A finish adds colour only where it covers grey

An open room hands over sooner moved a census of seventy-two wall paints through five rooms, asking where a satin finish’s cost to the room’s colour changes character between dark walls and pale ones. On the way it found one room where the finish does not cost colour at all. In a low wide room — two units by two, 0.6 high, its two side walls painted and its floor, ceiling and end walls mid grey — 21 of the 72 paints make a more colourful room in satin than in matt, by as much as 13.8 per cent. Every other room it measured lost colour to the finish at every paint, as a lobe takes colour out of a bounce had found the first satin room doing, and a gloss finish takes colour out of the whole room had confirmed with a ledger of every flux between the faces. The one earlier sighting of a gain, in a finish can add colour, had turned out to be a reading against daylight’s white rather than the lamp’s; this one survives being read against the lamp.

The explanation it offered was dilution. The low room’s colour is diluted by its large grey faces, which return the room’s light without adding any; the finish’s extra return, much of which has come off the painted walls at a glancing angle, is diluted less. It predicted 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, a low room with its floor painted as well should not, and the gain should vanish when floor and ceiling are painted too.

Where the finish goes decides it #

Painted area does not decide the gain, and all three of the prediction’s tests fail: the tall narrow room with small painted walls gains for 4 paints, the low room with its floor painted still gains for 6, and with every face painted for 11. What decides it is which faces carry the finish. With the finish on the painted faces alone, no paint gains in 29 of 30 arrangements of shape and paint; with it on the grey faces alone, paints gain in 28 of the 30, and every paint in 13. The low room gains because its grey faces are most of it: its painted walls’ finish costs 7.0 per cent of the room’s colour, its grey faces’ finish adds 2.7, and the whole finish is close to the sum of the two.

• A finish over paint costs colour, almost without exception, whatever the room.
• A finish over grey adds it, in almost every room, sometimes a great deal.
• A room gains when the second outweighs the first, and neither painted area nor painted light says when that is.
• The lamp’s face matters: the same paint on the floor or on the ceiling of the same cube gains at every paint or at none.

Six rooms, five ways to paint them #

The census keeps everything the earlier rooms used — the same seventy-two paints, each a band of reflectance centred at one of six wavelengths from 450 to 650 nanometres, 15 to 100 nanometres wide and peaking at 0.3, 0.6 or 0.85, on a base of three per cent; the same mid-grey faces reflecting half the light at every wavelength; a lamp on the ceiling; a satin finish at a roughness of 0.2 with a refractive index of 1.5 — and varies two things the earlier census varied together. The shape runs from the low room through a flat room and a wide one to the cube, a tall room and a tall narrow slot. The painted faces are the two side walls, all four walls, the side walls and the floor, the floor alone, or the ceiling alone. Thirty arrangements in all, from a slot with five per cent of its area painted to a tall room with eighty-three.

Each arrangement is solved four times for every paint: matt, and satin three ways — the finish on every face, on the painted faces only, and on the grey faces only. The last two are not a thought experiment. A room with coloured walls in flat paint and a satin ceiling is an ordinary room, and so is the reverse. The loss is one minus the satin room’s chroma over the matt room’s, both read against the lamp’s white, so a negative loss is a room made more colourful.

The prediction’s three tests #

The tall narrow slot, its small side walls painted, gains for 4 paints. Its painted walls are 23 per cent of its area against the low room’s 19, and on the prediction it should have gained about as often; it gains a fifth as often. The low room with its floor painted as well gains for 6. And with every face of the low room painted, where the prediction had the gain vanish, it gains for 11 — and so does every shape painted all over, for between 4 and 11 paints.

Across the thirty arrangements the gain does not follow painted area. The cube with its floor painted and the wide room with its floor painted cover 17 and 21 per cent of their areas with paint, and gain for every paint and for none.

The rank correlation over the thirty between painted area and the number of paints that gain is −0.43; with the painted faces’ share of the room’s reflected light — the quantity dilution actually concerns — it is −0.23. More paint tends slightly towards less gain, as dilution says, but a room’s painted share leaves most of the variation unexplained, and two rooms at the same share can sit at opposite ends of the scale.

The finish over the paint, and the finish over the grey #

The grids above answer the question the first grid cannot. With the finish on the painted faces alone, 29 of the 30 grid cells are zero. The room loses colour on average in every arrangement, by between 4.5 and 43.6 per cent. With the finish on the grey faces alone, 28 of the 30 cells are not zero, and 13 are 72. The grey finish costs less than the paint finish in every arrangement but one.

That is what a lobe takes colour out of a bounce and a tenth of the return arriving white would lead one to expect, read carefully. A finish over paint replaces part of the paint’s own coloured return with a reflection of whatever arrives at the wall — the room’s ambient, which is always less coloured than the paint itself. It can only whiten the wall’s contribution. The one exception in the census is the slot with all four walls painted, where the painted walls face each other across a third of a unit and what arrives at each is mostly the other’s colour; there four paints gain.

A finish over grey trades part of a grey face’s diffuse return — an even mixture of everything arriving at it — for a reflection weighted towards the light arriving at grazing angles, where an interface reflects most. A grey face adds no colour of its own either way. What the finish changes is which of the room’s light it hands on, and in a room where the paint is what the grey faces see at a slant, that is the paint’s light. The finish turns a neutral surface from a diluter into something closer to a mirror of the coloured one. That reading is a statement about Fresnel reflection, not a measurement of which directions carried the gain; what is measured is that the finish over grey adds colour in 28 rooms of 30.

The low room, paint by paint #

In the low room the painted walls’ own finish costs every paint between 4.6 and 8.6 per cent of the room’s colour, and the grey faces’ finish adds colour for 51 of the 72, by up to 17.9 per cent. The paints that gain from the whole finish are the ones for which the second outweighs the first. Fifteen of the 21 lie below the line where the two partial finishes cancel, six lie just above it, and no paint that loses lies below it. The paints that gain most are those whose walls lose least to their own finish and whose grey faces gain most from theirs, and they are the broad, pale bands: seven of the eight largest gains are paints at the highest peak of reflectance, and six are at the widest band. The paints that lose most are narrow bands at the lowest peak. A pale broad paint sends the most light round the room, so it is the paint whose colour the grey faces have most of to hand on — the side of the reversal the finish adds the room’s own colour found among pale walls, where the amount of light a wall returns decides the loss rather than its saturation.

Two halves that add #

The whole finish is close to the sum of its halves in every arrangement, and most lie a little towards zero, as though each half slightly blunts the other. In the low room the two halves sum to 4.3 per cent and the whole finish costs 2.7; in the cube with its floor painted, −2.2 and −1.5. The rank correlation between the sum and the whole over the thirty arrangements is above 0.95.

This is what makes “the low room gains because its grey faces are most of it” a statement rather than a story. The gain is a balance of two terms that can be computed apart, and the low room is where the grey faces’ term is largest against the painted walls’ — a room that is mostly floor and ceiling, whose painted walls are narrow strips seen at a slant from both.

The lamp’s face #

Painting the floor or the ceiling of the same cube puts the same area of paint in the room, opposite each other, and the two rooms go opposite ways. With the floor painted, every paint gains from the whole finish; with the ceiling painted, none does. The paint’s own finish costs about the same in both — 5.9 and 4.8 per cent. The difference is the grey faces’: they add 8.1 per cent of colour when the paint is on the floor and 3.6 when it is on the ceiling.

The ceiling is the face that carries the lamp. A grey wall’s finish hands on the light arriving at a slant from the ceiling, and a painted ceiling’s light arrives mixed with the lamp’s own, which is white; a painted floor sits opposite the lamp, lit directly by it, and nothing white shares its face. Where the paint sits relative to the lamp is a variable the painted share cannot see, and it is as large as any other in the census.

The same arrangement in the tall room is the census’s extreme. With only the floor painted, the grey walls’ finish raises the room’s chroma by 90 per cent on average and more than doubles it for the most favoured paint, and the whole finish by 77 per cent. A tall narrow room with glossy grey walls behaves like a light pipe down to a coloured floor. In the low room the same floor gains nothing, and the ceiling gains for 31 paints.

What a decorator can do with it #

Put the sheen on the neutral surfaces, not on the coloured ones. A satin coloured wall costs its room colour in every shape measured; a satin ceiling, floor or trim in a neutral, around matt coloured walls, adds colour in almost every shape. The advice runs against the habit of choosing a finish for the feature wall first.

Expect the effect to depend on what the neutral surfaces see. A low room’s floor and ceiling see its walls at a slant and gain most from their paint; a tall room’s walls see its floor at a slant. The census gives the direction and the order of size, not a rule a specifier can apply to a room with furniture in it.

Do not read a room’s gain off how much of it is painted. Dilution is real — more paint tends slightly to less gain — but two rooms at the same painted share can be at opposite ends of the scale, and the lamp’s position decides as much as the paint’s area.

How the rooms were computed #

Every room is solved with the collection’s directional radiosity: each face of the box carries a matt body under an optional interface, the interface reflecting by Fresnel’s equations for an index of 1.5 into a lobe of roughness 0.2, the light between faces resolved into rings of directions and the whole solved for energy conservation, band by band. The paint’s reflectance is a Gaussian band of the stated centre, width and peak on a three per cent base; the grey faces reflect half the light at every wavelength; the lamp is daylight, emitted from the ceiling. A face without the finish is the same body with no interface. The room’s colour is its total reflected light, the lamp’s own emission taken out, read in CIELAB against the lamp’s white. The painted share of the light is the matt room’s flux leaving painted faces over its total.

What this leaves out #

The room’s colour is its total reflected light, not what a viewer sees. A viewer standing in the room sees particular faces from particular directions, and a finish sends its reflection one way rather than another; a probe at the wall prices the finish measured what arrives at one wall, and what arrives at an eye is a different integral again. The balance here is of totals.

The rooms are boxes. Six shapes and five paintings are thirty rooms, not a sample of real ones, and furniture, windows and a floor that is not mid grey all change what the grey faces see.

The reading is Fresnel’s, not a measurement of direction. The census shows that a finish over grey adds colour and over paint removes it; that the grey faces’ gain comes from the light arriving at a slant is the physics of an interface, stated rather than separated out.

Still open: whether the slant predicts the grey faces’ gain #

The reading offered for the finish over grey is that it re-weights each grey face’s return towards the light arriving at grazing angles. That is computable without solving any room: for every grey face, the Fresnel-weighted share of its incoming view taken up by painted faces, against the plain cosine-weighted share. Where the first exceeds the second, the finish should make that face hand on more paint than it did matt.

The prediction is that the excess, summed over the grey faces and weighted by their area, orders the thirty arrangements’ grey-finish gains with a rank correlation near 0.9, with the ceiling-painted rooms as the exception — because there the lamp shares the painted face and the weighting hands on white as well. If it does, a decorator’s version of this essay is a single geometric number for a room: how much of its paint its neutral surfaces see at a slant.

A mechanism is tested by separating it, not by varying its proxy #

The habit is about how to test an explanation that names a quantity.

The explanation offered for the low room named painted area, and the natural test was to vary painted area and watch the gain. That test failed, and a failed test of a proxy says little about why. What found the answer was separating the two things the finish does — to the paint and to the grey — and computing each alone, which the model allows and a real room would allow too.

The failure mode is to test an explanation through the variable it happens to mention. Painted area was where the explanation was stated, not where it acted; the finish acts on surfaces, one at a time, and it was only once the surfaces were separated that the gain became a balance of two terms rather than a mystery of one room.