A room bounds its own bounces

The light in a room has bounced off the walls some number of times, and the number is not the same at every wavelength.

The claim #

The census models a room as a wall applied a whole number of times. A room applies a geometric mixture of every number of times, the mixture differs at every wavelength, and it is bounded where the integer is not.

• The census has two surface rows for the same paint — one bounce and two — and a search over that family treats the count as a free integer, which it always takes to the largest value offered.
• A closed room comes to a steady state. The light that survives has bounced some number of times; the spectrum of the whole is a geometric sum over all of them, weighted by how much survived each.
• The mean number of bounces is a function of wavelength, running from 0.33 to 5.67 across the band for the census’s own green — a factor of 17 — because the light that survives many bounces is the light the wall reflects.
• What a room does is not what any whole number of bounces does. The nearest integer is one, and it is 22 per cent away.
• And the room’s version is bounded by ρ < 1, which is not a choice anybody makes, unlike a box’s floor or an integer’s ceiling.

What an integer bounce count is doing there #

The same wall applied twice is one of this collection’s better findings: a bounce is elementwise multiplication, a metameric match is an identity between integrals linear in the reflectance, and squaring the reflectance breaks the match. It needed a wall applied twice, so the census has a wall applied twice.

That was a modelling decision made for a good reason and it has a consequence nobody looked at. When the same family was searched for a worst change of light, the bounce count came along as a fifth parameter — an integer, searched separately from the four continuous ones because a simplex would interpolate it into a wall applied 2.4 times.

In every search this collection has run, the answer takes the largest count offered. Three of three, at every band width, under every bound. That is the signature of a parameter with no bound on it, and it is the same signature the box’s walls had before physics was brought in.

What a room actually does #

A closed cavity whose walls reflect a fraction ρ(λ) of what falls on them comes to a steady state. Light arrives, some of it is absorbed, the rest bounces and arrives again; the total irradiance is the source plus the source once reflected plus twice and so on, which is a geometric series and sums to E₀ / (1 − ρ).

That is the interreflection gain this collection already computes, and it is exact rather than approximate for a cavity whose walls are all one reflectance and whose geometry is fully enclosed.

The point is that it is not ρⁿ for any n. It is a mixture: a share (1 − ρ) of the light has bounced zero times, ρ(1 − ρ) has bounced once, ρ²(1 − ρ) twice, and so on. Mixtures of powers are not powers, and the difference has a spectrum.

The mean of that distribution is ρ / (1 − ρ), per wavelength. For the census’s green wall — a Gaussian band at 540 nanometres, depth 0.6 on a base of 0.25 — it runs from 0.33 where the wall absorbs to 5.67 where it reflects.

Why the spread is the mechanism #

A factor of seventeen between the least- and most-bounced wavelengths is the whole content of the difference, and the reason for it is worth stating slowly because it is the mechanism behind every interreflection effect.

The light that survives many bounces is the light the wall was reflecting all along. At a wavelength the wall reflects 0.85 of, five per cent of the original light is still there after twenty bounces; at a wavelength it reflects 0.25 of, essentially nothing survives two. So the deep-bounce population is drawn overwhelmingly from the band the wall likes, and the steady state is a spectrum sharpened far past what one bounce does.

An integer power does something similar and not the same. Squaring a reflectance also sharpens it, and it sharpens uniformly: every wavelength’s reflectance is squared. The geometric sum sharpens unevenly, weighting the strong band’s contribution by how long its light survives. The two are different operations that happen to agree in direction.

There is a third operation worth naming so that the three are not confused. A gain is not a sharpening at all. Dimming a lamp multiplies every wavelength by the same factor, and an adapted observer removes it exactly; that is the census’s control row and the reason it exists. A bounce multiplies wavelengths by different factors, which is a change of colour a gain cannot undo. And a geometric sum multiplies them by different factors drawn from a distribution, which is a third thing again.

Setting the three side by side makes the family visible: the census’s rows are all multiplications by something, and what distinguishes them is how far that something is from being flat. A room is further from flat than one bounce and nearer than two, which is what the numbers below say and what the mechanism predicts.

What the room’s answer is #

Scored the same way as every census row — the colour difference an adapted observer is left with over this collection’s own reflectance family — the numbers are:

• one bounce: 1.72 ΔE00
• two bounces: 3.37
• three: 4.97
• four: 6.45
• the room itself: 2.22

The room sits between the census’s two existing surface rows, which is reassuring and is not the finding. The finding is that it equals neither: the nearest whole number is one, and the room’s answer is 22 per cent away from it.

And a room is bounded. The geometric sum converges for any wall that absorbs anything at all, so there is no way to make a room worse by turning a dial: a wall that reflects 0.99 gives a gain of a hundred and a wall that reflects 1.0 is not a wall, it is a perfect mirror enclosing a space and never coming to equilibrium.

That is the difference the title is about. The integer’s ceiling is a modelling decision and the room’s ceiling is the second law, near enough.

What was computed, and how #

The room’s filter is the interreflection gain normalised to its own mean, so that the comparison with an integer bounce is a comparison of colour rather than of level.

That normalisation is not cosmetic. A gain applied uniformly to every wavelength is invisible to an adapted observer by construction — it is the census’s own control row, the same lamp dimmed by half — so leaving it in would put a large level change into every number here and flatter the geometric model for a reason having nothing to do with its spectrum.

The reflectance is clamped just below one before the sum, which is a numerical guard rather than a physical claim: at exactly one the gain is infinite and the model has nothing to say. Nothing in the census’s own walls comes near it — the highest reflectance in the green wall is 0.85 — so the clamp never binds on any figure here.

One more property of the sum is worth extracting, because it explains why the room lands where it does. The mean number of bounces over the whole band is 1.15, so a room with these walls is on average not much more than a single bounce — most of the light has bounced zero or once, and the long tail at the reflected band is a small share of the total energy.

A room is nearer to one bounce than to two because most of the light does not survive two. Two bounces off the same wall is a corner, which is a real place in a real room and is exactly where the light has been reflected twice with nothing in between — so the census’s second surface row is a description of a geometry rather than of a whole room, and reading it as either is fine as long as it is not read as both. That is a fact about the wall’s mean reflectance rather than about its shape: a white room would sit much further along, and a black one would be almost exactly one bounce. So the room’s position between the census’s two rows is a property of the paint’s lightness, and its spectral sharpening is a property of the paint’s colour — two effects the integer count fuses into one parameter.

The mean bounce count is a statement about the band #

A fact about the wall’s mean reflectance rather than about its shape is offered as the explanation of why the room sits near one bounce, and the shape is doing most of the work.

The mean bounce count is ρ/(1−ρ) averaged over the band, and that function is convex — so the average of it is strictly larger than its value at the average reflectance, by a gap that is entirely the band’s doing. A flat wall with a mean bounce count of 1.15 would have to reflect 0.535 at every wavelength, which is more than twice the green wall’s base of 0.25 and not far below its peak of 0.85. The green wall’s own mean reflectance, at the band width that reproduces the stated 1.15, is about 0.41, and a flat wall reflecting 0.41 bounces 0.68 times.

So the wall’s shape raises the mean bounce count by nearly seventy per cent over what its mean reflectance would give. That does not overturn the essay’s conclusion — 1.15 is still much nearer one than two — but it changes what the number is evidence of. It is not a summary of a typical reflectance; it is a figure dominated by the narrow band where the light lives longest, which is the same mechanism the seventeen-fold spread reports, arriving in the one statistic the essay uses to place the room.

The room scores as more bounces than it has #

The four integer rows are nearly linear — 1.72, 3.37, 4.97, 6.45, with increments of 1.65, 1.60 and 1.48 — so the census’s own scale can be read off directly, and it says where a 1.15-bounce mixture should land.

Interpolating between the one- and two-bounce rows at 1.15 gives 1.97 ΔE₀₀. The room scores 2.22, which is 13 per cent higher, and inverting the same scale puts the room at an effective bounce count of 1.30 against a mean of 1.15.

That gap of 0.15 bounces is the finding the seventeen-fold spread was for, made into a number the census can use. The deep-bounce tail carries more colour per unit of energy than its share of the energy, because it is drawn entirely from the wall’s own band and is therefore the most saturated light in the room. A mixture weighted towards zero and one bounce by energy is weighted further along by effect, and the difference between those two weightings is 13 per cent of the score.

It also explains why the room can sit above a linear reading of its own mean while remaining far below two bounces. Neither the mean nor the effective count is the whole story, and the two differing is the signature of exactly the non-uniform sharpening this essay separates from an integer power. An integer count cannot have this property, because a whole number of bounces has no distribution to be weighted against.

Two survival figures that cross #

The mechanism paragraph illustrates the spread with two survivals, and read against each other they say the opposite of what they are offered for.

At a reflectance of 0.85, twenty bounces leave 3.9 per cent — described as five per cent of the original light is still there, which is the level reached at eighteen bounces rather than twenty. At a reflectance of 0.25, two bounces leave 6.25 per cent — described as essentially nothing survives two. The second figure is larger than the first, so the same fraction of light is called still there in one place and essentially nothing in the other.

The mechanism is right and the illustration inverts it. What separates the two wavelengths is not how much survives a stated number of bounces; it is how many bounces it takes to fall to a stated level. Reaching five per cent takes 2.2 bounces at ρ = 0.25 and 18.4 at ρ = 0.85 — a factor of 8.5 in depth, which is the quantity the seventeen-fold spread in mean bounce count is a summary of.

Stated that way the two numbers stop competing and the sentence they support gets stronger: the deep-bounce population is not merely dominated by the reflected band, it is exclusively it, because by the time the absorbed band has fallen to a tenth the reflected band has barely begun.

Where the model stops #

A single cavity with one reflectance is a caricature of a room. A real room has a floor and a ceiling and a window, several reflectances, and a geometry that decides how much of each surface each other surface sees. This collection solves that properly elsewhere, with form factors and a radiosity solve, and the answer there is a different spectrum on every face.

The caricature is used here deliberately, because the question is about the census, and the census’s surface rows are one wall applied n times. Replacing an integer power with a geometric sum over the same wall is the smallest correction that makes the model a room; replacing it with a solved five-surface enclosure would be a different essay and is one this collection has already written.

That last figure carries a small consolation for the census. The walls a worst-case search settles on are dark — base near 0.02 and a narrow band — so their mean reflectance is low, their geometric sums converge fast, and the difference between three bounces and a room is largest exactly where the search does not go. The free integer was unbounded and it was also, on the answers that mattered, not doing much.

And the source is assumed to enter and not to leave. A room with a window loses light through it, which is an absorption with a peculiar spectrum — the outdoor light comes back in through the same aperture — and a room with an open door is not a closed cavity at all.

What it changes about the census #

The immediate consequence is small and worth being exact about.

Nothing published from the census’s two surface rows is wrong. A wall applied once and a wall applied twice are legitimate objects: the first is a wall, the second is a corner, and the finding that a corner breaks a metameric match that a wall does not is about applying the same reflectance twice, which is what a corner does.

What changes is the reading of the search. When a worst-case search takes the bounce count to three, it is choosing a configuration that no single room produces — a light that has bounced exactly three times and no other number — and the number it returns is therefore an upper bound on what a room does rather than a description of one.

The room’s own answer, 2.22 ΔE00, is the number a census of rooms would carry, and it is smaller than the two-bounce row the census already has. So the census is, in this one respect, slightly pessimistic — which is the opposite of what the free integer suggested and is the sort of thing worth checking before assuming a model errs in the convenient direction.

The generalisation #

The pattern is the phase’s, in a form that is easy to state and easy to miss.

A discrete parameter with no upper bound is the same defect as a box with a wall, and it is harder to see — because a box announces itself as an arbitrary range and an integer looks like a fact about the world. One bounce, two bounces, three reads as a description of situations; it is a description of a modelling convenience.

The repair here was not to bound the integer but to notice that the physical process does not have one. The right question is rarely how large a parameter may be; it is what the mechanism actually produces, and a mechanism with a convergent series in it comes with its own ceiling.

The check that finds it is to ask, of every parameter in a search, what the search does with it — and to treat always takes the extreme as a report about the parameter rather than about the answer. Four parameters here sat on a wall and were bounded by physics one essay ago; the fifth sat on a wall and turned out not to be a parameter at all.

Who found it, and when #

The geometric series for interreflection is old and is the basis of the integrating sphere, which is the same arithmetic in a device: a cavity of high uniform reflectance whose gain is 1/(1 − ρ) and whose whole purpose is to average a source over angle. Ulbricht built one in 1900 and the sphere multiplier has been in every photometry textbook since.

The spectral version is equally old and less often drawn. That interreflection colours a room — that a red room is redder than its paint — is common knowledge among people who paint rooms and is quantified in the architectural lighting literature as a saturation effect. What is unusual here is only putting it beside an integer bounce count and asking which one a census should contain.

Where the ladder goes next #

Four parameters of a painted wall are bounded by physics and the fifth turns out to be a property of a room rather than a parameter at all. That leaves the question of what bounds the market rather than the mechanism — how dark and how saturated a wall anybody actually sells is.

That bound turns out not to bite, and the reason it does not is the most counterintuitive thing this thread produced: the worst wall for an adapted observer is a dark one rather than a colourful one, and a purity ceiling never comes near it.