The wedge of loss comes to a point

A painted floor in a room of grey walls and ceiling takes on a little more colour when the grey faces are given a satin finish instead of a matt one. The paint itself gains all or nothing found that the rule: the finish’s glossy lobe returns the floor’s own light to it at glancing angles, and light that has been to a coloured surface and back comes back more coloured. The exception is a narrow room near square, where the floor loses a few thousandths of a per cent of its chroma — for every one of seventy-two paints at once, since whether a box gains or loses has been a property of the box and not of the paint.

The exception has been mapped one box at a time. The window of loss closes from both ends found the loss in a pocket centred near a floor 0.4 units wide in a box 2.5 tall, and a room scaled whole loses its floor to the last digit found the transport indifferent to size, so that a box’s verdict depends only on its floor’s length and width over its height. On that map the losses fill one wedge, between about 0.14 and 0.2 of the height in width and at lengths up to about the height; the wedge’s deepest loss, −0.0056 per cent, sits at 0.16 of the height, and the row there crosses nought somewhere between floors 1.0 and 1.2 times the height long — 1.036 by a straight line between those two lengths.

A straight line drawn between samples a fifth of the height apart says little about where a curve crosses between them. The prediction was that in the 2.5-tall box a floor 2.6 units long — 1.04 times the height — still loses at a shorter side of 0.4, and a floor 2.7 units long — 1.08 times — gains at sides of 0.375, 0.4 and 0.425, placing the tip between them, a little above 1.036, because the pocket narrows as the floor lengthens and its deepest point stays near a sixth of the height. If the 1.08 floor had lost, the wedge would have a longer tip than any reading so far.

The longer floors gain everywhere; the 1.04 floor sits on the tip #

At 1.08 and 1.12 times the height, the floor gains chroma at all three widths, every paint with it: +0.0108, +0.0054 and +0.0078 per cent at 1.08, +0.0191, +0.0123 and +0.0147 at 1.12. At 1.04 times the height it gains +0.0052 per cent at 0.15 of the height and +0.0053 at 0.17, and at 0.16 changes the floor’s chroma by +0.00002 per cent — two parts in ten million of the floor’s chroma, about a two-hundred-and-seventieth of the loss at 1.0.

• The 1.08 half of the prediction holds, at every width.
• The 1.04 half fails by the thinnest margin the census can report. The floor does not lose; it gains by a quantity that is nought to the precision any earlier box was read at.
• The tip lies a little above the coarse crossing, as predicted: the row at 0.16 of the height crosses nought at 1.0399 times the height.

On the map of proportions the new boxes close the wedge from above. The floor as long as the room is tall loses only around 0.16 of the height, between about 0.15 and 0.18; one row up, at 1.04, only the centre box is at nought and its neighbours a hundredth of the height either side already gain; two and three rows up, every box gains. The wedge comes to a point within a few hundredths of 1.04 in length and at 0.16 in width, which is to say at one box.

The row crosses at 1.040 #

Along a sixth of the height the change in chroma runs −0.0056 per cent at 1.0 times the height, +0.00002 at 1.04, +0.0054 at 1.08 and +0.0123 at 1.12. Each tenth of a unit of length raises it by 0.0056, 0.0054 and 0.0069 per cent; the crossing, by a straight line between 1.0 and 1.04, lies at 1.0399. The coarse crossing of 1.036 was a line drawn between 1.0 and 1.2 across a curve that steepens a little past the tip, and so fell four thousandths short.

The row is close to straight across the whole range, which says something about what the tip is. It is not a feature of the transport — no change of mechanism, no corner — but the place where a smoothly rising quantity passes nought. Below it the satin finish’s glancing return of light to the floor carries less colour than the light it scatters away; above it, more; at it, the two balance, and they balance for the average paint at one length.

A sixth of the height at every length #

At every length from 1.04 to 1.12 times the height, the side of 0.16 of the height gains least of the three: +0.0052, +0.00002 and +0.0053 per cent at 1.04; +0.0108, +0.0054 and +0.0078 at 1.08; +0.0191, +0.0123 and +0.0147 at 1.12. The pocket’s deepest point does not move with the length; the whole pocket rises past nought, deepest point last.

The pocket is not symmetric about its centre. At 1.08 and 1.12 the narrower side, 0.15 of the height, gains more than the wider, 0.17 — by 0.003 and 0.004 per cent — while at 1.04 the two match to a ten-thousandth. Narrowing a floor brings its long walls nearer, and the width moves the turn only when it is the narrower side found a room’s turn sensitive to its narrower dimension in exactly this way; here the asymmetry grows as the floor lengthens past the tip.

At the tip, the paints part company #

At the floor 1.04 times the height long and 0.16 of it wide, fifty-five of the seventy-two paints gain chroma and seventeen lose. In every other box near the tip, and in every box either census has solved, the paints gain or lose together. All or nothing was a rule about boxes away from the tip, where the mean change dwarfs the differences between paints; at the tip the mean is nought, and what is left is those differences.

That is the expected way for an all-or-nothing rule to break, and it says the rule was never quite exact. The finish’s effect on a paint’s chroma depends a little on the paint — on how its reflectance is spread across the spectrum and how bright it is — and that dependence is a small share of the mean effect away from the tip. Anywhere the mean is large beside that share, every paint shares its sign. Only where the mean itself is nought does the paint-to-paint spread decide each paint’s sign, and that happens on a line through the map of proportions — the wedge’s edge — which the census had never sampled until a box landed on it.

Brightening and tilt through the tip #

The finish brightens the floor at its darkest reflectance by 48.98 per cent at 1.0 times the height, 48.80 at 1.04, 48.62 at 1.08 and 48.41 at 1.12 — falling by nearly two tenths of a point each tenth of a unit, with no feature at the tip. The brightening is the finish’s whole effect on the floor’s light; the chroma change is a small difference between how much it brightens the floor’s coloured and its grey parts of the spectrum. That difference crosses nought while the brightening it rides on does not change its pace.

The tilt of the brightening across reflectance — how much more the finish brightens the floor’s lightest reflectance than its darkest — runs −0.0357, −0.0001, +0.0341 and +0.0776 points along the row, crossing nought at the tip with the chroma. In every box both censuses solved the tilt has carried the chroma change’s sign, and here the two cross nought together, both a hundredth or less of their row’s size at 1.04. The tilt is the chroma change seen in grey: a finish that brightens a floor’s light reflectances more than its dark ones raises the contrast between a paint’s peak and its trough, which is chroma, and one that brightens the dark more lowers it.

How the wedge was found, and what the tip completes #

The tip closes a map that was assembled from questions asked one dimension at a time. A square floor needs more room than a long one found the turn from loss to gain on a floor’s shorter side at 0.64 when its longer side was one unit and about 0.51 when it was two, and a later turn for a square floor; a long floor never loses its colour to the finish found floors three and four units long, in the same 2.5-tall box, gaining at every shorter side. Read on the map of proportions those are the wedge’s long flank and its far side: a floor short enough to lose at narrow widths, a floor long enough never to lose at any.

The wedge’s tip is where those two descriptions meet. A floor whose length lies below about 1.04 times the height loses in a band of widths around a sixth of the height; a floor longer than that gains at every width; and the band of widths shrinks to a point as the length rises to 1.04. Every earlier essay on this room was measuring a cross-section of that shape. None of them needed the tip, because each held one dimension where the wedge is wide, and the tip is the one place where both dimensions matter at once.

It also bears on the question of resolution the census has carried throughout. The slot’s loss survives every finer division found a tall slot’s loss — about a twentieth of a per cent of chroma — holding its sign from three strips a side to eight; that loss was large beside anything a finer division moved. The tip is the opposite case: by construction the change in chroma there is nought, so any change in how finely the faces are divided can push it either way. The tip’s position, 1.040, is a property of the transport at six strips a side. That the row along a sixth of the height is so nearly straight through it says the position would move little with a finer division; that the change there is two parts in ten million says its sign would.

The pocket’s own resolution is a separate matter. Its deepest point at a sixth of the height rests on three widths a hundredth of the height apart, and the finish stops costing the paint six tenths deep found a slot’s verdict turning from loss to gain within a hundredth of a unit of depth — at 0.62 of the box’s width with the floor painted and 0.63 with the ceiling — a reminder that wherever the change in chroma is near nought, a very small change of proportion decides its sign, and a dip sampled at three points a hundredth of the height apart has not been sampled finely enough to say what shape it is. Boxes at widths a hundredth of a unit apart around 0.4 would show whether that deepest point is a smooth bottom or the sample of something sharper, and the census has not yet solved them.

What a lighting designer should take from this #

Under this model a satin finish dulls a painted floor only in rooms whose floor is shorter than about 1.04 times the room’s height and about a sixth of it wide. Past that length every paint gains; the loss is a narrow wedge that comes to a point.

The loss near the tip is unmeasurably small. At 1.04 times the height the floor’s mean chroma changes by two parts in ten million; even the deepest loss anywhere in the wedge is a few thousandths of a per cent, invisible to any observer. What the map gives is the boundary of a rule, not a visible effect.

At the boundary the paint matters. On the wedge’s edge some paints gain and some lose; a designer who needed to know which side of the line a room fell on would have to ask which paint.

How the boxes were solved #

The box is 2.5 units tall with its floor 2.6, 2.7 and 2.8 units long and 0.375, 0.4 and 0.425 wide, painted with each of the census’s seventy-two paints in turn — six hues from 450 to 650 nm, four band widths and three peaks, on a three-per-cent floor of reflectance — its walls and ceiling grey and the whole ceiling emitting. Each face is divided into six strips a side and solved by the directional transport with satin of roughness 0.2 on the grey faces and matt everywhere, band by band, the floor’s mean exitance read under each finish at a set of reflectance nodes and interpolated to each paint. The change in chroma is the mean over the paints of the satin floor’s CIELAB chroma over the matt floor’s, less one; the brightening and tilt are read from the ratio of the two floors’ exitance at the darkest and lightest reflectances. Each box takes six to ten minutes.

What this does not reach #

Finer divisions. Six strips a side is one resolution of the transport, and a change of two parts in ten million is far inside what a finer division could move.

The pocket’s true bottom. The deepest point sits at 0.16 of the height by three samples a hundredth of the height apart; where between them the bottom lies, and how deep it is, the census has not read.

Other finishes. Roughness 0.2 is one lobe; the wedge’s tip for a glossier finish is a different point.

Still open: which paints lose at the tip #

At the wedge’s tip seventeen paints lose and fifty-five gain, the first box in either census whose verdict depends on the paint. The paints vary in three ways — the hue of their reflectance band, its width and its peak — and a finish that returns light at glancing angles treats a narrow band differently from a broad one.

The calculation is each of the seventy-two paints’ own change in chroma in the box at the tip, set against its hue, band width and peak, beside the same paints in the losing box at 1.0 times the height and the gaining box at 1.08. The prediction is that the seventeen losers are the paints with the narrowest bands, because a narrow band’s chroma depends most on how much of the finish’s grey return mixes into it, and that their changes in the boxes either side are the largest losses and smallest gains of the seventy-two. If the losers are spread across band widths and sorted instead by hue or peak, the finish’s effect depends on where the paint’s colour sits in the spectrum rather than how pure it is.

A rule that held because nothing sat on its edge #

The habit here is about the reach of an all-or-nothing rule.

Every box either census had solved was all or nothing: the paints gained together or lost together, and the rule was recorded as a property of boxes. The first box placed exactly at the wedge’s edge split the paints seventeen to fifty-five. The rule had held because every earlier box sat far enough from the edge that the mean effect outweighed every paint’s departure from it; none had sat where the mean was nought.

The failure mode is to read a rule that holds across a sample as a law, when every sample lies away from the boundary where the rule’s mechanism would fail. The test is to place a sample on the boundary itself and read whether the rule survives there.