The finish stops costing the paint six tenths deep

A satin finish laid on a room’s grey faces changes the colour of the painted surfaces they surround, and in the painted census it nearly always changes it for the better — a finish adds colour only where it covers grey set up the census to find out where and why. The paint itself gains all or nothing read thirty rooms — six box shapes, each painted five ways — on their painted faces alone, the light a viewer looking at the paint receives, and found the finish adding chroma to every one of seventy-two paints in twenty-eight rooms and to none in two. Both exceptions were the same box: the tall narrow slot, one unit wide, two and a half tall and only 0.35 deep, painted on its floor and painted on its ceiling.

The slot’s loss survives every finer division asked whether that loss was real and found it was: solved with every face cut into one to eight strips a side, the slot’s paint gains at the coarse divisions, loses from four strips on, and converges to a loss of about six hundredths of a per cent. It left the obvious next question. The tall room in the census is the slot made a full unit deep — the same width, the same height, the same lamp — and it gains. Somewhere between 0.35 and 1 the sign turns. Where it turns says whether the slot’s loss is a property of shallow boxes in general or a quirk of one shape. The prediction was that it turns early, between 0.45 and 0.55, because the slot’s loss is small and the tall room’s gain is large.

The sign turns at six tenths, later than predicted #

Solved at eight depths with every face cut into four strips a side, the box’s painted face loses chroma to the finish at 0.35, 0.45, 0.55 and 0.6 deep and gains it at 0.65, 0.7, 0.85 and 1. Interpolated between 0.6 and 0.65, the turn falls at 0.62 with the floor painted and at 0.63 with the ceiling. At every depth all seventy-two paints change the same way. The finish brightens the painted floor by between 48 per cent and 112 per cent and the painted ceiling by under two per cent, yet the two turn within a hundredth of a unit of each other. And the sign of the change at each depth is the sign of one number: how much more the finish brightens the paint’s bright reflectances than its dark ones.

• The prediction fails. The turn sits past the middle of the scan’s range, not before it.
• The loss is not the slot’s alone. Every box shallower than about six tenths of its width loses, a range covering nearly half the depths between the slot and the tall room.
• Colour follows the tilt, not the brightness. The floor is brightened enormously and the ceiling hardly at all, and they agree.

The two lines run almost together from a loss at the slot to a gain at the tall room. They are not straight. From 0.35 to 0.45 the loss shrinks sharply, from −0.031 to −0.012 per cent with the floor painted; from 0.45 to 0.6 it barely moves, to −0.009 and then −0.005; between 0.6 and 0.85 it climbs steeply, through nought, to +0.047; and from there to a full unit it nearly levels off at +0.052. The turn sits in the steep middle stretch, where a twentieth of a unit of depth moves the mean change by a hundredth of a per cent.

The ends reproduce what the two earlier essays measured, and the second of them is itself a refinement: the tall room gains once its walls are divided showed the tall room’s verdict moving with the division of its faces before it settled. At 0.35 the scan is the slot at four strips a side, paint for paint to twelve decimal places; at a full unit it is the tall room, and every paint gains there with the floor painted and with the ceiling painted. The census’s two neighbouring shapes are connected by a curve that crosses once.

What the prediction got wrong #

The prediction leaned on the sizes at the ends. The slot’s loss at four strips is three hundredths of a per cent; the tall room’s gain is five. A smooth curve between a small negative number and a larger positive one crosses nearer the small one, and the guess put the crossing a quarter of the way along. The curve is not smooth in that way: it flattens just short of the turn, so the small loss lingers through three depths before the climb begins, and the crossing is pushed out to more than four tenths of the way from the slot to the tall room.

What the failure says is the more useful part. The earlier essay framed the alternatives sharply: if the sign turned only near the tall room, the slot’s loss would be a quirk of the slot’s one shape; if early, a property of narrow rooms. At 0.62 it is neither extreme, and the better reading is that the loss belongs to a band of shapes. A box whose two big walls stand closer together than about six tenths of their width — a corridor, an alcove, a cupboard, a lift shaft — costs a painted floor or ceiling colour when its grey walls are finished in satin. A box squarer than that adds it. The slot happened to be the census’s only member of the band, and the band is broad.

The flat stretch between 0.45 and 0.6 is worth noting and not worth explaining yet. It is where the change is smallest, and at four strips a side the strips on the side walls are little more than a tenth of a unit wide, so it is also where a coarse solve is most likely to misplace a crossing. The finer check later in the essay tests exactly the two depths either side of it.

The finish brightens the floor and hardly touches the ceiling #

The obvious way to think about a satin finish in a narrow box is as a light pipe. Two glossy walls facing each other across a small gap carry light along the gap by repeated reflection, the way a polished duct carries daylight down from a roof. In the slot the lamp is at the top of the gap and the painted floor at the bottom, and the finish should pipe the lamp to the floor.

It does, and by a great deal. With the floor painted, the finish more than doubles the light leaving it in the slot — an increase of 112 per cent — and the increase falls at every deeper box, to 72 per cent at 0.6, 66 at 0.7 and 48 at a full unit. The shallower the box, the closer its glossy walls, and the more of the lamp’s light they carry down instead of scattering it back up. This is the mechanism a grey finish mirrors the lamp onto the paint put at the centre of the census: the finish turning the lamp’s light towards the paint at a slant.

With the ceiling painted the picture is entirely different. The ceiling is the lamp, and a flat face cannot light itself, so the only light a painted ceiling reflects is what comes back to it off the other faces. The finish changes that by between −1.2 and +1.4 per cent over the whole scan — less than a hundredth of what it does to the floor.

If brightness decided the colour, the floor and the ceiling would behave very differently: the floor’s colour would move with a curve falling from 112 to 48, the ceiling’s would hardly move at all. They behave the same. The ceiling’s mean change sits within half a hundredth of a percentage point of the floor’s at every depth, and its turn is a hundredth of a unit later. Whatever decides the sign, it is not how much light the finish adds.

That is the earlier essay’s twin result, extended across a range of shapes, and it fits the finding that the paint’s own light does not order the rooms: no single count of where the light goes, taken from the room, says how the paint’s colour moves. The paint itself gains all or nothing found a box painted on its floor and the same box painted on its ceiling gaining the same colour to a hundredth of a point while the light they gained differed by up to eighty-two points. Here the gap in light is more than a hundred points and the colours stay twins at all eight depths, losses and gains alike.

Colour follows which reflectance the finish favours #

A paint’s colour is the shape of its spectrum: how much more light it returns at the wavelengths it reflects than at the ones it absorbs. Any change that multiplies every wavelength by the same factor leaves the shape, and so the chroma, alone. The finish changes chroma only to the extent that it multiplies the paint’s light by different factors at different wavelengths — and since the only thing distinguishing one wavelength from another on a given paint is its reflectance there, that means by different factors at different reflectances.

The solver makes that directly measurable. It is run at twenty reflectances between 0.02 and 0.90 for every depth, and the brightening at each can be read off separately.

In the slot the finish brightens a dark reflectance more than a bright one — on the floor by 0.28 percentage points more, on the ceiling by 0.13. From 0.65 deep it brightens the bright reflectance more, by 0.33 points on the floor at a full unit and 0.21 on the ceiling. The difference crosses nought between 0.6 and 0.65 for both faces, the same two depths between which the paint’s mean change crosses. At every one of the sixteen readings, eight depths by two faces, the sign of this tilt is the sign of the paint’s change in chroma.

A finish that brightens the dark wavelengths of a paint more than its bright ones raises the floor of its spectrum relative to its peak, and the paint looks a little greyer. One that brightens the bright wavelengths more deepens the contrast between peak and floor, and the paint looks a little more colourful. The sizes are tiny — a quarter of a percentage point on a brightening of a hundred — and they are the whole of the effect.

Why the dark and the bright are brightened differently #

At a reflectance of 0.02 the painted face sends almost nothing back into the room, so nearly all the light arriving on it is light that has never touched the paint: the lamp’s light, carried to it directly or by way of the grey faces. At a reflectance of 0.9 a large part of what arrives has been to the paint already — left it, crossed to a grey face, and come back. The brightening at the dark end is therefore very nearly the factor by which the finish multiplies the light that has never touched the paint, and the tilt says whether the light that has been back to the paint is multiplied by more or by less.

In the slot it is multiplied by less. The finish pipes the lamp’s light to the floor so efficiently that the light that has never touched the paint more than doubles, and the paint’s own returned light, which also passes along the glossy walls, gains slightly less. The share of the floor’s light that carries the paint’s colour twice falls, and the paint greys. As the box deepens, the piping weakens — the lamp’s factor falls from 2.12 to 1.48 — and the factor on the paint’s returned light falls more slowly, so that past six tenths it is the larger of the two. The share of twice-coloured light rises, and the paint gains.

This is what the slot essay suspected, now with a number attached. It read the floor and ceiling’s twinship as a sign that the paint’s own light returning across the grey faces decides the change, since that path is the same length from either face. The tilt is that return measured against everything else, and it carries the sign.

It also connects to the earliest finding about gloss in this collection. A lobe takes colour out of a bounce found that a glossy wall’s mirror reflection carries the lamp’s spectrum rather than the paint’s, and a tenth of the return arriving white found that a few per cent of uncoloured light compounds through an interreflection into a large shift. The same arithmetic runs here, at the scale of hundredths of a per cent: what the finish does to the colour is decided by how it re-weights coloured light against uncoloured, not by how much light it moves.

No paint turns at a different depth #

The mean crosses nought, and so does every paint. Up to 0.6 deep, both the paint that changes most and the one that changes least lose; from 0.65, both gain. No paint is on the other side of the line at any of the sixteen readings. The spread between them is widest at the ends — in the slot with the floor painted, from −0.096 per cent to −0.011, and at a full unit from +0.019 to +0.161 — and narrowest at the turn, where every paint’s change is close to nought together.

The paint that gains most at a full unit, a broad band of a hundred nanometres centred at 530 with a peak reflectance of 0.85, is the paint that loses most in the slot. The one that gains least, a band twenty-five nanometres wide at 570 with a peak of 0.3, is the one that loses least. The order of the paints does not change across the turn; only the sign does.

One pattern over the paints, scaled by the depth #

The paints turn together for a stronger reason than their sharing a sign. Divide each paint’s change by the mean change over all seventy-two and the ratio is the same at every depth to within four per cent. The dark narrow paints sit at 0.56 of the mean whether the box is the slot or the tall room; the bright broad ones sit at 2.23. The depth sets a single number — how strongly, and in which direction, the finish tilts the light — and the paint sets its own share of it, independently.

That separation is what a small effect should look like. The tilt is a quarter of a percentage point; a paint’s response to it is, to first order, proportional to it, with a constant set by how much of the paint’s spectrum sits at high reflectance against low. Bright broad paints have the most spectrum to tilt and respond most; dark narrow paints the least. The largest drift from the pattern, 3.6 per cent, comes at 0.6 deep with the floor painted, where the mean change is smallest and second-order terms have the least to hide behind.

It also explains why the earlier essay found the paint’s own brightness ordering the size of its gain in every gaining room. The same factor, the paint’s share, orders the size of its loss in every losing one.

Moved through the depths, the twelve bars shrink together towards the turn, pass through nought between 0.6 and 0.65, and grow again on the other side, keeping their order and their proportions throughout: the broadest, brightest paint always moves most and the dark narrow ones least. What the depth changes is a single number; what the paint changes is its share of it. At 0.6, the last losing depth, the largest bar is a hundredth of a per cent, and every paint is within a few thousandths of a per cent of grey’s effect on it — the turn is not a place where some paints gain while others lose, but where every paint’s change passes through nought at once.

Finer strips keep the turn where it was #

The slot essay showed that four strips a side gives the slot’s sign right and its size about half: −0.031 per cent at four strips against −0.058 converged. A turn found at four strips could move under a finer division, and in which direction is not obvious, since the finer solves deepen the slot’s loss and might also change the tall room’s gain.

Solved again at six strips a side with the floor painted, the box at 0.6 deep still loses and the box at 0.7 still gains, every paint with it. The loss at 0.6 doubles, from −0.0045 to −0.0091 per cent, and the gain at 0.7 shrinks, from +0.0215 to +0.0149. The turn stays between the two depths, and both changes push it the same way: deeper. A straight line between 0.6 and 0.7 crosses nought at 0.617 at four strips and at 0.638 at six.

That is the direction the slot essay’s convergence suggested. Finer division deepened the slot’s loss at every step, and here it deepens the loss just short of the turn and trims the gain just past it, so the band of shapes in which the finish costs the paint widens as the solve improves. Six strips moved the turn by two hundredths of a unit; the slot’s loss was still changing by a sixth between six strips and its extrapolated limit, so the converged turn is probably near 0.65, and very unlikely to fall back below 0.6. The prediction of 0.45 to 0.55 fails by more under the finer solve, not less.

Each six-strip solve took about seven minutes against forty seconds at four, which is why the finer check is confined to the two depths that bracket the turn. It is enough to say which way the turn moves and that it does not cross either of them.

How the scan was solved #

The box is one unit wide and two and a half tall, its whole ceiling emitting, its floor or its ceiling painted and its other faces grey at a reflectance of a half, with a satin finish — a microfacet lobe of roughness 0.2 on a dielectric of index 1.5 — on the grey faces. Its depth takes eight values: 0.35, 0.45, 0.55, 0.6, 0.65, 0.7, 0.85 and 1. Every face is divided into four strips along each side, so the two end walls, one unit by two and a half, carry sixteen patches each whatever the depth, and the side walls narrow with the box.

The unknowns are the radiances leaving each patch towards each patch it sees, the lobe averaged over the cones the two patches subtend, solved by Gauss–Seidel to a residual of 10⁻¹¹ with the finish and without it, at twenty reflectances on Chebyshev nodes between 0.02 and 0.90. A paint’s light at each wavelength is the lamp’s spectrum times the painted face’s response at the paint’s reflectance there, interpolated between the nodes; its chroma is read in CIELAB against the lamp. The seventy-two paints are six hues, four band widths and three peak reflectances, as in every essay on the census.

The turn is the straight-line crossing of nought between the two depths either side of it. The scan cannot place it closer than that, and the steep stretch the crossing sits in makes the straight line a fair guess: the mean change rises by about a hundredth of a per cent between 0.6 and 0.65, from −0.005 to +0.007.

What the scan does not reach #

One width and one height. Every box in the scan is one unit wide and two and a half tall. The turn is reported as a depth, but whether it is really a ratio of depth to width, or depth to height, or something else, cannot be read off one family.

One roughness. The finish is the census’s satin. A glossier finish pipes the lamp more efficiently and might push the turn deeper; a rougher one might pull it in. How far the solver can follow a glossier lobe is its own question, which the boundary belongs to the quadrature priced.

A closed box. Every box here is sealed, its lamp the only source. An open room hands over sooner found an open wall moving a gloss finish’s turning point by whitening the ambient, and an alcove open on one side is the commoner real case.

Painted floor and ceiling only. The census’s other painted arrangements — side walls, four walls, walls and floor — are not scanned in depth, and in a box with painted walls the walls are themselves part of the light pipe.

Four strips. The turn’s position is checked at six strips only at the two depths either side of it, where it moves two hundredths deeper. The flat stretch between 0.45 and 0.6 is not checked, and the converged turn is an estimate, not a solve.

Still open: whether the turn is a ratio of depth to width or to height #

The scan found the turn at about 0.62 in a box one unit wide and two and a half tall, and reported it as a depth. In the light-pipe reading, what matters is how close the two glossy end walls are against how far the lamp’s light has to travel between them — the gap against the height. In the reflectance-tilt reading, what matters is how much of the paint’s own light comes back to it, which depends on the gap against the width as much as the height. The two readings make different predictions for a box of another height.

The calculation is the same scan in boxes one unit wide and 1.5 and 4 tall, with the floor painted, at four strips a side — three or four depths around each box’s turn — and the turn read off as before. The prediction is that the turn scales with the height: deeper in the taller box, shallower in the shorter one, because the piping that greys the paint grows with the length of the pipe. If the turn stays near 0.62 of the width at every height, it is a property of the floor’s shape and not of the pipe, and the light-pipe picture of the slot was the wrong one.

A turn is a measurement of a family #

The habit here is about what to do once a single anomalous case is settled.

The slot’s loss was established at one shape, by refining one solve until its sign stopped moving. The temptation at that point is to treat the anomaly as a fact about that shape and file it. The scan treats it as one end of a family and finds where the family turns, and the turn turns out to be more informative than the anomaly. It says the slot was not special: it was simply the census’s only box on one side of a line that runs through nearly half the range of shapes between it and its neighbour. A census sees its boundaries only where its shapes happen to fall, which is the lesson divided faces keep the census’s count and lose its rule drew when the census’s count survived a re-solve and its rule did not.

The failure mode is to explain an outlier without asking how far its condition extends. The test is to pick the one dimension along which the outlier differs from its nearest well-behaved neighbour — here, depth — and walk along it. If the change is confined to a sliver near the outlier, the outlier is a quirk; if it spans the range, the census was sampling a boundary too sparsely to see it. This census was the second kind, and only the walk could say so.