A long floor never loses its colour to the finish

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 space with its floor or ceiling painted. The scene essays since have walked the edge of that place: the finish stops costing the paint six tenths deep found a box one unit wide and two and a half tall turning from loss to gain at about 0.64 of its depth, and a taller box must be deeper before the finish adds colour found the turn growing with the height.

The account that has survived each scan is a light pipe: glossy walls facing each other carry the lamp’s light down between them, reflection after reflection, and the paint loses colour when that piped light, which has never touched the paint, outweighs light that has been back to it. The width moves the turn only when it is the narrower side found the turn set by the floor’s shorter side, and a square floor needs more room than a long one found it falling as the longer side grew: 0.78 for a square, 0.64 with a longer side of one unit, about 0.51 with two.

Three turns that fall and flatten suggest a limit, and the closing section predicted the limit of a slot: two satin walls facing each other across the floor’s shorter side, the other pair so far off that they pipe nothing. With the longer side at three and four units, the turn should settle near 0.45 — below the two-unit floor’s 0.51 by about half the step from one unit to two — and be reached by four, because a wall pair four units apart pipes the lamp to the floor about as weakly as an open end. If the turn kept falling past four units, the far walls would still matter at a distance sixteen times the near pair’s.

No turn at all #

Solved at six strips a side in the same box two and a half tall, floors three and four units long gain chroma under the satin finish for every one of seventy-two paints at every shorter side from 0.35 to 0.65 — by 0.026 to 0.058 per cent on average at three units, and by 0.098 to 0.151 per cent at four. The floor two units long, read from the earlier six-strip boxes, gains at a shorter side of 0.25, loses at 0.35 and 0.5, and gains again from 0.65: its loss is a window, not a threshold. At every shorter side the four-unit floor gains more than the three-unit one. The finish brightens all three long floors’ darkest reflectance by about the same amount at each shorter side — from 55 to 58 per cent at 0.35 down to about 31 at 0.65 — and the tilt across reflectance, which carries every floor’s sign, turns from −0.19 percentage points on the floor two long at 0.35 to +0.33 at three units and +1.01 at four.

• The prediction fails at the root. There is no slot limit near 0.45; there is no turn in the range at all once the floor is three units long.
• The question it asked is answered the other way. The far walls do matter, and what they contribute is the loss.
• The light-pipe account needs a repair. A single pair of close walls does not cost the paint its colour.

The four curves fall in order from the bottom: the square, which loses to 0.7 on a side; the floor two long, which dips just below nought between 0.3 and 0.51; and the floors three and four long, which never reach it. Each lengthening of the floor lifts the whole curve, and the lift from two units to three is enough to clear the dip. The square’s and the two-unit floor’s losses are tiny in absolute terms — a few hundredths of a per cent of each paint’s chroma — and so are the long floors’ gains; what the census reads is their sign, as every scene essay has, because the sign is what says whether a glossy finish makes the room’s paint duller or brighter.

Five floors to scale #

Drawn to scale the five floors make the account’s geometry plain. In every box the near pair of walls — the long walls — stands the floor’s shorter side apart, between 0.45 and 0.78 across these five. What changes from the top of the figure to the bottom is mostly the distance between the end walls: 0.78 for the square at its turn, one unit, two, and then three and four for the long floors. The verdict follows that distance. A square floor at 0.78 and a floor four units long at 0.45 have their near walls within a third of a unit of the same spacing; the square’s end walls are as close as its side walls, and it is the square that loses.

The census has therefore varied, across five essays, one geometric quantity it had not been watching. The width scan held the box one unit wide and varied its depth; the square scan made the two equal; this scan stretched one far. Each was read as a statement about the near walls, the shorter side, and each was also a statement about the end walls’ distance, which grew with every longer floor. The turn’s fall was the end walls’ retreat.

The two-unit floor’s window #

The floor two units long was the last point of the earlier scan, and read again here beside the longer floors it looks different. It loses only between shorter sides of about 0.3 and 0.51, by under three hundredths of a per cent at worst, and gains on both sides: at 0.25, the narrowest box the scan solved, and from 0.65 up. The square essay’s turn for it, 0.51, is the upper edge of the window, and the lower edge sits between 0.25 and 0.35.

At three units long the curve keeps the window’s shape — least at 0.45, rising on either side — and sits above nought throughout. The dip is the same physics weakened: something that costs the paint colour in a box of those proportions is still at work, and three units of length leave it too little to outweigh the gain. At four units the dip has nearly flattened, and the gain is twice the three-unit floor’s.

What lengthening does #

Read against the longer side, the change is a steady climb. At a shorter side of 0.35 it goes from a loss of 0.028 per cent with the floor two long to gains of 0.049 and 0.151 per cent at three and four; at 0.65, from 0.034 to 0.058 to 0.104. Lengthening the floor moves the end walls away and adds gain at every shorter side, faster for the narrow box than the wide one. Nothing here flattens towards a limit, and the prediction’s slot — the box in which only the near pair of walls matters — is a box that gains.

That reverses the reading the turns suggested. A turn that falls as the floor lengthens looked like the near walls’ pipe weakening relative to the floor’s own light; the turns were in fact the far walls’ loss shrinking. A square floor has both pairs of walls as close as possible and loses most; a floor one unit long has its end walls a little further and loses less; a floor two long loses only where its proportions concentrate the end walls’ effect; and a floor three long has its end walls too far to cost it anything.

The finish brightens the long floors alike #

The light the finish adds to the floor tells a different story from the colour. At a shorter side of 0.35, the finish brightens the floor at its darkest reflectance by 57 per cent when it is two units long, 55 when three and 58 when four, and all three fall together to about a third at 0.65. The amount of light the near pair of walls pipes down does not depend on the floor’s length past two units — the near pair is what pipes it, as the light-pipe account said. The square floor, with both pairs close, is brightened far more, by a hundred per cent at half a unit.

So the long floors receive the same piped light and still gain colour, where the square, receiving more, loses. The colour does not follow the quantity of piped light; it follows something about how that light is spread across the paint’s reflectance, which the next figure shows.

The sign is in the tilt #

A paint is colourful because it reflects some wavelengths strongly and others weakly, and the finish changes its colour by brightening those two kinds of band differently. If the finish brightens the floor more where it reflects more, the paint’s strong bands gain on its weak ones and the paint grows more colourful; if less, the reverse. Every box the census has solved has had the sign of this tilt for the sign of its colour change, and the long floors are no exception. The floor two long at 0.35 tilts down, by 0.19 percentage points across the range of reflectance, and loses; at three units it tilts up by 0.33 and at four by 1.01, and both gain.

The piped light is what tilts. Light that reaches the floor straight from the lamp brightens every reflectance alike; light that has been round the room via the floor itself brightens the strongly reflecting bands more, because they sent more of it out. A tilt down means the finish has added mostly light that never touched the floor; a tilt up means it has added light that did. Why close end walls tilt it down is not separated by the census. One reading, untested here, is that a satin end wall sends the floor’s own light on along the box at a glancing angle where a matte one would have scattered some of it back, so that the finish takes from the floor the light that had been to it and adds light that had not.

What a lighting designer should take from this #

A long, narrow room does not lose its painted floor’s colour to a glossy wall finish. The finish adds the room’s own colour found how much colour a gloss finish costs a paint depending on the walls around it; here what decides is how far away the room’s end walls stand. The cost the scene essays found is a property of near-square plans, and of two-to-one plans only in a narrow band of proportions. A corridor or a gallery three or more times as long as it is wide gains a little colour on its floor from satin walls, at any width the census solved.

The sizes are small; the signs are robust. Every gain and loss here is under a sixth of a per cent of the paint’s chroma, well below what a viewer would notice in a single paint. What a designer can take is the direction: in a long room a satin wall finish nudges every floor paint towards more colour, in a near-square narrow one towards less, and the census has found no paint of seventy-two that disagrees with its box’s verdict. Small effects that never change sign across seventy-two paints are effects of the room, not of any paint.

What matters is the end walls. A gloss finish takes colour out of the whole room and a grey finish mirrors the lamp onto the paint found the finish’s effect set by which faces see which; here the faces that see the floor along the room’s length are the ones that cost it colour. In a room whose plan is near square, finishing the walls at its ends matte and its long walls satin would, on this account, keep the floor’s gain; the census has not solved that mixed finish, and it is the obvious next case.

How the boxes were solved #

Each box is two and a half units tall with a floor three or four units long and 0.35, 0.45, 0.55 or 0.65 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. The grey faces are matte or satin of roughness 0.2, and the box is lit by a ceiling that emits uniformly. Light is transported between the faces, each divided into six strips a side, by the census’s directional solver, which carries each patch’s outgoing light as a function of direction so that a satin face’s glancing reflection is kept; the floor’s light is solved at the census’s 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 matte one, less one, averaged over the paints. The floor two units long is the earlier scan’s six-strip boxes, read here the same way.

What this does not reach #

Six strips a side, at every length. A floor four units long is divided into the same six strips as one 0.35 wide; its strips are eleven times as long as they are wide. The scene essays found four strips misreading long, thin faces; six may misread a face this elongated, and the long floors’ gains, though uniform in sign, are as small as the losses that four strips once got wrong.

One height. Every box is two and a half units tall. A taller box made the turn grow faster than its height; whether a tall long box loses again is open.

One ground and one roughness. The grey faces are one grey at one roughness.

Still open: where between two and three units the window closes #

The two-unit floor loses between shorter sides of about 0.3 and 0.51, and the three-unit floor nowhere. Somewhere between the window narrows to nothing, and where — and whether it closes from both edges at once or from one — says how the end walls’ cost scales with their distance.

The calculation is the floor two and a half units long, solved at six strips a side at shorter sides of 0.3, 0.4 and 0.5, the mean change read as here. The prediction is that the window closes from below: at two and a half units the floor gains at 0.3 and loses only near 0.4–0.5, because the lower edge is where the end walls’ pipe is weakest relative to the near pair’s, and the upper edge moves little between two and three units. If the floor two and a half long loses at all three shorter sides, the window widens before it closes, and the end walls’ cost does not shrink smoothly with their distance. The window of loss closes from both ends makes that calculation.

The limit was not where the trend pointed #

The habit here is about extrapolating a trend to its limit.

Three turns fell and flattened, and a limit was the natural reading: the curve approaches an asymptote, and the asymptote is the slot. The calculation went to the limit and found no turn at all. The three turns were the shrinking of one effect, and the effect could shrink to nothing.

The failure mode is to read a flattening trend as approaching a non-zero limit without asking what the trend is made of. A series that falls and flattens may be heading for a floor or for zero, and only the case past the end of the series says which. The test is to solve the case the extrapolation describes — here, floors three and four units long — rather than to fit the series and read off its asymptote.