An open room hands over sooner

The finish adds the room’s own colour explained a reversal. Over seventy-two painted rooms, the share of a room’s colour a satin finish takes is ordered first by how much light the wall returns — a dark wall pays for a finish — and second, inside a band of lightness, by the wall’s chroma, in opposite directions at the two ends. Among dark walls a more saturated paint loses less; among pale walls it loses more.

The explanation was two terms handing over. The light a gloss finish adds is not the lamp’s white but the room’s ambient, which has already bounced off the walls, and among dark walls the colour of that addition orders the loss: a more saturated dark paint colours its own ambient more, the finish returns something nearer the wall’s colour, and the room loses less. Among pale walls that term has run out of room — the addition is already mostly the wall’s colour — and the amount of added light takes over, pointing the other way. In a closed cube the sign changes at a wall returning about a third of the light.

The essay named the number a decorator could use — where that change falls — and predicted how it would move. “Less interreflection means a whiter ambient at every lightness, so the colour term saturates later”: the handover should move up the lightness scale as a room opens, and a room with a window should have no pale end at all.

The first half of that is right. The rest runs the other way.

Opening a wall moves the handover down #

In five rooms, the handover falls at a luminance factor of 0.36 in the closed cube, 0.37 in a corridor four times as deep, 0.38 in a low wide room, 0.29 in the cube with one wall open and 0.24 with two open. The ambient whitens as the room opens, as predicted, and a whiter ambient makes the colour term weaker rather than later, so the other term takes over at a darker wall.

• Opening one wall moves the handover down by 0.07 and a second by a further 0.05. The prediction was up.
• Changing the shape of a closed room moves it up, and by little — two hundredths in the corridor and the low room.
• Among the darkest walls the light the finish adds is 26 per cent as coloured as the room in the cube, 10 per cent with one wall open and 3 with two. The ambient does whiten.
• The five rooms fall in the same order by that number as by where they hand over, from two walls open at 3 per cent and 0.24 to the low room at 68 per cent and 0.38.
• A room with a window keeps its pale end: with one or two walls open, the palest walls’ partial correlation is +0.68, as it is in the corridor.
• In the low wide room, 21 of the 72 paints make a more colourful room when glossy, by up to 13.8 per cent.

Five rooms #

The census is the one that found the reversal: seventy-two paints, from six band centres, four band widths and three peak reflectances, each painted on the two side walls of a room whose floor, ceiling and end walls are a mid grey, lit from the ceiling by daylight, solved matt and at a satin roughness of 0.2 by the directional transport solver the gloss essays built.

The closed rooms vary the proportions. The cube is the reference. The corridor is four times as deep as it is wide, with its long walls painted, so a large share of every bounce lands on colour. The low wide room is twice as wide and deep as the cube and six tenths as high, so its painted walls are small bands between a large grey floor and a large grey ceiling.

The open rooms vary the enclosure. An open face returns nothing: a window at night, or a doorway into an unlit hall. With one wall open, a quarter of the lamp’s light leaves the room through it; with two, nearly half. Everything that leaves is light that would have come back coloured by another bounce.

Where the reversal changes sign, room by room #

The figure is the reversal itself, read continuously. The paints are sorted by how much light the wall returns, and in each window of eighteen neighbouring paints the rank correlation of the loss with the wall’s chroma is taken with the wall’s luminance factor held still. Below zero, a more saturated paint loses less; above zero, more. The point where each curve crosses is the handover.

The three closed rooms cross within two hundredths of each other, at 0.36, 0.37 and 0.38, the less cubic rooms slightly higher. Their curves differ in detail — the low room’s stays deeper below zero through the middle, the corridor’s rises more gently at the pale end — and they agree about where the sign changes.

The open rooms cross earlier. One open wall puts the crossing at 0.29; two put it at 0.24. With two walls open, a wall returning a quarter of the light, which in the closed cube is still firmly in the regime where saturation protects, is already on the other side.

And both open rooms have a pale end. Among the palest walls, the partial correlation is +0.68 with one wall open and +0.68 with two, against +0.81 in the cube. A window does not remove the half of the reversal the prediction said it would; it reaches it sooner.

The ambient does whiten #

The prediction’s first step was that less interreflection means a whiter ambient. That is directly measurable: the light a satin finish adds to a room is the satin room’s reflected light minus the matt room’s, and its colour can be set against the room’s own.

In every room the addition grows more coloured as the wall gets paler, because a paler wall sends more of the light round the room again before the finish returns it — the mechanism the earlier census found, visible in all five geometries. And opening walls whitens it at every lightness. Among the darkest walls the addition is 26 per cent as coloured as the room in the cube, 10 per cent with one wall open and 3 per cent with two; among the palest, 62, 36 and 18.

So the prediction’s premise holds. An open room’s finish adds something much closer to the lamp’s white, because the light that would have bounced a second and third time off the painted walls has left through the opening instead.

The closed rooms go the other way: the corridor’s addition is more coloured than the cube’s, 42 per cent among dark walls, and the low room’s more coloured still, 68 per cent — because in both, a larger share of what the finish reflects has come off a painted wall. The low room’s number is inflated by its own room, whose colour is diluted by the grey floor and ceiling, so its addition is large as a share of a small thing.

A weaker term, not a later one #

The prediction’s second step was that a whiter ambient would make the colour term saturate later. That would be right if the handover were the point where the colour term runs out of room at the top. It is not. It is the point where the colour term and the amount term balance, and a whiter ambient does not move the first term’s ceiling; it shrinks the whole term.

The five rooms fall in exactly the same order on both axes. Two walls open: an addition 3 per cent as coloured as the room, handing over at 0.24. One wall open: 10 per cent, 0.29. The cube: 26 per cent, 0.36. The corridor: 42, 0.37. The low room: 68, 0.38. Where the finish adds nearly white light, how saturated a paint is barely changes how coloured that white light is, so saturation stops protecting a wall sooner and the amount of added light, which penalises saturation, wins the balance at a darker wall. Where the addition is strongly coloured, saturation keeps protecting further up the scale.

The relation is steep at the bottom and flat at the top: going from an addition 3 per cent as coloured as the room to one 10 per cent as coloured moves the handover by 0.05, and going from 26 to 68 per cent moves it by 0.02. That is the shape of a term that matters most when it is small, which is the other half of why the prediction pointed the wrong way — it pictured the colour term as large and running out of room, and in an open room it is small and easily outweighed.

Five rooms are five points, and a shared order among five is not a law. What it does establish is that nothing in these rooms moves the handover up as the enclosure falls, and that the quantity the earlier essay said was doing the work — how coloured the added light is — is the quantity that tracks where the handover lands.

What a finish costs in each room #

The handover is a statement about which paints lose more than which; the loss itself is the number a specifier pays.

Opening walls raises the loss at every lightness. Averaged over the census, the cube loses 10.5 per cent of its colour to a satin finish, the room with one wall open 13.0 and with two 16.8. The whiter addition dilutes more, which is the first-order rule of a gloss finish takes colour out of the whole room arriving through a different door: a room that loses light through an opening keeps less of its own colour in circulation, and what the finish adds is closer to the lamp.

And in the low room, a finish can add colour. Its average loss is 2.7 per cent, and for 21 of the 72 paints the satin room is more colourful than the matt one, by as much as 13.8 per cent. The earlier census of thirty-six rooms under six lamps, a finish adds colour only to a daylight meter, found no room gaining colour once each was read against its own lamp’s white — every one of them a cube. In a room whose painted walls are narrow bands between a large grey floor and ceiling, the room’s colour is mostly the grey faces’ return, diluted, and a finish whose reflection has come off the painted walls can carry more colour than the room it is added to. The mechanism is not separated here; the number is.

What a specifier can take from it #

The earlier essay’s practical rule was that among dark walls a more saturated paint survives a gloss finish better and among pale walls a less saturated one does, with the change somewhere around a third of the light returned. Three refinements follow from the five rooms.

The rule’s direction is the same in every room; its dividing line is not. In a closed room of any proportions the line sits near a third. In a room that loses light — a window uncurtained at night, an open plan whose far end is dark — it sits lower, near a quarter with two walls’ worth of opening. A mid-tone wall that would be on the protected side in a closed room can be on the exposed side in an open one, and the same paint choice then goes the other way.

The loss itself is larger in an open room, at every lightness. A satin finish costs the cube a tenth of its colour and the room with two walls open a sixth. A tenth of the return arriving white first measured how a small white contribution compounds through interreflection; an opening takes away the interreflection that would have coloured it, so the contribution arrives whiter and compounds less in the room’s favour.

And a viewer in the room sees more of it than the numbers here. Every loss above is measured on the room’s light against its lamp’s white. What an adapted viewer loses is set by the wall found an appearance model multiplying that loss by 1.4 to 2.4, depending on the wall’s colour, for a viewer adapted to the room. Nothing about the geometry changes that multiplier’s direction, and the open room’s larger loss is the one that multiplier acts on.

The finish itself is held at one roughness throughout. A glossier finish sends more of its return into a narrow lobe, which a lobe takes colour out of a bounce showed carries the lamp’s colour rather than the paint’s; whether a glossier finish moves the handover further down, as a whiter addition should, is the natural next measurement and is limited by where the directional solver stops resolving a lobe.

What was computed #

The room ledger is the one every gloss census here has used, extended in two places. The room’s width, height and depth are arguments of the directional solver’s geometry, whose form factors are exact for any box; and a face can be opened, which gives it zero reflectance at every wavelength and no lobe, so that anything arriving at it is lost. The lamp is D65 on the ceiling in every room.

For each room and paint, the loss is one minus the satin room’s chroma over the matt room’s, both read against the lamp’s white. The added light is the satin room’s total reflected spectrum minus the matt room’s, band by band, and its colouredness is its distance from the lamp’s white on the 1976 chromaticity diagram divided by the matt room’s. The sliding windows hold eighteen paints and step by three; the partial rank correlation holds the wall’s luminance factor by the first-order formula on the ranks; the handover is the linear interpolation of the last crossing from negative to positive.

What this leaves out #

An open face is a hole, not a window. A real window admits light as well as losing it, and in daytime the light it admits — skylight, not the ceiling lamp — has its own colour. A window at night is what is modelled here; a window by day would add a second lamp, and a room with two lights has no white is the standing reminder that the room then has no single white to read the loss against.

The rooms are boxes with two painted walls. A real room has furniture, a floor that is rarely mid grey and ceilings that are rarely grey at all, and all of them change how much light goes round again before it reaches the finish. The finding that shape matters little is a finding about three boxes.

The handover is read from windows of eighteen paints. Its position moves with the window’s size — the cube’s from 0.33 at twelve paints to 0.36 at twenty-four, the low room’s from 0.34 to 0.41 — and the effect of shape grows with it, to five hundredths at the widest window. What does not move is the ordering of the five rooms, which is the same at every window from twelve to twenty-four, and the direction of each change: every change of shape moves the handover up, and every opened wall moves it down.

Still open: why a low room gains colour #

The low wide room is the one geometry in which a satin finish makes a room more colourful rather than less, and it does so for a third of the paints. The candidate mechanism is dilution: the room’s colour is diluted by the large grey faces’ body return, and the finish’s addition, much of which has come off the painted walls at a glancing angle, is diluted less. If that is right, the gain should grow as the painted walls shrink relative to the grey ones and vanish when the floor and ceiling are painted too.

The prediction is that the gain is a property of how much of the room is painted, not of its height. A tall narrow room with small painted walls should gain as the low room does, and a low room with its floor painted should not. The computation is this census in rooms that vary the painted area and the proportions independently, with the gain recorded for each; and the measurement that would test it is a white room with one small coloured feature wall, in matt and in satin, read with a spectroradiometer at the viewing position rather than at the wall.

A prediction about a balance needs both sides #

The habit is about which part of a mechanism a prediction moves.

The earlier essay’s mechanism had two terms whose sum changed sign, and its prediction reasoned about one of them — the colour term — as though the handover were a property of that term alone: the point where it saturates. A change to the room that weakened the colour term was then read as delaying its saturation. But a sum changes sign where the two terms balance, and weakening one term moves the balance towards the other, whatever it does to the first term’s ceiling.

The failure mode is to predict where a balance point moves from one side of the balance. Both terms move when the room changes — here, opening two walls raised the amount term by a third among dark walls and shrank the colour term by a factor of nearly nine — and the only safe prediction is one that says what happens to each. Measuring the ambient first, as the essay recommended, would have confirmed the premise and still left the conclusion wrong, because the conclusion depended on a picture of the handover the census had not tested.