A grey finish mirrors the lamp onto the paint

A finish adds colour only where it covers grey took seventy-two wall paints through thirty rooms — six box shapes, each painted five ways — and split a satin finish into its two halves. On the painted faces alone the finish costs colour in twenty-nine rooms of thirty, for the reason a lobe takes colour out of a bounce first gave: an interface reflection carries no pigment, so it can only whiten a coloured wall’s return. On the grey faces alone it adds colour in twenty-eight rooms of thirty, and at every one of the seventy-two paints in thirteen.

The reading it offered for the second half was about angles. A grey face adds no colour of its own. Matt, it hands on an even mixture of everything arriving at it; satin, part of that mixture is replaced by a reflection that favours light arriving at grazing angles, because an interface reflects most there. If the paint is what a grey face sees at a slant, the finish should make it hand on more paint. That is computable without solving any room, and the essay predicted that the area-weighted excess of paint in the slant-weighted view would order the thirty rooms’ gains at a rank correlation near 0.9, with the painted-ceiling rooms as the exception.

The slant orders the rooms only once it is pointed at the lamp #

The prediction fails. How much paint the grey faces see at a slant orders the thirty rooms at 0.24, and the twenty-four with a wall or the floor painted at 0.47. The number that works is the same slant pointed the other way — where the finish sends the light it reflects — weighted by where the lamp’s light actually arrives from. It orders the twenty-four rooms at 0.92, and its sign alone sorts them: every room where the finish sends more of the lamp’s light to the paint than the matt surface did gains for 29 paints or more, and every room where it does not gains for 22 or fewer.

• Seeing and sending are one number until light is put in, because a surface’s reflection obeys reciprocity; through the true box the two agree to the last digit.
• The lamp is what separates them, and it is more than half the answer: unlit, the send number orders the rooms at 0.51.
• The mechanism is a mirror. A grey wall lit from the ceiling at a slant sends most of its gloss to the floor, so the finish helps when the floor is painted and hurts when the walls beside it are.
• Two things fall outside the rule. The painted-ceiling rooms read negative and gain anyway. And the rule belongs to the solver’s rooms, whose faces are single patches: through the true box it still leads, but its sign stops sorting, and the tall room is where the two disagree.

The figure above is the whole result in one picture. Every filled point is a room with a wall or floor painted, and a vertical line at zero divides them into two groups that do not overlap on the vertical scale. The hollow points are the six rooms with only the ceiling painted, and they sit on the wrong side of the line at the top of the scale. Everything below is how that number was arrived at, why the predicted one could not have worked, and what the two exceptions say.

Four numbers, read off an empty box #

The rooms, paints and finish are those of the earlier census, unchanged. Each box has a lamp on its ceiling; its painted faces carry one of seventy-two band-shaped paints and the rest reflect half the light at every wavelength; the satin finish is a microfacet interface at a roughness of 0.2 and a refractive index of 1.5, solved by the directional method that thirty unknowns instead of six checked against radiosity. The gain being predicted is the census’s own: for how many of the seventy-two paints a finish on the grey faces alone leaves the room’s reflected light more colourful, read against the lamp’s white.

What is new is four numbers per room, each computed from the box, the set of painted faces and the lamp’s position, with no paint chosen and no room solved with colour in it. Each is worked out for every grey face and then averaged over the grey faces by area.

The view number is the prediction’s. For light arriving at a grey face from each other face, weight it by the solid angle that face subtends and by how much of it the interface reflects — the Fresnel-weighted albedo, which rises steeply at grazing arrival. The painted faces’ share of that weighted view, minus their share of the plain cosine-weighted view, is how much more paint the finish “sees” than the matt surface does.

The send number asks the question the other way round. For light arriving from each face, where does the interface’s reflection go? The lobe of a satin surface points roughly at the mirror direction, so light from the ceiling arriving at a wall is sent mostly away from the ceiling. The painted faces’ share of where the lobe sends it, minus their share of where a matt surface would have sent it, is how much more the finish hands to the paint, averaged over the arriving directions by the same albedo weights.

The lit versions of both weight every arriving direction also by how much light actually comes from it. That needs a room, but not a painted one: the same box, grey everywhere, matt, under the ceiling lamp. A grey face in it receives the lamp’s direct light from the ceiling and a weaker, evenly spread light from everything else, and those proportions are fixed before any paint is chosen.

All four are computed in the solver’s own discretisation first — one patch a face, directions from centre to centre, the lobe averaged over the cone each patch subtends — because that is the room the gains were measured in. They are then computed again through the true box, by casting rays from a grid of nine points on every grey face.

What the grey faces see does not decide what they add #

At 0.24 over the thirty rooms, the predicted number orders almost nothing. Setting the painted-ceiling rooms aside, as the prediction did, lifts it to 0.47, which is a tendency rather than an ordering: the scatter above holds rooms at nearly the same value of the view number that gain for no paint and for all seventy-two. The low wide room with its floor painted and the same room with its ceiling painted read exactly the same, −14 per cent, because floor and ceiling are mirror images of each other in any box; one gains for fifteen paints and the other for all seventy-two.

The number was not wrong in the sense of being miscomputed. The share of paint in a grey face’s view really does rise when the finish is applied, in exactly the rooms where the view number is positive. What it cannot say is whether the light coming from that paint is worth anything. A painted side wall seen at a slant from a grey floor is still a coloured surface lit only by what the room passes to it, and in most of these rooms the light a grey face receives is dominated by one face: the ceiling that carries the lamp.

Seeing and sending are the same number until the lamp is lit #

The unlit send number was computed as a separate quantity, with its own weights and its own directions. Through the true box it agrees with the unlit view number in every one of the thirty rooms to better than a part in a billion.

That is not a coincidence, and it is the most useful thing in the whole computation. The reflection of any physical surface obeys reciprocity: light going from direction A to direction B is reflected with the same strength as light going from B to A. Summed over a hemisphere, “how much of the paint does this face see through its lobe” and “how much of its lobe lands on the paint” are the same integral written in two orders. A room is not a sphere used the same symmetry to explain why an integrating sphere reads a surface’s reflectance exactly.

So the prediction, stated as a view, was already a statement about where light is sent — for a room in which light arrives equally from every direction. That is the one condition no lit room meets. In the solver’s patches the two unlit numbers differ slightly, because averaging a lobe over a patch’s cone is not exactly symmetric; they order the twenty-four rooms at 0.47 and 0.51.

Once the lamp’s light is put in, the two come apart in opposite directions. The lit view number falls to 0.26: weighting the view by where light comes from mostly weights it towards the ceiling, which is grey in these twenty-four rooms, so the lit view mostly reports how little paint is overhead. The lit send number rises to 0.92. The lamp’s light is the largest single thing a grey face receives, and where the finish sends that light is where the room’s extra colour does or does not come from.

That difference, from 0.51 to 0.92, is the lamp’s position. Nothing about the paint enters it; the same number applies to a blue room and an orange one.

A grey wall mirrors the lamp onto the floor #

The tall room, one unit by two and a half by one, shows the mechanism most plainly. Its grey back wall receives the lamp’s light from the ceiling at 68 degrees from its own normal, which is a slant at which an interface reflects well and its lobe is sharply aimed. Of what the satin reflects, 94.5 per cent goes to the floor; the matt body sends the floor 9.5 per cent of its return. The two painted side walls, which the matt body sends half of everything to, receive 2.6 per cent of the lobe between them.

Paint the side walls of that room, then, and a finish on its grey faces takes the lamp’s light the end walls would have spread across the paint and sends it down to a grey floor instead. The census’s answer for that room is that not one of seventy-two paints gains. Paint the floor instead, and the same end walls become the room’s most effective route for lamp light to reach the paint. That room gains at every paint.

This is also the plainest statement of what the highlight is the lamp established about a single glossy object, carried into a room. What an interface reflects is the lamp’s own light, aimed; whether that makes the room more colourful depends on what the aim lands on.

Face by face #

Taken face by face, the room-level number decomposes into contributions that can be read as a decorator would read a room. A wall lit from above sends its gloss down, so in every room the grey walls push the lamp’s light towards the floor. A grey ceiling lit from below — by the light the walls and floor return — sends its gloss back down and across, towards whatever it sees at a slant.

The low wide room with its side walls painted is the room the whole question started from. Its grey ceiling, two units square and only 0.6 above the floor, receives light mostly from the floor straight below it and sends its lobe at a slant towards the side walls, 13.2 per cent more to the paint than matt. Its grey floor does the same at 3.2 per cent. Its grey end walls, lit from the ceiling almost straight ahead of them, send 5.4 per cent less. The area-weighted sum is positive, 5.1 per cent, and 51 of 72 paints gain.

The cube with its side walls painted is the same arrangement in a taller room, and it goes the other way: its grey end walls see the lamp at 45 degrees and send the lobe to the floor, 15.2 per cent less to the paint than matt. The ceiling and floor cannot make that up, and seven paints gain. The same painted faces, moved into a box of different proportions, change which way each grey face aims the lamp. That is exactly what painted area could not see, and why an open room hands over sooner found one low room gaining among several that did not.

The painted ceiling gains for a different reason #

The prediction set the painted-ceiling rooms aside, on the grounds that the lamp shares the painted face, and it was right to. All six read negative, from −6.8 to −2.5 per cent, and five of them gain at every one of the seventy-two paints. The tall narrow slot, painted on its ceiling, gains at none.

The negative reading is what the mirror predicts. A grey wall lit from the ceiling aims its lobe at the floor, not back at the ceiling, so the finish sends less light to a painted ceiling than matt paint did. By the rule that sorts every other room, none of the six should gain.

The rule fails here because of what “the lamp’s light” means when the lamp and the paint share a face. In every other room the lamp’s direct light is white and the paint’s light is coloured, and sending one to the other is how the colour grows. With the ceiling painted, what arrives at the grey faces from the ceiling is the lamp’s white and the paint’s colour together, in one direction, and the send number cannot separate them. Whatever the grey finish does in those rooms, where it aims the ceiling’s light does not capture it, and why five of the six gain at every paint while the slot gains at none is not settled by any of the four numbers.

The solver’s patches and the true box disagree about the tall room #

The ordering of 0.92 is an ordering of the census, and the census is solved with one patch a face. That is how the directional solver works, the one built after the solver had no slot for gloss showed radiosity had no place for a lobe. The lobe is averaged over the cone each patch subtends, which the boundary belongs to the quadrature found reliable at a roughness of 0.2. But every direction in the room is still taken from the centre of one face to the centre of another. A tall wall is treated as though every point on it saw the ceiling from its middle.

Through the true box, the lit send number still leads every other number by a wide margin, at 0.77 against 0.25 for the view. But its sign no longer sorts the rooms: twenty-two of the twenty-four read positive. The largest disagreement is the tall room with its side walls painted, the room the mechanism above was illustrated with. In the patches its grey end walls send 32.3 per cent less of the lamp’s light to the paint than matt; through the true box, casting rays from nine points on each wall, they send 5.5 per cent more.

The reason is visible in the geometry. From the middle of a tall end wall the ceiling sits at 68 degrees and its mirror direction is the floor. From a point near the top of the same wall the ceiling arrives at grazing incidence, and its mirror runs down the wall at a shallow angle, where much of it reaches the side walls first. One patch cannot hold both.

So the number is exact about the census and approximate about the room. Its correlation with the census’s gains is high partly because it is computed in the same approximation the census was, and that is the right test of whether it explains the census. Whether a real tall room with painted side walls also loses colour to a grey finish is not settled by either computation. The census says it does, for every paint; the true-box number says the finish should help it a little.

Aim the gloss where the lamp’s mirror meets the paint #

The practical version of the earlier finding was that a satin finish belongs on the grey surfaces of a room with coloured walls, not on the coloured walls. This one adds where the grey finish does the most good. Put the gloss on a neutral surface whose mirror direction, seen from the lamp, lands on the paint. In these rooms that means satin end walls when the floor carries the colour, where each wall sends the paint 59 per cent more of the lamp’s light than matt in the tall room; and a satin ceiling and floor in a low room with coloured side walls, where the ceiling’s lobe runs across to the paint at a slant. It also names the arrangement to avoid: satin end walls in a tall room with coloured side walls, which in the census aim the lamp’s light at a grey floor and whiten every paint. A gloss neutral surface aimed at another neutral one can only whiten the room.

A figure worth having for any real room is the one this computation makes: for each neutral surface, the share of the lamp’s light its mirror sends to the coloured surfaces, against the share its matt version would. It needs a plan of the room and the lamp positions, and nothing about the paint. It is a statement about aim rather than exposure, and a probe at the wall prices the finish is its counterpart on the painted side: there, what arrives at the paint decides what a finish on the paint costs; here, where the neutral surfaces send the lamp’s light decides what a finish on them adds.

Still open: whether the tall room gains in a finer room #

The census’s rooms are one patch a face, and the true box disagrees with them most about the tall room with its side walls painted. The census has every paint losing colour to a grey finish there, while the lamp-lit send number, computed through the true box, reads positive. Both cannot be the room’s answer. The computation that settles it is the directional solve on subdivided faces — four or nine patches each — for the tall room and the slot, with the census’s seventy-two paints. The prediction is that the tall room’s end walls, split into upper and lower halves, disagree in sign: the upper halves send the lamp’s light across to the painted side walls and the lower halves down to the floor. If so, the room gains for some paints, and the sorting line of the patch census moves.

The painted-ceiling rooms need a different number, one that separates the lamp’s white from the paint’s colour arriving from the same face. The candidate is the send number weighted not by luminance but by the ceiling’s reflected light alone, so that it measures where the finish aims the paint rather than the lamp.

An explanation names a direction as well as a quantity #

The habit is about what to compute when an explanation mentions a geometry.

The explanation offered for the grey finish named a quantity — how much paint the grey faces see at a slant — and the quantity was computed and failed. What it had left unstated was a direction: seeing the paint and sending light to it are the same integral only when light arrives from everywhere equally, and in a room with one lamp it does not. The fix was not a better estimate of the same quantity but the same quantity pointed the other way, with the light put in.

The failure mode is to test an explanation in the form it was stated rather than the form in which it could act. A reflection acts on light, and light has a source; any account of what a surface does in a room has to say where the light it acts on comes from before it can say anything about where that light goes.