect a rule about blocking light to be reliable.

And it says which measurement to take instead. For a close pair the useful number is not the mean but the range over one period of registration — a maximum and a minimum — which is two readings on a sample slid by one thread spacing, and which nothing in the trade’s practice asks for.

Three layers, and why the rule survives #

The natural next question is whether the same thing happens with three, and the answer is a little different in a way that is worth a paragraph.

With n layers the open fraction is the overlap of n slit trains, which is the minimum-width intersection of n shifted intervals. Its mean over independent offsets is still the product of the n open fractions — the same expectation argument, applied n times — so the rule survives at every depth.

What changes is the spread. Two layers can be as open as one; three layers can too, but only if all three are in register, which is a measure-zero coincidence for independent offsets and an ordinary occurrence for a stack folded from one piece. And the probability that a stack is shut rises steeply with n once the cloth is more closed than open, because it takes only one badly registered pair to close a path.

So a deep stack is nearly always at its mean and a shallow one is nearly never — which is the ordinary behaviour of an average over many independent draws, arriving in a place where nobody is thinking about draws at all. The product rule is a law of large numbers for plies.

Where the model stops #

The threads are opaque and the cloths are flat. Both of those matter more here than in a single layer, because two layers of a semi-transparent thread transmit the product of two thread transmittances in the covered region — so the real curve sits above this one everywhere, by an amount that depends on a quantity nothing here computes.

The layers are treated as coplanar and they are not. Two fabrics in contact are separated by their own surface roughness, and a ray at any angle other than normal walks between them — which means the effective offset is a function of the viewing angle as well as of the registration. That is the oblique argument applied to the gap between layers rather than to the thickness of one, and it makes the true two-layer behaviour more averaged than this and therefore closer to the product.

The setts are taken as exactly periodic. A real cloth’s threads are not evenly spaced — that is what the reed’s own mark is about, and what a drifting end does to a filter’s rating — so the registration wanders locally as well as drifting globally. Every departure from periodicity pushes the answer towards the average, which is the product.

And nothing here is about colour. Two coloured layers subtract as well as block, and the interaction of a transmitted colour with a blocked one is a different problem.

The generalisation #

An expectation over a nuisance parameter is not a property of an object, and the two get confused whenever the expectation happens to be a clean closed form.

The product rule is clean, memorable and exactly correct as an average, which is precisely what makes it dangerous: nothing about it announces that it is an average. A rule with an ugly constant in it invites the question of what; a rule that says multiply the two does not.

The diagnostic is to ask what was integrated out. Here it is the registration, and the registration is not random in any real assembly — a quilted panel is stitched at a fixed offset, a folded curtain settles, and a laminated pair is bonded at whatever offset it was bonded at. In every one of those the object has a definite value and it is not the average.

And the second lesson is about variance. The mean being exactly right is compatible with the object being anywhere in a range from zero to one cloth’s worth, which here is a factor of infinity at one end and 2.6 at the other. A quantity quoted as a mean with no spread beside it is a quantity whose spread nobody has computed, and the spread here is larger than most of the effects the mean is used to argue about.

Who found it, and when #

The convolution of two periodic apertures is elementary optics and the triangular overlap function is the autocorrelation of a rectangle, which is a standard result. Moiré between two gratings is nineteenth-century.

The product rule for stacked fabrics is trade practice and appears wherever plies are being specified.

What appears to belong to this collection is the observation that the product rule is exactly the mean rather than approximately the answer, and the identification of the threshold at a cover of a half — below which no registration shuts a doubled cloth and above which one does. That threshold is a statement about ordinary fabrics: a shirting is above it, a voile is below it, and the difference between a lining that works and one that does not is on either side of a cover factor.

Where the ladder goes next #

The threads being opaque is the assumption both light essays have been leaning on, and for a fine white cotton it is the wrong one: opacity is not cover, and the fraction of the surface that is thread turns out to be a lever rather than a barrier.

Sideways, the same registration argument decides something quite different in a filter: two layers stacked to improve a rating do not improve it at every registration, and the largest hole through a pair is a maximum over offsets rather than a mean — which is the question a filter’s rating is about.