The tall room gains once its walls are divided

A grey finish mirrors the lamp onto the paint found one number, read off an empty box, that ordered the thirty rooms of the painted census by how many of seventy-two paints a satin finish on their grey faces makes more colourful: how much more of the lamp’s light the finish sends to the painted faces than the grey faces’ matt return did. In the census’s own rooms its sign sorted them. Through the true box — rays cast from nine points on every face — it still led, but stopped sorting, and it disagreed with the census most about one room.

That room was the tall room with its side walls painted: a box one unit wide and deep and two and a half high, lit by its whole ceiling, its two side walls carrying the paint and the other four faces mid grey. The census has a grey finish there colouring the room for none of the seventy-two paints. In the census’s own patches, the grey end walls send 32.3 per cent less of the lamp’s light to the paint through their satin than through their matt return; through the true box, 5.5 per cent more. Both cannot be the room’s answer. The essay asked for the directional solve on subdivided faces, and predicted that the end walls, split into upper and lower halves, disagree in sign — the upper halves sending the lamp’s light across to the painted side walls, the lower halves down to the floor — so that the room gains for some paints.

Divided, the tall room gains, and it was the patches that said it could not #

The prediction holds, and the census’s answer for that room was an artefact of its one patch a face. Split into halves, the grey end wall’s upper half sends 7 per cent more of the lamp’s light to the paint than matt, its lower half 27 per cent less. Divided more finely the pattern sharpens — the top of the wall gains, its bottom and centre lose — and the room’s send number climbs from −23.4 per cent with one patch a face to +0.9 at six a side, towards the true box’s +4.1. A grey finish colours the tall room for 0, 4, 8, 11, 15 and 15 of the seventy-two paints as the faces are divided one to six times a side, and the mean cost falls on a straight line in one over the division towards a small gain. The narrow slot, painted the same way, goes from 22 paints to all 72.

• The census’s zero belonged to the discretisation. One patch a face points every direction from the middle of a wall, and from the middle of a tall end wall the lamp’s mirror direction is the floor.
• The top of the wall is where the colour comes from: it sees the lamp at grazing incidence, and its mirror runs across the room to the paint.
• The gainers are the light, broad paints — peak reflectance 0.6 or 0.85, bands fifty or a hundred nanometres wide.
• The slot has not settled by six patches a side, and its number is a direction rather than an answer.

The figure is the result. The left edge is the census: the tall room at zero and the slot at twenty-two. Each step to the right divides every face more finely, and the tall room’s count rises steadily until it holds at fifteen from five patches a side to six. The slot falls before it rises, and reaches all seventy-two by five. The tall room with all four walls painted, where there is no grey end wall to send anything, barely moves.

The census’s solve, with its faces divided #

The solver is the census’s own, changed in one thing only. The solver had no slot for gloss is why it exists — radiosity gives each face one number and a lobe needs a direction — and thirty unknowns instead of six built it: a directional transport in which every face sends a separate radiance towards every other face, so that a glossy surface can send more light one way than another, checked against radiosity in the matt limit to a part in a million billion. With one patch a face that is thirty unknowns. Here every face is divided into n × n rectangles, each rectangle becomes a patch, and each patch sends a separate radiance to every patch it can see — 2,430 unknowns at three a side, 7,680 at four, 38,880 at six.

Everything else is carried over. The form factor between two patches is the exact point-to-polygon formula averaged over the emitting patch, and every patch’s form factors sum to one within a part in ten million million; with one patch a face they are the census’s closed forms to within a hundred-thousandth. The satin lobe is averaged over the cone each patch subtends, exactly as the boundary belongs to the quadrature found reliable at a roughness of 0.2 — the averaging where a patch stops being a point introduced after a point-sampled lobe returned a chroma of thirty-four thousand. The lamp is still the whole ceiling. The room’s colour is read as the census read it, from the sum of each face’s reflected exitance against the lamp’s white, so that a count at six a side means what a count at one meant. With one patch a face the divided solver gives every one of the census’s thirty arrangements exactly the census’s count.

A paint enters the solve only through one number in each band, the painted faces’ reflectance there. So each room is solved at twenty reflectances spread across the range any paint reaches, and each of the seventy-two paints is read off the curve through them; at two reflectances between those nodes the curve agrees with a direct solve to a part in a million.

Where each part of the end wall sends the lamp’s light #

Split in two, the wall behaves as the prediction said. Its upper half sends 7.2 per cent more of the lamp’s light to the painted side walls through its satin than through its matt return, and its lower half 26.8 per cent less. Split in three, the top row sends 13 to 16 per cent more, the middle row a little more at the edges and 8 per cent less at the centre, the bottom row about 20 per cent less. At six a side the picture is a gradient in two directions: the top row gains 17 to 20 per cent, the columns nearest the painted walls gain almost all the way down, and the centre of the wall’s lower two thirds loses up to 21 per cent.

The geometry explains every part of it. A point near the top of the end wall sees the lamp — the whole ceiling — at grazing incidence, where an interface reflects most and its lobe is sharpest — the aimed reflection of the lamp that the highlight is the lamp found in a single glossy object — and the mirror of a ceiling seen at a graze runs down and across the room at a shallow angle, where much of it meets a side wall before it reaches the floor. A point near the bottom sees the ceiling nearly face on, two and a half units up, and its mirror points at the floor at its feet. A point near a side edge of the wall sees the painted wall beside it at a slant, and its lobe sends a share of whatever it reflects there. One patch a face reads the whole wall from its centre, where the lamp sits at 68 degrees and its mirror is the floor, and so every part of the wall is taken to send the lamp’s light down.

The room’s send number walks towards the true box’s #

The room’s send number — the area-weighted average over the grey faces of how much more of the lamp’s light the finish sends to the paint — follows. In the tall room it goes from −23.4 per cent with one patch a face to −7.2, −1.3, +0.2, +0.7 and +0.9, crossing zero between three and four patches a side. The true box, reading the same quantity by rays from nine points on each face, put it at +4.1. The divided solver is not there at six a side, and it is moving towards it at a slowing rate. The slot’s number moves the same way, from −10.3 to +7.4, against the true box’s +12.8.

This is the resolution of the disagreement the earlier essay left. The true box was right about the tall room’s grey faces, and the census’s patches were wrong about them, for the reason the essay suspected: one patch cannot hold both the top of a tall wall and the bottom of it. The earlier essay’s rule — rooms where the grey finish sends more of the lamp’s light to the paint gain, and rooms where it sends less do not — bends here without breaking: the send number rises with the count at every division and is positive by the time the room gains for eleven paints, though the first four gain while it is still below zero.

The mean cost falls on a line towards a gain #

A count is a coarse summary of seventy-two paints, and a count that holds at fifteen from five to six patches a side could still be moving underneath. The mean cost says more. In the tall room it is 14.8 per cent with one patch a face and 4.5, 2.3, 1.5, 1.1 and 0.9 per cent after. Plotted against one over the division, so that an infinitely fine division is the left edge, the last three points lie on a straight line to within four hundredths of a per cent, and the line meets the left edge at −0.4 per cent: a small average gain.

That straight line is an assumption, the usual one for a discretisation whose error falls in proportion to the patch size, and the three points’ fit to it is the evidence for using it here. It says the tall room’s refined count of fifteen is close to the room’s own and may be a little low; it does not say how many paints the undivided room would colour, since a mean near zero can hide a count anywhere between none and all.

The slot does not lie on a line. Its mean goes from 3.9 per cent to 5.2, 3.1, −0.8, −2.5 and −2.9, and a line through its last three points misses them by nearly three tenths of a per cent. The slot is a box a third of a unit deep and two and a half high, whose two large end walls face each other at a third of a unit, and dividing each face into squares-in-proportion makes patches on those walls larger than the gap between them. Its count of seventy-two at five and six patches a side is where the solver has got to, not where the slot is: it says the census’s twenty-two was too low, and not by how much.

The light, broad paints gain #

Every paint loses less colour to the grey finish when the faces are divided, and the fifteen that gain at six a side are all paints of peak reflectance 0.6 or 0.85 in bands fifty or a hundred nanometres wide. The narrow bands and the dark paints still lose. The census found the same division in the rooms that gained partly, and the finish adds the room’s own colour explained it: what a finish adds has already bounced off the paint, and a light broad paint has more of its own colour in the room’s light for the finish to add, while a narrow dark one’s colour is spent in a bounce or two.

In a room lit from above, the paints that gain are the ones for which the lamp’s white, sent across by the top of the grey wall, arrives at a wall that returns most of it coloured.

The census’s other answers are due the same question #

The tall room was the census’s largest disagreement with the true box, and dividing its faces settled it. But the census solved every room with one patch a face, and this room’s answer changed from none to fifteen. The same solver, divided, gives answers for the other twenty-nine arrangements, and some of them move as far: in the same tall room with its ceiling painted, the census’s seventy-two paints gaining become none at every division from two patches a side. What the painted census’s conclusions become when every room is solved finely enough to trust is a larger question than this essay’s, and it has its own.

For the tall room itself the practical advice a finish adds colour only where it covers grey gave is strengthened rather than reversed. It said a satin finish belongs on the neutral surfaces of a room with coloured walls, and the census made the tall room the exception, where satin end walls aimed the lamp’s light at a grey floor and whitened every paint. Divided, the tall room is not an exception: the top of its end walls aims the lamp at the paint, and for light broad paints the room gains. A decorator’s version is to put the gloss where it sees the lamp at a slant — the upper part of a neutral wall below a bright ceiling — rather than to judge a wall by its middle.

How the solve was run #

Each face of the box is divided into n × n rectangles in its own two directions, and each rectangle is a patch with the face’s normal. The form factor from one patch to another is the exact contour formula for the view from a point to a polygon, averaged over the emitting patch by a twelve-by-twelve Gauss rule; patches on the same face see nothing of each other. The satin lobe — a Trowbridge–Reitz interface of roughness 0.2 and refractive index 1.5 over a Lambertian body — is averaged over the cone each patch subtends at each other, as in the census, with the interface’s albedo taken out of the body. The transport is solved band by band by Gauss–Seidel iteration to a hundred-billionth, each solve started from the last. The room’s light per unit lamp is solved at twenty Chebyshev nodes in the painted faces’ reflectance from 0.02 to 0.9 and interpolated for each paint in each of eighty-one bands; the census’s paints are its seventy-two Gaussian bands, six centres by four widths by three peaks. A paint gains when its chroma against the lamp’s white, with the grey faces finished, exceeds its chroma matt. The send number is the census’s, computed patch by patch in the divided geometry against the radiance a matt grey room delivers from each patch. Divisions of one to six a side are solved for the tall room and the slot; a division of six a side is 38,880 unknowns a band.

What this leaves out #

The division is uniform. Every face is cut into squares in its own proportions, which makes long thin patches on a tall wall and patches larger than the gap between them in the slot. A division that followed the geometry — finer where two faces meet, finer across the slot’s gap — would converge faster, and for the slot it is needed.

Six patches a side is where the solve stopped, at under ten minutes a box. The tall room’s count held from five to six and its mean lies on a line; neither is proof of convergence, and both are stated as what they are.

The lamp is the whole ceiling, glowing evenly, as in the census. A lamp that is a smaller patch of the ceiling would change what the top of the end wall sees at a graze, and so change this answer as much as the division did.

Still open: whether the slot converges once its gap is divided #

The slot is where the uniform division fails, and it fails in a known way: its end walls, one unit by two and a half, face each other a third of a unit apart, and patches a fifth of a unit wide on those walls are each seen from the opposite wall at an angle a single cone cannot average. The calculation is the slot solved with its faces divided in proportion to the gap rather than to their own size — patches no larger than a quarter of the gap, so about a dozen across each end wall’s width and thirty up its height — with the count and the mean read as the division refines. The prediction is that the slot’s mean settles near −4 per cent, a gain for every paint, and that its count stays at seventy-two: the slot’s two grey end walls, seeing each other and the lamp at grazing angles everywhere, are the room in which a grey finish does the most. If its count falls back below seventy-two, the uniform division’s seventy-two was as much an artefact as the census’s twenty-two.

A patch is a claim about everything inside it #

The habit is about what a discretisation assumes and where it stops being true.

One patch a face assumes that every point of a face sees the room the way its centre does. For light that arrives evenly from everywhere, that assumption costs little; for a lobe aimed by a lamp, it is the whole answer, because the lobe’s direction depends on where on the face the light arrives, and the centre of a tall wall sees the lamp at an angle nothing else on the wall shares. The census’s answer for the tall room was a statement about the centre of the end wall.

The failure mode is to trust a solver’s answer about a geometry it cannot represent. The test is cheap in principle and was here: divide, and see whether the answer moves. It moved from none to fifteen.