Each direction falls short by one band's ratio

Under a clear mercury lamp the light arrives in a few narrow lines, and a surface colour is decided almost entirely by how much of each line a pigment reflects. The directions of the object-colour solid that ask a pigment to reflect the green line at 546 nm and absorb the yellow pair at 577 and 579 nm are the hardest the lamp poses, because the two lines are thirty-one nanometres apart and a real pigment’s reflectance cannot change from one to nought in that distance unless something in it is sharp. A band pigment must be narrower than the gap it splits set the census: twenty-five splitting directions, a pigment built from up to three absorption bands no narrower than forty nanometres, and a direction counted lost when the best pigment’s reach falls more than one per cent short of the ideal.

The run of essays since has been about the steep side of those bands. A band is only as wide as its steep side gave a band a steep side and a wide one, and at one half-width a flank can lose nothing or nearly everything found a Gaussian steep side’s count set by its half-width alone. Real absorption bands are not Gaussian at their edges. A hundredth of Lorentzian undoes a Gaussian flank mixed a little Lorentzian into the steep side, whose wing falls only as the inverse square of distance, and found it eating the gap; a narrower flank moves its wing in with it and a five-nanometre flank carries a fiftieth of Lorentzian narrowed the steep side to pull the wing back. At a twentieth of Lorentzian, the pigment loses eighteen directions with a steep side fourteen nanometres wide at half depth, thirteen at ten, twelve at seven, and five at five — after a second search that found two of the five-nanometre census’s seven losses to be searches that stopped short.

Thirteen at ten and twelve at seven, then five at five, is an uneven fall, and the obvious suspicion was the search. The seven- and ten-nanometre counts came from the same first search that had stopped short at five. The prediction was that each falls by one or two when searched again from the best pigments found at the neighbouring half-widths, to about ten and eleven, and that the count then falls nearly linearly in the half-width. If either fell by four or more, the first search would have been weakest exactly at the intermediate widths, and every count between the Gaussian and the full Lorentzian would need searching again.

Searched again, both counts stand #

Searched again from the census’s best pigments at five and ten nanometres and at seven as a Gaussian, 4,000 steps from each, the seven-nanometre pigment loses the same twelve directions; searched again from seven and fourteen nanometres and ten as a Gaussian, the ten-nanometre pigment loses the same thirteen. No direction at either width gains as much as six tenths of a point.

The prediction fails in the less interesting direction: the first search was right. Ten of the seven-nanometre directions gain something, most by a few thousandths of a point, and the largest gain is direction 30’s, from 98.39 per cent of the ideal to 98.98. That one comes within two hundredths of a point of the one-per-cent line and stays on the wrong side of it, which says something about how close a direction can sit to the threshold without a search’s precision mattering. At ten nanometres five directions gain, by nothing visible.

So the five-nanometre census was the exception. Why its search alone stopped short is not established here; the census’s starts are the same at every half-width, and a five-nanometre band must sit nearer the passed line than a wider one, which is one reading, untested. The count’s uneven fall is the pigment’s, and the question becomes why a pigment would lose one direction between ten and seven nanometres and seven between seven and five.

Every direction falls short by one band’s ratio #

The search’s answer can be put beside a much simpler calculation. A single band twelve deep, its wide side a Gaussian and its steep side the same pseudo-Voigt, placed as well as it can be between the two lines, has an error — the larger of what it fails to absorb of the yellow and what it fails to pass of the green — that the essays on the flank have computed at every half-width. At a twentieth of Lorentzian that single band’s error is 4.2 per cent at fourteen nanometres, 2.8 at ten, 1.74 at seven and 1.05 at five. It falls by a factor of 0.62 from ten to seven and 0.60 from seven to five.

From ten nanometres to seven, every one of the 25 directions’ shortfall falls by a factor between 0.55 and 0.67; from seven to five, between 0.54 and 0.69. One band’s error falls by 0.620 and 0.600. Each direction’s shortfall is, to within a tenth, a fixed multiple of the single band’s error.

That is a strong statement about what the three-band pigment is doing. Whatever the direction — whatever mixture of the lamp’s other lines it also asks to be reflected or absorbed — the pigment’s failure in it is the failure of the one band that splits the green from the yellow, multiplied by a constant the direction sets. The other bands, the base absorbance and the placement do their work at every half-width equally, and only the splitting band’s wing at the passed line changes. A narrower flank moves its wing in with it found the mechanism for one band; here the whole pigment inherits it, direction by direction, in proportion.

Every direction narrows down one fan #

The four hardest directions fall short by 6.5 to 7.1 per cent at fourteen nanometres and by 1.0 to 1.2 at five; the easiest by under a third of a per cent at fourteen and under a twentieth at five. Drawn across the half-widths the directions form a fan that converges toward nought, and the lines keep nearly their order: of the 300 pairs of directions, 287 rank the same way at fourteen nanometres as at five.

A fan that keeps its order and narrows in proportion has a simple consequence for a count. The one-per-cent line crosses the fan at a different place at every half-width, and the count is how many lines lie above it there. Where the lines happen to be spread out at the height the threshold sits, a step in half-width moves few across it; where several lie bunched at that height, one step moves them all.

A ranking of constants sets the count #

The constant for each direction can be read once, as its seven-nanometre shortfall over the single band’s seven-nanometre error, and the count at any other half-width then predicted with no search at all: a direction is lost wherever its constant times the single band’s error exceeds one per cent.

The constants run from 1.20 down. A direction is lost at ten nanometres if its constant exceeds 0.36, at seven if it exceeds 0.57, at five if it exceeds 0.96 — and seven constants lie between 0.57 and 0.96, while only one lies between 0.36 and 0.57. Predicted this way the count is thirteen at ten nanometres and five at five, both exactly what the searches found.

That is the whole of the uneven fall. Between ten and seven nanometres the threshold rises from 0.36 to 0.57 through a stretch of the ranking with one step in it; between seven and five it rises from 0.57 to 0.96 through a stretch with seven. The pigment’s failure falls by the same three fifths at both steps. What differs is where the directions’ constants happen to sit, and those are properties of the lamp’s lines and the solid’s geometry, not of the steep side.

At fourteen nanometres the prediction gives sixteen where the search found eighteen, and there the reason is different. A fourteen-nanometre Gaussian steep side already falls half a per cent short with no Lorentzian at all, as a hundredth of Lorentzian undoes a Gaussian flank found when it counted four directions lost at that width as a pure Gaussian, so the shortfall there is the wing’s and the core’s together, and a constant fitted where the core is negligible underestimates it.

What makes a direction hard #

A constant a direction carries through every half-width is a property of the direction, and the obvious candidate is how much the direction depends on the line the splitting band’s wing reaches. A direction that asks for the green line passed and the yellow absorbed, or the reverse, also asks for other things — the violet and blue lines, the deep red — and its ideal reach is spread over all of them. The splitting band can only cost it the passed line.

Across the 25 directions, each one’s constant follows the share of its ideal reach that the passed line carries, at r = 0.91. The five hardest directions owe 44 to 63 per cent of their reach to that one line; the five easiest, under a twentieth. The three hardest of all split the lines in both senses: two pass the green line and absorb the yellow, one passes the yellow and absorbs the green, and each has most of its reach riding on the line it passes.

So the ranking is not arbitrary, and it is not a property of the pigment either. A direction’s constant is roughly the single band’s error converted into reach — the share of the passed line the wing absorbs, weighted by how much of the direction’s whole reach that line is. The correlation is not perfect, because the band must also absorb the other line and its own error there differs, and because the other two bands are not free at every direction. But it is close enough to say that the count at any half-width is decided by how many of the solid’s splitting directions lean heavily on one of mercury’s two lines, and that a lamp whose lines were spread more evenly through a direction’s reach would lose fewer directions at the same flank.

The count falls by one, then by seven #

The ranking’s prediction runs through every searched count from ten nanometres down, and between them it says what no census has searched: nine directions lost at six nanometres, three at four and a half, none at four. The step from twelve to five is not a cliff between seven and five but a staircase of seven risers, most of them between six and five.

The mean shortfall over the 25 directions shows what the count hides. It falls from 3.0 per cent at fourteen nanometres to 1.6, 0.91 and 0.55 at ten, seven and five — by 0.58 and 0.60 a step, nearly as one band’s error does. The pigment’s performance improves smoothly and in proportion; only the number of directions over an arbitrary line moves in jumps. A count of lost directions is the right measure for asking whether a pigment can reach a colour, and the wrong one for asking how its reach changes, and the run of essays on this flank has reported both without always saying which was meant.

What a colour scientist should take from this #

A count over a threshold inherits the threshold’s arithmetic. When every case’s shortfall scales by one factor, the count’s steps are set by the spacing of the cases’ constants near the threshold, and a count that falls by one and then by seven says nothing about the physics changing between the steps.

One band’s error is the pigment’s, scaled. The three-band pigment’s shortfall in every splitting direction is the single splitting band’s error times a constant the direction sets, to within a tenth, across ten nanometres of half-width. The single band is cheap to compute at any width, and the expensive search was needed once, at one width, to fix the constants.

A search that comes within two hundredths of the line is not evidence of a weak search. Direction 30 gains six tenths of a point on a second search and stays lost by a hair, and no third search would move it the hundredths it needs, because its constant is what places it there.

How the pigments were searched #

The pigment is up to three absorption bands no narrower than forty nanometres and a base absorbance, each band’s steep side a pseudo-Voigt — a twentieth Lorentzian, the rest Gaussian — of a stated half-width at a forty-nanometre band, scaled with the band’s width, and its wide side a Gaussian. For each of the 25 directions that split mercury’s green line from its yellow, the pigment’s reflectance under the lamp is weighted by the direction’s combination of the 1931 matching functions on a one-nanometre grid and maximised by Nelder–Mead from the census’s starts; the second search starts again, for 4,000 steps, from the best pigments found for that direction at the neighbouring half-widths and at the same width as a Gaussian, and keeps the better of the census’s answer and these. A direction is lost when the best reach falls more than one per cent short of the ideal. The single band is twelve deep with a 26 nm Gaussian wide side, placed on a grid of a fiftieth of a nanometre.

What this does not reach #

The constants are measured at seven nanometres and tested at ten and five. That they hold at six and at four and a half, where the ranking predicts nine and three, is a prediction, not a measurement.

One share of Lorentzian. The proportionality is shown at a twentieth; whether each direction’s constant is the same at a hundredth or a fiftieth, where the single band’s error is smaller and the core’s share of the shortfall larger, is not read.

One lamp. The constants belong to mercury’s lines and the 1931 observer; a lamp with lines at other spacings has its own ranking.

Still open: whether the ranking predicts the widths no census has searched #

The ranking of constants reproduces the searched counts at ten and five nanometres from the seven-nanometre shortfalls alone, and it predicts nine lost at six nanometres, three at four and a half and none at four. The census holds no pigment at those widths.

The calculation is the forty-nanometre pigment searched at steep half-widths of six, four and a half and four nanometres with a twentieth of Lorentzian, from the census’s starts and from the best pigments at five and seven nanometres, the counts read as here. The prediction is that six nanometres loses nine directions, four and a half loses three and four loses none, and that every direction’s shortfall at each width lies within a sixth of its constant times one band’s error there. If the count at six differs from nine by more than one, the constants are not fixed between seven and five — the other bands’ share of the shortfall changes with the half-width — and a count between searched widths cannot be read off one band.

The ranking reads six nanometres to the direction made that calculation: six nanometres loses the nine named, four loses none, and four and a half loses one of three, the other two a hair under the line.

A search that was right #

The habit here is about checking the search before believing a count, and then checking the count’s arithmetic when the search turns out to be right.

The uneven fall looked like a search artefact, because one had been found at five nanometres. Searching again found nothing at seven or ten. The tempting conclusion at that point would have been that something about the physics changes between seven and five — a regime, a threshold in the wing — and the evidence for it would have been the count itself. Setting each direction’s shortfall beside one band’s error showed that nothing changes: every direction falls by the same factor at both steps, and the unevenness lives entirely in how many directions sit near the line.

The failure mode is to read a count’s jumps as the measured quantity’s, when the count is a threshold applied to something that varies smoothly. The test is to plot the quantity under the count — here each direction’s shortfall — and to ask whether its own steps are even before explaining the count’s.