The window of loss closes from both ends

A satin finish laid on the grey faces of a room changes the colour of the painted surfaces they surround, and the paint itself gains all or nothing found that it nearly always adds a little colour. The exception is a tall, narrow room with its floor painted, and a run of essays has mapped where that exception lives. The width moves the turn only when it is the narrower side found the shorter side deciding where a box turns from loss to gain, and a square floor needs more room than a long one found the turn falling as the longer side grew — 0.78 for a square floor, 0.64 for one a unit long, 0.51 for two.

A long floor never loses its colour to the finish solved floors three and four units long and found them gaining at every shorter side, and read the earlier turns again: the floor two units long loses only in a window of shorter sides, from about 0.30 to 0.51, and gains on either side of it. The loss belonged to the room’s end walls, and moving them away removed it. Somewhere between two units and three the window must close, and how it closes says how the end walls’ cost falls with their distance.

The prediction was that it closes from below: that the floor two and a half units long gains at a shorter side of 0.3 and loses only near 0.4 to 0.5, because the lower edge is where the end walls’ pipe is weakest against the near pair’s and should give way first, while the upper edge moves little between two units and three. The alternative named was that the floor two and a half long loses at all three shorter sides, which would mean the window widens before it closes and the end walls’ cost does not shrink smoothly with distance.

A gain, a loss, a gain #

Solved at six strips a side, the floor two and a half units long gains chroma under the satin finish for every one of seventy-two paints at a shorter side of 0.3 — by 0.034 per cent on average — loses for every one at 0.4, by 0.0056 per cent, and gains for every one again at 0.5, by 0.021. Its window of loss runs, by straight lines between the solved sides, from 0.39 to 0.42, where the floor two units long lost from 0.30 to 0.51. The deepest change each length reaches over its solved shorter sides rises at every lengthening — −0.028 per cent at two units, −0.0056 at two and a half, +0.026 at three, +0.098 at four — and crosses nought, by straight line, at 2.59 units. Every box’s verdict is all or nothing, and every box’s change has the sign of its tilt across reflectance.

• The prediction holds at the lower edge and fails at the upper. The floor gains at 0.3 as predicted; it gains at 0.5 too, where the prediction had it losing.
• The alternative fails. The window does not widen; it narrows at once from both ends.
• The closing is smooth. The deepest dip rises steadily through nought a little past two and a half units.

Drawn as a map of verdicts, the losses form a wedge. At two units they span three solved boxes’ worth of shorter sides, at two and a half one, at three none, and the wedge’s point sits near a shorter side of 0.4. Around it every box gains, at every length. The map is coarse — the census has solved only these boxes, and each is several minutes of the directional solver — but it is enough to show the shape: not a threshold sliding to one side, but a pocket shrinking to a point.

The curves #

The three curves have one shape. Each falls from the narrowest box solved to a minimum near a shorter side of 0.4 and rises again, and each lies above the one for the shorter floor. The floor two long dips to −0.028 per cent at 0.35; the floor two and a half long to −0.0056 at 0.4; the floor three long only to +0.026 at 0.45. Lengthening lifts the whole curve, and the window of loss is wherever the curve is still below nought — wide when the dip is deep, a sliver when it is shallow, nothing once the curve clears.

That is why the window closes from both ends. A dip that rises closes from both sides at once, and the window’s edges move towards the dip’s lowest point, which barely moves itself: 0.35 at two units, 0.4 at two and a half, 0.45 at three. The prediction treated the two edges as separate mechanisms, a weak lower edge and a fixed upper one; the curves say they are two sides of one dip.

The edges #

By straight lines between solved boxes, the window is 0.21 wide at two units and 0.04 wide at two and a half — a fifth of its width, halfway to three units. Its lower edge rises from 0.30 to 0.39 and its upper edge falls from 0.51 to 0.42, the upper edge moving by about as much as the lower. The edges are interpolated, and the window’s width at two and a half units rests on a loss only six thousandths of a per cent deep; a box at 0.45 would place the upper edge more surely. But the gain at 0.5 is not marginal — 0.021 per cent, nearly four times the loss at 0.4 — and it is what refutes the prediction.

How the dip rises #

Plotted against the floor’s length, the deepest dip rises nearly in a straight line from two units to three — by about 0.05 per cent per unit — and keeps rising, more slowly, to four. It crosses nought at 2.59 units, a little past halfway between two and three. The crossing is the length at which the end walls stop costing any shorter side of floor its colour, and it says the end walls’ cost falls off with their separation steadily rather than switching off at some distance: in this box of height 2.5, a floor about 2.6 units long — a little longer than the room is tall — is the last whose paint the finish can dull.

The minimum’s position is only roughly known, since each length’s dip is read at the solved sides alone; the true minimum of each curve may lie between them and a little deeper. The census’s sides at two and a half units straddle the dip closely enough that the crossing would move by a few hundredths of a unit at most.

The tilt follows the colour #

Every scene essay has found the paint’s change in colour carrying the sign of one quantity: how the finish’s brightening of the floor varies with the floor’s own reflectance. Where the finish brightens strongly reflecting bands more than weakly reflecting ones, the paint grows more colourful; where less, the paint dulls. The tilts here trace the same dip as the colour: the floor two and a half long tilts by +0.24 points at 0.3, −0.04 at 0.4 and +0.13 at 0.5, and every one of the twelve boxes in the figure has its tilt and its colour change of the same sign.

The tilt is the more physical quantity and the colour change the consequence. The tilt says which light the finish adds: light that has been back to the floor, which favours its bright bands, or light that has not, which brightens every band alike and so washes the paint towards grey. The dip is where the end walls’ satin sends the floor more of the second kind than the first, and it is centred, plausibly, where the room’s proportions bring the end walls’ glancing reflections to the floor most directly; the census locates the dip and does not trace the rays.

The light the finish adds #

The quantity of light the finish adds hardly depends on the floor’s length. All four lengths lie on nearly one curve, the brightening at the floor’s darkest reflectance falling from about four fifths at a shorter side of 0.25 to a third at 0.65. The near pair of walls, a shorter side apart, decides how much light the finish pipes down to the floor; the end walls, much further off, add little to that total. What they change is the balance within it — a shift of a few tenths of a percentage point across the floor’s reflectance, invisible in the total and decisive for the sign.

That separation is the scene essays’ most useful result for a designer. The brightening a satin finish gives a floor depends on the room’s narrow dimension and is large; the change in the floor’s colour depends on the room’s length as well and is tiny. A designer worried about a narrow room looking dark should reach for the finish; a designer worried about its floor looking dull need not, unless the room is near square.

The earlier turns, reread #

The scene essays reached this pocket by walking along one dimension at a time, and each walk crossed it differently. The finish stops costing the paint six tenths deep walked a box one unit wide from shallow to deep and found one turn, near 0.64; with the floor’s longer side fixed at one unit, the walk left the pocket through its far edge and never saw a near one, because at one unit long the pocket reaches all the way to the narrowest box the census could solve. A taller box must be deeper before the finish adds colour found the turn growing with the room’s height — the pocket’s far edge moving outward as the room grows taller, which the height scan in the open question below would place.

The slot’s loss survives every finer division had established the loss itself, in the tall narrow slot, as real rather than an artefact of how coarsely the faces were divided, and every essay since has relied on that. The two-unit floor’s window was the first sign that the loss was not a region bounded on one side; the floor two and a half long shows it closing. Each earlier turn was one edge of the pocket seen from inside it, and the square, one-unit and two-unit floors were cross-sections taken at three lengths, each wider than the next.

Read that way, the turns’ fall with the floor’s length — 0.78, 0.64, 0.51 — is the pocket’s far edge retreating as the end walls move away, and its near edge, invisible until the floor was two units long, advancing to meet it. The square floor sits where the pocket is widest, its end walls as close as its side walls; the floor two and a half long sits at its tip.

What the far walls change #

A grey finish mirrors the lamp onto the paint found the satin finish’s effect on a painted surface set by which grey faces see the paint at a slant, re-weighting each face towards what it sees. In a long room the end walls see the floor only at grazing angles, across the whole length of the room, and what they mirror onto it is mostly the near walls and the lamp; in a short room they see the floor nearly face-on and mirror onto it, among other things, its own reflection from the opposite end. The census does not decompose which exchanges carry the loss, but the pocket’s shape — deepest near a shorter side of 0.4 at every length — says the end walls’ effect depends on the room’s proportions more than on any one distance.

What a lighting designer should take from this #

A satin finish dulls a painted floor only in a small region of room proportions: rooms 2.5 units tall whose floors are less than about 2.6 units long and, at the longer lengths, a shorter side near 0.4. In a room of this height, any floor longer than about 2.6 units — a little more than the room is tall — gains colour from satin walls at every width the census solved.

The effect is tiny everywhere. Losses and gains alike are hundredths of a per cent of the paint’s chroma. What is robust is the sign — every one of seventy-two paints agreeing in every box — and the sign is what decides whether the finish works with the paint or against it.

How the boxes were solved #

Each box is two and a half units tall with a floor two and a half units long and 0.3, 0.4 or 0.5 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, matte 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, so that the satin’s glancing reflection is kept; 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 matte one, less one, averaged over the paints; the tilt is the finish’s brightening at the floor’s brightest reflectance less at its darkest. The floors two, three and four units long are the earlier census’s boxes, read the same way. Window edges and the crossing in length are straight lines between solved values.

What this does not reach #

Three boxes at two and a half units. The window’s edges are interpolated across a tenth of a unit of shorter side, and its upper edge rests on one loss of under a hundredth of a per cent.

Six strips on long faces. At two and a half units long a floor 0.3 wide is divided into strips eight times as long as they are wide; the earlier scene essays found four strips misreading elongated faces, and six may misread these, though every box’s verdict here agrees with its neighbours’.

One height. The crossing at 2.59 units is for a box 2.5 tall; how it scales with height — whether the end walls stop mattering at a fixed ratio of length to height — is open.

Still open: whether the crossing scales with the room’s height #

The end walls stop costing the floor its colour when they are about 2.6 units apart in a box 2.5 tall, roughly the room’s own height. If the cost is set by the angle at which the end walls’ glancing reflections reach the floor, the crossing should scale with the height, so that a box twice as tall needs its floor twice as long before the window closes.

The calculation is the box 4 units tall with its floor’s shorter side at 0.4, solved at six strips a side with its floor three, four and five units long, the change read as here and the length at which it crosses nought found. The prediction is that the crossing lies near 4.2 units — 1.04 times the height, as here — because the end walls’ reflections reach the floor at the same angle in a box of the same proportions. If the crossing barely moves from 2.6 units, the end walls’ cost is set by their distance from the floor’s centre rather than by the room’s shape, and a tall room’s floor is no more at risk than a low one’s. A taller room keeps its floor losing in proportion makes that calculation.

A pocket, not a threshold #

The habit here is about the shape assumed for a region before it is mapped.

The earlier essays found turns — single shorter sides at which loss became gain — and the language of turns carried into the prediction: a threshold that moves, an edge that gives way first. The two-unit floor had already shown a window, a loss with gains on both sides, and the two-and-a-half-unit floor confirmed it as a pocket closing to a point.

The failure mode is to describe a region by the first boundary found and to assume the region lies wholly on one side of it. A turn found while walking from narrow to wide says where loss became gain on that walk, not that loss filled everything before it. The test is to solve on both sides of every boundary — here, narrower and wider than each window — and to draw the region as a map before naming its edges.