Divided faces keep the census's count and lose its rule

The tall room gains once its walls are divided took the one room where the painted census and the true box disagreed most and divided its faces into patches. The census had a satin finish on the room’s grey faces colouring it for none of seventy-two paints; divided six times a side, it colours it for fifteen, because the top of a tall grey end wall sees the lamp at a graze and sends its light across to the paint. The census’s answer for that room belonged to its one patch a face.

The census has twenty-nine other rooms, and they were solved the same way. A finish adds colour only where it covers grey built its findings on them — above all that a finish on a room’s grey faces adds colour in 28 of 30 rooms, and at every paint in 13 — and a grey finish mirrors the lamp onto the paint built a rule on those counts: one number read off the empty box, the share of the lamp’s light the grey finish sends to the paint over what the matt return sent, which ordered the twenty-four rooms with a wall or floor painted at a rank correlation of 0.92, and whose sign sorted them without an exception. That essay noted the rule’s weakness itself: its correlation with the census’s gains was high partly because the number was computed in the same approximation the census was solved in. What the census and the rule say when every room is solved finely enough to trust is the question here.

The count survives; the rule was the approximation agreeing with itself #

Divided five or six times a side, the census keeps its headline and loses its rule. A grey finish still colours some paint in 28 of the 30 rooms, though two rooms trade places — the tall room with its side walls painted now gains, the tall room with its ceiling painted now does not — and it colours every paint in 17 rooms instead of 13. Twelve of the thirty counts do not move; nine move by twenty paints or more. The send number, computed on the same divided faces, now agrees with the true box’s own to a rank correlation of 0.96 where the one-patch number agreed at 0.70. But it orders the divided counts at only 0.63, where the one-patch number ordered the one-patch counts at 0.92, and its sign no longer sorts them: three rooms on the wrong side of zero gain for sixty-four paints or more. Of the six rooms with only the ceiling painted, five gained at every paint and three still do.

• A satin finish on the grey faces adds colour in almost every room — as robustly after division as before.
• Which rooms, and how much, moved, and the moves are large where a face is long or a lamp is seen at a graze.
• The send number converged on the right value and away from the counts. Its high correlation in the census was two approximate numbers sharing an approximation.
• The painted-ceiling exception is a property of wide rooms, not of painted ceilings.

The grid is the whole census, before and after. Each cell holds the one-patch count and the divided count, and the shading marks the nine cells that moved by twenty paints or more — darker where the divided room gains more. The moves are not random. The low and flat rooms with their floors painted go from twelve and fifteen paints to all seventy-two; the tall room and the cube with their ceilings painted fall; the slot with its side walls painted rises to every paint. Where a box is squat and its paint low, division helps the finish; where the lamp shares a face with the paint in a deep box, it hurts.

The census re-solved #

The solver is the census’s own directional transport — the one thirty unknowns instead of six checked against radiosity — with every face divided into n × n patches, exactly as in the tall room: form factors between patches from the exact point-to-polygon formula, the satin lobe averaged over the cone each patch subtends, the lamp still the whole ceiling, and the room’s colour read as the census read it. With one patch a face it returns every one of the census’s thirty counts exactly. The four squat and middle boxes are solved at five patches a side; the tall room and the slot, whose long faces need it more, at six. The seventy-two paints, the finish and the lamp are unchanged.

A count is how many of the seventy-two paints a finish on the grey faces makes more colourful, against the lamp’s white. It is a coarse measure, and a room whose paints all sit near the boundary can move its count by many paints for a small change in the physics. The mean cost across the paints is the steadier companion, and both are reported.

What survives: a grey finish adds colour almost everywhere #

The census’s central finding is that a finish over grey adds colour and a finish over paint costs it, so that a room’s gain is the balance between the two. The divided census here solves only the grey faces’ finish — the half of the balance the later essays built on — and its answer is the census’s: some paint gains in 28 of 30 rooms, and every paint in 17. The rooms where nothing gains are different: with one patch a face they were the tall room with its side walls painted and the slot with its ceiling painted; divided, the tall room with its ceiling painted and, again, the slot with its ceiling painted.

The means say what kind of change division makes. Twenty-two of the thirty rooms move towards zero: the census’s large gains shrink — the slot with its floor painted from a 129 per cent gain in chroma to 57, the tall room with its floor painted from 91 to 29 — and its losses shrink too. One patch a face exaggerates whatever a grey finish does, because a single patch’s lobe, aimed from the face’s centre, sends all of a face’s gloss one way — the aimed return the highlight is the lamp found on a single object, applied to a whole wall at once. Divided, a face’s gloss goes several ways at once and the effect on the room averages down. The exceptions are the painted-ceiling rooms of the deep boxes, whose losses grow, and a handful of rooms that cross from a small loss to a small gain.

So the practical advice from the census stands, and an open room hands over sooner’s low wide room, which started the question, stands with it: with its side walls painted it gained for 51 paints under a grey finish with one patch a face and gains for all 72 divided. A satin finish belongs on a room’s neutral surfaces, not its coloured ones, and in most rooms it adds colour; the amount it adds is smaller than the census said.

The send number reaches the true box #

The grey-finish essay computed its send number twice: in the solver’s patches, one a face, and through the true box, by rays cast from nine points on every face — the same reciprocity-bound quantity that a room is not a sphere relied on to explain an integrating sphere. The two disagreed, and the essay could not say which was right. Divided, the solver’s send number lies on the true box’s: a mean 1.7 points from it over the thirty rooms, where the one-patch number was 10.1 points off, and ranking with it at 0.96 where the one-patch number ranked at 0.70. The true box was right, as the tall room already suggested, and the patches converge on it.

That settles which geometry to believe, and it is the reason the next result is not a failure of the division. The send number computed on divided faces is the send number of the real box, to within a couple of points.

The rule does not survive #

On divided faces the send number orders the twenty-four rooms with a wall or floor painted at a rank correlation of 0.63. With one patch a face, the same two quantities — each computed in the census’s approximation — ordered them at 0.92. And the sign no longer sorts them: the low wide room with its floor painted sends 1.0 per cent less of the lamp’s light to the paint through its grey finish than matt, and gains for all seventy-two paints; the flat room with its floor painted, 4.7 per cent less, and all seventy-two; the flat room with its sides and floor painted, 1.8 per cent less, and sixty-four.

The grey-finish essay warned of this in its own terms: the send number’s ordering of the census was high partly because it was computed in the same approximation the census was solved in. Divided, each number is closer to the truth and they agree less. The one-patch census and the one-patch send number shared an error: from a face’s centre, a lobe aimed at the paint is also the lobe that makes the paint’s colour grow, and both numbers inherited the same misplaced aim. Correct the aim, and what remains is a real but weaker tendency — rooms whose grey finish sends more of the lamp’s light to the paint tend to gain for more paints — with rooms that gain everywhere while sending a little less.

Those rooms have something in common, and it suggests why. They are the low and flat boxes with their floors painted, where the lamp’s light reaches the painted floor mostly directly, and the likeliest thing a grey finish adds is not the lamp’s light at all but the floor’s colour, reflected back down by a low grey ceiling at a slant. The send number counts only the lamp’s light, and in those rooms it may be the wrong light to count — the gap the grey-finish essay already found for painted ceilings, where the lamp and the paint arrive from one face and the number cannot separate them, and the one the finish adds the room’s own colour anticipated when it found a finish returning light that has already bounced off the paint.

The painted-ceiling exception belongs to wide rooms #

The grey-finish essay set the six painted-ceiling rooms aside: all six read negative on its send number, five gained at every paint anyway, and the slot at none. Divided, three keep every paint — the low, the flat and the wide room — while the cube falls to 24 and the tall room to none, and the slot stays at none. The exception was never about painted ceilings as such, and the pattern of which rooms keep it points at the walls. In a wide room the grey walls are short and see the painted ceiling close to overhead, so their gloss can return the ceiling’s colour across a large floor; in a deep room the grey walls are tall, and divided, their lower parts send the ceiling’s light — lamp and paint together — down to the grey floor, where the paint’s colour is diluted rather than concentrated. That reading fits the six rooms; it has not been tested face by face.

What the earlier essays now need to say #

The painted census and the essays built on it were computed with one patch a face and say so in their methods. Their quantitative results are the results of that model, and several are now known to move when the model is refined:

• A finish adds colour only where it covers grey: its grey-finish counts — 28 rooms with a gain, 13 at every paint — survive as 28 and 17; the individual rooms’ counts move, by twenty paints or more in nine rooms.
• A grey finish mirrors the lamp onto the paint: its send number is right about the real box once computed finely, but its ordering of the rooms falls from 0.92 to 0.63 and its sign stops sorting them; the rule was substantially the approximation agreeing with itself.
• The tall room, which both essays treated as the room a grey finish cannot colour, gains for fifteen paints.

The mechanism the grey-finish essay found — a neutral surface’s gloss aimed by the lamp — is still how a grey finish adds colour, and the tall room is its clearest demonstration. What does not survive is a single number that predicts a room’s outcome from its box and lamp alone; the counts depend on how the aimed light meets the paint across the whole of each face, and a number computed per face, whether from its centre or averaged over it, does not capture that well enough to sort rooms.

How the census was re-solved #

Each of the six boxes — the low wide room two by two by 0.6, the flat room three by three by 0.5, the wide room three by one by one, the cube, the tall room one by two and a half by one, and the slot one by two and a half by 0.35 — is solved with every face divided into n × n patches, n = 5 for the first four and 6 for the last two, under the whole ceiling as lamp. The grey faces carry a Trowbridge–Reitz satin interface of roughness 0.2 over a Lambertian grey body of reflectance 0.5; the painted faces carry each of the census’s seventy-two Gaussian paints, matt. For each room the transport is solved at twenty values of the painted reflectance and interpolated per band, and a paint gains when the room’s chroma against the lamp’s white is higher with the grey faces finished than matt. The send number is computed patch by patch in the divided geometry from the radiance a matt grey room delivers from each patch, and area-weighted over the grey faces; the true box’s is the grey-finish essay’s, by rays from nine points on each face.

What this leaves out #

Only the grey faces’ finish was re-solved. The census’s other halves — a finish on the painted faces, and on every face — were not, and the balance between them, which is how the census explained the low room, has not been checked on divided faces.

Five and six patches a side are not proof of convergence. The tall room’s count held from five to six and its mean lay on a line; the slot’s did not, and the four boxes solved at five were not taken further. The census’s headline numbers, 28 and 17, were steady from four patches a side on; individual counts in rooms near their boundary may still move.

The lamp is the whole ceiling, glowing evenly, as it was in the census; a smaller lamp would change what every grey face sees at a graze.

Still open: the number that orders the divided rooms #

The lamp-lit send number counts only the lamp’s own light arriving at the grey faces, and the rooms it now misplaces are the ones where a grey finish’s contribution is mostly the paint’s light, reflected once already. A number that counts both is the natural next candidate: for each grey patch, the light arriving from each direction weighted by how coloured it is — its chroma, or its difference from the lamp’s white, in a grey room with the paint in place at a single mid reflectance — and the share of that coloured light the finish sends back towards the paint, against the matt share.

The calculation is that coloured-light send number for the thirty divided rooms, and its rank correlation with their counts. The prediction is that it orders the twenty-four wall and floor rooms above 0.85 and brings the painted-ceiling rooms into the same order, because it separates the lamp’s white from the paint’s colour arriving from one face, which is exactly what the lamp-lit number cannot do. If it orders them no better than 0.63, the counts are set by something no single geometric number of the empty room carries, and the census has to be solved room by room.

An agreement inside one approximation is not evidence for either number #

The habit is about what a correlation between two computed quantities can show.

The send number and the census’s counts were computed with the same one-patch-a-face approximation, and they agreed at 0.92. That agreement looked like an explanation of the counts. It was partly the approximation’s own error appearing in both: a lobe aimed from a face’s centre sets both where the gloss goes and what the room gains, so the two moved together whether or not the real room would. Refining both removed the shared error, and the agreement fell to 0.63, while each number, separately, moved towards the truth.

The failure mode is to treat the agreement of two quantities computed in the same model as confirmation of either. The test is to refine the model and see whether the agreement survives; here the count survived and the rule did not.