The width moves the turn only when it is the narrower side

A satin finish laid on the grey faces of a room changes the colour of the painted surfaces they surround. The paint itself gains all or nothing found that it nearly always adds a little colour to the paint, and that the one place it takes colour away is a tall narrow slot with its floor or ceiling painted. The finish stops costing the paint six tenths deep walked the slot’s depth out and found the painted floor turning from loss to gain at about 0.62 of the width in a box one unit wide and two and a half tall — 0.64 at a finer division — and a taller box must be deeper before the finish adds colour found that turn growing faster than the box’s height.

The account that survived both scans was a light pipe. Glossy walls facing each other carry the lamp’s light down between them, reflection after reflection at nearly the angle it arrived, the way a polished duct carries daylight into a windowless room. The finish multiplies the light reaching the floor, and the paint loses colour when that multiplication favours light that has never touched the paint over light that has been back to it. A taller box is a longer pipe; a deeper box separates the glossy walls and weakens it.

Every box in both scans was one unit wide, and the turn was reported as a depth. The height essay’s closing section asked whether the width matters once the height is fixed, and predicted that it does not: the pipe is formed by the two end walls whose separation is the depth, and their width only adds more of the same pipe, so the turn should move by less than a tenth of a unit between a box two units wide and one half a unit wide. If instead the turn scaled with the width, the paint’s own returned light would be the controlling term.

The width matters when it is the narrower side #

Solved at six strips a side, the 2.5-tall box two units wide makes every one of seventy-two paints on its floor less colourful at 0.35 and 0.5 deep and every one more colourful from 0.65, turning between 0.5 and 0.65 — about 0.51, within 0.15 of the one-unit box’s 0.64. The box half a unit wide loses for every paint at 1.25 and 1.5 deep, and its quarter-turned twin, two wide and half a unit deep, still loses: its turn lies past two, more than a unit and a quarter deeper. A box w wide and d deep is the same box as one d wide and w deep — solved both ways they agree within three per cent — so in the floor’s plan the finish takes colour wherever the shorter side is under about half a unit and adds it once the shorter side passes about 0.65. The first pass, at four strips a side, had the half-unit box turning at 1.22 and the two-unit box gaining at every depth; four of the seven boxes solved at both divisions change sign between them.

• The prediction holds on one side and fails on the other. Widening the box past its depth barely moves the turn; narrowing it below its depth moves the turn by more than a unit.
• There is no end wall. With only the floor painted, the two pairs of satin walls are interchangeable, and whichever pair is closer does the work.
• A coarse division misreads long thin boxes, and the scan nearly reported a rule that six strips do not support.

Placed in the plan, the six-strip boxes make a picture symmetric about the diagonal, as they must. The losses sit in two arms along the axes — floors half a unit across and between 1.25 and 2 long, floors half a unit or 0.35 across and two long, and the one-unit box at 0.6 — and the gains lie further from both axes: the one-unit box at 0.7, the two-unit box at 0.65 and 0.8. Read across the arms rather than along them, the line between loss and gain is a line in the floor’s shorter side, at about half a unit when the longer side is two and about 0.64 when it is one.

One point does not fit. The two-unit box a quarter of a unit deep gains, at six strips as at four, though its shorter side is the narrowest in the scan. It is the shallowest box solved, its quarter-unit faces cut into strips four hundredths of a unit wide, and its gain shrank by three quarters between four strips and six; it is reported here as unresolved, as the height essay reported its shallowest boxes.

Four strips get four signs of seven wrong #

The scan’s first pass solved every box at four strips a side, as the depth and height scans had, and it told a different story. The half-unit box seemed to lose to a full unit deep and turn at 1.22; the two-unit box seemed to gain at every depth from a quarter to two. Read together, those turns lay at floor areas of 0.61 and above, near the one-unit box’s 0.62, and it looked as though the floor’s area, not its depth, set the turn.

Solved again at six strips, four of the seven boxes change sign, all from gain to loss: the half-unit box at 1.25 and 1.5 deep and the two-unit box at 0.35 and 0.5. The half-unit box at 1.25 goes from +0.0016 to −0.0156 per cent; the two-unit box at 0.35 from +0.036 to −0.028. The three that keep their sign — the two-unit box a quarter deep and at 0.65 and 0.8 — keep it with sizes that move by up to four fifths. Division has overturned a reading before — the tall room gains once its walls are divided found a census of one patch a face wrong about a tall room’s sign — but nothing like this happened in the earlier scans of the slot: the slot’s loss survives every finer division found its sign steady from three strips to eight, and the depth scan’s six-strip check moved its turn by two hundredths.

What the long, thin boxes have that the earlier ones did not is long, thin faces. Four strips cut a two-unit wall into patches half a unit wide and 0.625 tall, and the floor of a box half a unit wide and 1.5 deep into patches an eighth of a unit by three eighths. A direction-resolving solver averages each lobe over the cone one patch subtends at another, and a patch much longer than it is wide subtends a cone that is not much like the face it stands for. The six-strip division makes the patches squarer and smaller. Whether six is enough for the longest of these faces is not proven here; what is proven is that four is not, and that the four-strip reading of these boxes was the wrong one.

The finish’s brightening shows the same thing from the other side. Six strips brighten every one of the seven floors less than four: the two-unit box a quarter deep from 128 per cent to 80, the half-unit box at 1.25 deep from 74 to 59. The coarse division overstated how much of the lamp the satin walls pipe down in these boxes, and overstated it most where the faces are longest and thinnest. A floor that is piped less is, on the light-pipe account, a floor where the paint’s own light keeps pace more easily — so the corrections in brightness and in sign are not the same correction, and the six-strip losses are not simply the four-strip gains with less light.

The shorter side sets the turn #

The solid points are the result. At six strips the one-unit box loses at 0.6 deep and gains at 0.7; the two-unit box loses at 0.35 and 0.5 and gains at 0.65 and 0.8; the half-unit box loses at 1.25 and 1.5. Making the box twice as wide moves its turn from 0.64 to about 0.51 — a little shallower, as if the wider pipe were a slightly weaker one — and within the change the prediction allowed for. Making it half as wide moves its turn past 1.5, and its twin moves it past 2.

The light-pipe account explains the asymmetry once the end walls lose their special place. In a box two units wide and less than a unit deep, the pipe is formed by the walls a depth apart, the closer pair, and the depth decides the turn as it did at one unit wide. In a box half a unit wide and more than half a unit deep, the closer pair is the side walls half a unit apart, and no amount of depth separates them: the pipe stays strong, and the floor keeps losing until the box is long enough that the strong pipe carries a smaller share of the lamp’s light to the floor than the paint returns to itself. The width moves the turn exactly when the width is the narrower side.

That also reframes the height essay’s square law. Every box in it was one unit wide and shallower than wide near its turn, so its depth was always its shorter side; the turn growing faster than the height is the shorter side’s threshold rising with the length of the pipe. Whether the threshold depends on the longer side as well — 0.64 with a longer side of one, about 0.5 with two — is suggested by two points and not established.

The two points also say which way the dependence runs, if it is real. A longer side that grows without limit turns the box into a slot — two satin walls facing each other across the shorter side, open at neither end, the shape the first of these essays found to be the one place a finish takes colour away. If the turn falls from 0.64 to about 0.5 as the longer side goes from one to two, it is falling towards the slot’s own threshold, the shorter side at which a long corridor’s floor stops losing. That threshold is what a builder of real corridors would want, since no corridor is as short as it is wide, and the two-unit box is already a fair approximation to it. The other direction, a longer side shrinking to equal the shorter, is the square floor, where both pairs of walls pipe at once and neither can be called the end; the closing section asks what that does.

The tilt still carries the sign #

At a reflectance near nought almost everything arriving on the floor has never touched the paint; near one, much of it has been to the paint and back. The tilt says which of the two the finish favours, and the six-tenths essay found its sign always the sign of the paint’s change. At six strips as at four, in every box of the width scan, it still is: the four boxes whose sign the finer division reversed had their tilt reversed with it. The mechanism survives the correction; what the correction moved was how much each box pipes, and so which side of the balance it falls on.

What this means for a real room #

A corridor, a stair core, a lift lobby and a deep cupboard are all tall boxes lit from one end. The census says that a glossy grey surround greys a coloured floor when the floor’s shorter side is under about a fifth of the height — half a unit in a box two and a half tall — however long the longer side is, and colours it once the shorter side passes about a quarter of the height. A stairwell a metre and a half across and three storeys, nine metres, tall is on the greying side; a lobby three metres square and three tall is well on the colouring side. The changes stay small — a few hundredths of a per cent of chroma — and the room’s own reading moves by tens of per cent under the same finish, as a grey finish mirrors the lamp onto the paint found. What the scan maps is the boundary in the space of shapes where a satin surround stops helping the paint’s own colour.

How the boxes were solved #

Each box is 2.5 tall, its width half a unit or two units, its depth from 0.25 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 earlier scans’ were, and seven boxes 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 one-unit box’s numbers are the depth scan’s, at four strips across its scan and six at 0.6 and 0.7.

A box’s quarter-turned twin is the same box with its width and depth exchanged, and in the plan each solved box is drawn at both. The two-unit box’s turn is the straight-line crossing of nought between 0.5 and 0.65; the half-unit box’s is bounded by its twin.

What the scan does not reach #

Six strips everywhere. The six-strip boxes are seven chosen around the four-strip turns; the plan is filled in only where they fell, and the long faces may need more than six strips too.

One height. The shorter-side reading comes from boxes 2.5 tall; the height essay’s boxes, all one unit wide, fit it, but the threshold’s dependence on the longer side rests on two points.

The shallowest box. The two-unit box a quarter deep gains at both divisions, against the reading, and its size changes too much between them to call.

The floor only, one roughness, a closed box. The ceiling, a glossier finish and an open wall all moved the earlier turns; none is solved here.

Still open: whether a square floor turns at the same shorter side #

The six-strip picture puts the turn on the floor’s shorter side, at about 0.64 when the longer side is one and about 0.5 when it is two. A square floor is the one case with no longer side to speak of, and it decides whether the threshold is a property of the shorter side alone or of the two together.

The calculation is the 2.5-tall box with a square floor, solved at six strips a side at sides of 0.5, 0.6, 0.7 and 0.8, the floor painted, and the turn read off as here. The prediction is that the square floor turns at about 0.7, later than the one-unit box’s 0.64, because both pairs of walls are then equally close and both pipe the lamp down, so the floor needs more room before its own light keeps pace. If the square turns at 0.64, the longer side does not matter once it is at least the shorter, and the two-unit box’s shallower turn belongs to its coarser six-strip division rather than to its length. A square floor needs more room than a long one solves it.

A scan is only as fine as its coarsest face #

The habit here is about carrying a numerical setting from one family of shapes to another.

Four strips a side had been checked, and checked again, in the boxes the earlier scans solved: the slot’s sign held from three strips to eight, the depth scan’s turn moved by two hundredths. That was a statement about those boxes. The width scan made faces several times longer than they are wide, and four strips a side cut them into patches of a shape the earlier checks never tested. The first pass looked clean and produced a tidy rule about floor area, and four of its signs were wrong.

The failure mode is to treat a convergence check as a property of the method rather than of the cases it was run on. The test is to re-solve, at a finer division, the cases that differ most in shape from the ones the check covered. Here that removed a rule the coarse pass had suggested and replaced it with a reading — the shorter side decides — that the account the scan was testing could explain.