The pale paints cross the tip last

A painted floor in a room of grey walls and ceiling gains a little colour when the grey faces are given a satin finish instead of a matt one, because 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 paint itself gains all or nothing found that rule holding for every one of seventy-two paints together in every room it solved: a room’s verdict was a property of the room. The exception, a few thousandths of a per cent of chroma lost, lies in a narrow wedge of proportions — floors about as long as the room is tall and about a sixth of its height wide.

The wedge of loss comes to a point found where it closes. Along a floor width of 0.16 of the height, the mean change in chroma runs −0.0056 per cent at a floor as long as the room is tall, +0.00002 at 1.04 times the height and +0.0054 at 1.08, crossing nought at 1.0399. At 1.04 itself — and in no other box either census has solved — the paints split: fifty-five gain and seventeen lose. All or nothing was a rule about boxes away from the tip, where the mean effect dwarfs every paint’s departure from it. At the tip the mean is nought and what remains is the departures.

The paints differ in three ways: the centre of their reflectance band, from 450 to 650 nm in six steps; its width, from 15 to 100 nm in four; and its peak, 0.3, 0.6 or 0.85, on a reflectance floor of three per cent. The prediction was 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 at 1.0 and the smallest gains at 1.08. If the losers were spread across the widths and sorted by hue or peak instead, the finish’s effect would depend on where a paint’s colour sits in the spectrum rather than on how pure it is.

The losers are the pale paints #

Of the seventeen paints that lose at the tip, none has the narrowest band, three have bands 25 nm wide, five 50 nm and nine 100 nm. Fourteen have the highest peak, 0.85, three a peak of 0.6, and none the lowest. Every band centre has a loser — three at 450 nm, four at 490 and 530, three at 570, two at 610 and one at 650.

• The prediction fails at the opposite end. The narrowest bands are the one width with no loser; the broadest is the width with the most.
• The peak sorts the losers more sharply than the width. Fourteen of the seventeen are the brightest paints, and no dark paint loses.
• The hue does not sort them. Every centre has losers and none holds more than a quarter.

A broad band at a high peak is a pale paint: its reflectance is high across much of the spectrum and its colour is a tint rather than a hue. The seventeen losers are the census’s tints — pale greens, pale blues, pale yellows — and the fifty-five gainers include every saturated paint. The broadest band at the highest peak, centred at 530 nm, is the deepest loser at the tip, at −5.5 millionths of its chroma.

The peak’s effect is the cleaner of the two. None of the twenty-four paints with a peak of 0.3 loses at the tip, three of the twenty-four at 0.6, and fourteen of the twenty-four at 0.85; their mean changes at the tip are +0.94, +0.42 and −0.74 millionths. A plausible reading, not traced here: a light paint sends more of the lamp’s light back into the room than a dark one, so more of the interreflected light reaching the floor has already been to the floor, the finish’s glancing return is a larger share of that floor’s light, and its effect, in both directions, is larger.

By hue the losers spread out. The paints centred at 530 nm have the lowest mean change at the tip, −0.31 millionths, and those at 650 nm the highest, +0.62; but every centre has at least one loser and at most four. The finish’s effect depends a little on where a paint’s band sits — a green band under the census’s daylight lamp sits where the lamp and the eye both peak — but much less than on how wide and how bright the band is.

The losers are the paints the finish moves most #

Of the seventeen paints that lose at the tip, at least twelve are among the seventeen with the largest losses at 1.0 times the height, and at least twelve among the seventeen with the largest gains at 1.08. Across the seventy-two paints the change at 1.0 runs from −181 to −20 millionths and at 1.08 from +21 to +164. The prediction’s first half holds: the losers lose most below the tip. Its second half fails: above the tip they gain most, not least.

So the losers are not paints the finish treats differently in kind. They are the paints it moves most in either direction. On a plot of each paint’s change below the tip against its change above it, the seventy-two paints lie along a line running from small losses and small gains to large losses and large gains, and the seventeen losers sit at its far end.

Every paint has its own tip #

The mechanism follows. Each paint’s change in chroma rises through nought as the floor lengthens, as the mean’s does, and each paint crosses at its own length. Read by straight lines between the three boxes, the seventy-two paints’ own crossings lie between 1.0387 and 1.0413 times the height. Each paint’s swing — its change at 1.08 less its change at 1.0 — runs from 0.37 to 3.12 times the mean’s, and swing and crossing correlate by 0.85: the paints the finish moves most cross last. The palest paint, the broad 530 nm band at a peak of 0.85, crosses at 1.0413; the narrow, dark paints at the other end cross at 1.0387.

That is why the pale paints are the losers at 1.04. The box sits at 1.04, between the earliest crossing and the latest; the paints whose own crossing lies below 1.04 have already turned to gain, and those whose crossing lies above it have not. The latecomers are the paints whose change rises steepest — and a paint whose change rises steeply across the tip, with its losses and gains both large, crosses later because its loss below the tip is deeper in proportion than the mean’s.

The wedge’s tip is therefore not a point but a band two and a half thousandths of the height wide — about six millimetres in a room two and a half metres tall. Outside it every paint shares the mean’s sign; inside it, the paint decides. A box placed anywhere in that band would split the paints, and the census happened to place one there only because the tip it was looking for lay at a round number of the room’s height.

At the tip the paints spread over a few millionths #

At the tip the seventy-two paints’ changes run from −5.5 to +1.2 millionths of their chroma; at 1.0 times the height, from −181 to −20; at 1.08, from +21 to +164. The spread at the tip is a twentieth of the spread either side, and centred on nought. Away from the tip, the paints’ changes scale with the mean’s: every paint loses at 1.0 and gains at 1.08 in proportion to its swing. At the tip the proportional part vanishes, and what is left is the small part of each paint’s response that does not scale — of the order of a millionth.

These are vanishingly small quantities. A change of five millionths in a paint’s chroma is invisible by four orders of magnitude, and even the deepest loss anywhere in the wedge is a few thousandths of a per cent. What they decide is not anything a viewer would see but the reach of a rule: the census’s all-or-nothing rule is the statement that the proportional part dominates, and at the tip it does not.

What every earlier room was measuring #

Every earlier reading of this room reported the change in chroma as one number, the mean over the seventy-two paints, and every one could, because the paints agreed. The slot’s loss survives every finer division found a tall slot’s painted face losing about a twentieth of a per cent of its chroma at every resolution from three strips to eight; the finish stops costing the paint six tenths deep found the slot’s loss turning to gain at 0.62 of its width; a taller box must be deeper before the finish adds colour found the same turn moving with the box’s height. In each, a mean was a fair summary of seventy-two numbers of one sign.

The pale paints were there all along, as the paints whose share of each mean was largest. Where a room lost, they lost most; where it gained, they gained most. The window of loss closes from both ends found a floor two units long losing only in a window of widths from about 0.30 to 0.51, and every edge of that window was a place where the mean crossed nought — a place where, read paint by paint and if the tip is any guide, the pale paints would have crossed last and the dark ones first, a band of crossings rather than a line. A square floor needs more room than a long one placed the square floor’s turn between solved widths a tenth of a unit apart, which is coarse beside a band a few thousandths of the height wide; none of those turns would move by more than the census’s sampling if read paint by paint, but every one of them is a band of seventy-two turns, not one. Every boundary on the map of proportions is such a band, and its width is set by how differently the finish moves the palest paint and the darkest.

That gives the boundaries a precision they did not have. The tip’s band runs from 1.0387 to 1.0413 times the height, and the mean’s crossing at 1.0399 lies inside it, about where the median paint crosses, at 1.0395. A boundary read from the mean is the boundary for the typical paint; for a pale tint it lies a few thousandths of the height further out, and for a saturated dark paint a few thousandths further in.

A room scaled whole loses its floor to the last digit showed that the transport carries no length, so that a room multiplied by any factor gives the same numbers to eleven figures. That holds for every paint as for the mean: each paint’s band of crossings is a band in proportions, not in metres, and a tall room’s pale floor crosses at the same share of its height as a low room’s.

What a lighting designer should take from this #

Near the wedge’s tip, pale paints are the last to stop losing. A tinted floor — a broad band at high reflectance — is the paint a satin finish moves most, in loss below the tip and in gain above it.

Saturated, dark paints are the most stable. They swing least between rooms and cross nought first; at the tip, every one of them gains.

None of this is visible. The effects here are millionths of a paint’s chroma. The value of the reading is in knowing which paint decides a room’s verdict when the room sits on the boundary, not in any visible difference.

How the paints were read #

The census’s box is 2.5 units tall with its floor 2.5, 2.6 and 2.7 units long and 0.4 wide — 1.0, 1.04 and 1.08 times the height long, 0.16 of it wide — its walls and ceiling grey and the whole ceiling emitting, each face divided into six strips a side and solved by the directional transport under a matt finish and a satin finish of roughness 0.2 on the grey faces, band by band, the floor’s mean exitance read at a set of reflectance nodes. Each paint is a Gaussian reflectance band of stated centre, width and peak on a three-per-cent floor; its exitance under each finish is interpolated from the nodes band by band, its colour read in CIELAB against the D65 white, and its change is the satin floor’s chroma over the matt floor’s, less one. The seventy-two paints’ changes average to each box’s census mean to the twelfth decimal. A paint’s own crossing is a straight line between the two boxes whose changes straddle nought for it; its swing is its change at 1.08 less its change at 1.0, over the mean paint’s.

What this does not reach #

More widths and peaks. Four widths and three peaks are a coarse sample of paints; whether the crossings vary smoothly with the paint’s pallor, or step, is read at twelve points of each hue.

Finer lengths. Each paint’s crossing is a straight line between boxes a twenty-fifth of the height apart; within that, the paints’ crossings are interpolated, not solved.

The transport’s resolution. At six strips a side, a band two and a half thousandths of the height wide is far inside what a finer division could move.

Still open: whether the pocket has a bottom #

The tip is where the pocket of loss at a sixth of the height closes as the floor lengthens; the pocket itself, at a floor as long as the room is tall, has been read at three widths a hundredth of the height apart, deepest at 0.16. If the pocket is a smooth dip, boxes between those widths will trace a curve with one bottom; if it is the trace of something finer — the way the strips fall on a narrow floor — they will not.

The calculation is the box with its floor as long as the room is tall at widths of 0.38, 0.39, 0.41 and 0.42 units, between the census’s 0.4 and its neighbours, read as here. The prediction is that the change in chroma traces one smooth dip with its bottom between 0.39 and 0.41, every box between 0.375 and 0.425 losing. If a box inside that range gains, the pocket is not one dip but a pattern set by how the faces are divided, and every boundary drawn on this map at six strips a side is a boundary of the division as much as of the room.

A rule that broke where it was weakest #

The habit here is about reading the exception to a rule.

The paints’ all-or-nothing rule broke at the tip, and the natural reading was that the losers there were special — paints the finish treats differently, narrow bands most sensitive to grey. They were the reverse: the paints the finish treats most strongly, losing most below the tip and gaining most above it. The rule broke at the tip not because some paints behave differently but because there the common behaviour, which every paint shares in proportion, cancels.

The failure mode is to read the exceptions to a rule at its boundary as a different kind of case, when they are the same kind of case with the shared effect cancelled. The test is to read the exceptions on both sides of the boundary and ask whether they are the extremes of one distribution rather than members of another.