Colour Science & Perception
what the eye settles, and what it cannot
Concept

Convergence — where it appears

Whether an estimate approaches a fixed value as its budget is increased, and how fast. It is what separates a quantity that has been measured from one that has merely been computed: an estimator whose answer keeps moving as the sample grows has not found anything, and the rate it settles at identifies which error dominates.

Named by 37 essays across 8 fields — each of them below, with the objects they name alongside it.

The points a ratio needs are proportional to the ratio. A scatter of 133 points on logarithmic axes, one per MacAdam ellipse under each of six coordinate systems. The horizontal position is that ellipse's true axis ratio; the vertical is the smallest sample size, from a sequence of doublings, at which the sampled ratio comes within one per cent and stays there. A line of slope 0.94 runs through them, against a predicted 1 — the minimum's notch is 0.88 σ₂/σ₁ radians wide, so resolving it takes a number of points proportional to σ₁/σ₂, and nothing about the basis or the ellipse enters beyond that. An ellipse with a ratio of two needs seventeen points and one with a ratio of twenty-six needs a hundred and ninety-two.

An extremum is not a sample

Three separate measurements in this collection took a maximum or a minimum over a sample of a set — forty-eight points round an ellipse, twenty-four directions out of an optimum, fourteen changes of light off a list. All three are wrong, all three are wrong in the same direction, and the error in each grows with the very quantity being measured.

limits · Limits Three numbers for one set of ellipses, and which of them is which. Two curves and a horizontal line, against the size the ellipses are drawn at. The line is the analytic axis ratio — the ratio of the singular values of the map's own derivative, which is what "does this space make discrimination contours circles" means. The upper curve is a very finely sampled ring, which sits 0.6 per cent above the line at full size and converges onto it as the ellipse shrinks, because the gap between them is the second-order distortion of the map across a real ellipse rather than an error. The lower curve is the forty-eight-point sample used for this until now: it does not converge onto anything, because its error is set by the sample and not by the size.

Three numbers for one ellipse

How far a colour space is from making a discrimination contour circular has three different answers — what a coarse sample of the boundary reports, what a converged sample of a contour of stated size reports, and what the map's own derivative says. They differ by up to a factor of two, they mean different things, and only one of them is what the question is asking.

difference · Metric Every width at the wide end of its span, and at the narrow end. One line per published quantity, each spanning the value it takes when all four declared widths are read at the narrow end of their reported ranges to the value at the wide end, with a marker at the value as declared. The largest span is the deutan margin at a factor of 2.62; the smallest is 1.22. This is the reading the population model's own documentation promised for four phases and nothing ever took. It is not a confidence interval — the four ends are not quantiles and the widths are not independent draws — it is what a reader who distrusts all four at once sees.

A width nobody varied

Five numbers say how much people differ from one another, and every conclusion drawn here about a population rests on them. Each was written down with the range the literature reports beside it, so that a result could be re-read at the pessimistic end. Nothing ever was.

eye · Cones How wrong the ellipses would have to be for a pair to change places. One bar per adjacent pair in the uniformity table: the relative error on each ellipse's own axes at which that pair changes places in one draw in twenty. No error on the data is quoted anywhere — the question is inverted, so what is reported is how large an error would have to be, and a reader with an opinion about MacAdam's experiment can compare it with their own number. The nearest pair goes at 0.171; 2 of the 7 pairs do not reverse under any error this search covers.

How wrong would the data have to be

Twenty-five ellipses measured on one observer in 1942 are the ruler every colour space here is judged against, and they have never been given an error. Rather than invent one, the question is turned round, and asks how large an error would have to be before the ranking changed.

difference · Metric Twenty-five ellipses is a sample, and the score has an error bar. One row per colour space this collection ranks: the mean axis ratio its ellipses come out at, with the standard error of that mean over the twenty-five ellipses it was computed from. No literature is quoted — a mean of twenty-five numbers has a standard error those twenty-five numbers determine. The bars are far from equal: the best space carries ± 0.07 and the worst ± 1.56, because a space that makes the ellipses nearly circular makes all of them nearly circular and one that does not is dominated by whichever ellipse it handles worst.

Twenty-five is a sample of the diagram

A colour space's uniformity score is the mean of twenty-five numbers, and a mean of twenty-five numbers has a standard error those twenty-five numbers determine. Nothing has to be quoted to compute it, and three of the seven adjacent pairs in this collection's ranking survive it.

difference · Metric A quadratic is believed least far at the one place anybody takes one. One bar per basis: the radius, in the nine coefficients, within which the second-order model predicts the objective to within ten per cent in every one of eighteen directions. The shortest bar is the objective's own optimum, at 2.3×10⁻², and the longest is XYZ scaling at 1.1×10⁻¹ — several times further. The reason is not that the model is worse at a minimum but that it has less to do there: away from one the linear term is exact and carries most of the change, so a ten per cent error in the prediction takes longer to accumulate. It does not make a Hessian at a minimum wrong; it says the picture drawn from it describes the smallest neighbourhood in the table.

How far a quadratic can be believed

A second-order model has a radius inside which it describes a surface and outside which it does not, and that radius can be measured. Measured at eight places on one objective, it is smallest at the optimum — the one place anybody ever takes a Hessian.

limits · Limits Room is not safety: two orderings of the same three claims. Three pairs of bars, one pair per published statement about the confusion points. The upper bar in each pair is the margin — how far the measured number is from the threshold that makes the statement true, as a ratio. The lower bar is the headroom — the factor by which one declared width of the population model would have to be wrong for the statement to fail. Both start at one, which is the line. Ordered by margin the three read the protan margin, the tritan margin, the deutan margin; ordered by headroom they read the protan margin, the deutan margin, the tritan margin, and the middle two change places. Every one of the three is inside a factor of two of failing, which the margins do not say.

What would have to be wrong

A great many statements here have thresholds written into them, which turns out to make an audit possible — for each one, the smallest change in a declared input that would stop it holding. Most are unreachable. One is inside a factor of one and a third.

limits · Limits Every census row under five constructions of the same test set. A slope chart with 5 columns — lattice, coarse, fine, uniform, natural — and one line per change of light in the census, each line joining that row's mean residual under each construction. Four of the five columns describe the same region of surfaces walked at different densities or against different measures; the last is the clamped, realistic family, which is not linear in its parameters and is therefore answering a slightly different question. The levels move: between the coarse and fine lattices every row shifts by seven to nine per cent, in the same direction, which is a common-mode factor no published residual here has ever carried. The order almost survives. Inside the region exactly one pair crosses, and it is the pair the standard error had already flagged; under the clamped set two more cross, including one the error separates by nearly nine standard errors. The crossing lines are drawn heavy.

A lattice is a quadrature rule

Walking a set of test surfaces more finely does not converge on a better answer, because refining a lattice under a constraint changes which corners of the region get sampled and not only how densely. The lattice used here turns out to be a two per cent biased estimate of the integral it stands for.

difference · Metric A published residual is a mean, and the worst object in the room costs twice it. Three bars for each of the 14 changes of light in the adaptation census, ordered by how uneven the change is across surfaces. The first bar is the published mean residual. The second is the worst single surface in the audit's published test set. The third is the worst surface anywhere in the region that set is drawn from, found by search rather than by reading a maximum off a lattice. The mean-to-worst ratio runs from 1.90 to 4.02 and averages 2.43, so every published adaptation number has a worst case about twice it that no essay had ever quoted. The gap between the second and third bars is the other finding: a maximum over 125 sampled points understates the region's own maximum by up to 34 per cent.

An extremum is still not a sample

Two rounds ago three measurements turned up that took a maximum over a sample of a set and were short by up to a factor of two. The same error was live in a fourth place the whole time, on the set of surfaces every adaptation number is averaged over, and it is short by up to a third.

matching · Gamut What three fill-in rules cost when a five-nanometre table is read at one. A five-nanometre table read at one nanometre by three rules — hold the value, straight lines, a cubic through four points — each compared with the same tenth-nanometre reference. The dashed rule is the answer obtained by summing the table as it stands, at 0.00000 ΔE₀₀. Two of the three interpolations are worse than not interpolating. That is not a paradox: the summation's error is already small because the normaliser cancels most of it, and an interpolator introduces a shape the original curve did not have, which the cancellation cannot touch. A finer grid is not more resolution when the table is not finer.

A finer reading of a coarser table

Interpolating a five-nanometre spectrum to one nanometre helps a daylight calculation by a factor of five and harms a three-emitter LED by a factor of a hundred and twenty thousand. Both are the same operation on the same table, and which one happens is decided by a property of the light nobody records.

light · Light How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.

Where the grid starts

Holding the step at five nanometres and sliding the grid's origin through one cell moves a fluorescent tube's computed colour by 3.18 ΔE₀₀ and a laser projector's by 35.0. Refining the step does not fix it and averaging over origins hides it. It is the one tabulation fault with no smooth error to cancel against.

light · Light The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.

The endpoint term has a name

The five-nanometre error on a smooth light falls linearly with the step, which is not what a sampling error does. It is the half-cell at each end of a truncated range, it is first order where the sampling is second, and halving two weights removes fourteen fifteenths of it for nothing.

light · Light How far this collection's analytic observer is from the tabulated one. The construction every figure in this family uses is three pigment absorptances through one fitted 3×3, and this is the residual of that fit against the CIE's 1931 functions: root-mean-square error as a percentage of each curve's peak, and the colour difference it produces over forty-two surfaces. The short-wavelength function is the worst at 16.4 per cent, which is where a pigment template is weakest and where the ocular media are doing most of the work. The median colour difference is 1.42 ΔE₀₀, so this is an observer of the right shape rather than a copy of the table — and every departure in this family should be read beside that number rather than against zero.

The reference had to be built

A five-nanometre error cannot be measured with five-nanometre data. Interpolating the tables and integrating finely measures the interpolator, not the grid — so the audit of this collection's index had to be run against an observer made of formulae, and the price of that is a residual of 1.42 ΔE₀₀ that every number in the section is read beside.

limits · Limits The directional solver reduces to the radiosity solver exactly. A solver with a new unknown in it is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ/π collapses all thirty ordered-pair radiances onto their patch's radiosity divided by π, and the answer agrees with this collection's existing radiosity solution to 9.8e-16 relative — the floating-point floor. That is the check that makes every other number in this family a statement about lobes rather than about a new piece of arithmetic, and it is the reason the reduction is drawn rather than mentioned.

Thirty unknowns instead of six

A directional transport solver is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ over π collapses thirty ordered-pair radiances onto six radiosities and reproduces this collection's existing answer to 9.8 × 10⁻¹⁶ relative — which is the only reason anything else it says can be believed.

scene · Scene Where a point-sampled patch stops resolving a lobe. The horizontal axis is the wall's roughness, logarithmic; the vertical is how far the answer moves when the cone quadrature is refined from eight directions to sixteen, also logarithmic. A patch in a cube subtends a cone of angular radius 25.8° at the face opposite, and a microfacet lobe of roughness a is about a radians wide, so a lobe below about 0.15 is narrower than the quadrature that samples it. The movement at 0.05 is 21.2 ΔE₀₀ and at 0.3 it is 0.033. This is where the method stops, not where the paint does: matt, eggshell and satin finishes are inside it and a high-gloss varnish is not.

Where a patch stops being a point

A patch in a cube subtends a cone of 25.8° at the face opposite. A microfacet lobe of roughness 0.05 is about three degrees wide. Point-sampling the second inside the first returned a chroma of thirty-four thousand and a negative lightness, and the boundary between working and not working is measured rather than declared.

limits · Limits Which of the collection's published quantities a departure can be pushed through. The six quantities the previous round recomputed under six different colour-difference units, and whether the same treatment works for a departure. Two do: the adaptation census and the metameric pair both take reflectances and a light, which is what a departure acts on. Four do not, and the reasons are different in each case rather than a single obstacle. A unit is a function applied to the answers, so it can be swapped at the end of any computation; a departure changes the object at the start, so it has to be accepted by every stage in between. That is the practical difference between auditing a convention and auditing a structure.

What this round could not reach

Three audits, about twenty departures and ten conditions, and a longer list of things that were named and not measured. Every item on it is specific, most of them are an afternoon's work, and the reason none was done is the same in every case — the round ran out of round.

limits · Limits The straight piece under the cube root, and where it stops. CIELAB's lightness against relative luminance, over the bottom 5.0 per cent of the range. Below Y = 0.008856 it is a straight line of slope 7.787; above it, a cube root. The two meet at L* 8 in value and in slope, exactly — the CIE's two constants are chosen to make that true. The dashed curve is the pure cube root, which reaches negative lightness before it reaches zero luminance and has an infinite slope there. The break is marked, and the axis runs to Y = 0.050.

The straight piece under the cube root

CIELAB's lightness is described everywhere as a cube root and below a stated luminance it is a straight line, spliced on with two constants chosen so the join is exact in value and in slope. Every black a delivery chain reaches is inside that straight piece — a press black at L* 2.4, a projected black at 1.1 — where the compression does not compress, the price of a deviation is flat to two parts in a thousand, and the second derivative the composition of two departures needs does not exist.

matching · Gamut Where averaging the model's answers is and is not averaging its argument. Each row takes a spread of situations, averages the model's predictions across them, and compares that against the model's prediction for the average situation. The bar is the gap as a share of the spread itself. Over the differences between observers it is 1.4 per cent — the model is very nearly linear there. Over the range of adapting luminance one room covers in a day it is 59 per cent, and from indoors to outdoors 74.

Where the model's curve does not matter

This round has been about what a nonlinearity does to an average, and CIECAM16 is the most nonlinear thing in the collection. Over the spread a population of observers produces, the average of its predictions is its prediction for the average to within 1.4 per cent — the nonlinearity is there and the excursion is too small to reach it. Over the range of adapting luminance one room covers in a day, the same gap is 59 per cent of the spread, and from indoors to outdoors 74.

brain · Appearance Out to the device and back, once and again. Each intent applied 3 times in succession to the same ramp. The upper bar is what the first pass moves and the lower is what the second moves. relative-colorimetric is idempotent — the second pass moves nothing, to the floating-point floor — and the other two are not: perceptual moves another 2.73 and saturation moves another 3.09. A file converted twice is not a file converted once.

A second conversion is not a repeat

A colour converted to a device and back is described as lossy where the device cannot hold it and exact where it can, which suggests that a colour surviving one round trip survives any number. One of the four intents is idempotent to the floating-point floor. The other two are not — a perceptual conversion moves a ramp 3.70 colour differences on its first pass and another 2.73 on its second, and after three passes it is still moving by 2.42.

applied · Delivery The length of one grey ramp, cut into more and more steps. A neutral ramp from L 1 to L 100 cut into between one and ten thousand equal steps, each step measured and the steps added, in four units, both axes logarithmic. ΔE*ab and the model's Euclidean J′a′b′ give the same length at every step count, 99 and 96. ΔE₀₀ settles at 74.6 once the steps are small. The power-corrected ΔE′ does not settle: 25 in one step, 137 in a hundred, 755 in ten thousand, growing as the number of steps to the power 0.37.

A distance raised to a power has no length

CAM16-UCS's colour difference is its Euclidean distance raised to the power 0.63 and multiplied by 1.41. That is still a metric — the triangle inequality holds on every one of four thousand random triples — and it has no length. A grey ramp from black to white measures 25 units in one step, 137 in a hundred and 755 in ten thousand, growing as the number of steps to the power 0.37, and halving the size of a step triples the number of steps that fit.

difference · Metric The glossiest finish the solver can report, against what it costs to report it. Six quadratures, each with the roughness at which its answer stops being stable, on logarithmic axes. The line is a fit and its slope is -0.350: the reachable roughness falls as the cost to the power of about a third, so reaching a finish twice as glossy costs about 7 times the work. The solver used here sits at 108 directions and reports down to a roughness of 0.145, which is where its own note put the boundary by inspection.

The boundary belongs to the quadrature

The directional solver stops at a roughness of about 0.15, and below that its answers are not imprecise but unphysical. The boundary is where the lobe stops being resolved by the sampling, so it belongs to the discretisation rather than to the room — and moving it is a purchase. Measured across six quadratures the reachable roughness falls as the cost to the power of a third, so a finish twice as glossy costs seven times the work and a polished varnish costs two hundred and thirty-six times.

scene · Scene The same gradients, priced by a penalty and by a projection. Each row is one gradient held inside a coated press from six starts. The pale dots are the penalised relaxation — a free step, with a price for leaving the press — and the dark ones are the projected relaxation, which takes the free step and then moves each point to the nearest printable colour. Across is how much longer than the free path each result is, logarithmic. On the 10 gradients whose straight line leaves the press, the penalty leaves 9 starts at more than twice the free length and the projection leaves none. The projection's best route costs a median 0.02% against the penalty's 0.13%, and its spread across starts is 0.95% against 132.5%.

A projection has no reason to detour

Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.

matching · Gamut Divide the faces and the tall room gains colour from a grey finish. How many of the census's 72 paints a satin finish on the grey faces makes more colourful, with every face divided into one to six patches a side. One patch a face is the census, where the tall room with its side walls painted gains for 0; from one to six patches a side it gains for 0, 4, 8, 11, 15, 15. The tall narrow slot, painted the same way, goes 22, 7, 11, 45, 72, 72, and the tall room with all four walls painted barely moves.

The tall room gains once its walls are divided

The room census solves each face as one patch, and it says a satin finish on the grey faces of a tall room with painted side walls makes the room more colourful for none of seventy-two paints. The true box disagreed: rays from points on the grey end walls said the finish sends the lamp's light towards the paint. Dividing every face into patches settles it. The end wall's upper patches, which see the lamp at a grazing slant, send its light across to the painted walls; its lower patches send it to the floor. The room gains for 0, 4, 8, 11, 15 and 15 paints as the faces are divided one to six times a side — the light, broad paints — and the narrow slot beside it goes from 22 to all 72.

scene · Scene The painted census's counts with one patch a face and with the faces divided. For each of the thirty rooms — six boxes, each painted five ways — how many of 72 paints a satin finish on the grey faces makes more colourful with one patch a face, as the census solved it, and with every face divided five times a side (six for the two tall boxes). 12 counts do not move; 9 move by twenty paints or more, shaded — darker where the divided room gains more. Some paint gains in 28 rooms both ways, and every paint in 13 with one patch and 17 divided.

Divided faces keep the census's count and lose its rule

The painted-room census, solved with one patch a face, found that a satin finish on a room's grey faces adds colour in 28 of 30 rooms, and a later essay found one geometric number whose sign sorted which rooms gain. Re-solved with every face divided into twenty-five or thirty-six patches, the count survives exactly — 28 rooms, though two of them trade places — and the finish colours every paint in 17 rooms instead of 13. The rule does not: the send number, now matching the true box's own to a rank correlation of 0.96, orders the divided counts at 0.63, and rooms on the wrong side of its zero gain for every paint. Nine of the thirty counts move by twenty paints or more.

scene · Scene Counting the paint's light instead of the lamp's orders the rooms not at all. The thirty rooms of the painted census on divided faces, each placed twice: by the send number weighted by the paint's own light arriving at the grey faces, and by the send number weighted by the lamp's light, against how many of 72 paints a grey finish colours. The lamp-lit number ranks the wall and floor rooms at 0.63; the paint-lit one at -0.07, and over all thirty at -0.02.

The paint's own light does not order the rooms

A satin finish on a room's grey faces colours some paint in 28 of the painted census's 30 rooms, and the number meant to say which rooms gain most — how much of the lamp's light the finish sends to the paint — orders the rooms solved on divided faces at only 0.63. The rooms it misplaced return mostly the paint's own light, so the proposal was to count that instead. Counted, the paint's light orders the rooms at −0.07. In every room with its floor painted the finish sends the floor's light away from the floor, and every paint gains; with the ceiling painted the paint is the lamp and the two numbers are the same number. Even the room's own response, read off the solve with the paint in place, orders the counts only at 0.71. The census has to be solved room by room.

scene · Scene The slot's paint gains at a coarse division and loses at a fine one. In the tall narrow slot — one unit wide, two and a half tall, 0.35 deep, its ceiling the lamp — with its floor painted and with its ceiling painted, the mean change over seventy-two paints in the chroma of the painted face's light when a satin finish is laid on the grey faces, solved with every face divided into one to eight strips a side. Undivided the paint gains 0.28 per cent (off the scale, clipped); at two and three strips it still gains, barely; from four strips it loses, and the loss grows at each finer division, to -0.055 and -0.059 per cent at eight.

The slot's loss survives every finer division

Read on the painted faces alone, a satin finish on a room's grey faces makes the paint more colourful in twenty-eight of the painted census's thirty rooms and less colourful in two: the tall narrow slot with its floor painted and with its ceiling painted, where the paint loses about a twentieth of a per cent of its chroma. A loss that small could belong to how finely the slot's faces were divided, and the prediction was that it does not — that the sign holds at seven and eight strips a side. It holds. Solved from one strip a side to eight, the slot's paint gains at one, two and three strips and loses from four on, every one of seventy-two paints crossing at the same division; the loss deepens at every finer division, to −0.055 and −0.059 per cent at eight strips, and run out to no strip width it settles near −0.058 and −0.065. The coarse census had the sign wrong; the census at six strips had it right. The bright, broad paints lose most, as they gained most everywhere else.

scene · Scene The slot's loss turns to a gain about six tenths of the way deep. A box one unit wide and two and a half tall, lit by its whole ceiling, with its floor painted and with its ceiling painted, solved at eight depths from the slot's 0.35 to the tall room's full unit at four strips a side. The mean change over seventy-two paints in the painted face's chroma when a satin finish is laid on the grey faces: a loss of 0.031 per cent at 0.35, shrinking through 0.45, 0.55 and 0.6, a gain from 0.65 that grows to 0.052 per cent at a full unit. The floor turns at a depth of 0.62 and the ceiling at 0.63.

The finish stops costing the paint six tenths deep

In the tall narrow slot a satin finish on the grey faces costs a painted floor or ceiling a little of its chroma; in the tall room, the same box a full unit deep, it adds some. Solved at eight depths between, the sign turns at 0.62 of the box's width with the floor painted and at 0.63 with the ceiling — later than the prediction of 0.45 to 0.55, and far enough along that the loss belongs to a range of shallow boxes rather than to the slot alone. A finer division moves the turn a little deeper, not back. Every one of seventy-two paints turns at the same depth, and each paint's share of the change is the same at every depth, loss or gain. Brightness does not decide it: the finish brightens the painted floor by between half and more than double while hardly touching the painted ceiling, yet the two turn together. What decides it is which reflectance the finish favours — whether it multiplies the light that has been back to the paint by more or by less than the light that has never touched it.

scene · Scene A taller box turns deeper. Boxes one unit wide and 1.5, 2.5 and 4 tall, lit by the whole ceiling, floor painted, at depths from 0.2 to 2: the mean change over seventy-two paints in the painted floor's chroma when a satin finish is laid on the grey faces. The 1.5-tall box gains at every depth from 0.35, turning near 0.26; the 2.5-tall box turns at 0.62; the 4-tall box loses at every depth to 1.5 and turns at 1.67, deeper than it is wide.

A taller box must be deeper before the finish adds colour

In a box one unit wide and two and a half tall, a satin finish on the grey faces costs a painted floor a little of its chroma when the box is shallower than 0.62 of its width and adds some when it is deeper. Whether that turn is set by the width or by the height was the open question, and the prediction was that it scales with the height, because the satin walls pipe the lamp down to the floor and a taller box is a longer pipe. It does, and faster than in proportion. A box 1.5 tall gains at every depth from 0.35 and turns shallower than that; a box 4 tall loses at every depth to half again its width and turns near 1.67. As a share of the height the turn rises from about 0.17 to 0.25 to 0.42 — close to a depth growing as the square of the height. The taller box pipes far more of the lamp to the floor, tripling its light at 0.35 deep, and at every depth of every box the paint's colour still follows which reflectance the finish favours. Six strips a side keep both new boxes' signs.

scene · Scene Where in the plan a satin surround takes colour, at six strips. Every box 2.5 tall solved at six strips a side with its floor painted, placed by its floor's width and depth and again turned a quarter, since only the floor is painted and the turned box is the same box. The finish takes colour from the paint wherever the floor's shorter side is under about half a unit — the half-unit box at 1.25, 1.5 and, by its twin, 2 deep, the two-unit box at 0.35 and 0.5 — and from the one-unit box at 0.6; it adds colour once the shorter side passes 0.65, and at the two-unit box a quarter deep.

The width moves the turn only when it is the narrower side

In a box one unit wide and two and a half tall, a satin finish on the grey faces costs a painted floor a little chroma when the box is shallower than about 0.64 and adds some when it is deeper. The prediction was that the width hardly matters — that the two end walls a depth apart form the pipe that carries the lamp down, and wider walls only add more of it — so the turn would move by under a tenth between widths. Solved at six strips a side, a box two units wide turns between 0.5 and 0.65, within the tenth or so predicted. A box half a unit wide still loses at 1.5 deep, and its quarter-turned twin says it still loses at 2: its turn is more than a unit and a quarter deeper. With only the floor painted a box w wide and d deep is the box d wide and w deep, so depth and width are one question, and the answer is that the floor's shorter side decides: under about half a unit the finish greys the paint, past about 0.65 it colours it. The scan's first pass, at four strips, said otherwise, and got four of seven signs wrong in the long, thin boxes.

scene · Scene A square floor turns later than an oblong one. Boxes 2.5 tall solved at six strips a side with the floor painted and a satin finish on the grey faces: the mean change over seventy-two paints in the floor's chroma, against the floor's shorter side, for square floors and for floors whose longer side is one unit and two. The square floor loses at every side to 0.7 and gains from 0.8, turning near 0.78; the floor one unit long turns at 0.64 and the floor two long at 0.51.

A square floor needs more room than a long one

A satin finish on a room's grey faces costs its painted floor a little chroma when the room is narrow and adds some when it is wide, and in a box two and a half tall the turn was found on the floor's shorter side — 0.64 when the longer side is one unit, about 0.51 when it is two. The prediction for a square floor, with no longer side to speak of, was a later turn, near 0.7, because both pairs of walls are then equally close and both carry the lamp down. Solved at six strips a side, the square floor loses for every one of seventy-two paints at 0.5, 0.6 and 0.7 on a side and gains for every one from 0.8, turning near 0.78. A floor 0.7 by 0.7 loses where a floor 0.7 by one gains. The finish brightens the square's darkest floor by more than an oblong's at every shorter side — by a hundred per cent against forty-six at half a unit — and the turn falls steadily as the longer side grows: 0.78, 0.64, 0.51. The longer side matters, and the light-pipe account says why.

scene · Scene A floor three units long never loses its colour to the finish. Boxes 2.5 tall solved at six strips a side, floor painted, satin on the grey faces: the mean change over seventy-two paints in the floor's chroma against the floor's shorter side, for a square floor and floors two, three and four units long. The square loses to 0.7 on a side and the floor two long between 0.35 and 0.5; the floors three and four long gain at every shorter side from 0.35, by 0.026% to 0.151%.

A long floor never loses its colour to the finish

A satin finish on a room's grey faces costs its painted floor a little chroma when the room is narrow, and the shorter side at which the cost turns into a gain fell as the floor grew longer — 0.78 for a square, 0.64 for a floor one unit long, 0.51 for two. The prediction was that it settles near 0.45 by four units, the limit of a slot whose far walls pipe nothing. Solved at six strips a side, floors three and four units long gain chroma for every one of seventy-two paints at every shorter side from 0.35 to 0.65: there is no turn to find. The floor two units long already loses only in a window, shorter sides from about 0.3 to just past 0.5, and gains on either side of it; by three units the window has closed, and a four-unit floor gains more than a three-unit one everywhere. The loss was never the slot's. It belonged to the far pair of walls, and moving them away removes it.

scene · Scene Every box's verdict, in length and width. Every solved box with its floor two to four units long, placed by its shorter and longer sides and marked by its verdict: all seventy-two paints gaining chroma under the satin finish, or all losing. The losses form a wedge that narrows from 0.35 to 0.5 at two units to 0.4 alone at two and a half, and is gone by three.

The window of loss closes from both ends

A satin finish on a room's grey faces costs a painted floor a little chroma only in narrow rooms near square, and a floor two units long loses it only in a window of shorter sides, from about 0.30 to 0.51, while a floor three units long never does. The prediction for the floor two and a half units long was that the window closes from below: a gain at a shorter side of 0.3 and a loss only near 0.4 to 0.5, because the lower edge is where the far walls' pipe is weakest. The floor gains at 0.3, loses at 0.4 — by under six thousandths of a per cent — and gains again at 0.5. The window closes from both ends at once, to 0.39–0.42, and its deepest point rises through nought near 2.6 units long. Every box is still all or nothing, every one of seventy-two paints with the same verdict, and every verdict has the sign of the finish's tilt across the floor's reflectance. The far walls' cost shrinks smoothly with their distance and centres on a shorter side of about 0.4.

scene · Scene Scaled by the height, the two rooms close their windows together. The deepest change in the painted floor's chroma over each length's solved shorter sides, against the floor's length over the room's height, for boxes 2.5 and 4 units tall. The 2.5-tall box's dip crosses nought at 2.59 units, 1.04 times its height; the 4-tall box's at 4.04, 1.01 times.

A taller room keeps its floor losing in proportion

A satin finish on a room's grey faces nearly always adds a little colour to a painted floor, and in a box 2.5 units tall it takes some away only in a pocket of proportions: floors shorter than about 2.6 units, with a shorter side near 0.4. The prediction was that the pocket scales with the room's height — that a box four units tall keeps losing until its floor is about 4.2 units long, 1.04 times its height as in the lower box. It loses until 4.04 units, 1.01 times its height: the crossing scales, a little short of the prediction. The pocket moves too. At a shorter side of 0.4, where the lower box lost, the taller box's floor gains at every length; its losses sit at 0.6 — at three and four units long — and at 0.8 when three units long, and its deepest loss lies near a sixth of its height, as the lower box's did. Every box is still all or nothing across seventy-two paints, and every box's change carries the sign of its finish's tilt across reflectance.

scene · Scene Two rooms' floors as long as the room is tall, read as one row. The mean change in the painted floor's chroma over seventy-two paints under a satin finish, against the floor's shorter side as a share of the room's height, for the floor four units long in the box four units tall and two and a half long in the box 2.5 tall. Where the two rooms were solved at the same proportions their points coincide. Read together the floor loses from 0.148 to 0.176 of the height, the dashed lines; the deepest loss is 0.0056% at 0.16.

A room scaled whole loses its floor to the last digit

A satin finish on a room's grey faces costs a painted floor a little of its chroma only in a pocket of proportions, and a box four units tall was found losing in proportion to a box 2.5 tall — its crossing at 1.01 times its height against the lower box's 1.04, its pocket near a sixth of the height in both. The prediction was that the answer is the room's proportions alone: that the four-tall box built as the 2.5-tall box multiplied by 1.6 gives the lower box's loss to the solver's precision, and that the floor four units long loses from about 0.55 to 0.65 wide. The scaled box gives the lower box's numbers to eleven significant figures, and a pair the census had already solved agrees to twelve. The window sits higher than predicted at both edges, from 0.59 to 0.70. And the gap between the two rooms' crossings was the census's sampling: read at the pocket, the taller room crosses at 1.029 times its height against 1.036.

scene · Scene The wedge of loss comes to a point at 1.04 times the height. Every box of the 2.5-tall room near the wedge, by its floor's sides as shares of the height, marked by whether every one of the seventy-two paints gains, every one loses, or they split, with the crossing along the row at 0.16 of the height. The floors 1.08 and 1.12 times the height long gain at 0.15, 0.16 and 0.17; the floor 1.04 times as long gains at 0.15 and 0.17 and splits at 0.16, 55 paints gaining; the row crosses nought at 1.0399.

The wedge of loss comes to a point

A satin finish on a room's grey walls and ceiling adds colour to a painted floor in almost every proportion of room, and costs it a few thousandths of a per cent of its chroma in one narrow wedge of proportions: floors about as long as the room is tall and a sixth of its height wide. The wedge was known to close between floors 1.0 and 1.2 times the height long. The prediction was that a floor 1.04 times the height long still loses at a sixth of the height and one 1.08 times as long gains at every width near it, placing the tip between them a little above the coarse crossing of 1.036. The longer floor gains everywhere, as predicted. The 1.04 floor sits on the tip itself: its change in chroma is two parts in ten million, the row crosses nought at 1.0399, and there — and in no other box of the census — the seventy-two paints part company, fifty-five gaining and seventeen losing.

scene · Scene Every paint has its own tip, and the pale ones cross last. Each of the seventy-two paints' own crossing from loss to gain along a sixth of the height, against how far its change swings between 1.0 and 1.08 times the height relative to the mean's, by band width. The crossings run from 1.0387 to 1.0413 of the height; the swings from ×0.37 to ×3.12, and the paints that swing most cross last.

The pale paints cross the tip last

A satin finish on a room's grey faces costs a painted floor its chroma only in a narrow wedge of proportions, and at the wedge's tip — a floor 1.04 times the room's height long and a sixth of it wide — the mean change is nought and, alone in the census, the seventy-two paints part company: seventeen lose, fifty-five gain. The prediction was that the losers are the paints with the narrowest reflectance bands, whose chroma depends most on how much grey light mixes in, and that they lose most below the tip and gain least above it. The opposite holds. No narrow band loses; half the broadest do, and fourteen of the seventeen have the highest peak — the pale paints. They lose most below the tip and gain most above it: they are the paints the finish moves most, up to three times the mean, and each paint has its own tip, from 1.0387 to 1.0413 times the height, the palest last.

scene · Scene The loss deepens to 0.16 and jumps to a gain a hundredth wider. The same row between 0.15 and 0.175 of the height. The loss deepens at every step from −0.0016% at 0.150 to −0.0056% at 0.160, turns to +0.0010% at 0.164, sits at +0.00004% at 0.168 and loses again, −0.00076%, at 0.175.

The pocket has a step where its bottom should be

Along a floor as long as the room is tall, a satin finish on the grey faces costs the floor a few thousandths of a per cent of its chroma in a pocket around a sixth of the height wide, deepest at 0.16 of the height by three samples a hundredth of the height apart. The prediction was that boxes between those samples trace one smooth dip with its bottom between 0.39 and 0.41 units, every box from 0.375 to 0.425 losing. The loss does deepen smoothly, at every step from 0.375 to 0.40. Then, a hundredth of a unit wider, at 0.41, every one of the seventy-two paints gains; at 0.42 they split; at 0.4375 every paint loses again. The row changes sign three times in a tenth of a unit of width, and the one step from 0.40 to 0.41 rises three times as fast as the pocket falls. The finish's brightening, which the chroma change rides on, falls smoothly straight through. What the census has called the pocket's bottom is a step.

scene · Scene

Named alongside it

The objects these essays reach for when they reach for this one.

Modelling assumptionSpecularInterreflectionRadiosityChromaSamplingQuadratureDeclared inputAnisotropyChromatic adaptationCIELABColour difference

All concepts