Chroma — where it appears
Named by 66 essays across 8 fields — each of them below, with the objects they name alongside it.
A grey edge arrives coloured
A sensor site measures one channel and the other two are interpolated from neighbours that sat somewhere else. Across a black-and-white step that reconstruction gives an achromatic scene a chroma of 144 per cent of its own local mean, and nothing in the scene or the sensor was coloured.
No mapping preserves everything
More than half the sRGB solid is outside a press's gamut, so something has to be done with it, and there are two things that can be done. One leaves every reproducible colour exactly where it was and delivers a gradient as a flat area. The other keeps the gradient and moves every colour that needed no help by about three.
How fine a colour edge can be
The eye resolves a lightness pattern to about fifty cycles per degree and a red–green one to twelve. Every colour difference here is quoted as though a patch had no size, and the same difference is plainly visible at one scale and gone at another.
A gradient is a path
Two colours fix the ends of a blend and nothing else. The middle is decided by the space the interpolation happens in, and four spaces in daily use put the halfway point of one ordinary gradient as much as thirty-seven units of colour difference apart.
A catalogue is not a vocabulary
A colour order system is a regular lattice, because a person has to be able to find a page. Sorted by what its chips would be called, that lattice comes apart into piles differing by a factor of three — and a fifth of all one-step moves in it change the name.
Three constants nobody quotes
CIEDE2000 is defined with three parametric factors in it, the CIE leaves them to the industry using the formula, and the two settings in ordinary use differ by a factor of two in one of them. Twenty-nine per cent of acceptance decisions change between the two — a larger disagreement than any between the formulae themselves.
There is no word for that colour
The naming model built this phase gives a saturated cyan the name green, calls part of a chromatic ring grey, and puts one of the eleven focal colours outside what a display can show. Each failure is a measurement rather than a disclaimer, and together they say what a vocabulary is that eleven points and a distance are not.
A brighter white still looks white
Colorimetry says a brightened sheet is nine units of b* from neutral, which sounds like a visible blue. An appearance model with the observer adapted to the room says it carries eleven units of chroma at a hue of 295 degrees — genuinely blue-violet — and that the number falls by nearly half between a bright room and a dark one. What it cannot say is why anybody calls the result white.
Saturation is nearly everything
The set of test surfaces has three numbers describing it, and only one of them matters. How saturated the surfaces are carries an elasticity of about 0.7 on every result computed over them; how bright they are carries 0.10. A test chart's chroma range decides its answer and its lightness range does not.
A chart decides what a camera scores
A camera profile's reported error changes by a factor of five when the test chart's saturation changes, with the camera and its matrix untouched. The elasticity is 0.67 — the same figure, to two per cent, that an entirely unrelated measurement over an entirely unrelated set of surfaces gives.
The weighting is the disagreement
Five colour-difference formulae, three decades and two committees, and the single property that predicts which of them agree is whether a chroma difference gets divided by the chroma it was measured at. It sorts the menu exactly, it cuts across the distinction between a matching difference and an appearance one, and it halves the census's largest sensitivity.
A dial through a discrete menu
ΔE*94 is ΔE*ab with two weighting constants in it, and at zero those constants make every weight exactly one — so the two ends of the oldest disagreement in colour difference are joined by a line rather than separated by a choice. Walking it gives a derivative where a menu gives only a spread, and the derivative says the published weighting is on the far side of the interesting part.
The disagreement is at the near end
Every colour-difference formula on the menu was fitted to threshold data, so the expectation is that they agree about pairs an observer can only just tell apart and diverge on large differences. They do the opposite. Proportionally the disagreement is largest at the near end, by a factor of six for the appearance unit, and the cause is an exponent of 0.63.
A unit rests on a space that was ranked
This collection ranks three colour spaces by how nearly they make MacAdam's ellipses circles, and CIELAB comes last. It then publishes every difference it computes in a formula built on CIELAB. Turning the same instrument on the formulae rather than the spaces shows the repair works — and that it buys roundness by paying in evenness.
A tolerance is a boundary through pairs
Twenty-four pairs of surfaces built to sit exactly on a ΔE2000 tolerance of one read from 0.69 to 1.93 in the other five units. A contract quoting "one unit" without naming the formula does not become slightly wrong in another; it accepts and rejects a different set of deliveries, and in one case rejects every single one.
The coincidence was a mechanism
Two sensitivities from two libraries with no shared code came out two per cent apart, and the claim made about them was that they share a mechanism rather than a number. That claim has a colour-difference formula inside it, so it can be tested by changing the formula — and both curves move together across the whole menu, from 0.5 to 1.15.
An aperture is a filter
A measuring aperture throws away the light that came back outside it, and the light that comes back furthest is the light at the wavelengths the sample absorbs least. So an aperture does not attenuate a translucent sample evenly — it attenuates its peaks more than its troughs, which is a filter whose transmission curve the sample itself decides.
The hue the hole decides
A piece of pale marble measured through a wide aperture is faintly yellow. Measured through a narrow one it is faintly blue, and between the two there is an aperture at which it is exactly neutral. Nothing about the stone changes; the aperture is a filter with a colour, and what decides that colour is the size of the particles rather than the pigment between them.
A departure is straight in the excitations
Walk a sample a quarter of the way from the light towards its own reflectance and exactly a quarter of the observer disagreement remains — in cone excitations, to two parts in a hundred. In ΔE₀₀ the same quarter leaves 0.347 where proportionality wants 0.428, and the discrepancy belongs entirely to the unit.
A lobe takes colour out of a bounce
A gloss wall sends more light to the floor and less colour. The extra light is a Fresnel reflection at the interface, it carries the lamp's spectrum rather than the paint's, and it arrives at the next surface white — so a green room in satin paint is less green than the same room in flat paint by nearly a colour difference of chroma.
Two ways to put a lobe on a wall
Take the interface's return out of the body term and a gloss wall makes the room less colourful. Add it beside the body term and the same wall makes the room more colourful. Same solver, same room, one line of energy accounting, and the two answers differ by nearly three units of chroma at the glossy end.
A tenth of the return arriving white
Nine per cent of what leaves a satin wall is a Fresnel reflection carrying no pigment. That nine per cent moves the room's colour by 4.89 ΔE₀₀ and its chroma by five per cent, because an interreflection multiplies and a small contribution with a different spectrum compounds into a large one.
A contrast control is three controls
A tone curve is a function of one number at a time and knows nothing about hue. Applied to each channel separately it rotates a hue circle by up to seventeen degrees, raises chroma by a factor of 1.27 at ordinary strength, and lifts lightness — so a photographer who moves a contrast slider has moved three things and the interface names one of them. All three scale with the curve's strength, monotonically, and the hue rotation depends on which hue it is.
A stop is not a stop afterwards
Doubling the light is exactly a factor of two in raw values and in tristimulus values, which is the one thing about exposure everybody is sure of. A stop taken after the tone curve is a factor of something else, and matching the two on an eighteen per cent grey leaves the rest of the frame between three and four colour differences apart at the mean and up to eleven at the worst patch. The gain that matches one stop is 2.47 rather than 2.
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.
A gloss finish takes colour out of the whole room
A gloss wall makes the floor's return less colourful seen from the front of a room and more colourful seen from the coloured walls, which leaves open whether the lobe removes colour or only moves it. A ledger of every flux between the room's faces answers it. At an eggshell finish the room's reflected light gains 16 per cent in quantity and loses 8.3 per cent of its chroma, and the loss is nearly the same for blue, green and orange walls while the walls' own losses range from 13 to 25 per cent. Only the painted walls receive light as colourful as before.
A tolerance has no light level
Twenty-three pairs built at exactly one colour difference stay at exactly one in every room, because the formula has no argument for the room. Read in the unit that does have one, the same pairs are 0.72 in a cinema, 1.04 in an office and 1.30 in direct sun — and inside any one room they spread by half again, so no single conversion between the two units exists at all.
A difference has no rate
A colour difference formula answers for two patches that are both there and stay. Alternate the same two colours and the difference is not scaled but taken apart: the colour half is gone by fifteen hertz and the lightness half is four times louder at eight, so twenty-three pairs the formula calls identical run over a factor of six at the rate the eye is best at, and are worth nothing at all above sixty.
The reversals have a straight edge
Twenty-one of 276 pairs change places in chroma between a dark background and a light one, and the reason given was that chroma is a product of a term carrying the exponent and a term that does not. That is true and it is not a description of which pairs. Chroma at one background is a single common factor times a power of lightness at another, so a pair reverses exactly when its chroma ratio and its lightness ratio point opposite ways and the first is the smaller — a wedge with two straight edges, which names every reversal and nothing else.
Two rooms with one lightness scale
CIECAM16's surround and its background both reach lightness, and they reach it through one product. So the rooms fall into classes: a television in a lit living room against a mid grey returns exactly the lightness a print on a desk against a background of 2.8 does, for every sample, to the last bit of a double. It returns 0.76 of its chroma and 0.81 of its brightness. Two of the model's four viewing-condition parameters are one parameter, and only colour tells them apart.
A finish adds colour only to a daylight meter
Measured against daylight's white, a satin finish under six lamps and six wall colours takes anything from −6 to 42 per cent of a room's colour, and one room reads as more colourful glossy than matt. Measured against the white of the lamp each room is actually lit by, the same thirty-six rooms lose between 7.7 and 12.9 per cent and none gains. The whole spread was the lamp's own colour, and the finish does one simple thing to every room: it pulls the room's colour a tenth of the way towards the lamp's white.
What an adapted viewer loses is set by the wall
A satin finish takes about a tenth of a room's colour, measured on the room's light against its lamp's white. Read through an appearance model by a viewer adapted to the room, the same thirty-six rooms lose between 12.8 and 30.9 per cent — 1.4 to 2.4 times as much — and the wall colour decides the multiplier: a deep red room loses twice what a green one does, under every lamp. A test on the bare wall predicts it. Add a little of the lamp's white to the wall's own colour and ask the model what that costs: the answer orders the thirty-six multipliers at 0.95.
A dark wall pays for a finish
A satin finish takes about a tenth of a room's colour, and the tenth varies. The natural explanation is that a weak colour loses a larger share of itself to the white light a glossy surface adds. Over seventy-two paints under one lamp that explanation orders nothing: the loss runs from 2.5 to 15.5 per cent, it follows how much light the wall returns at a rank correlation of −0.89, and it follows how colourful the wall is at −0.04. Two rooms equally colourful matt lose shares nearly six times apart, and the one that loses more is the darker.
White turns a hue, and the models part at blue
Mix a saturated light with white and its hue changes as well as its saturation — the Abney effect, and the reason lines of constant perceived hue curve in a chromaticity diagram. Asked what adding white does to the twenty-four most saturated colours a display makes, CIECAM16, CIELAB and Oklab all turn the hue. About reds they agree: CIECAM16 turns within a degree of Oklab, whose hue was fitted to observers' constant-hue judgements. About the display's blue they do not: Oklab turns sixteen degrees, CIECAM16 four, and CIELAB nine the other way.
A tint in paint turns the other way
A pale colour can be made two ways: by stirring white pigment into a coloured one, or by adding white light to it — which is what a halftone on white paper and a display both do. At the same luminance the two tints are not the same colour. For seven of eight pigments they turn the hue in opposite directions, an orange by +13.5 degrees in paint and −13.9 in light, and the paint tint is the more colourful of the two every time. The cause is in the reflectance: diluting a pigment moves its absorption edge, and adding light does not.
The finish adds the room's own colour
Among dark walls a more saturated paint loses a smaller share of its colour to a gloss finish, and among pale walls a larger one. The mechanism proposed for that reversal was spectral concentration — a narrow tall band — and the census that found it already contained four families of paints whose bands are all the same width. Holding lightness still, the band's width orders the losses at a rank correlation of 0.01. What does order them is that the light a finish adds has already bounced off the walls.
The units part by hue, not by light level
Two colour differences with a light level in them were compared on a display running from 5 to 5,000 cd/m², and a prediction was drawn from how each divided a difference between lightness and chroma. Built into pairs that differ in lightness alone and in chroma alone, both units say the same thing about light level: as a display brightens, chroma differences gain on lightness differences — ΔEITP by a tenth, CAM16-UCS by a quarter. Where they part is hue. The appearance model moves every colour's balance by the same factor; ΔEITP moves the violets, reds and cyan-blues the other way, and at nine of twelve base colours the two units disagree about the direction.
A strategy keeps the refit only where black prints
Refitting an inverse table's nodes against their own error halves it, and in a four-ink table the refit has to leave the separation strategy's choice of black alone. Held to the strategy, it keeps the whole of its gain wherever the strategy prints black and almost none of it where the strategy does not — and the share it keeps equals the share of the grey ramp printed with black, to within three hundredths. The start of black generation, predicted to be where the refit would fail, is where the table needs it least.
An open room hands over sooner
A gloss finish takes the least colour from a saturated dark wall and the most from a saturated pale one, and in a closed cube the sign changes at a wall returning about a third of the light. The prediction was that a less enclosed room would move that point up the lightness scale and that a room with a window would have no pale end. Opening one wall moves it down, from a luminance factor of 0.36 to 0.29, and opening a second to 0.24; stretching the room into a corridor or flattening it nudges it slightly up. The ambient does whiten, as predicted. A whiter ambient does not delay the colour term — it weakens it, so the other term takes over sooner.
A probe at the wall prices the finish
The light arriving at a painted wall from the rest of its room is what a gloss finish hands back, and a small probe held against the wall reads it. It lies on the line from the lamp's white to the wall's own colour, a fraction of the way along, and the fraction is set by how much light the wall returns, not by how colourful it is — as predicted. It is not the fixed fraction the prediction said: it runs from 2 to 46 per cent across seventy-two paints in one room. That variation is what makes it useful. Read in the matt room, it orders what a satin finish would cost more tightly than the paint's own lightness does, in every room tried.
White drains the blue last in every model
CIECAM16 says a dab of white costs a deep red wall four times the colour it costs a dark blue one, and the experiment proposed to test it asked whether observers lose colour in that hue order. Held at equal luminance and given equal doses of white, twenty-four display hues are ordered alike by CIECAM16, CIELAB and Oklab — an orange loses fastest and a blue slowest in all three. So the order cannot tell the models apart. What does is how much slower the blue is: CIECAM16 has it losing a sixteenth of what the orange loses, CIELAB and Oklab about a quarter. That ratio is the number an observer study has to measure.
A lit room brings the units' medians together
As a display brightens, CAM16-UCS says chroma differences gain a quarter on lightness differences and ΔEITP says a tenth. All of the appearance model's movement comes from what the viewer is adapted to, and every calculation had the viewer adapted to the display alone. Give the room its share of the adaptation and the model's movement shrinks at every step: with a 20 cd/m² room supplying two thirds of it, the two units' medians fall by the same amount. What does not shrink is their disagreement about direction. The model still moves every colour the same way, ΔEITP still moves violets, reds and cyan-blues the other way, and in a lit room that becomes the whole of what separates them.
The hue disagreement is the quantiser's
As a display brightens, the appearance model shifts every colour's balance between lightness and chroma by one factor, and ΔEITP shifts violets towards lightness and yellow-greens towards chroma. The proposed source was the perceptual quantiser acting on three cone-like signals at unequal levels. Replacing the quantiser with a pure power law and changing nothing else removes the disagreement exactly: every colour moves by a factor of one. With the quantiser, how far a colour moves follows the spread of its three signals, and which way follows where its S signal sits — above the other two, towards lightness; below, towards chroma.
The blue resists through the model's denominator
CIECAM16 says a saturated blue is nearly impervious to added white: at equal luminance and an equal dose it loses a sixteenth of what an orange loses, where CIELAB and Oklab say a quarter. The model has two obvious places for that to come from — a hue factor fitted to make colourfulness equal across hues, and the sum of cone signals it divides its opponent signals by. Setting the hue factor to one changes nothing. Replacing the sum with the model's own achromatic signal makes the blue lose 0.233 of the orange's loss, beside CIELAB's 0.221 and Oklab's 0.223. The disagreement is one denominator.
A little black ends the cast
A four-ink inverse table errs by a warm cast along the grey axis wherever the separation strategy prints only three inks, and almost only in lightness wherever it prints black. Real strategies start black gently, and the gentle start was expected to keep the cast until black carried much of the grey. It does not. The cast falls to a tenth of its three-ink size once black supplies a tenth to a sixth of the grey's darkening, at every toe from nothing to thirty units wide — because inside the toe the table's chord runs along the trade of black against the other three inks, which errs in lightness. What the toe does cost is the segment where black begins, which goes from the table's best to its worst.
A finish adds colour only where it covers grey
One low wide room was found in which a satin finish makes the room more colourful rather than less, and the explanation offered was dilution: the more of a room is grey, the more a finish's glancing return stands out. Painted area turns out not to decide it. Across six room shapes and five ways of painting them, a finish on the painted faces alone never adds colour, and a finish on the grey faces alone almost always does. The low room gains because its grey faces are most of it, and the whole finish is close to the sum of its two halves.
The cast returns where black is steep
Below a black-generation toe a four-ink table's grey picks up a small cast again, up to a third of the three-ink cast at the widest toe, and the explanation offered was the jump in black's curvature where the toe ends. It is not. A toe whose curvature is nothing at its end returns exactly the same cast, segment for segment, and a kink put into straight black casts in its own segment once in twelve tries. What the returned cast follows is how steep black has to be once the toe is done — how far the chromatic inks climbed while black was held back, and how fast they then have to come down.
At the gamut's edge the reds move as far as the violets
As a display brightens, ΔEITP shifts a saturated colour's balance between lightness and chroma by an amount set by its three quantised signals, and CAM16-UCS shifts every colour alike. The largest shifts found so far were a yellow-green at ×0.63 and a dark blue at ×1.62, both at the census's edge — and the blue turns out to be a colour no BT.2020 display can show. Walked out to the display's real boundary, the yellow-greens go lower, to ×0.54, and the dark blues never pass ×1.35, because BT.2020's blue edge is at low chroma. The top of the range goes to the reds and red-purples at ×1.6: at the edge the L-minus-M term grows to two thirds of the S term. Pairing the two extremes still needs a third of the observers the violet experiment needs.
An ink limit takes the shadows' cast away
A cap on total ink forces black up and the chromatic inks down where it is reached, which is the descent that brought a small cast back below a black-generation toe, and the prediction was that a cap would bring one back too. Measured against what the same strategy prints with no cap, it does the opposite. Below where it binds the cap decides the separation outright — nine strategies print one table there — and in 32 of 33 cases its shadows cast less than the strategy's own would have.
A display's red takes the reds' move and leaves the pair
At BT.2020's edge ΔEITP moved a dark red's balance between lightness and chroma furthest of any colour, ×1.64 from a dim display to a bright one, and the reason offered was BT.2020's red primary, a line far out at 630 nanometres. Moving that primary in along the spectral locus confirms it: the reds fall faster than any other hue, to ×1.40 at 615 nanometres and ×1.25 at 605, and Display P3, whose red is a 615 nm line to three ten-thousandths, gives the reds the same. The top of the range goes to a red-purple, which rises as the red moves in because BT.2020 had pushed it past its own peak. And the forced-choice pair the experiment needs is the same two colours on every display.
The tolerance unit opens every edge
On a finely sampled press every corner moves under a CIELAB margin as a perfect corner of its own shape would, the shape read off the press model's derivatives. A print buyer states the margin in ΔE₀₀, and the census in that unit had been explained one corner at a time. Carried into ΔE₀₀'s local metric, the same shapes explain all nine at once: each of the six edges' predicted move lies within a tenth of the margin of the census's, and each vertex's within a seventh. ΔE₀₀ opens the edges from 67–141 degrees to 123–157, nearly faces, because a unit along chroma is three to five CIELAB units at these colours. The solid yellow's cone opens only to 59 degrees and stays the one corner sharper than a cube's. Without its chroma term, ΔE₀₀ would leave every corner as sharp as CIELAB does, or sharper.
The paint itself gains all or nothing
The painted census counts how many of seventy-two paints a satin finish on a room's grey faces makes more colourful, reading the room's colour as all of its light, and eleven of its thirty rooms sit part-way. Read on the painted faces alone — the light a viewer looking at the paint receives — no room sits part-way: twenty-eight gain for every paint and two for none. The change on the paint is never more than a per cent and a half, where the room's reading swings from −18 to +57 per cent, so the census's middle ground was the grey faces all along. A box with its floor painted and the same box with its ceiling painted gain the same colour to a hundredth of a point while the light they gain differs by up to eighty-two points; neither send number orders the size of the gain, and the paint's own brightness does.
A warm room tints the black, not the halo
A local-dimming panel's black can be declared as two floors, the room's light reflected off the screen folded in, and the earlier censuses gave that light a hundredth of a candela, grey. Under a warm lamp it is not grey. The prediction was that a warm room would hide the halo's misreading in the long and middle cone signals, leave its short-wave part exposed, and tint the halo's dark greys towards blue. The tint is real and it is none of those things. With the floors declared in grey, a 2,700 K room putting a tenth of a candela on the screen tints every zone of a dark picture by more than one ΔEITP — up to 17.7 at the darkest grey — and the tint points towards the lamp's own orange, opposite blue. It is not the halo's: it is the whole black's, because a grey declaration has the floor's colour wrong everywhere. The prediction's conclusion holds by a different road. A grey step laid on a warm floor is partly a change of colour, so the warm room leaves seven tenths of the moon picture's grey steps misread where a grey room leaves none; and declaring the floors in the room's colour removes all of it but the halo's own.
A knee in the ink limit quiets the shadow
A cap on total ink coverage quiets a four-ink table's shadow, and the one segment where the cap is first reached is quiet or loud depending on where the binding point falls among the table's nodes: inside a segment it is quiet, within a quarter of a segment of a node it is not. A real cap is approached through a knee. The prediction was that a knee as wide as a segment would make the binding segment quiet wherever it fell, and that a knee much wider would turn the cap into a steep straight black that casts like the heavy-black strategy. The first half holds only in part: a five-unit knee halves the binding segment's cast and quiets three of the nine loud ones, but those whose binding point sits within a quarter of a segment of a node stay louder than a neighbour. The second half is wrong in sign. A wider knee leaves the whole capped shadow quieter at every step — its mean cast falls from 11.5 thousandths to 3.0 at fifteen units — because a knee spreads the chromatic inks' forced descent, and a slower descent is one the table's straight lines between nodes can follow.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
InterreflectionSpecularModelling assumptionRadiosityLightnessColour differenceSpecificationCIEDE2000ConvergenceCIECAM16ΔERank correlation