A taller box must be deeper before the finish adds colour

A satin finish on a room’s grey faces changes the colour of the painted surfaces they surround. In the painted census, which a finish adds colour only where it covers grey set up, it nearly always adds colour, and the paint itself gains all or nothing found the one exception: the tall narrow slot, one unit wide, two and a half tall and 0.35 deep, whose painted floor loses a twentieth of a per cent of its chroma. The slot’s loss survives every finer division confirmed the loss, and the finish stops costing the paint six tenths deep walked the slot’s depth out to the tall room’s full unit and found the floor turning from loss to gain at 0.62 deep — 0.64 at a finer division.

That essay also found what decides the sign. The finish brightens the painted floor a great deal — more than doubling its light in the slot — because two glossy walls facing each other carry the lamp’s light down between them, as a polished duct carries daylight. But brightness alone does not colour the paint. The paint gains chroma when the finish multiplies the light that has been back to the paint by more than the light that has never touched it, and loses when the order reverses; the sign of that tilt was the sign of the change at every depth.

It left the scan’s one dimension unexamined. Every box was one unit wide and two and a half tall, and the turn was reported as a depth. If the light pipe is what greys the paint, a taller box is a longer pipe and should need to be deeper before the paint’s own returned light can keep pace. The prediction was that the turn scales with the height: deeper in a taller box, shallower in a shorter one. If instead it stayed near 0.62 at every height, the turn would be a property of the floor’s shape, depth against width, and the light-pipe reading would be the wrong one.

The turn scales with the height, faster than in proportion #

Solved at four strips a side with the floor painted, the box one unit wide and 1.5 tall gains for every one of seventy-two paints at every depth from 0.35 to a full unit, and its first crossing from loss to gain sits near 0.26. The box 4 tall loses for every paint at every depth from 0.35 to 1.5 — half again as deep as it is wide — and gains at 2, turning near 1.67. With the 2.5-tall box’s 0.62 between them, the turn is 0.17, 0.25 and 0.42 of the height: a larger share at every taller box, close to a depth that grows as the square of the height. The finish brightens the floor more in every taller box, tripling it in the 4-tall box at 0.35 deep, and at every depth of every box the sign of the paint’s change is the sign of the finish’s tilt across reflectance. Solved again at six strips, the 4-tall box still loses a full unit deep and the 1.5-tall box still gains at 0.35.

• The prediction holds in direction, and understates the size. The turn moves with the height, and more than proportionally.
• Neither a share of the width nor of the height sets it. One unit of width fixed, the turn runs from under a third of a unit to more than one and a half.
• The light-pipe reading survives its test. The longer the pipe, the more lamp light it carries and the deeper the box must be before the paint keeps up.

The three curves are the same box at three heights, and they sit one above another. At a full unit deep — a box square in plan — the short box gains 0.20 per cent, the middle one 0.05 and the tall one loses 0.05. The short box climbs steeply from a third of a unit and is still rising at one. The tall box drops to a loss of about four hundredths of a per cent by half a unit deep, holds it nearly flat to a full unit, and only then recovers, crossing nought between 1.5 and 2.

The shallowest boxes are not resolved #

The short box has one feature that is not part of the pattern. At 0.2 deep it gains 0.023 per cent; at 0.25 it loses 0.002; at 0.35 it gains again. The tall box at 0.35 sits at −0.003, close to nought, where at half a unit it loses fourteen times as much. Those are the shallowest boxes in the scan, and at four strips a side their side walls are cut into strips only five to nine hundredths of a unit wide while their end walls’ strips are a quarter to a whole unit tall. The slot essay found coarse division getting the sign wrong at exactly this kind of mismatch, and these two readings are too close to nought for four strips to settle.

So the short box’s turn is reported as under 0.35 rather than at 0.26. The finer check below confirms that it gains at 0.35, by less; where between 0.2 and 0.35 it turns, or whether it turns twice, would need the very shallow boxes solved at many more strips than the solver comfortably holds.

Neither the width nor the height alone #

The width is fixed at one unit in all three boxes, so a turn set by the floor’s shape — depth against width — would sit at the same depth in each. It moves from under 0.35 to 0.62 to 1.67. A turn set by the depth against the height would lie on the dashed line through the middle box; the short box turns shallower than that line and the tall box far deeper. The dotted curve, depth growing as the square of the height, passes close to all three. Three points do not establish a power law, and the short box’s turn is only bounded; what they establish is that the turn grows faster than the height.

Plotted against depth over height, the three curves do not collapse. They cross nought at about 0.17, 0.25 and 0.42 of the height. The taller the box, the larger the share of its height it must be deep before a satin surround adds colour to its floor.

A taller box pipes more of the lamp #

The reason a taller box needs to be deeper shows up first in brightness. At 0.35 deep the finish adds 51 per cent to the short box’s floor, 112 per cent to the middle box’s, and 298 per cent to the tall box’s — nearly quadrupling its light. At a full unit deep the three are 22, 48 and 94 per cent. In a matt box, a floor four units below its lamp receives the lamp’s light mostly after many diffuse bounces, each sending as much of it back up as down; satin walls send each reflection on downward at nearly the angle it arrived at, and the further the floor is from the lamp, the more that matters. The pipe’s gain grows with its length.

The tall box’s plateau makes the same point from the other side. Between half a unit and a full unit deep its loss barely moves, from −0.042 to −0.046 per cent, while the finish’s brightening of its floor falls from 178 per cent to 94. The tilt barely moves either, from −0.48 to −0.37 points. Half the boost goes and the colour stays where it was, because what the finish takes away from the lamp’s share it takes away in nearly the same proportion from the paint’s. The paint’s own light does not order the rooms found no single count of where the light goes predicting the paint’s change across the census; within one box the brightening does not predict it either.

Every box brightens less as it deepens, because a wider gap between the glossy walls lets more of the lamp’s light reach the floor directly in the matt box too, leaving less for the finish to add. The paint’s chroma depends on how that boost compares with what the finish does to the paint’s own light.

The tilt still carries the sign #

At a reflectance near nought almost everything arriving on the floor has never touched the paint, so the brightening there is very nearly the factor by which the finish multiplies the lamp’s light on the floor. At a reflectance near one a large part of what arrives has been to the paint and back. The tilt — the bright reflectance’s boost less the dark one’s — says whether the finish favours the paint’s own returned light over the lamp’s, and at every depth of all three boxes, at six strips as at four, its sign is the sign of the paint’s change in chroma. In the short box it is positive throughout the resolved range, rising to more than a percentage point at a full unit; in the tall box it is negative to 1.5 deep and positive at 2.

That is the mechanism the light-pipe reading proposed, tested where it could have failed. A taller pipe multiplies the lamp’s light more, and the paint’s returned light, which also travels along the glossy walls but starts at the floor and goes nowhere near the lamp, cannot keep pace until the box is wide enough in depth for the walls’ piping to weaken. The deeper box weakens the pipe; the taller box strengthens it; the turn is where the two balance, and because the pipe’s gain grows faster than linearly with its length, the turn does too.

Six strips keep both signs #

The tall box needs the check more than any box in the series. The tall room gains once its walls are divided found the 2.5-tall box’s verdict moving with division before it settled, and divided faces keep the census’s count and lose its rule found re-solving every room changing which rooms gained. At four strips a side its end walls, four units tall, are cut into strips a full unit high — coarser than the 0.625 at which the earlier scans’ signs were settled, and the slot essay showed coarse strips getting a sign wrong. Solved at six strips, the 4-tall box a full unit deep loses 0.0443 per cent against 0.0458 at four: the same sign and nearly the same size. The short box at 0.35, the shallowest depth at which it was claimed to gain, gains 0.0073 per cent at six strips against 0.0184 at four — the same sign, smaller by three fifths.

The two moves are in the direction the previous scan’s check found. Finer division deepened the slot’s loss and moved the 2.5-tall box’s turn two hundredths deeper; here it shrinks the short box’s shallow gain, which would move its turn deeper too, towards 0.35, and leaves the tall box’s loss where it was. Neither finer solve reverses a sign, and the turn’s growth with height is if anything slightly understated by four strips at the short end.

What this means for a real room #

A corridor, a stairwell, a lift shaft and a deep cupboard are all tall, narrow boxes lit from one end. The census says that in any of them a glossy grey surround takes colour from a painted floor or ceiling unless the space is at least as deep as its height requires — about a quarter of the height in a room two and a half times as tall as it is wide, about four tenths in one four times as tall, and less than a fifth in a squat one. A stairwell four storeys tall with satin walls will grey a coloured floor at the bottom whatever its plan.

The sizes are small — at most a twentieth of a per cent of chroma — and no one would see them as a change in the paint. The same census found the room’s own reading, all of its light rather than the paint’s, swinging from −18 to +57 per cent under the same finish, and a grey finish mirrors the lamp onto the paint traced most of that to the grey faces themselves. What the scan describes is the paint’s own colour, read on the painted face, and the boundary in the space of shapes where a satin surround stops helping it.

How the boxes were solved #

Each box is one unit wide, its height 1.5, 2.5 or 4, its depth from 0.2 to 2, its whole ceiling emitting, its floor 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. Every face is divided into four strips along each side, as the depth scan’s were, and the 4-tall box at a full unit and the 1.5-tall box at 0.35 again into six. 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 and without the finish at twenty reflectances on Chebyshev nodes, and each of seventy-two paints read in CIELAB against the lamp on the painted floor. The 2.5-tall box’s numbers are the depth scan’s.

The turn is the straight-line crossing of nought between the depths either side of it; for the 1.5-tall box it is the first crossing from loss to gain, between 0.25 and 0.35, and for the 4-tall box the crossing between 1.5 and 2.

What the scan does not reach #

One width. Every box is one unit wide, so the scan separates the height from the depth but not from the width. A box twice as wide at each height would say whether the turn is the height alone or the height against the width.

The shallowest boxes. Below about a third of a unit the four-strip readings are near nought and unstable in sign; the short box’s turn is bounded, not located.

The floor only. The ceiling, which the depth scan found turning within a hundredth of the floor, is not solved at the new heights.

One roughness. A glossier finish is a better pipe and should move every turn deeper; how far the solver can follow a narrower lobe is the limit the boundary belongs to the quadrature priced.

A closed box. An open room hands over sooner found an open wall moving a gloss finish’s turning point; a stairwell open at each landing is not the sealed box solved here.

Still open: whether the width matters once the height is fixed #

The scan held the width at one unit and found the turn growing faster than the height. In the light-pipe reading what matters is the pipe’s length against its bore — the height against the gap between the glossy walls — and the width of those walls should matter little; in the reflectance-tilt reading the width sets how much of the paint’s light comes back to it, and should matter as much as the depth.

The calculation is the 2.5-tall box made two units wide and half a unit wide, its floor painted, solved at four strips at four or five depths around each one’s turn. The prediction is that the turn moves less than a tenth of a unit between widths, because the pipe is formed by the two end walls whose separation is the depth, and their width only adds more of the same pipe. If the turn instead scales with the width, the paint’s returned light is the controlling term and the light pipe is the lesser half of the balance.

A scaling is the test of a mechanism #

The habit here is about how to test an explanation once it has fitted the case it was built on.

The light-pipe reading explained the depth scan: glossy walls carry the lamp to the floor, more strongly in a shallower box, and the paint greys when that carriage outpaces its own returned light. An explanation that fits one family of shapes is easy to come by. The useful test is to change a dimension the explanation says should matter, by a factor large enough that the answer cannot be mistaken for noise, and see whether the result moves as it says.

The failure mode is to accept a mechanism because it reproduces the measurements it was proposed to explain. The test is a scaling it predicts in advance. Here the prediction was a direction, deeper with height, and the scan returned the direction and more: a turn that grows faster than the height, a brightening that triples, and a tilt whose sign follows the change in every box. The mechanism made a claim about boxes it had not seen, and the boxes agreed.