The paint itself gains all or nothing

The painted census asks thirty rooms one question. Six box shapes — a low wide room, a flat one, a wide one, a cube, a tall room and a tall narrow slot — are each painted five ways: the two side walls, all four walls, the side walls and the floor, the floor alone and the ceiling alone. The rest of each room is grey, the whole ceiling is the lamp, and a satin finish is laid on the grey faces. For how many of seventy-two paints does the finish make the room more colourful? A finish adds colour only where it covers grey asked it first, and divided faces keep the census’s count and lose its rule solved every room again with each face divided into strips, five or six to a side, and found that some paint gains in twenty-eight of the thirty rooms and every paint in seventeen.

What no one has managed is to say which rooms from anything simpler than the room’s own solve. A grey finish mirrors the lamp onto the paint proposed the send number — how much of the lamp’s light a grey face’s finish sends towards the paint, over what matt would send — and on divided faces it ordered the counts at 0.63. The paint’s own light does not order the rooms weighted the same number by the paint’s light instead of the lamp’s and got −0.07. Its reading of that failure was about what the census counts. The census reads a room’s colour as all of its light together, grey faces included, and a finish can colour a room in two ways: by concentrating the paint’s light back onto the paint, or by spreading it onto grey faces that then carry the paint’s colour too. A send number sees only the first.

So the proposal was to read the same solves on the painted faces alone, which is what a viewer looking at the painted wall actually receives. Then only concentration counts, and the prediction was that the lamp-lit send number would order the counts above 0.85, and that the rooms with painted floors — which spread the floor’s light onto the grey faces — would fall from seventy-two paints to few.

On the paint, no room sits part-way #

Read on the painted faces, every one of the thirty rooms gives an all-or-nothing verdict: in twenty-eight the finish makes all seventy-two paints more colourful, and in the other two — the slot with its floor painted and the slot with its ceiling painted — it makes none of them more colourful. The census, reading the whole room, left eleven rooms part-way, from ten paints to sixty-four. The paint’s own change is small, between −0.05 and +1.52 per cent of its chroma on average, while the whole room’s reading of the same finish runs from −18 to +57 per cent. A box with its floor painted and the same box with its ceiling painted gain the same colour within a hundredth of a point in all six shapes, although the light the finish adds to them differs by up to eighty-two points. Neither send number orders the size of the gain — 0.32 and 0.47 — and inside any one room a brighter paint always moves further than a darker one of the same band.

• The prediction fails in a way that answers the question behind it. There is nothing on the painted faces for a send number to order: the count is seventy-two almost everywhere.
• The census’s middle ground was the grey faces. Every room it left between none and all is a room where the paint gains everywhere and the grey faces decide the count.
• What sets how much the paint gains is the paint’s return to itself, which is why floor and ceiling rooms agree and why brightness of the paint matters more than any number of the empty room.

The figure sets the two readings side by side for every room: the census’s count, reading the room as a whole, in the upper bar, and the painted faces’ count in the lower. The upper bars take every value from none to all — the tall room’s ceiling at none, the slot’s four walls at ten, the tall room’s side walls at fifteen, the cube’s side walls at thirty-two, the low room’s side walls and floor at thirty-nine, the low room’s four walls at fifty-six. The lower bars take two values. The low room’s four walls, which the census said gain for fifty-six paints, gain for all seventy-two on the paint; the tall room’s ceiling, which gains for none in the census, gains for all of them on the paint. The two rooms at none are the slot’s floor and ceiling, which the census put at seventy-two and none respectively.

A count that is always all or none is no longer a census of paints; it is a verdict on the room. The seventy-two paints were chosen to span centres from 450 to 650 nm, widths from 15 to 100 nm and peak reflectances from 0.3 to 0.85, so that a room’s count says how general its gain is. On the paint the answer is: completely general, or completely absent, room by room. The paints that sat near the census’s threshold did so because of what the finish did to the grey faces’ light, and the grey faces’ light carries the paint’s colour in proportions that differ from paint to paint.

Reading the paint instead of the room #

The solves are the divided census’s, unchanged. Each box is divided into five strips a side for the four lower rooms and six for the tall room and the slot; each patch carries directional radiance, the finish is a satin lobe at roughness 0.2 with its coefficients averaged over the cone each patch subtends, and the transport is the one thirty unknowns instead of six built for whole faces, extended to patches. The census solves each room at a set of reflectances for the painted faces — the grey faces fixed at 0.5 — and interpolates the room’s light between them, band by band, so that a paint’s reflectance curve can be pushed through the room without solving it seventy-two times.

What changes is the sum. The census adds every face’s reflected light, grey faces included, and takes the chroma of the total. Here the same solves are summed over the painted faces only, area-weighted, with the lamp’s own emission taken out so that a painted ceiling reports only what it reflects. The chroma is taken of the resulting spectrum against the lamp’s white at that spectrum’s own luminance, so that it measures how coloured the paint’s light is and not how much of it there is.

Before any of this was trusted, the new solves were summed the old way. Read on the whole room, they give every one of the thirty rooms exactly the count the divided census gave, from the tall room’s ceiling at none to the seventeen rooms at seventy-two. That check matters more than it might seem: it is the same solve read two ways, so any difference between the two readings belongs to the reading and not to a change in the solver.

A per cent on the paint, tens of per cent in the room #

The bars are the mean change in chroma, over the seventy-two paints, that the finish makes on the painted faces; the whole room’s mean change is written beside each in brackets. On the paint the finish adds between a sixth of a per cent and one and a half per cent of chroma in twenty-six of the rooms, a few hundredths of a per cent in the tall room’s floor and ceiling, and costs five hundredths in the slot’s. In the room the same finish moves the reading from −18 per cent (the slot’s ceiling) to +57 (the slot’s floor), and six rooms read as losing colour on average while their paint gains.

The contrast has a plain cause. A grey face is grey only while its light is the lamp’s; once it receives light from the paint it carries the paint’s colour, and a room of mostly grey faces is a room whose chroma is decided mostly by how much coloured light the grey faces receive. The finish changes that a great deal — a satin grey face reflects more of what arrives from one direction and less from the rest — while the paint’s own light changes little, because the paint’s colour is fixed by its reflectance and only the proportions of its light that arrive once, twice or more are open to change. The census was measuring the grey faces’ share of the paint’s light, and the grey faces’ share is what a finish moves most.

The slot shows it at the extremes. With the floor painted, a satin finish on its tall narrow walls carries the lamp’s light down the slot: the painted floor receives eighty per cent more light, nearly all of it the lamp’s white, and loses a twentieth of a per cent of its chroma. The room as a whole gains fifty-seven per cent, and since the paint does not, the gain is in the light of the grey walls, which see the brighter floor and carry its colour. The paint is paler and the room is more colourful, both by the same finish, and the census counted the room.

Floor and ceiling gain the same colour #

Each box painted on its floor and the same box painted on its ceiling are, geometrically, one room turned upside down. In the first the finished grey faces are the four walls and the ceiling; in the second, the four walls and the floor. Reflected through the box’s middle, each arrangement becomes the other — except for the lamp, which is the ceiling in both. So whatever path the paint’s light takes from the paint to a finished face and back is the same path in both rooms; the path the lamp’s light takes to the paint is not.

The chroma the paint gains agrees between the two rooms within a hundredth of a point in every one of the six boxes: 0.49 and 0.50 per cent in the low room, 0.85 and 0.85 in the flat one, 0.44 and 0.44 in the wide one, 0.27 and 0.26 in the cube, 0.05 and 0.04 in the tall room, −0.05 and −0.05 in the slot. The light the finish adds to the paint does not agree at all: 3.1 and 6.9 per cent in the low room, 41 and 1.2 in the tall room, 80 and −1.8 in the slot.

That is a measurement of what sets the paint’s colour, and it is the cleanest result of the census. The chroma is taken at the light’s own luminance, so how much light arrives does not enter it; what enters it is the balance between light that meets the paint once and light that has met it before. Light arriving from the lamp, directly or off grey faces that have seen only the lamp, is white; light that has left the paint and come back carries the paint’s reflectance twice. The finish changes the paint’s colour only through that second kind, and the second kind is the mirror-symmetric part. The lamp can deliver eighty per cent more light to a floor than to a ceiling and it does not move the colour, because what the lamp adds is not coloured by the room at all.

The two rooms that lose colour on the paint are the slot’s two. Its floor and ceiling are 2.5 units apart across a gap a third of a unit deep, so the paint sees almost none of itself directly, and a satin finish on the long walls sends what it receives from the paint onward along the slot rather than back. The tall room, a unit square in plan, sits between them and the cube: its gain is a twentieth of a per cent, positive, and the same both ways up.

Neither send number orders the gain #

The count cannot be ordered, so the question the prediction asked has to be put to the size of the gain instead. Placed against the mean chroma the paint gains, the lamp-lit send number ranks the thirty rooms at 0.32 and the paint-lit one at 0.47. Without the six painted-ceiling rooms, whose paint is the lamp’s own face, the two numbers rank the twenty-four others at 0.17 and 0.54. The share of the room’s surface that is painted ranks them at 0.39, and the light the finish adds to the paint at 0.22.

The paint-lit number doing better than the lamp-lit one is what the symmetry above would lead one to expect — it weights the grey faces by the light that has come from the paint, which is the light whose return sets the paint’s colour — but 0.47 is not an ordering. Both numbers were built from the empty room, a room with no paint in it and every face grey, and the path that matters here is paint to finish to paint, whose weight depends on how much of the paint each finished face can see and at what angle. A send number averages over the grey faces’ view of the whole room; the paint’s gain lives in the part of that view that is paint.

So on the paint, as in the room, the size of the gain belongs to the solve. The difference is that on the paint the sign does not: in every room but the slot’s two, it is positive for every paint.

Brighter and broader paints gain most #

Within a room, the paint matters more than any number yet tried for the room itself. In the low room with its four walls painted, a paint whose band peaks at 0.85 gains 1.78 per cent of chroma on average, one at 0.6 gains 1.17 and one at 0.3 gains 0.62 — close to proportional to the peak. In every one of the thirty rooms, raising a paint’s peak reflectance moves the change the finish makes further from zero, in the same direction: all 1,440 such steps. That is the second-kind light again: a paint that reflects more of what it receives sends more of it round to itself, and the finish only works on what comes round.

Width matters in the same room by a factor of two: 1.90 per cent for a 100 nm band against 0.83 for a 25 nm one, with the 15 nm band slightly above the 25. A broad band reflects over more of the spectrum, so more of the light the paint returns to itself lands where it will be reflected again. Centre matters least, from 0.96 per cent at 610 nm to 1.60 at 450, and this essay does not separate how much of the blue’s lead belongs to the room and how much to how CIELAB spaces blues. The first two are what the mechanism predicts, and each holds across rooms: the flat room’s side walls and floor gain in the same proportions, the cube’s floor in smaller ones.

This is where paint is not a filter and the room is the illuminant meet the finish. A room’s surfaces illuminate each other, and a painted surface is partly lit by itself; a satin finish on its neighbours changes how much of that self-lighting returns. A room bounds its own bounces showed how interreflection saturates rather than runs away, and the paint’s gain here is the same saturation seen from the finish’s side: it rises with reflectance and never exceeds a couple of per cent.

What this settles about the census #

The census asked the right question of the wrong light. Its central finding — a satin finish belongs on a room’s neutral surfaces — is untouched and in one sense strengthened: on the paint a viewer looks at, the finish adds colour for every paint in twenty-eight rooms of thirty. What falls is the reading of the census’s partial counts as a property of the paints. They were a property of the grey faces, which receive the paint’s light in proportions the finish reshapes a great deal.

The painted-floor rooms were never an exception on the paint. The previous reading predicted that, read on the paint, they would fall from seventy-two paints to few, since the finish sends the floor’s light away from the floor. They stay at seventy-two in five boxes of six. Sending light away from the floor on average does not stop some of it coming back, and on the paint the part that comes back is all that counts for colour.

And the send numbers are finished with. Two geometric numbers of the empty room have now been asked to order two readings of the census and have ordered neither better than 0.63. On the paint the reason is visible: the gain follows the paint’s view of itself through the finish, which the empty room does not have.

A viewer’s sense of a room’s colour is neither reading alone. A gloss room looks less colourful than it measures found that adaptation discounts the colour a room shares and keeps what varies from face to face; a room whose grey faces carry the paint’s colour after finishing shares more of it and may look less coloured for it, while the paint itself grows slightly richer. Which of the two a person reports is an experiment, and the solves here say how far apart the two can be.

How the numbers were computed #

The rooms, paints, finish and solver are the divided census’s. The boxes are 2 × 0.6 × 2 (low), 3 × 0.5 × 3 (flat), 3 × 1 × 1 (wide), 1 × 1 × 1 (cube), 1 × 2.5 × 1 (tall) and 1 × 2.5 × 0.35 (slot); the paints are Gaussian bands centred every 40 nm from 450 to 650, with standard deviations of 15, 25, 50 or 100 nm and heights of 0.3, 0.6 or 0.85 over a floor of 0.03; the grey is 0.5; the lamp is D65 over the whole ceiling. Each room is solved at twenty painted reflectances placed at Chebyshev nodes, matt and with the grey faces finished, and each face’s reflected exitance is recorded with emission removed. A paint’s spectrum on the painted faces is its reflectance times the lamp times the interpolated exitance of the painted faces at that reflectance, band by band. Chroma is CIELAB’s, computed at the spectrum’s own luminance against the lamp’s white; a paint gains when the finished chroma exceeds the matt. The light gain is the ratio of luminances, finished over matt. Rank correlations are Spearman’s. The send numbers are those of the preceding essay, unchanged.

What this leaves out #

The paint is read as one face. A painted face’s chroma varies across it — the corners of a painted floor see more of the walls than its middle does — and a viewer at one position sees some patches more than others. The area-weighted mean is one choice; a view-weighted one might move the size of the gain, though the mirror symmetry would hold for any weighting symmetric top to bottom.

One finish, one grey. A glossier lobe on the grey faces would concentrate more sharply and could change the slot’s sign; a darker grey would carry less of the paint’s light and shrink the room reading’s swings towards the paint’s.

The lamp is the whole ceiling. With a smaller lamp the painted-ceiling rooms would not be the lamp’s own face, and the floor-and-ceiling symmetry would compare two rooms that differ by more than where the light comes from.

Still open: the slot’s sign #

The slot is the one box where the finish costs the paint, and its size — a twentieth of a per cent — is small enough that the division could decide it. The calculation is the slot’s floor and ceiling at seven and eight strips a side, to see whether the sign holds or is the division’s. It is also the box least suited to the solver: its long walls are divided as finely as its short ones, and the division that would put more strips along the slot’s length than across its depth — the refinement the tall room gains once its walls are divided points towards — needs more patches than the current solver can reach. The prediction is that the sign holds, since the tall room’s gain is positive and only a twentieth of a per cent, and the slot sees even less of itself.

Read the thing the viewer sees before asking a number to rank it #

The habit is to check what a census counts before explaining its counts.

Three numbers were built and tried against the painted census — the lamp-lit send number, the paint-lit one and the solved room’s response slope — and each failed; each was a refinement of the one before. What none of them could do was see that the census’s partial counts were set by faces that are not the paint. The test that found it was not a better number but a different sum over the same solves — the light of the faces a viewer looks at, rather than of the room — and on that sum the thing to be explained disappeared.

The failure mode is to refine a predictor while the quantity it predicts is a mixture. When a count resists every simple ordering, the first check is whether it is one quantity or the sum of two that move in different directions.