A room scaled whole loses its floor to the last digit

A painted floor in a room of grey walls and ceiling takes on a little colour when the grey faces are given a satin finish instead of a matt one. The paint itself gains all or nothing found the gain to be the rule — the finish 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 — and the exception is a narrow room near square, whose floor loses at most a few hundredths of a per cent of its chroma, for every one of seventy-two paints at once.

The exception has been mapped one measurement at a time. A long floor never loses its colour to the finish found floors three and four units long, in a box 2.5 units tall, gaining at every shorter side; the window of loss closes from both ends found the loss in a pocket centred near a shorter side of 0.4, closing when the floor is 2.59 units long. A taller room keeps its floor losing in proportion built a box four units tall and found the same pocket moved up with the height: a crossing at 4.04 units, 1.01 times the height, against the lower box’s 1.04 times, and the deepest loss near a sixth of the height in both rooms.

Three hundredths is close, and not equal. Either something in the transport carries a length — so that a taller room is not just a larger copy of a lower one — or the two rooms’ answers are the same and the census read them at different places. The prediction was the second: that a four-tall box built as the 2.5-tall box’s floor two and a half by 0.4 multiplied by 1.6 in every dimension reproduces the lower box’s loss to the solver’s precision, because every exchange of light between two faces depends on their shapes and not their size, and that the floor four units long loses from about 0.55 to 0.65 wide. If the scaled box differed, the pocket would have to be found room by room.

The same number at two sizes #

The box four units tall with its floor four by 0.64 changes the floor’s chroma by −0.0055974 per cent, the 2.5-tall box with its floor two and a half by 0.4 by −0.0055974 per cent; they agree to eleven significant figures, and so do the finish’s brightening of the floor, 49.0 per cent at its darkest reflectance, and its tilt across reflectance, −0.036 points. All seventy-two paints lose in both.

• The prediction holds, as exactly as a computer can say it. A relative difference of four parts in a million million is rounding in the solver’s arithmetic, not a property of rooms.
• The census already held a scaled pair and had not noticed. The tall room’s floor four by 0.8, solved for the earlier essay, is the lower room’s two and a half by 0.5 multiplied by 1.6, and the two agree to twelve figures.
• So the two censuses are one. Every box the tall room’s census solved is a box of the lower room at proportions the lower census never reached.

The reason is in what the transport computes. Light leaves a patch of a face, crosses the room and lands on another patch; how much of it lands depends on the angle each patch makes with the line between them and on the solid angle one subtends at the other, and both are ratios of lengths. The finish’s lobe is described by a roughness, an angle’s spread, not a size. The lamp is the whole ceiling, emitting a fixed amount per unit area, so a larger ceiling sends more light to a larger floor in the same proportion. And each face is divided into the same six strips a side whatever its size, so the two solutions divide two similar rooms into similar patches. Nothing in the census has a unit of length in it. A room multiplied by any factor is the same room to this solver, and the finish’s effect on the floor is a function of two ratios: the floor’s length and width over the height.

A window higher than predicted #

Read with the new boxes, the floor four units long loses from a shorter side of 0.59 to 0.70: at 0.6, 0.64 and 0.7 it loses, at 0.5 and 0.8 it gains. The prediction was 0.55 to 0.65; both edges are four or five hundredths higher. The window is also wider than any earlier reading of it. The tall room’s first census, solved at 0.4, 0.6 and 0.8, put it between 0.60 and 0.61, because its only losing point sat at 0.6 and straight lines from there to the gains on either side reached nought within a hundredth or two. The lower room’s own census, solved at 0.3, 0.4 and 0.5 and scaled by 1.6, put it between 0.62 and 0.67.

All four placements come from the same curve. They differ only in where it was sampled and in the straight lines drawn between samples, and the prediction took the lower room’s window, rounded it and placed it a little low. The window’s edges in this census are no better than its spacing, and every reading with three points across a window of a tenth has been a guess at its width.

Two rooms, one row #

Put in shares of the height, the lower room’s floor two and a half long and the taller room’s floor four long are the same row of boxes. The lower room’s points fall at 0.12, 0.16 and 0.2 of the height; the taller room’s at 0.1, 0.125, 0.15, 0.16, 0.175 and 0.2, two of them on the lower room’s own points. Read together the row loses between 0.148 and 0.176 of the height, and its deepest loss, −0.0056 per cent, is at 0.16. The upper edge is where the most was learnt: the lower room’s census placed it at 0.169 and the tall room’s first census at 0.154, and it lies above both.

The row is still not smooth. Between 0.15 and 0.175 the change in chroma goes −0.0016, −0.0056, −0.0008 per cent, a dip with its bottom near 0.16 that samples a hundredth or so of the height apart have caught at three points, and whose true bottom could sit a little either side.

The crossing gap was the sampling #

The taller room’s crossing was read from the deepest loss at each floor length, and at four units that was −0.0016 per cent at 0.6, the only losing side its first census had. Read at all six sides it is −0.0056 per cent at 0.64, and the crossing moves from 1.009 times the height to 1.029, against the lower room’s 1.036. Three quarters of the gap was the taller room’s census missing the pocket’s bottom.

The rest belongs to the lengths. The lower room’s crossing is a straight line from its deepest loss at 1.0 times the height to its deepest at 1.2, the taller room’s from 1.0 to 1.25, and the two lines are drawn through different points of the same surface. Neither is the crossing; both bracket it from samples a fifth or a quarter of the height apart. With the solver indifferent to size, the remaining seven thousandths is a statement about which lengths were solved and nothing else.

Brightening and tilt on one curve #

The finish brightens the floor at its darkest reflectance by 78 per cent at a tenth of the height and by 43 per cent at a fifth, falling at every step, and the lower room’s three values, 66, 49 and 43 per cent, lie on the taller room’s curve. The brightening has no feature at the window: it falls smoothly through it. A taller box must be deeper before the finish adds colour read the satin walls as a pipe carrying the lamp down, and found the turn moving faster than in proportion as the box grew taller at a fixed width of one unit; that box changed its proportions as it grew, and what it measured was proportion, not height.

The tilt is where the window shows. It is +0.63 points at a tenth of the height, +0.17 at an eighth, −0.011 at 0.15, −0.036 at 0.16, −0.005 at 0.175 and +0.13 at a fifth, and its sign is the chroma’s at every box, as it has been in every box both censuses solved. Where the brightening’s tilt across the floor’s reflectance turns below nought, the chroma falls with it, and it turns over the same few hundredths of the height as the window.

One map of proportions #

On one map the two rooms’ twenty-eight boxes fill one wedge. The losses sit between 0.14 and 0.2 of the height in width and at lengths up to 1.0 of it; every box longer than 1.04 times its height gains, from either room. The lower room’s census reached floors 0.8 to 1.6 times its height and widths from a tenth to a third of it; the taller room’s added lengths of 0.75 and 1.25 and four new widths at 1.0. Neither room needed the other’s height to be read; they needed each other’s proportions.

That changes what a square floor needs more room than a long one and the width moves the turn only when it is the narrower side were measuring. Each held the height at 2.5 and varied a width; each found a turn, and each turn is a point on this same map — a place where a line of proportions crosses the wedge’s edge. The rooms they solved were particular sizes only for convenience.

What scaling leaves alone, and what it does not #

One property of the census survives the scaling untouched, and it is worth separating from the ones that do not. The six strips a side are a count, not a size, so a room scaled by 1.6 is divided into patches 1.6 times as large and the two solutions are identical. A room divided more finely is a different solution. The tall room gains once its walls are divided found a verdict reversing as the faces were divided from one patch to six, and the slot’s loss survives every finer division found the slot’s loss holding its sign from three strips to eight. Those are statements about how finely the transport is resolved, and the scaled pairs say nothing about them: the two rooms here agree because they were resolved alike, not because six strips is fine enough.

So there are now two independent questions about any box in this census, and they should not be confused. Whether its answer depends on its size is settled: it does not, to eleven figures. Whether its answer depends on how it was divided is settled only where it has been tested, and the pocket at 0.16 of the height has been solved at six strips alone. A loss of −0.0056 per cent is small enough that a finer division could move it, and nothing here says which way.

The scaling also shows which earlier comparisons were comparisons of proportion all along. Every earlier essay on this room that changed one dimension and held the others changed the room’s shape, and each attributed what it found to the dimension it changed. A taller box at a fixed width, a wider box at a fixed height, a square floor beside a long one: in each, the dimension named in the title was one of three ratios moving at once. Read again on the map of proportions, none of their numbers changes. What changes is the variable the numbers belong to.

There is a saving in that too. Each box costs twenty-five to thirty minutes to solve at six strips a side, whatever its size, and a census that indexes its boxes by three lengths solves some of them twice under different names — the tall room’s floor four by 0.8 was the lower room’s two and a half by 0.5, solved again. A census indexed by two ratios cannot repeat itself that way, and the next one should be.

What a lighting designer should take from this #

Under this model a room’s size does not decide whether a satin finish dulls its floor; its proportions do. A narrow room whose floor is about as long as the room is tall, and whose width is about a sixth of its height, loses a few thousandths of a per cent of the floor’s chroma, whether it is a cupboard or a stairwell.

The loss is too small to see and the map is what matters. The deepest mean loss either census has found is about three hundredths of a per cent; what the map says is where the finish stops adding colour, which in every other proportion it does.

What breaks the rule in a real room is what the model leaves out. A lamp of fixed size in a room of growing size, a finish whose texture is coarse at the scale of a small room, air that absorbs over long paths — each puts a length into the transport, and each would let a large room and a small one part.

How the boxes were solved #

The box four units tall has its floor four units long and 0.5, 0.64 and 0.7 wide, painted with each of the census’s seventy-two paints in turn, its walls and ceiling grey, the whole ceiling emitting. Each face is divided into six strips a side and solved by the directional transport with satin of roughness 0.2 on the grey faces and matt everywhere, band by band, the floor’s mean exitance read under each finish; the change in chroma, the brightening at the darkest reflectance and the tilt across reflectance are read as in every box of both censuses. The window’s edges are straight lines between adjacent solved sides; the crossing is a straight line between the deepest losses at adjacent solved lengths.

What this does not reach #

The tip of the wedge. The longest losing floor lies between 1.0 and 1.2 times the height, and every crossing so far is a line drawn between samples that far apart.

Other finishes. Roughness 0.2 is one lobe; a glossier or a duller finish is another function of the same two ratios, not a scaling of this one.

Lengths the model omits. Every conclusion about size is a conclusion about this solver, which has none.

Still open: where the wedge’s tip lies #

The wedge closes between floors 1.0 and 1.2 times the height long, and its deepest loss at 1.0 sits at 0.16 of the height. Size no longer matters, so the tip can be found in either room at the cost of a handful of boxes.

The calculation is the 2.5-tall box with its floor 2.6, 2.7 and 2.8 units long — 1.04, 1.08 and 1.12 times the height — at shorter sides of 0.375, 0.4 and 0.425, 0.15 to 0.17 of it. The prediction is that the floor 1.04 times the height long still loses at 0.4 and that the 1.08 floor gains at all three, placing the tip between them, a little above the crossing the coarse lengths gave, because the pocket narrows as the floor lengthens and its deepest point stays near 0.16. If the 1.08 floor loses, the wedge has a longer tip than any reading so far, and the crossing at 1.036 was a line drawn below the curve.

A difference that was asked of the wrong object #

The habit here is about reading a small disagreement between two measurements.

Two rooms gave crossings three hundredths apart, and the natural reading was that the rooms differed a little, as rooms of different heights might. The test was to build the one box that should give exactly the lower room’s number if the solver had no length in it. It did, to the last digit, which moved the disagreement out of the rooms and into the census: one census had sampled the pocket’s bottom and the other had sampled beside it.

The failure mode is to read a disagreement between two results as a property of the two things measured before asking whether the two measurements were the same measurement. The test is to construct the case where the two must agree if the difference is in the method, and read whether they do.