What a white wall costs

Paint manufacturers sell dozens of whites and the differences between them are small — a percent or two of reflectance, a slight warmth or a slight coolness, differences a customer holding two cards side by side may struggle to see at all.

Put one of them on every surface of a room and the difference stops being small. Not because the paint changed, and not because of anything perceptual: because a room multiplies.

The claim #

A closed cavity of spectral reflectance $\rho(\lambda)$ returns $1/(1-\rho(\lambda))$ times the light that entered it, and because that is a per-wavelength quantity it amplifies a wall’s colour more than its brightness.

The formula is a geometric series and there is nothing subtle in deriving it. What is worth attention is that the geometry disappears from it entirely, and what that means.

The geometry drops out #

Light enters a closed cavity. Some fraction meets a surface, is multiplied by $\rho$, and leaves again; some of that meets a surface and is multiplied again. The total is

$$1 + \rho + \rho^2 + \rho^3 + \cdots = \frac{1}{1-\rho}$$

The step that deserves noticing is why no form factor appears. In a closed cavity $\sum_j F_{ij} = 1$ for every surface — everything leaving a surface arrives somewhere, because there is nowhere else to go. That is closure, it is the identity the whole transport library is checked against, and it means that when the reflectance is uniform the light’s next destination does not matter. Every path is multiplied by $\rho$ regardless of which surface it hits.

So a room’s colour cast is a property of its paint and not of its shape — while the rate at which the room approaches that cast is entirely a property of its shape. A long corridor and a cube of the same paint end up the same colour and get there through different numbers of bounces.

That is a clean separation and it is unusual. Most quantities in this field are entangled with the geometry: a metameric match breaks by an amount the enclosure decides, and a three-channel renderer’s error depends on how many bounces the geometry delivers. The equilibrium colour of a uniform cavity is the one place the geometry cancels exactly.

The numbers are worse than they look #

The series is not linear in the albedo and the top end is where all the action is. Going from 0.4 to 0.6 buys a factor of 1.5. Going from 0.8 to 0.9 buys a factor of 2 — from a change in albedo half the size.

For a lighting designer that is the familiar and welcome half: a white room needs fewer lamps, and the saving accelerates. For anybody who cares what colour the room is, it is the unwelcome half, because the same acceleration applies to the tint.

Consider what “five percent off neutral” means after amplification. The wall in the figures reflects 0.8500 at the peak of its bump and 0.8111 away from it — a ratio of 1.0480, which is the 4.8% a paint card would show. Its gain runs from 6.67 down to 5.29, a ratio of 1.2593. A 4.8% spectral difference in the paint has become a 26% one in the light, and the room’s equilibrium white sits 41.8 units of $\Delta E_{00}$ from the D65 lamp that lit it.

Why this is not “the walls look coloured” #

The claim is about the illuminant rather than about the walls, and the distinction is the practical content.

If a slightly warm wall merely looked slightly warm, nothing here would need saying. What happens instead is that the wall becomes the dominant light source for everything else in the room, so every object in a white room is lit by the wall paint. A grey card on a table in that room measures warm. A sample being colour-matched measures warm. A photograph taken there is warm in a way no single white balance recovers, because the mixture varies with what each surface can see.

This is why colour-matching booths are painted neutral grey rather than white, and why the standards specify the grey. A white booth would be brighter and would contaminate every judgement made in it with its own paint, amplified by the very gain that made it bright. The grey is deliberately giving up light to avoid the amplification — and the same argument is why the chrome on this site is exactly neutral rather than merely tasteful.

What was computed, and how #

The gain is per band and the series is closed-form. Nothing is iterated: $1/(1-\rho(\lambda))$ is evaluated at each of the 81 wavelengths, and the function throws rather than returning infinity if handed a reflectance of 1, because a cavity that reflects everything never comes to equilibrium and a figure implying otherwise would be wrong rather than merely extreme.

The tint is a stated rule. The wall reflectance is a flat albedo with a broad Gaussian bump of stated amplitude — the 5% in the figures — rather than a measured paint. That keeps the argument about the amplification rather than about any manufacturer’s product.

The assertion is monotonicity in albedo. The generator computes the shift of the room’s white point away from the lamp at each albedo and refuses to draw unless each is larger than the last. The failure it guards against is a sign error or a reciprocal taken the wrong way round, either of which would produce a smooth plausible family of curves running the wrong way.

The white point comparison is $\Delta E_{00}$ against the lamp, under the CIE 1931 2° observer, with both normalised — so the number is about chromaticity rather than about the room being brighter, which it obviously is.

The interior-design consequence nobody states #

There is a practical rule buried here that is worth pulling out, because it contradicts standard advice.

The advice is that a small room should be painted white to make it feel larger and brighter. The arithmetic says a small room painted white is also the geometry in which the paint’s own colour most dominates the light — small rooms have higher form factors between their surfaces, so the equilibrium is reached in fewer bounces and there is less direct light in the mixture.

So the whitest paint available is the one whose residual tint will most strongly colour the room, and the effect scales with how enclosed the space is. A cupboard painted in a warm white is, spectrally, a warm-light box. This is not a large effect in absolute terms and it is a completely reliable one, and it is the reason that choosing between two whites from cards in a shop is not a well-posed exercise: the cards are being viewed at $q \approx 0$ and the room will be viewed at $q$ close to 1.

A worked room #

It is worth walking one case all the way through, because the intermediate numbers are more surprising than the conclusion.

Take a cubical room, every surface painted at albedo 0.85 with a broad warm bump making the paint 5% more reflective in the yellow than in the blue. One lamp in the ceiling, D65.

Direct light. Anything on the floor receives the ceiling’s share of the lamp. The form factor between opposite faces of a cube is 0.199825, so a fifth of what the ceiling sends goes straight down and the rest goes to the four walls.

First bounce. Those walls return 0.85 of what they got, tinted. So the floor’s second helping is the lamp times the wall’s reflectance — the lamp already reddened once.

Equilibrium. The series sums to 6.67 at the bump and 5.29 away from it. The 4.8% spectral difference in the paint has become 26% in the light, and the room’s white has moved 41.8 units of $\Delta E_{00}$ from the lamp.

What arrives at an object. Only the fraction that came directly from the ceiling is untinted, and in a cavity at 0.85 that fraction is small: the gain is 6.67, so roughly 15% of the light on any surface is first-hand and 85% has been through at least one wall.

That last number is the one worth carrying. In a bright room, most of the light on any object has been coloured by the paint, and it is not a subtle contribution being teased out of a dominant direct term — it is the dominant term, with the lamp as the minority contributor.

Forty-one units is the brightness, not the colour #

The 41.8 is quoted twice and described as a chromaticity difference — both normalised, so the number is about chromaticity rather than about the room being brighter, which it obviously is. It is the unnormalised number.

Rebuilding the wall from the two reflectances the essay prints, 0.8500 at the bump and 0.8111 away from it, and running the cavity gain against D65: with the room’s sixfold brightness left in, the difference from the lamp is 40.5 to 42.2 ΔE₀₀ across every plausible bump centre and width. With the result normalised to the lamp’s own luminance, it is 5.3 to 8.9, clustering near 7.5.

Only the first family contains 41.8, and the second is what the sentence beside it describes.

CIEDE2000 says so on its own, without any spectra. Its chroma term saturates: for a difference from neutral it is C/(1 + 0.0225C), which rises to a hard ceiling of 44.4 however saturated the colour gets. So 41.8 is 94 per cent of the largest value a pure chromaticity difference can ever take, and reaching it requires a chroma of about 700 — against roughly 150 for the most saturated colour anybody can produce. A number that near the ceiling of a saturating scale is a number with something else in it, and here the something else is a room six and a half times brighter than its lamp.

What the corrected number is worth #

The correction removes nothing from the argument and it changes which comparison the number invites.

Seven and a half ΔE₀₀ is still a very large illuminant shift. It is seven times the tolerance a print contract is written in, and two thirds of the distance between two lamps at opposite corners of a single 4000 K bin. A room whose light sits that far from its lamp is a room in which no colour judgement carried out of it will hold.

And the amplification is the finding, measured the same way both times. The paint applied once — a single bounce, which is what a card in a shop shows — moves the light 1.8 ΔE₀₀ from D65. The same paint as a closed cavity moves it 7.5. So the room multiplies the paint’s chromatic effect by about 4.2, which is the essay’s own 4.8 per cent becomes 26 per cent stated in the unit the rest of this collection argues in.

Four is a better headline than forty-one, because it is the number that survives being asked compared with what. Forty-one compares the room’s light with the lamp and includes the six-fold gain that a lighting designer counts as the whole point; 4.2 compares the room with the paint card, which is the comparison the shop is offering and getting wrong.

Why the two normalisations answer different questions #

The distinction is worth keeping rather than treating as a slip, because both quantities are real and they belong to different readers.

Unnormalised is what an instrument in the room reports against a reference outside it: the room is brighter and warmer, and 41.8 is both facts together. It is the right quantity for asking whether a measurement taken indoors can be compared with one taken under the lamp alone, and the answer is no by a wide margin.

Normalised is what an adapted observer’s instrument reports, and it is the right quantity for everything the essay actually argues — the grey card that measures warm, the sample being colour matched, the booth painted grey rather than white. Every one of those comparisons is made inside the room, against another surface in the same light, so the gain is common to both terms and cancels.

The essay’s own conclusions are all in the second column, which is why the correction leaves them standing: a booth is painted grey because of the 7.5 and not because of the 41.8. The 41.8 would be the argument for a dimmer booth, and nobody makes that argument, because the brightness is the one part of the gain everybody wants.

What the pictures cannot show #

Three limits of the figures, all of which push in the same direction.

The equilibrium illuminant is often outside the display’s gamut, and where it is the swatches are hatched rather than filled. That is the marking this site uses instead of the nearest available lie, and on this page it appears at the high-albedo end, which is exactly where the argument is strongest.

A swatch is a patch of one colour and a room is not. The real phenomenon is a gradient: the light near a wall carries more of that wall than the light in the middle of the floor, so what a photograph of the room shows is a smooth field of slightly different illuminants rather than a single cast. The figures report the equilibrium, which is the limit rather than the average.

Nothing here says what a person in the room would see. A reader adapts to the room’s own white, discounts a great deal of the cast, and will report the room as white. The measurement is what an instrument reports and the two are different quantities — which is the whole reason matching and appearance are separate fields on this site. The cast is real, measurable, and largely invisible to the person standing in it; it becomes visible the moment a camera is involved, or a sample is carried out of the room.

Where the model stops #

Uniform reflectance. The clean cancellation of the geometry needs every surface to be the same. A room with a white ceiling, mid-grey walls and a dark floor has an equilibrium that depends on the form factors, and there is no closed form.

Closed cavity. A real room has windows, doorways and a great deal of furniture. An opening is a surface with reflectance zero and it lowers the effective albedo substantially — which is why the gains quoted here are the upper bound for a given paint rather than a prediction for a given room.

No participating medium and no fluorescence. Air is treated as empty, which is fine, and the walls are treated as non-fluorescent, which for modern white paint is not quite true: optical brighteners absorb in the ultraviolet and emit in the blue, and a brightened white wall in a room with any UV in the light is a source of blue that this model cannot express at all.

The band is 380–780 nm. Real paint reflects strongly in the near infrared, and the gain applies there too. Nothing on this site sees it, and for a thermal question rather than a colour question it would matter more than everything above.

Two more settings say that the tint and the albedo are two independent decisions taken by the same person on the same afternoon.

The generalisation #

The transferable statement is about feedback rather than about colour: a system with gain amplifies the shape of its response, not merely its magnitude, and the amplification of the shape goes as the same series as the amplification of the magnitude.

Anywhere a signal is recirculated through a frequency-dependent element — a reverberant room, a resonant cavity, a feedback loop with a non-flat return path — the equilibrium is the input divided by one minus the loop response, evaluated at each frequency independently. Small departures from flatness in the loop become large departures in the equilibrium, and the closer the loop gain is to one, the more so.

The engineering habit that follows is to specify the flatness of a high-gain return path far more tightly than a low-gain one. A wall at $\rho = 0.4$ can afford a 5% tint. The same tint at $\rho = 0.95$ is a different product.

Who found it, and when #

The integrating sphere is the same physics used deliberately, and it is Ulbricht’s, from 1900. A sphere coated with a high-reflectance diffuse white and used to collect all the light leaving a sample works precisely because the gain is high and the geometry cancels — the whole point is that the reading is independent of where in the sphere the light entered.

Sphere designers discovered every limitation on this page the hard way, because for them the tint is not an aesthetic matter but a calibration error. The standard coatings — barium sulphate, and later sintered PTFE — were chosen for spectral flatness at high reflectance rather than for reflectance alone, and sphere manufacturers quote the flatness because the amplification is understood. A sphere at 0.98 has a gain of 50 and the coating’s residual absorption bands are multiplied by it.

The room-lighting form entered illuminating engineering through the lumen method and the concept of “room surface reflectance”, standardised in the mid-twentieth century. There the calculation is done scalar and photopic — one number for the whole visible band — which is entirely adequate for asking how many lamps a room needs and discards exactly the information this page is about.

The reason the scalar version survived is that for most of the twentieth century it was sufficient. Under an incandescent lamp the spectrum is smooth, the paint’s spectrum is smooth, and there is little structure for the amplification to bite on. Narrow-band sources changed that: a phosphor-converted LED has a spiky spectrum, and what a lamp cannot give back it cannot be given credit for — the gain multiplies whatever structure the source and the paint have between them, and both now have more of it than they used to.

Where the ladder goes next #

The neighbour on this rung takes up the geometry that was cancelled here: where the form factors come from, how they are checked, and what closure guarantees — the identity that made the cancellation possible in the first place.

The rung above moves to materials, where the multiplication model itself stops being right: paint is not a filter, because a mixture of pigments is one scattering layer rather than two in series.