Opacity is not cover

Hold a single cotton thread up against a window. It is not a bar. It glows a little at its edges, it is brighter than the wooden frame behind it, and a fine one is noticeably translucent along its whole length.

The covering rule treats it as a bar. Cover factor is a fraction of area covered, and the covering it computes is total: a point of the plane is either under a thread or it is not, and one minus the cover is what gets through. That is the assumption in every openness argument this collection has made, including thread count is not quality and the oblique correction to it.

For a heavy indigo drill it is nearly right. For a bleached voile it is not, and the error is in the direction that makes an argument about openness weaker.

The claim #

What a cloth transmits is the open area plus the thread’s own transmittance times the covered area, so the covering rule’s error is proportional to the cover — and is therefore largest exactly where the rule is being used to argue that a cloth is closed.

Write τ for the fraction a thread passes. Then

$$T = A + \tau(1-A)$$

where A is the open area. At τ = 0 this is the covering rule. At τ = 1 the cloth is a window. And the correction term carries (1 − A), which is the cover.

τ is not computed here and cannot be. It is a scattering problem in a bundle of birefringent cylinders with air between them, and this collection has no optics in it at all. So τ is an argument with a stated default, every number is quoted at the value it was computed at, and what is claimed is the shape.

The argument #

The shape is the whole content, and it has three parts.

The correction is largest where the cloth is closest. At a cheesecloth’s 65 per cent open area, a thread transmittance of 0.15 adds 5.3 points — a relative error of eight per cent. At a sheeting’s 24.5 per cent it adds 11.3 points, a relative error of forty-six. The rule’s fractional error is τ(1 − A)/A, which grows without bound as the cloth closes.

So the two corrections this collection now carries pull opposite ways. The oblique argument says the covering rule overstates openness, by a factor that grows as the cloth closes. This one says it understates transmission, by a factor that also grows as the cloth closes. They are not the same quantity — one is about straight paths and one is about light — and a fabric held to a window is showing both at once.

And it is a lever, not an offset. A treatment that changes τ without changing the geometry moves the transmission by τ(1 − A), which for a close cloth is most of the answer. That is exactly what a finish does.

What moves a thread’s transmittance #

Nothing here computes τ, but the geometry does say what τ ought to depend on, and the dependences are the useful output.

A thread is a packed bed of fibres and light going through it crosses many interfaces. The number of fibre–air interfaces along a diameter goes as the diameter over the fibre diameter, so a coarse yarn of a given fibre should scatter far more than a fine one — which is the ordinary observation that fine cottons are translucent and heavy ones are not, and it is a statement about the count rather than about the cotton — the same kind of statement as a yarn’s diameter coming from its count rather than from its fibre.

The interfaces are what scatter, so filling the air between the fibres should raise τ sharply. A wet cloth is markedly more transparent than a dry one, and the reason is not that the fibres changed: water’s refractive index is much closer to cellulose’s than air’s is, so the same fibres in water present a far smaller index step at each interface. That is a prediction this arithmetic makes without computing anything — the effect should be largest for the yarns with the most interfaces, which are the coarse ones.

And raising the packing factor should do the same. Mercerising is a packing factor in this collection’s arithmetic: caustic soda swells the fibres, they fill their own voids, and the packing rises. Fewer air gaps means fewer interfaces means less scattering — and the lustre mercerising is actually sold for is a surface effect this collection has been careful to say it does not compute. The transmission consequence is a second one, it follows from the same packing change, and it is available for free.

What was counted, and how #

Very little, and the accounting is the point.

The open area is computed as always, from the sett and the diameters through the cover factor. The transmission is one line of arithmetic on top of it. What the function returns beside the total is the fraction of the transmitted light that came through a thread rather than through a hole, because that is the quantity that says whether τ matters at all for a given cloth.

At τ = 0.15 — a placeholder chosen only because it is plausible for a fine bleached cotton and is quoted with every number — a muslin transmits 47.2 per cent against the covering rule’s 37.9, and twenty per cent of what gets through came through a thread. For a sheeting the same τ gives 35.9 against 24.5, and thirty-two per cent came through the threads.

The function refuses a τ outside [0, 1] and an open area outside it, which are the only two refusals available when there is nothing else to check.

Where the two corrections meet, and what it takes #

This collection now carries two amendments to the covering rule and they point opposite ways. It is worth asking what value of τ would make them cancel, because the answer says which of the two dominates for a real cloth.

The geometric openness averaged over the sky is far below the covering rule. The thread’s contribution is above it. Setting the two equal gives

$$\tau^{*} = \frac{A - A_{\text{sky}}}{1 - A}$$

and for the eight cloths that comes to:

A cheesecloth’s cancelling τ is above one, which is impossible: no thread transmits more light than reaches it. So for an open cloth the geometric correction wins outright and the covering rule overstates transmission however translucent the threads are.

A sheeting’s is 0.30, which is not obviously impossible for a fine bleached cotton. So for a close cloth the two corrections are of comparable size and the covering rule could be right by accident — for the wrong reasons, in the wrong way, and by an amount nobody can compute.

That is a peculiar and honest place to end up. The rule is most defensible where it is least justified, and the reason is that two errors it does not account for happen to be pointing at each other.

The correction is an affine map, and that is more useful than it sounds #

Rearranged, the one line of arithmetic above says something sharper than it does as written. Collect the open area:

$$T = \tau + A\,(1-\tau)$$

Transmission is an affine function of open area, with intercept τ and slope 1 − τ. It is not a correction that acts differently on different cloths; it is one straight line that every cloth in the table sits on, and the whole effect of a translucent thread is to take the scale of open areas and squeeze it towards τ.

Three consequences follow, and each of them is about what can and cannot be learnt without knowing τ.

Order is preserved exactly. The slope is positive for every τ below one, so ranking a set of fabrics by transmission gives the same ranking as by open area, whatever the threads do. That is a strong negative result: no experiment that only ranks cloths can detect thread transmittance at all, and a great deal of practical fabric assessment is ranking.

Differences are scaled and ratios are not preserved. A gap of ten points of open area becomes a gap of 8.5 points of transmission at τ = 0.15, which is a mild compression. A ratio of open areas becomes something much closer to one — the cheesecloth and the sheeting stand at 2.65 to one on open area and at 1.96 to one on transmission at that τ, and at 1.4 to one by τ = 0.5. So an argument built on a ratio, which is what the thread-count argument is, weakens as τ rises, while an argument built on a difference barely moves.

And the slope is measurable without any absolute calibration. Two cloths whose open areas are known from their setts and diameters, measured on the same instrument in the same light, give a line through two points; its slope is 1 − τ and its intercept is τ, and the two are a redundant pair that has to agree. Nothing about the lamp, the detector or the geometry survives into the slope, because both cloths saw the same one.

That is a real experiment, it needs no photometry anybody would call careful, and it would settle the largest unknown in this essay. What it needs that this collection cannot supply is the cloths — two fabrics of the same yarn at genuinely different setts, whose geometry is known well enough that the open areas are not themselves the fitted quantity. The parameter this essay declines to invent is not unmeasurable. It is unmeasured.

What the affine form says about the fitting temptation #

There is a reason to state the line explicitly rather than leave the arithmetic in its first form, and it is about how a fit would go wrong.

Any measurement of transmission against open area over a set of cloths will lie near a straight line, because the model says it must. So a fit will succeed, it will return a τ, and the fit’s quality will say nothing whatever about whether the model is right — a set of points on a line is exactly as consistent with the covering rule plus a translucent thread as it is with several other mechanisms that also produce a linear relation.

The test that could fail is the intercept. The line’s slope and its intercept both encode τ, separately, and they are required to sum to one. A fit whose slope gives 0.15 and whose intercept gives 0.28 has falsified the model rather than measured it — which is the check to run, and it is available in the same two measurements as the fit.

The oblique correction is what would produce that disagreement, and it would produce it in a known direction. A real measurement under diffuse light is reading the straight-through geometry averaged over the sky rather than the plan’s open area, and that average is smaller for the closer cloths, so the points would tilt: the line’s slope would come out too steep and its intercept too low. The two amendments this collection carries are separable by that tilt, which is the only handle on them there is.

Where the model stops #

τ is invented. Everything above is a shape, and the numbers are illustrations of the shape at a value nobody measured. This is the largest thing this collection has ever declined to compute in the middle of an argument, and it is declined rather than fudged because a fitted τ would make every number look like a result.

Transmission is not opacity. What a person calls opaque is about whether a shape behind the cloth can be resolved, and a translucent thread scatters, so light that gets through carries no information about where it came from. A cloth can transmit half the light and hide everything behind it — which is what a lampshade is — and the two properties are computed by different physics. This essay is about the first.

The two paths are treated as independent and the interesting case is not. Light entering a hole at an angle can hit the side of a thread on its way through, so the geometric and the transmissive paths meet inside the cloth. That is a real coupling and it goes in the direction of less transmission.

The transmittance is treated as a single number for a thread and it is a function of path length. A ray crossing a thread near its edge passes through less of it than one crossing at the centre, so τ varies across the width of every thread, and the effective value is an average weighted by the chord. Nothing here does that, and it is a computation that could be done — it needs only the chord distribution across a circle, which this collection already has for the channel’s own profile.

And nothing here is about colour. A dyed thread’s τ is a function of wavelength and so is what gets through it, which is why a red curtain against the sun is red and a white one is white. Everything above is written as though light had one colour.

The layers multiply differently too. Two layers of an opaque-thread cloth average to the product of their open areas; two layers with a τ in them average to something with cross terms, because light through a hole in the first and a thread in the second is a path the geometric arithmetic does not have.

The generalisation #

A binary model of a continuous quantity fails hardest where its answer is smallest, because the fraction it neglects is the fraction it says nothing gets through.

The covering rule is a binary model: covered or not. Its error term is the neglected leakage times the covered fraction, so it is proportional to the very thing that makes the answer small, and its relative error diverges as the answer goes to zero. Any model of the form this fraction passes and the rest does not has this shape — an aperture ratio, a fill factor, a duty cycle, a coverage map — and in each case the residual is invisible until the main term has nearly vanished.

The diagnostic is to ask what the model says about its own extreme. The covering rule says a cloth at cover 1 transmits nothing. A cloth at cover 1 is a sheet of cotton, which is a piece of paper, which is not black. That is a two-second check and it exposes the assumption without any measurement at all.

The second lesson is about the honest response. The temptation with an unknown parameter is to fit it, and a fitted τ would reproduce any measurement of any cloth and would predict nothing. What is worth having instead is the structure — which way τ moves with the count, the packing and the wetness — because those are testable without ever knowing its value, and a paired comparison cancels it entirely.

Who found it, and when #

That fibres and yarns are translucent is not a discovery; it is the basis of the whole optical treatment of textiles, and Kubelka and Munk’s 1931 two-flux theory is the standard tool for turning a scattering layer into a reflectance and a transmittance. Applying it to fabrics is routine in colour science.

The covering rule with an opaque thread is Peirce’s, and Peirce was computing a geometric cover rather than an optical one; the confusion is in the later use rather than in the original.

What appears to belong to this collection is the framing of the error term as a lever proportional to the cover, and the observation that the two corrections it now carries — the thickness one and the transmittance one — grow together as a cloth closes and point in opposite directions. So the covering rule is at its worst for close cloths in both senses, and the two errors partially cancel in a way that nobody can quantify without the parameter this essay declines to invent.

Where the ladder goes next #

Everything in this pair of essays has been about a cloth’s holes as the weave left them. The next question is what holds them there, and in every ordinary weave the answer is friction: a leno’s hole cannot drift and nothing else’s is safe.

Sideways, the shape of a hole matters as much as its size for anything that has to flow: a satin’s hole is a slot, and at constant open area a slot passes less than two fifths of what a square does.