The limits assume a pigment that switches instantly

The set of tristimulus values a reflecting surface can produce is the hardest boundary in colorimetry: no pigment chemistry, no manufacturing process and no amount of money moves it, because it follows from the fact that a surface cannot return more light than falls on it. A white that is not a reflectance used it to catch a brightened sheet sitting outside, and a third of the appearance box is no surface used it to find that a third of the space a paint is specified in has no paint in it.

Its boundary is reached by optimal colours: reflectances that are nought or one and switch between them at one or two wavelengths. Nothing in that construction asks whether a material could behave that way, and none can. A real colorant’s absorption edge rises over tens of nanometres.

This essay first appeared with a census that could not see most of its own result. It reported that the median direction of the solid loses under a thousandth of its reach at every transition width up to 160 nanometres, and that the whole cost sits in one saturated corner. Nearly a fifth of the directions it swept point into the black corner, where the ideal reach is exactly nothing; each of them was a ratio of nought over nought, and a sort with those in it left the median wherever it happened to fall. Two of its six widths were also not the widths computed. What follows is the census with both repaired, and its conclusion is different.

A corner at a dye’s sharpness, most of the boundary at a pigment’s #

Forbidding a reflectance to switch instantly costs the object-colour solid almost nothing at the sharpness of a good dye, and a substantial share of most of its boundary at the sharpness of an ordinary pigment.

• At twenty nanometres the median direction keeps 0.997 of its reach and the tenth percentile 0.985; one direction in six loses more than a per cent, and none loses a tenth.
• At eighty nanometres the median keeps 0.95 and the tenth percentile 0.77; seven directions in ten lose more than a per cent and three in ten lose more than a tenth.
• At 160 nanometres the median keeps 0.83 and the tenth percentile 0.42.
• The greens pay most at every width, then the reds; the yellows, cyans and blues lose least.
• The perfect diffuser is untouched at every width, exactly, because a flat reflectance has no transitions in it — which is the check that the constraint does what it says.

The limit with a second argument #

The boundary is computed through its support function: in a direction d, the solid reaches as far as the largest d · XYZ any admissible reflectance produces. Unconstrained the answer is immediate — set the reflectance to one wherever the weighted matching function is positive and to nought elsewhere, which is the optimal colour and where the bang-bang shape comes from.

With a slope limit the reflectance cannot follow the sign changes, and the answer is a maximisation of a linear functional over a convex set. On a wavelength grid it is a dynamic program: quantise the reflectance into levels, walk the spectrum band by band, and allow the level to change by at most the number of steps the slope permits. Unconstrained, that program returns the optimal colour exactly — the disagreement over the directions checked is zero to rounding — which is what makes the constrained answers trustworthy.

The constraint is stated as a transition width: how many nanometres a reflectance needs to go from nought to one. Twenty nanometres is a sharp dye; eighty is an ordinary pigment; five is sharper than anything in a paint box and is included to show where the curve starts.

The figure above is the census at six widths. The three curves separate early and keep separating. The worst direction falls first, from one at five nanometres to 0.94 at twenty and 0.76 at forty. The tenth percentile follows, holding near one to twenty nanometres and falling to 0.77 at eighty and 0.42 at 160. The median is last and slowest, but it moves: 0.997 at twenty, 0.986 at forty, 0.95 at eighty, 0.83 at 160. Of the three, the worst direction is the one to hold loosely: it is set by directions that barely clear the black corner, and it moves with where that line is drawn, while the other two do not.

The shape of the loss at two widths #

The sorted curve is where the change in the conclusion shows most clearly.

At twenty nanometres the curve is nearly flat. Just over a quarter of the directions lose under a thousandth, a sixth lose more than a per cent, and none loses a tenth. At a dye’s sharpness the cost really is in a corner — the corner the ideal limit is quoted for, where the boundary is reached by a narrow band.

At eighty nanometres it is not a corner. A quarter of the directions still lose almost nothing, and they are not a lucky region of the boundary: 73 of those 78 are directions whose optimal reflectance has no transition at all, because the farthest the solid reaches in them is the perfect diffuser, a flat reflectance with nothing to blunt. The rest fall steadily: 218 of 305 directions lose more than a per cent and 93 lose more than a tenth. That is the width an ordinary organic pigment’s absorption edge takes, and at that width the ideal boundary overstates what a pigment can reach across most of its surface, not in one saturated extreme.

The hardest direction, followed down #

The worst direction is worth following through the widths, because the reflectance it asks for shows what the limit forbids.

The optimal reflectance here is a green pass-band about fifty nanometres wide, from 505 to 555, at one inside and nought outside. At ten, twenty and forty nanometres it can still reach one, with sloped sides that eat into the band’s edges. At eighty it cannot: two transitions of eighty nanometres do not fit inside a band of fifty, so it must turn round before it arrives, and it peaks at 0.63.

As colours, the retreat is from a green at lightness 72 and chroma 161 to lightness 69 and chroma 148 at forty nanometres, and to lightness 56 and chroma 121 at eighty. The limit a saturated green is said to reach is a limit only for a material that switches instantly; a pigment whose transitions take eighty nanometres reaches three quarters of the chroma, at a markedly lower lightness. Most of those patches are hatched, because the page cannot show them either — they are outside every display’s gamut as well as outside every pigment’s.

Which colours pay #

The census can be grouped by the hue of the colour the ideal boundary reaches in each direction, and the grouping says where the loss lives.

The greens lose most at every width — 2.9 per cent at twenty, 12.2 at forty, 39.0 at eighty — and the reds come second, at 1.6, 6.4 and 22.9, although only seven directions of this sweep end on a saturated red, so their average is the least certain in the figure. The yellows, cyans and blues lose between one and two per cent at forty and between five and eight at eighty; the purples sit between.

The reason is the shape of each optimal colour. An optimal green is a pass-band in the middle of the spectrum, with an edge on each side for the limit to blunt, and it is narrow because the matching functions change sign close together there. An optimal yellow, cyan or blue is mostly a single step — reflect everything above an edge, or below it — and a single step survives a slow transition with only its edge moved. The reds pay second because a saturated red is a step placed where the long-wavelength matching function is falling steeply, and moving the edge costs more there.

The practical reading is the one what a gamut costs has been making about displays, arriving at surfaces: the boundary is most generous in the region hardest to manufacture, and for a pigment the hardest region is a saturated green.

What a real colorant’s band looks like #

The widths above are chosen to bracket the chemistry rather than to flatter it.

A dye’s absorption band has a width of its own — an organic colorant’s main band is typically fifty to a hundred nanometres across at half its height, and the sharpest systems a colourist meets are still tens of nanometres. An edge’s transition is roughly half a band’s width, since a band has two sides, so an ordinary pigment sits between the forty- and eighty-nanometre rows and the sharpest systems near twenty.

That places the trade’s actual materials in the part of the table where the tenth percentile has lost between a sixteenth and a quarter of its reach and the median between one and five per cent. It does not mean any real pigment reaches even the constrained boundary — a brand colour is an ink is the reminder that a specified colour has to be made of something, and what it is made of falls short of any bound for reasons of scattering and purity as well as sharpness.

Why this does not rescue any gamut #

It would be convenient if this meant real gamuts are closer to the limit than they look. It does not, for two reasons.

The first is that the slope limit is the weakest possible statement of “cannot switch instantly”. It admits every triangular and trapezoidal reflectance, and no dye produces those either. A dye’s absorption band has a shape, closer to a Gaussian in absorbance with wings, and a constraint stated in band shapes would remove more, not less.

The second is that the constrained solid is still an upper bound. Pigments fall short of it for reasons that have nothing to do with slope — scattering, impurity, binder — so what the census establishes is a floor under the gap between the ideal boundary and real materials, not its size.

What the calculation does buy is a boundary with the assumption made visible. The model has six arguments is the standing account of what a reflectance leaves out — direction, place, the two it assumes equal — and this is one more: the reflectance is treated as an arbitrary function of wavelength, and a material’s is not.

What was computed, and how #

The solid is the support function under D65 through the 1931 observer, evaluated over a Fibonacci sweep of directions, with the perfect diffuser on its boundary as a check. The census takes every sixth direction of the sweep — 400 — and sets aside the 72 whose ideal support is exactly nought, because they point into the black corner where no reflectance reaches anything, and 23 more whose ideal support is under five units on a scale where the perfect diffuser’s luminance is 100. Those are nearly tangent to the solid at black, and a ratio of two supports that small exaggerates any change: set aside at one unit instead of five, the worst direction at forty nanometres keeps 0.50 of its reach rather than 0.76, while the median and the tenth percentile move in the third decimal place. That leaves 305.

The constrained support is a dynamic program over the 81-band grid with the reflectance quantised into 161 levels, which makes each of the six widths a whole number of levels per band: 160 at five nanometres down to five at 160. The program keeps back-pointers, so the reflectance achieving each support is recovered and can be drawn and converted to a colour. A hue sector counts the directions whose ideal boundary colour has a CIELAB chroma of at least 20 and a hue angle inside the sector.

What the first version got wrong, and why it looked right #

Two defects, and the second is the instructive one.

The quantisation was 41 levels, and a width was turned into a whole number of levels per band by rounding. At eighty nanometres the step came out as 2.5 and rounded to three, which is a width of 66.7; at 160 it came out as 1.25 and rounded to one, which is 200. Two rows of the table were labelled with widths they did not compute.

The census also divided by the ideal support in every direction of the sweep, including the black corner, where both numbers are nought. The ratio there is not a number, and a numerical sort does not fail on one — it simply stops sorting reliably around it. Every median the census reported came out as 1.0000, the essay read that as the median direction losing nothing, and the finding that the cost is confined to a corner followed. Every individual number was computed correctly; the summary of them was not a summary of anything. With the black corner set aside the same arithmetic gives the table above.

It looked right because it agreed with the expectation. A boundary reached by bang-bang reflectances should be robust to a mild constraint over most of its surface, and a flat median is what that expectation predicts — so a flat median at every width, even at 160 nanometres, was read as confirmation rather than as the sign of a summary that could not move.

What a specification can take from it #

Two things, one about writing colours down and one about arguing over them.

A colour specified inside the ideal solid may still be unmakeable, and the distance from the boundary is the wrong test. The usual check is whether a requested colour sits inside the object-colour solid at all. Being inside is not enough: a colour inside the ideal boundary but outside the boundary for the sharpness a chemistry can reach is equally unmakeable, and nothing in the first test says so. At a pigment’s sharpness the two boundaries separate across most of the solid, by the most where a saturated green is specified.

And an argument about how much gamut is theoretically available should say which limit it means. Most of this diagram cannot be shown made the point for displays and the chromaticity diagram: the figure everybody quotes is drawn for a set of stimuli nobody can produce. The same applies one level deeper — the object-colour solid is a bound over reflectances rather than over materials, and the difference is worth a quarter of the chroma in the green corner and several per cent of reach across most of the rest.

Still open: the constraint a real colorant imposes #

The useful version of this is not a slope limit but a band-shape limit: the set of reflectances a real chemistry can produce. A library of measured dye and pigment reflectances would give it empirically — fit each with a small number of absorption bands, find the narrowest band anybody achieves, and use that as the constraint. The computation is harder than the one here, because a family of band shapes is not convex and a search over it can stop short of the true boundary; a nested check, that allowing a third band never reaches less than two, is the minimum it would have to pass.

A second measurement is cheaper and already possible. The constraint here is stated per nanometre and applied uniformly, and a real band’s width is set in energy, which makes it wider in the red than in the blue. A limit written in energy charges the reds runs the census that way. And the lamp matters: three lines spare a slow pigment runs it under an LED whose power sits in three narrow lines.

An assumption inside a bound, and a summary inside a census #

The habit is about the difference between a bound and a possibility, and about the number that summarises a distribution.

A bound proves that nothing beyond it exists. It does not prove that anything up to it does, and the construction usually assumes a freedom the world does not grant — here, a reflectance that can switch instantly. The move is to name the freedom and charge for it, and to recover the unconstrained case first: the width of nought reproduces the optimal colour exactly, and a construction that cannot do that is wrong before any constrained answer is read.

The second half of the habit is what this essay learned from its own first version. A summary statistic is only as good as the set it summarises, and a set with undefined members in it produces a summary that looks stable because it cannot move. The check is cheap: count what went into the median, and confirm that the median changes when the thing being varied changes. A median that stays at one while the worst case falls to an eighth is not a finding about the solid.

The failure mode is to treat the untouched part as evidence that the assumption is harmless — or, worse, to treat a summary that never moved as evidence that the untouched part was large.