A taller room keeps its floor losing in proportion

A painted floor in a room of grey walls and ceiling takes on a little colour when the grey faces are given a satin finish instead of a matt one, and the paint itself gains all or nothing found that the gain is the rule: the finish reflects the floor’s own light back to it at glancing angles, and light that has been back to a coloured surface comes back more coloured. The exception is a narrow room near square, and a run of essays has mapped where it lives.

A long floor never loses its colour to the finish found floors three and four units long, in a box 2.5 units tall, gaining at every shorter side the census solved, and the window of loss closes from both ends found the loss confined to a pocket: the deepest loss rising through nought when the floor is 2.59 units long — about the room’s own height — and centred at a shorter side near 0.4. The pocket belongs to the room’s end walls; moving them away closes it.

One height cannot say what sets the pocket’s size. A taller box must be deeper before the finish adds colour had already found height mattering, a taller box losing colour to deeper shorter sides. The prediction was that the crossing scales with the room’s height: a box four units tall loses until its floor is about 4.2 units long, 1.04 times its height as before, because the end walls’ glancing reflections reach the floor at the same angle in a box of the same proportions. The alternative was that the crossing barely moves from 2.6 units, which would mean the end walls’ cost is set by their distance from the floor’s centre and a tall room’s floor is no more at risk than a low one’s.

The crossing scales; the pocket moves #

In the box four units tall, solved at six strips a side, the painted floor loses chroma under the satin finish at a shorter side of 0.6 when three and four units long — by 0.029 and 0.0016 per cent — and at 0.8 when three units long, by 0.006; it gains in every other box, and at 0.4 at every length, by 0.016 to 0.19 per cent. The deepest loss crosses nought, by straight line between the floors four and five units long, at 4.04 units, 1.01 times the room’s height, where the 2.5-tall box’s crossed at 1.04 times its own. Every box gains or loses for all seventy-two paints together, and every box’s change has the sign of the finish’s tilt across the floor’s reflectance.

• The prediction holds in its scaling, a little short in its number: 4.04 units against 4.2.
• The alternative fails. A four-tall room’s floor keeps losing well past 2.6 units.
• The pocket moves with the height as well as stretching with it: its losses now sit near 0.6, not 0.4.

Drawn against the floor’s length as a share of the room’s height, the two rooms’ deepest dips cross nought within three hundredths of each other. The lower box’s dip crosses at 1.04 of its height and the taller box’s at 1.01; in absolute terms one room stops losing its floor’s colour at 2.6 units and the other at 4.0. The end walls stop costing the floor its colour at about the same proportion of the room, not at the same distance.

The two curves are drawn through few points — the taller box’s through three lengths, its crossing interpolated between four units, where it loses by 0.0016 per cent, and five, where it gains — and the lower box’s through four. Neither rests on a finely sampled minimum, and the three hundredths between the ratios are within what the sampling allows.

Six tenths, not four #

The prediction asked about one shorter side, 0.4, where the lower box’s pocket was deepest; the census solved three, because a taller box had been found losing at deeper floors than a lower one. At 0.4 the taller box’s floor gains at every length, by 0.016 per cent three units long and 0.19 per cent five; at 0.6 it loses three and four units long; at 0.8 it loses three units long and gains at four and five. Had the census solved 0.4 alone, it would have reported no loss at all in a four-tall room and the prediction’s crossing as missing entirely.

Each length’s curve has the lower box’s shape: a fall from the narrowest box, a minimum, a rise. The minimum sits at 0.6 for every length solved, and the curves rise with the floor’s length, so that the window of loss — the shorter sides over which the floor loses — shrinks to a sliver around 0.6 at four units long and is gone by five. At 0.8 the three curves are already rising towards the gains every wider floor in the census has shown, and the floor five units long gains there by 0.043 per cent, only a little less than it gains at 0.6.

Where each verdict falls #

Mapped as verdicts, the taller box’s losses form the same wedge as the lower box’s. Three units long, two of three shorter sides lose; four units long, one; five units long, none. The wedge narrows towards a shorter side near 0.6, as the lower box’s narrowed towards 0.4, and at its tip the loss is a few thousandths of a per cent — the same smallness the lower box’s tip had at two and a half units.

That is the shape the lower box’s essays called a pocket rather than a threshold, found again a room’s height higher. It is still a coarse map — nine boxes, each several minutes of the directional solver — but coarse in the same way as the first, and it closes at the same proportion.

The pocket’s centre #

In the 2.5-tall box the deepest losses sit at shorter sides of 0.35 and 0.4 — 0.14 and 0.16 of the height; in the four-tall box at 0.6, 0.15 of it. The pocket’s centre scales with the height as its far edge does. A room scaled up in height and length together keeps its pocket of loss in the same place relative to its height, and a floor’s shorter side matters to the finish in proportion to how tall the room is.

The census’s shorter sides are a tenth or two apart in both rooms, so the centre is placed only that finely; the agreement to a hundredth of the height is partly luck of the sampling. What the sampling cannot hide is the move from 0.4 to 0.6: the taller room gains at 0.4 at every length, and the lower room lost there.

The tilt follows the colour #

Every scene essay has found the floor’s change in colour carrying the sign of one quantity: the tilt of the finish’s brightening across the floor’s reflectance, how much more it brightens a strongly reflecting band than a weakly reflecting one. In the four-tall box every one of the nine tilts has the sign of its box’s change in colour: negative at 0.6 three and four units long, −0.19 and −0.011 points, and at 0.8 three units long, −0.039; positive everywhere else, up to +1.44 points at 0.4 five units long.

The tilt is what the finish does; the colour change is its consequence. A positive tilt adds light that has been back to the floor, favouring its bright bands; a negative one adds light that has not, brightening every band alike and washing the paint towards grey. The taller room’s pocket is where its end walls’ satin sends the floor more of the second kind, and it sits, as the lower room’s did, at a shorter side that is a fixed share of the height.

More light from a taller pipe #

The quantity of light the finish adds depends on the height much more than the colour change does. In the four-tall box the finish brightens the floor’s darkest reflectance by 80 per cent at a shorter side of 0.4 three units long, where the lower box brightens it by 55 per cent at 0.35. A taller narrow room is a longer pipe, and the satin walls carry the lamp’s light down it in more bounces; a taller box must be deeper before the finish adds colour found the same piping when it raised the box from 1.5 to 4 units with its floor a unit wide.

In both rooms the brightening falls as the floor widens and barely depends on the floor’s length. The near walls set how much light reaches the floor; the end walls tilt it. That separation, which the window of loss closes from both ends drew for one height, holds at two.

The earlier walks, read in proportion #

The scene essays reached this pocket by walking along one dimension of a 2.5-tall box at a time, and each walk can now be read as a statement about proportions. The finish stops costing the paint six tenths deep walked a box one unit wide from shallow to deep and found the turn from loss to gain at 0.62 of its width; a square floor needs more room than a long one found the turn falling as the floor lengthened, 0.78 for a square, 0.64 at a unit long, 0.51 at two. Each of those turns was one edge of the pocket, and each was found at one height.

Read against the height, those turns are 0.31, 0.26 and 0.20 of it: a pocket whose far edge retreats towards about a sixth of the height as the floor lengthens towards the height. The four-tall box’s losses at 0.6 and 0.8 three units long — 0.15 and 0.20 of its height, three quarters of its height long — sit inside the same band. The census has not walked the taller box’s floor across its full range, but where it has solved, it lands where the lower box’s map, rescaled, says it should.

The losses also keep the scale they had. The slot’s loss survives every finer division established that the loss in a tall narrow box is the solver’s answer and not a product of how coarsely the faces are cut, and every loss since has been a few hundredths of a per cent of the paint’s chroma or less. The taller box’s deepest loss, 0.029 per cent three units long, is the same order as the lower box’s 0.028 two units long — nearly the same proportion of floor to height, four fifths and three quarters — and its tip, 0.0016 per cent, the same order as the lower box’s 0.0056.

A grey finish mirrors the lamp onto the paint offered the mechanism: each satin face re-weights what it sends the floor towards the faces it sees at a slant. Angles are what a box’s proportions fix and its size does not, which is why the pocket should scale; the next calculation tests that directly, by building one room as the other enlarged.

What a lighting designer should take from this #

Think of the pocket in the room’s proportions. A satin finish costs a painted floor colour only in rooms whose floor is shorter than about the room’s height and whose narrow side is near a sixth of it. A room 2.5 units tall is at risk with a floor 0.4 wide; a room four units tall with a floor 0.6 wide.

Height is the cheaper variable to check. A designer knows a room’s height before its furniture and its floor finish; the pocket’s place follows from it, and a floor wider than about a third of the height, or longer than the height, is outside it whatever the finish. The square floor in the lower box, the widest the census has found losing, turned at 0.31 of its height; nothing wider has lost in either room.

The effect is still tiny and the brightening is not. Losses and gains alike are hundredths of a per cent of the paint’s chroma; the finish’s brightening of a narrow floor is tens of per cent, and larger in a taller room.

How the boxes were solved #

Each box is four units tall with a floor three, four or five units long and 0.4, 0.6 or 0.8 wide, its floor painted with each of seventy-two paints — Gaussian reflectance bands at eight hues, three widths and three peak reflectances on a grey ground — and its walls and ceiling grey, matt or satin of roughness 0.2, lit by a uniformly emitting ceiling. The census’s directional solver carries each patch’s outgoing light as a function of direction between faces divided into six strips a side; the floor’s light is solved at Chebyshev nodes across reflectance and interpolated for each paint. The change in chroma is each paint’s chroma under the satin finish over its chroma under the matt, less one, averaged over the paints; the tilt is the brightening at the floor’s brightest reflectance less at its darkest. The 2.5-tall box’s floors are the earlier census’s, read the same way; each crossing is a straight line between the last losing length and the first gaining one.

What this does not reach #

Three shorter sides a room. The pocket’s centre and edges are placed to a tenth or two of a unit; a finer scan would move the crossing by a few hundredths.

Six strips on long thin faces. A floor five units long and 0.4 wide is divided into strips of more than eight to one; the earlier essays found four strips misreading elongated faces.

Two heights, not a scaling. The two rooms differ in height and in their floors’ shorter sides, which were not scaled with the height, so the census compares them but does not test whether the answer depends on proportions alone.

Still open: whether the answer is the room’s proportions alone #

The two rooms’ crossings agree as shares of the height within three hundredths, and their pockets’ centres within a hundredth, but the census’s floors were not scaled from one room to the other. If the finish’s effect depends only on the room’s proportions, a four-tall box built as the 2.5-tall box multiplied by 1.6 in every dimension must give the lower box’s numbers exactly.

The calculation is the box four units tall with its floor four units long and 0.64 wide — the 2.5-tall box’s floor two and a half by 0.4, scaled — and, to place the taller pocket’s edges at four units long, the same box with its floor 0.5 and 0.7 wide. The prediction is that the scaled box reproduces the lower box’s loss of 0.0056 per cent to the solver’s precision, because every exchange between faces depends on their shapes and not their size, and that the four-unit floor loses from about 0.55 to 0.65, the lower box’s window at two and a half units scaled. If the scaled box differs, something in the census carries a length, and the pocket’s place in a real room has to be read room by room. A room scaled whole loses its floor to the last digit makes that calculation.

One number, asked at the wrong place #

The habit here is about carrying a location from one case to the next.

The prediction carried the lower room’s shorter side, 0.4, to the taller room along with its ratio, and asked whether the crossing moved. At 0.4 the taller room never loses at all; the question only had an answer because the census also solved 0.6 and 0.8. The ratio scaled, and the place to look for it scaled too.

The failure mode is to test whether one quantity scales while holding fixed another that scales with it. A loss that is a property of a room’s proportions moves in every proportion at once, and a test that fixes one of them can find nothing to measure. The test is to scale every dimension the effect could depend on — or to search each one — before reading the one the question was about.