The hue the hole decides

An aperture makes a translucent sample read dark. That much is expected, and it is the reason every standard says to use a wide one. It also makes it read a different hue, and on one of the six materials in this collection’s table the hue ends up on the other side of neutral.

The claim #

There is an aperture at which a pale marble measures exactly grey, and on either side of it the sample reads a different hue.

• At no aperture the stone is warm — hue angle 70°, chroma 7.15 — because the iron in it absorbs weakly in the short wavelengths.
• At a four-millimetre radius it is cool — hue 253°, chroma 1.49 — which is very nearly the opposite direction.
• At 5.32 millimetres it is neutral, found by bisection along the sample’s own hue direction rather than by looking for the shallowest point of a dip.
• The cause is the scattering’s wavelength slope, not the absorption: remove the slope and the reversal disappears while the desaturation stays.
• And the aperture at which it turns is a function of that slope, moving from 2.19 millimetres at a slope of 0.5 to 8.90 at a slope of 2.

What an aperture takes away #

An aperture is a filter, and its transmission at each wavelength is the share of that band’s kernel that fits inside the hole. That share falls as the kernel widens, and the kernel widens where light travels furthest before absorption ends it.

Two things set how far light travels. Absorption is one, and it is where the sample’s own colour lives: a band the material absorbs strongly has a short kernel and loses little to the aperture. Scattering is the other, and it has a wavelength dependence of its own — small particles turn short wavelengths more sharply than long ones, which is the same mechanism that makes a clear sky blue.

Those two pull in opposite directions. Absorption makes the aperture’s filter follow the sample’s own reflectance, which desaturates. Scattering makes it favour the short wavelengths, which is a blue filter, and its strength depends on nothing about the pigment at all.

On most materials the first term wins and the result is a plain desaturation. On a weakly-absorbing one it does not, and the reason is arithmetic rather than physics: the aperture’s filter is a ratio of two kernels, so its strength does not fall with the absorption while the sample’s own colour does. Weaken the pigment and the filter stays where it was.

That is the same structure as several results elsewhere in this collection. The whiteness of a sheet is mostly the lamp because the sample’s contribution shrinks while the lamp’s does not; a metameric pair separates in a corner because the bounce multiplies a difference that a flat wall leaves alone. A term that does not scale with the thing it is being compared against wins somewhere, and finding where is usually the whole result.

The stone, in numbers #

The marble in the table is calcite grains with a trace of iron: an absorption of 0.006 per millimetre everywhere, plus a shallow band at 450 nanometres taking it to 0.026 there. Its reduced scattering is 1.5 per millimetre at 550, falling as λ^−1.3.

That gives a diffusion length of 2.52 millimetres at the blue end and 7.61 at the red — the iron absorbs blue, so blue does not travel; and the particles scatter blue more strongly, which also shortens it. Both effects run the same way, and the ratio between the two ends is threefold.

Under no aperture the sample is warm, because the iron band removes blue and leaves the rest: hue 70 degrees, chroma 7.15, lightness 84.7.

Through a four-millimetre radius, 63.2 per cent of the light comes back — but the share is not the same at both ends. The blue end, with its short kernel, keeps more; the red end, with its long one, keeps less. The transmission of the aperture’s filter runs from 59 per cent at 380 nanometres to 33 at 780, which is a strong blue filter, and it is stronger than the iron band it is fighting. The reading comes out at hue 253 degrees, chroma 1.49, lightness 71.0.

Somewhere between there is a crossing, and it is at a radius of 5.32 millimetres.

Finding a crossing rather than a dip #

A chroma that falls and rises again could be two things: a genuine passage through the neutral axis, or a dip that never reaches it. Those are entirely different claims and a plot of chroma against aperture cannot distinguish them, because chroma is a magnitude and has no sign.

So the crossing is found along a signed coordinate: the projection of the measured a*b* onto the direction of the sample’s own hue. That quantity is positive when the reading is on the same side as the sample, negative when it is on the other, and zero exactly at the crossing. Bisection on it either brackets a sign change or reports that there is none.

The distinction is not academic. Of the six materials, four have a crossing and two do not, and on the two without one the chroma still dips. A search for the minimum would have reported six crossings, five of them imaginary. A minimum is not a zero, and reporting one as the other is how a figure claims something it does not have.

Which input decides it #

A result this striking is worth attacking, and the obvious attack is that it depends on a modelling choice nobody measured. The scattering slope is such a choice: σₛ′ ∝ λ^−b with b set to 1.3 for this material, on the general grounds that calcite grains are of the right size for a Mie regime.

So the whole computation is run again at five values of b, with the absorption spectrum and everything else held.

With a flat scattering there is no crossing at any aperture. The chroma still falls — from 10.46 to 2.64 at a two-millimetre radius — so the desaturation is not the slope’s doing. The reversal is.

That is a much better result than an unconditional one, because it names the mechanism instead of merely reporting it. The desaturation is the absorption’s work, and it happens on every material. The reversal is the scattering’s work, and it happens when the particles have a size that cares about wavelength.

It also gives the finding a way to be wrong that can be checked: a real marble whose scattering slope were measured to be flat would have to show a dip and no crossing. Nothing here is a prediction about a particular quarry, and the table’s coefficients are constructed rather than measured, which is the standing caveat on everything computed in this collection from a constructed sample.

What was computed, and how #

Nothing in the computation is specific to the reversal, which is the point: the same code produces the desaturation on the other five materials and the crossing on this one.

The stone is eighty-one slabs, one per band of the site’s grid, each built from that band’s absorption and scattering coefficients. Each slab’s kernel is integrated twice — once over the whole plane, which is the reflectance the model wants, and once against the aperture’s weight, which is the reading. The two reflectances then go through the ordinary colour arithmetic: multiplied by D65, integrated against the 1931 observer, converted to CIELAB against the light’s own white.

The sweep runs eleven apertures from one millimetre to forty, and the crossing is found by bisection on the signed projection described above, on a logarithmic bracket because the answer is a length and lengths are better bisected geometrically.

Two properties of the arithmetic are worth having stated. The reading is monotone in the aperture at every wavelength, because the aperture’s weight increases everywhere as the hole widens, so nothing here depends on a numerical accident. And the crossing is a crossing of a continuous quantity, so it exists by the intermediate value theorem once the sign change is bracketed — the bisection locates it rather than establishing it.

What a reader can see without any of this #

A translucent object beside a window shows the same thing without a spectrophotometer anywhere.

The lit edge of a piece of alabaster or a thick sheet of white plastic is a different hue from its middle, and the difference has the same sign as the one computed here: the part of the object receiving light directly is reading its own colour, and the part a few millimetres away is reading what survived the journey. The journey is long for the wavelengths the material transmits best, so the far part is the transmitted colour and the near part is the surface one.

That is the same phenomenon as the aperture with the geometry turned inside out. An aperture asks how much of what came back landed inside this circle; an edge asks how much of what went in over there arrived here. Both are the kernel integrated against a shape, and both select on path length.

It is also why the collection’s essay on thickness and this one are about the same physics from two sides, and why a bounce off a coloured wall is a multiplication while a journey through a scattering one is not. There the path length is set by how far the light has to cross; here it is set by how far it wanders. The exponential dependence on path and the linear observer do the rest in both cases.

Candle wax is the material at the far end of this collection’s set, and its kernel on its own is what makes the crossing radius argument concrete.

The crossing radius is proportional to the slope #

The slope table is offered as a robustness check and it contains a law, which the closing paragraph comes within one sentence of and does not state.

Fitting the four crossing radii against their exponents gives an exponent of 1.001 — the crossing radius is proportional to the scattering slope, at about 4.3 millimetres per unit of slope, with every one of the four points inside five per cent of that line.

That converts the essay’s own caution into something sharper. The position moves by a factor of four across a plausible range of exponents is true and is not a separate fact: the exponents themselves span a factor of four, from 0.5 to 2.0, and a proportionality is exactly what makes the two spans equal. The position is not unstable in a way that undermines the finding; it is a linear readout of the one input the finding is about.

It also explains the flat row without needing a separate mechanism. If the crossing sits at 4.3 times the slope, then a slope approaching zero puts the crossing at an aperture approaching zero, so it does not disappear at some threshold — it retreats through the smallest hole anybody could drill, and never is what the sweep reports when the answer has left the range of realisable apertures.

And it turns the closing proposal into a formula. The aperture at which a sample goes grey measures its particle size is offered as a possibility; with a linear relation and a constant it is a calibration. The constant, 4.3 millimetres per unit of slope, depends on this stone’s absorption and would have to be established per material — but the linearity is what makes such a measurement worth attempting, because a nonlinear readout with a fourfold range would need the whole curve where a linear one needs two points.

Three numbers about the four-millimetre reading do not sit together #

The stone’s paragraph gives four quantities for the four-millimetre aperture, and they cannot all be measured the same way.

The per-band transmissions are quoted as running from 59 per cent at 380 nanometres to 33 at 780. The overall share is quoted as 63.2 per cent. A weighted average of numbers lying between 33 and 59 cannot be 63.2, whatever the weights.

The lightness pair says the same thing independently. Lightness 84.7 with no aperture and 71.0 at four millimetres are luminance factors of 0.654 and 0.422, a ratio of 64.5 per cent — again above the largest per-band transmission on the list.

So two of the four numbers agree with each other, at 63.2 and 64.5, and both are outside the interval the other two define. The likeliest reading is that the 59-and-33 pair is quoted for a different aperture than the rest of the paragraph, since a narrower hole would push both down while leaving their ratio much as it is.

Nothing the essay argues depends on which, and it is worth saying why. The argument needs the transmission to be higher at the blue end than at the red, because that is what makes the aperture a blue filter, and every version of the numbers says so — 59 against 33 is a ratio of 1.8 in the right direction. The hue reversal is driven by the ratio between the ends and not by their level, and the level is what the inconsistent numbers disagree about.

What an instrument recovers through the aperture it actually has is the practical form of the whole argument, and four millimetres is what a hand-held spectrophotometer offers.

One number that checks exactly #

Against those, it is worth recording that the model’s own arithmetic reproduces from the published coefficients.

The stone is given as absorption 0.006 per millimetre away from its band and reduced scattering 1.5 per millimetre at 550, falling with the 1.3 power of wavelength. At 780 nanometres that is a scattering of 0.9524 per millimetre, and the standard diffusion length — one over the square root of three times the absorption times their sum — comes to 7.613 millimetres against the essay’s 7.61.

The blue end needs an absorption of 0.0214 rather than the band’s peak 0.026 to give the stated 2.52, which is what a band centred at 450 has decayed to by 380 and is therefore a check rather than a discrepancy. The transport model is the ordinary one and the coefficients are the ones printed, which is the part of the computation the whole result rests on.

Where the model stops #

The crossing is a property of a construction and three of its assumptions are load-bearing.

The slab is semi-infinite. A thin piece of marble on a dark backing loses the light that reaches the back, which shortens every kernel and moves the crossing.

The scattering slope is a single exponent. Real Mie scattering off a distribution of grain sizes is not a clean power law, and its departure from one is largest where the grain size and the wavelength are comparable — which is exactly the visible band for a stone.

And the diffusion approximation is worst near the entry point, which is where a narrow aperture is doing most of its collecting. A model is a claim about what can be known, and this one claims nothing about the first mean free path. At a one-millimetre radius on a material whose mean free path is a fraction of a millimetre, the model is being used near the edge of where it holds. The crossing at 5.32 millimetres is comfortably inside; the one-millimetre column of the sweep is not, and should be read as the direction of travel rather than as a number.

The generalisation #

The transferable part is a warning about what an instrument’s design parameter can do to a measurement.

An aperture size looks like a decision about area — how much of the sample gets averaged — and on an opaque material that is all it is. On a translucent one it is a decision about which photons get averaged, and photons that travelled different distances have different spectra. Once a measurement selects on path length, it selects on wavelength, because path length and wavelength are coupled by the physics.

Colour measurement has one other parameter of this kind and this collection has already measured it: the bandpass, which selects on wavelength directly. What is new here is a parameter that does not mention wavelength anywhere in its definition and selects on it anyway.

The general test is worth stating: ask whether the instrument’s parameter is correlated with anything that varies across the band. An aperture is correlated with path length, path length is correlated with absorption, and absorption is the sample’s colour. Three steps, no wavelength in the first of them, and a hue reversal at the end.

A second material with a crossing is what says the turn is a property of the mechanism rather than of one stone, and it turns at a different hole size.

Who found it, and when #

That translucent samples read dark through small apertures is old and standardised. That they read a different hue is reported in the plastics and dental literatures as an observation — inter-instrument comparisons on translucent specimens show disagreements with a consistent direction — and it is usually attributed to differences in aperture and geometry together, without separating them.

The mechanism named here is not new physics. Wavelength-dependent scattering is Mie’s from 1908, and the coupling between transport length and spectrum has been the working stuff of tissue optics since the 1980s: the reason a hand held over a torch is red is exactly this, with the sign of the distance reversed.

This collection has been careful before about claiming a mechanism from a construction — a ranking between two constructed spectra is a fact about the constructions — and the caution applies here in a specific form. The reversal’s existence is a property of any material whose scattering has a wavelength slope and whose absorption is weak. Its position is a property of these particular coefficients.

What appears not to be written down anywhere is the crossing — that there is an aperture at which such a sample is neutral, and that its position is a measurement of the scattering’s wavelength slope rather than of anything about the pigment. If it holds for a real stone it is an instrument: the aperture at which a sample goes grey measures its particle size.

Where the ladder goes next #

The obvious next step is the one this collection cannot take, and it is worth saying why rather than leaving it implied. The crossing is computed from constructed coefficients; confirming it needs a real translucent sample measured at four or five apertures, which is an afternoon in a laboratory and impossible here.

What can be done from a desk is to check whether the crossing survives the model’s own weaknesses — a finite thickness, a backing, a non-power-law scattering — and whether its position is more stable than its existence. The slope table says the position moves by a factor of four across a plausible range of exponents, so the honest claim is that the crossing is robust and its location is not.