The pocket has a step where its bottom should be

A satin finish on a room’s grey walls and ceiling usually adds a little colour to a painted floor, and in one narrow wedge of proportions it costs the floor a few thousandths of a per cent of its chroma instead. A room scaled whole loses its floor to the last digit mapped the wedge in shares of the room’s height and found it deepest along a floor as long as the room is tall, at a width of 0.16 of the height. The wedge of loss comes to a point found it closing at a floor 1.04 times the height long, and the pale paints cross the tip last found that closing a band of seventy-two crossings, one for each paint, the palest last.

All of those readings rest on a pocket sampled coarsely across its width. Along the floor as long as the room is tall, the census had the change in chroma at widths of 0.375, 0.4 and 0.4375 units — 0.15, 0.16 and 0.175 of the height, two of them from the four-tall room scaled — at −0.0016, −0.0056 and −0.0008 per cent, and gains at 0.3 and 0.5. Three samples a hundredth of the height or so apart make a dip with a bottom at 0.16, and every boundary on the map was drawn through samples spaced at least that far.

The prediction was that boxes at 0.38, 0.39, 0.41 and 0.42 units trace one smooth dip with its bottom between 0.39 and 0.41, every box from 0.375 to 0.425 losing. If a box inside that range gained, the pocket would not be one dip but a pattern set by how the faces are divided, and every boundary drawn at six strips a side would be a boundary of the division as much as of the room.

The loss deepens, then jumps #

The change in chroma runs −0.0016 per cent at 0.375, −0.0027 at 0.38, −0.0042 at 0.39 and −0.0056 at 0.40, deepening at every step and every paint losing. At 0.41 it is +0.0010 per cent and every paint gains. At 0.42 it is +0.00004 and the paints split, fifty-nine gaining. At 0.4375 it is −0.0008 and every paint loses again.

• The first half of the prediction holds. The loss deepens smoothly from 0.375 to 0.40.
• The second half fails, by a box a hundredth of a unit past the deepest. At 0.41 every paint gains; the pocket’s other side is not a slope but a step.
• The alternative named arrives. The row changes sign three times between 0.40 and 0.5 — loss, gain, nought, loss, gain — which no single dip does.

Drawn at the scale of the pocket, the row is a slope down to 0.16 of the height, a cliff up to just above nought, a short level stretch, and a slope down again. Nothing about the shape is a bottom. A curve with a bottom would have its steepest falls at its edges and flatten towards its centre; this one flattens on the way down, 22 then 15 then 14 millionths a hundredth, and then turns up all at once. The deepest point the census has is the last box before the cliff, and the cliff is where the coarse samples had placed the pocket’s centre.

The step is three times the slope #

Per hundredth of a unit of width, the change in chroma falls by 22, 15 and 14 millionths of the floor’s chroma on the way from 0.375 to 0.40, rises by 66 from 0.40 to 0.41, and falls by 10 and then 5 per hundredth on to 0.42 and 0.4375. The one rising step is three times the steepest falling step in the pocket and opposite in sign; on either side of it the row falls at a pace that, extended, would carry it from one side of the step to the other in about four hundredths of a unit.

That pattern — a smooth trend interrupted by one jump — is what a quantity computed on a grid looks like when something in the grid changes discretely as a dimension crosses a value. The transport divides each face into six strips a side, so a floor 0.40 units wide is cut into strips 0.067 units across and a floor 0.41 wide into strips 0.068; nothing jumps there. But the finish’s lobe is integrated over a discretised cone of directions, and which patches of the opposite faces fall inside which ring of that cone changes discontinuously as the room’s proportions change. A box whose proportions carry one patch across a ring’s edge can see its exchange of light change by a step. That is a reading of the mechanism, not a computation of it: the census has not traced which ring or which patch, only that the row jumps once where a smooth quantity would not.

A second box splits the paints #

At 0.42 units the change in mean chroma is +0.00004 per cent — four hundred-thousandths of a per cent — and fifty-nine of the seventy-two paints gain. This is the second box in either census whose paints split, after the wedge’s tip at 1.04 times the height, and it splits for the same reason: its mean change is so near nought that the paints’ small departures from the mean decide their signs. A box placed anywhere the row crosses nought would split them, and the row crosses nought here twice within four hundredths of a unit.

The pale paints cross the tip last read the tip’s split as a band of crossings, each paint at its own length. The same reading applies across the width: each paint’s change, read along this row, would have its own crossings, and the split at 0.42 is the band of crossings on the far side of the step.

The finish’s brightening does not jump #

The finish brightens the floor at its darkest reflectance by 52.3 per cent at 0.375 units, 51.6 at 0.38, 50.3 at 0.39, 49.0 at 0.40, 48.5 at 0.41, 47.4 at 0.42 and 45.5 at 0.4375, falling at every step and no faster across the step in chroma than elsewhere. The brightening is the finish’s whole effect on the floor’s light; the chroma change is a small difference between how much the finish brightens the floor’s coloured part and its grey part — a few thousandths of a per cent of a quantity of about fifty per cent.

So the step is in a difference of two large, smooth quantities. Each of them may carry a small discretisation error of its own, invisible against fifty per cent; the difference is a ten-thousandth of either, and there the errors show. The slot’s loss survives every finer division found a loss of a twentieth of a per cent holding its sign from three strips a side to eight — a quantity ten times the pocket’s deepest, and large beside anything the division moved. The pocket is near the floor of what six strips a side can resolve.

The tilt of the brightening across reflectance runs −0.0107, −0.0178, −0.0270 and −0.0357 points from 0.375 to 0.40, jumps to +0.0062 at 0.41, sits at −0.00004 at 0.42 and returns to −0.0050 at 0.4375. The tilt has carried the chroma change’s sign in every box either census has solved, and it carries this step too: whatever jumps, it jumps in how the finish’s brightening is spread across reflectance, not in its size.

The whole row, at its own scale #

Across the whole row the change runs +0.034 per cent at 0.12 of the height, −0.0016 at 0.15, −0.0056 at 0.16, +0.0010 at 0.164, −0.0008 at 0.175 and +0.021 at 0.2. At the scale of the row, from 0.12 to 0.2, the pocket is a narrow notch below nought in a curve that otherwise sits well above it, and the step is a small feature in the notch. The coarse readings were not wrong about the pocket’s existence or its depth; they were wrong to call its centre a bottom.

That changes what the earlier boundaries mean. Every crossing drawn on the map of proportions — the window’s edges, the tip — was a straight line between samples, assuming a smooth curve between them. Where the quantity crossing nought is a few thousandths of a per cent and the grid can move it by a fraction of that, the crossing is a statement about the grid’s resolution as much as about the room.

What the earlier readings of this row stand on #

The row along the floor as long as the room is tall has been read four times before this, each time from samples further apart than the step. The window of loss closes from both ends solved it at 0.3, 0.4 and 0.5 units and drew the window’s edges by straight lines between them, the only edges three samples allow; a taller room keeps its floor losing in proportion added the four-tall room’s boxes, which put two more points on the row; a room scaled whole loses its floor to the last digit read them together as one row and placed its losing stretch between 0.148 and 0.176 of the height.

None of those readings is contradicted by the step; each is refined by it. The losing stretch does begin near 0.148 of the height — the box at 0.375 units, 0.15 of the height, already loses, and the census’s earlier gain at 0.3 lies well below it. But the losing stretch is not one interval. It is at least two: 0.148 to just past 0.16, and again around 0.175, separated by a gain at 0.164 that no reading spaced a hundredth of the height apart could have seen. The upper edge the scaled reading placed at 0.176 is the upper edge of the second stretch.

That matters for what the wedge’s tip measured. The wedge of loss comes to a point read the tip along 0.16 of the height — the box at 0.4 units, the last before the step — and found it at 1.040 times the height. If the step at 0.405 moves with the floor’s length, as a feature of the discretisation would, the row the tip was read along runs beside it at every length, and the tip’s position is a property of a row that sits on the edge of a jump. The tip’s own readings were smooth along the length — 0.0056, 0.0054 and 0.0069 per cent per tenth of a unit, steady through the crossing — so the step does not cross that row between 1.0 and 1.12; but the tip is the tip of the stretch before the step, and the stretch after it may close at a different length.

The tall room gains once its walls are divided is the census’s reminder that a verdict near nought can depend on the division: a tall room’s verdict there reversed between one patch a face and six. The pocket’s step is the same lesson at a finer scale, found by sampling the room’s proportions rather than its division. Both say that below a few thousandths of a per cent, the census’s map carries the transport’s fingerprints. Above that level — the finish’s brightening of fifty per cent, its tilt, the gains of a few hundredths of a per cent across most of the map — nothing in either reading suggests the division matters, and the conclusions drawn there stand as they were.

What a lighting designer should take from this #

The pocket of loss is real at this resolution and too small to see at any. A narrow room whose floor is as long as the room is tall and a sixth of it wide loses at most a few thousandths of a per cent of the floor’s chroma to a satin finish; whether a particular box loses or gains depends on its proportions to a hundredth.

Nothing visible depends on this. The finish’s brightening — about half again the floor’s light at its darkest reflectance — falls smoothly with the room’s proportions and is the effect a viewer sees.

The map’s fine structure belongs to the model. The wedge’s broad shape has held across every reading; its edges at the scale of a hundredth of the height are set partly by how the transport is discretised.

How the boxes were solved #

The box is 2.5 units tall with its floor 2.5 units long and 0.38, 0.39, 0.41 and 0.42 wide, painted with each of the census’s seventy-two paints in turn, its walls and ceiling grey and 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 at a set of reflectance nodes and interpolated to each paint. The change in chroma is the mean over the paints of the satin floor’s CIELAB chroma over the matt floor’s, less one; the brightening and tilt are read from the ratio of the two floors’ exitances at the darkest and lightest reflectances. The four-tall room’s boxes at 0.6 and 0.7 are this room’s at 0.375 and 0.4375, since the transport carries no length.

What this does not reach #

The step’s cause. The census reads that the row jumps between 0.40 and 0.41; it does not trace which part of the transport’s discretisation jumps.

Finer divisions. Every box here is six strips a side; whether the step moves, shrinks or vanishes at eight is the question that decides whether it is the room’s or the solver’s.

The other edge. The row’s return to loss at 0.4375 and its gain at 0.5 are sampled once each; whether a second step lies between them is unread.

Still open: whether the step survives a finer division #

The row along a sixth of the height falls smoothly to 0.40 and jumps to a gain at 0.41, while the brightening it is a small difference of falls smoothly through. If the step belongs to the transport’s discretisation, a finer division will move it or remove it; if it belongs to the room, it will stay.

The calculation is the boxes at 0.40 and 0.41 units solved again at eight strips a side, with the box at 0.39 for the slope. The prediction is that at eight strips the step shrinks to under half its size at six and the box at 0.41 loses, so that the pocket at the finer division is one dip again with its bottom near 0.40. If the box at 0.41 still gains at eight strips, the step is the room’s, and the wedge’s boundary near its centre has a structure at a hundredth of the height that no coarser map could show.

A bottom that was the edge of a step #

The habit here is about what a minimum of a few samples is.

Three samples a hundredth of the height apart made a dip, and the middle one became the pocket’s bottom. Four more samples between and beside them showed the middle sample sitting at the top of a step rather than the bottom of a curve: the loss deepened right up to it and turned to a gain a hundredth past it. The coarse samples had found the deepest point of what they sampled, which happened to be the last point before a jump.

The failure mode is to read the lowest of a few samples as the bottom of a smooth curve, when nothing guarantees the quantity is smooth at the scale of the spacing. The test is to sample between the lowest point and its neighbours and read whether the shape that appears is the one the coarse samples implied.