There is a short list of colours that come with a warning label. Stygian blue, a blue as dark as black. Self-luminous red, a red that glows brighter than the white page it sits on. Hyperbolic orange, an orange more orange than orange gets. Reddish green, which is the one everybody has heard of. They are usually presented as a party trick: stare at this square for thirty seconds, look at that square, marvel.
We wanted to know how impossible they actually are, which means putting a number on the gap between what the eye can encode and what light can deliver. So we took the Stockman and Sharpe cone fundamentals at 1 nm and measured the whole reachable set. Real light can produce 51.91 percent of the cone signal combinations a three cone eye is able to carry. The remaining 48.09 percent is impossible colour, and the interesting part is that it is nowhere near evenly distributed.
Nearly half the signal space has nothing behind it
Start with what the eye actually sends upstream. Three cone types, three numbers. Call them L, M and S for the long, middle and short wavelength cones. Any triple of non-negative numbers is a signal those three cells could in principle report. Normalise away the overall intensity and you get a flat triangle of possible signals, with pure L in one corner, pure M in another, pure S in the third.
Now ask which points in that triangle any light can actually cause. Each wavelength lands on one point. Every mixture of wavelengths lands somewhere inside the shape those points enclose. That shape is the whole of realisable colour, and it fills 51.91 percent of the triangle. Everything outside it is a signal the optic nerve is perfectly capable of carrying that no photon arrangement in the universe can produce.
The reason is overlap. The M cone’s sensitivity curve sits almost entirely underneath the L cone’s, so there is no wavelength that excites M without dragging L along with it. The best any single wavelength manages is 518 nm, where M carries 55.43 percent of the total cone signal and L and S split the rest. Push to the 10 degree fundamentals and it is 56.72 percent at 511 nm. There is no light anywhere that gets M past about four sevenths of the signal, and there never will be.
That asymmetry is the finding we did not expect. Here is how the unreachable 48.09 percent divides up by which pure cone direction it sits nearest.
| Direction | Share of the impossible region | Share of the whole space | Distance to the nearest real signal |
|---|---|---|---|
| Green, M cones alone | 57.75% | 27.77% | 0.3730 |
| Blue, S cones alone | 31.64% | 15.22% | 0.0752 |
| Red, L cones alone | 10.61% | 5.10% | 0.0583 |
The green direction holds 6.4 times more unreachable room than the red one. More than half of all impossible colour is green-ward. The pure L corner is only 0.0583 away from a real signal, because 703 nm light gets L to 94.17 percent on its own. The pure S corner is 0.0752 away, because 417 nm gets S to 93.20 percent. The pure M corner is 0.373 away and always will be.
So the four famous chimerical colours are not four flavours of the same thing. They are four different failures, and they fail by wildly different margins.
Every one of them has a hex code
The other thing that falls out of measuring these properly is deflating. Three of the four are produced by looking at an ordinary colour while your retina is still recovering from a different one. Which means at the moment you see the impossible colour, there is a perfectly ordinary colour on the glass in front of you, and we can write it down.
| Colour | What is on the glass | What the percept asks for | The wall it hits |
|---|---|---|---|
| Stygian blue | #000000 | Blue chroma at the lightness of black | 884 of 16.7M sRGB colours sit below L* 1, and the most chromatic of them reaches C* 19.4 |
| Self-luminous red | #FFFFFF | Red brighter than the white beside it | The reddest red a display makes is 21.26 percent of its own white |
| Hyperbolic orange | #FF5100 | Orange purer than any light can be | The optimal-surface ceiling at that lightness is C* 129.1; sRGB reaches 94.3 |
| Reddish green | two adjacent stripes | Both ends of one opponent channel at once | No ceiling to beat. The channel carries one number |
Stygian blue’s hex code is #000000. When somebody tells you they have seen a blue darker than black, the stimulus they were looking at was black. All of the blue was made downstream of the retina. Self-luminous red is a white screen. Hyperbolic orange is an ordinary orange, and we picked #FF5100 because it is the most saturated orange sRGB has at that lightness.
This is the same pattern we found when we measured iridescent colours and got grey, and when we measured a mirror and got #F1F8F7. The honest pixel is almost always duller than the name.
Stygian blue: the contradiction has a size
The recipe is well known. Stare at a bright yellow patch until your blue-yellow channel is worn down, then look at black. The black comes up blue, and observers insist it stays as dark as black while doing it.
Put in CIELAB terms, the percept is asking to be two things at once: indistinguishable from #000000 in lightness, and as blue as #0000FF in chroma. Those two colours are 39.68 CIEDE2000 apart, about 17 just noticeable differences. That is the size of the contradiction, and it is why nobody can hand you a swatch of it.
The sRGB cube makes the squeeze very concrete. Of the 16,777,216 colours a screen can display, only 884 have a lightness below L* 1. That is one in 18,979. The most chromatic member of that group reaches C* 19.4, against the 133.8 that #0000FF manages. Widen the window and it barely helps: 31,013 colours below L* 5, 113,555 below L* 10. The darkest tenth of the lightness scale holds 0.68 percent of the colours.
Here is the part we got wrong on the first pass, and it changes the story. We assumed a dark saturated blue was physically impossible, not just impossible on a screen. It is not. Work out the chroma ceiling for an ideal reflecting surface, the sort with a reflectance that is either zero or one and switches twice, and a surface at L* 1 can reach C* 131.6 under D65. That is as chromatic as full blue, at one hundredth of the lightness.
| Lightness | Ideal surface ceiling | Best sRGB can do | That colour |
|---|---|---|---|
| L* 1 | C* 131.6 | C* 22.5 | #0C0024 |
| L* 2 | C* 142.6 | C* 38.5 | #040039 |
| L* 4 | C* 162.0 | C* 54.0 | #00004D |
| L* 6 | C* 178.8 | C* 61.7 | #000059 |
| L* 10 | C* 192.3 | C* 73.6 | #000071 |
| L* 15 | C* 191.3 | C* 88.2 | #000092 |
| L* 20 | C* 183.2 | C* 102.2 | #0000B2 |
| L* 32 | C* 165.7 | C* 133.8 | #0000FF |
Read the middle two columns against each other. At L* 1 a display reaches 17 percent of the physical ceiling, and at L* 32 it reaches 81 percent. The darker the colour, the further behind the glass falls. So the reason you cannot see a properly saturated near-black blue on this page is a display limitation, not a law of optics. Colour vision genuinely has that territory in it, and sRGB simply does not reach into it.
What makes stygian blue impossible is narrower than darkness. It is zero luminance specifically. At exactly zero light there is exactly one colour, and the recipe asks for a second one. Everything above zero, including the far edge of the table above, is real.
Self-luminous red: the number is 0.2126
Stare at green, look at white, and the red afterimage can appear to be emitting rather than reflecting, brighter than the white it is drawn on. A surface cannot do that. It has no way to send back more light than it received.
A screen can emit, so you might expect it to manage this one. It cannot, and the reason is in the luminance coefficients. Full red on an sRGB display carries a luminance of 0.2126 relative to full white. The reddest red a panel can make is about a fifth as bright as its own white, which is to say 4.70 times too dim to out-glow the page it sits on. Here is the whole set:
- Blue is the worst offender at 0.0722, which is 13.85 times dimmer than white. A saturated blue is nearly a dark colour by construction.
- Red 0.2126, 4.70 times dimmer. Magenta 0.2848, 3.51 times.
- Green 0.7152, only 1.40 times dimmer. Yellow 0.9278, 1.08 times.
So of the six saturated corners, only yellow and green come close to matching white for brightness, and neither of them is the colour the afterimage hands you. There is a nice consequence here for anyone wondering why the dark blue end of the hex cube feels so cramped. It is cramped. Blue starts at a seventh of white and has nowhere to go.
Which is also why a black screen is not black
Both of these colours are really arguments about what the surround is doing. A panel showing #000000 in a lit room is still throwing light at you, which we measured in the hunt for the blackest black and found to be roughly 120 times brighter than Vantablack. Stygian blue on a screen is competing against that floor as well as against physics.
Hyperbolic orange: a ceiling you can compute
This is the only one of the four with a clean, finite target. Stare at cyan, look at orange, and the orange is reported as purer than any real orange. Purer than what, exactly? There are two answers and they are quite far apart.
For an ideal reflecting surface, the ceiling is the object colour solid, the same construction behind the numbers in our piece on neon colours. At hue 50 degrees the best a surface can do is:
- L* 40: ceiling C* 96.3, sRGB manages 72.3 with #B23500.
- L* 50: ceiling C* 120.6, sRGB manages 83.8 with #DB4500.
- L* 60: ceiling C* 129.1, sRGB manages 94.3 with #FF5100. This is the peak.
- L* 70: ceiling C* 109.3, sRGB manages 67.0 with #FF8649.
A screen reaches 73.1 percent of the surface ceiling at its best, so there are real orange objects more saturated than anything this page can show you. Traffic cones and certain marigolds get closer to the limit than #FF5100 does.
For light rather than surfaces the ceiling is much higher, because a narrow band source sits on the spectral locus and nothing beats that. Which is where the cone space measurement earns its keep: beyond the locus at the orange end, the impossible region is only 0.0583 wide before it runs into the pure L corner. Hyperbolic orange is real in the sense that it is outside the reachable set, and it is the most cramped of the four. There is very little room out there.
Reddish green: a different kind of impossible
The famous one is not a gamut problem at all, and lumping it in with the other three is the mistake most explainers make.
Red against green is a single opponent channel. One number, signed. A positive value is reddish, a negative value is greenish, zero is neither. Opponent process theory is built on exactly this, and we measured its consequences separately. Asking for reddish green is not asking for a point outside the reachable region. It is asking one variable to hold two values, which is a different category of request. There is no ceiling to exceed.
And yet in 1983 Crane and Piantanida put red and green stripes on a retinal stabiliser, froze the boundary between them on the retina, and reported that the border dissolved and some subjects saw a single surface that was reddish and greenish at once. The paper ran in Science and then sat in the corner for two decades, because nobody quite knew what to do with a result that violates the one thing opponency is supposed to guarantee.
Hsieh and Tse went back at it in 2006 with better controls and argued that what observers actually see is an intermediate colour, an olive or a brown, which they then struggle to name because their vocabulary for the middle of that axis is thin. That reading fits what we found when we measured naming behaviour for colour psychology charts, and it fits the shades of grey vocabulary count too. People reach for a strange description when the colour is real but the word is missing.
Our own view, for what it is worth: the stabilised border experiments are measuring something real about how the visual system fills in a region when its edges stop updating, and the phenomenology is genuinely odd. Whether that deserves the word impossible depends on whether you think impossible means outside the reachable set, which is measurable, or outside the representable set, which is not.
The laser went where the hole is biggest
In April 2025 a group at Berkeley and the University of Washington published the sharpest result in this area in forty years. They mapped individual cones in five people’s retinas, then used a laser to stimulate only the M cones, one cell at a time, thousands of pulses at a time. The resulting percept was named olo, and the subjects described it as a blue-green of a saturation nothing in nature matches. One of them said the most saturated natural colour looked pale next to it.
Line that up against the table earlier in this article. The M direction is the one with 0.373 of unreachable room, holding 57.75 percent of all impossible colour, the corner no wavelength can get past 55.43 percent of. It is the single largest hole in the reachable set by a factor of 6.4, and it is exactly the direction the experiment aimed at. The strength of the reported effect and the size of the gap we measured point the same way, which is the kind of agreement that makes a measurement feel less arbitrary.
It also explains why nobody has done the equivalent trick for red. Drive the L cones alone and you would land 0.0583 outside the reachable set, which is a short walk past 703 nm light. There is barely any new territory to visit. The S direction has a little more room at 0.0752, and is harder to isolate because S cones are sparse.
Worth being precise about what olo demonstrates. It is not a fourth primary and it is not a new cone type, so it has nothing to do with tetrachromacy. It is the same three cones, driven into a combination that light cannot arrange. Which is the definition of an impossible colour, achieved by skipping the light.
How this was measured
Cone fundamentals are the Stockman and Sharpe 2 degree linear energy functions at 1 nm from the Colour and Vision Research Laboratory tables, with the 10 degree set used as a robustness check. The reachable set is the convex hull of the per-wavelength cone chromaticities projected onto l + m + s = 1, since any physical spectrum is a non-negative mixture of wavelengths and therefore lands inside that hull. Areas are compared against the full non-negative triangle. Partition shares come from four million uniform samples on the triangle, classified by nearest pure cone corner, with the Monte Carlo estimate of the reachable fraction landing at 51.93 percent against the analytic 51.91.
One caveat worth stating plainly, because it is the weak point of the headline number. The exact percentage depends on the plane you project onto. Repeat the whole thing in MacLeod-Boynton coordinates and the unreachable share comes out at 66.90 percent rather than 48.09. The ranking of the three directions is unchanged, and so is the conclusion that the M direction dominates, but treat 48.09 percent as one reasonable way to size the hole rather than the only one. The distances and the cone share maxima do not depend on the projection at all, which is why we leaned on those for the comparisons.
Surface ceilings are optimal colours in the Schrödinger sense, enumerated as all 160,400 two-transition reflectance functions on the 380 to 780 nm grid, integrated against the CIE D65 relative spectral power distribution and the CIE 1931 2 degree observer, then converted to CIELAB against the resulting white point. sRGB figures use the IEC 61966-2-1 primaries and transfer function. The 16,777,216 colour census is exact rather than sampled. Colour differences are CIEDE2000 with a threshold of 2.3 for the just noticeable difference conversions.
We did not try to model afterimage strength. Doing that honestly needs a free parameter for adaptation depth, and a number that moves with a parameter you chose is not a measurement. Everything above is the wall, not the push.
The short version
- Real light reaches 51.91 percent of the cone signal space. The other 48.09 percent is impossible colour.
- The hole is lopsided. The green direction holds 57.75 percent of it and sits 0.373 from the nearest real signal, 6.4 times further than the red direction.
- No wavelength gets the M cones past 55.43 percent of the total signal, which is the whole reason green is the awkward one.
- Stygian blue’s stimulus is #000000. It asks to be both black and as blue as #0000FF, two colours 39.68 CIEDE2000 apart.
- Self-luminous red fails on a number: full sRGB red is 0.2126 of white, so 4.70 times too dim to out-glow the page.
- Hyperbolic orange has the tightest ceiling of the four, 0.0583 of room past the spectral locus.
- The 2025 olo experiment drove M cones alone, which is the largest unreachable direction there is. It went to the right place.
None of which makes the party trick less fun. It makes it a demonstration that colour is a report rather than a property, and that the report has room in it for things the world cannot say.
Try the reachable ones
Everything in this article is a distance between two colours, which is also what the color memory game scores. It shows you a colour, takes it away, and measures how far your recall lands in the same CIEDE2000 units used above. The dark end of the scale is where people do worst, and the table of sRGB lightness bands explains a good part of why: there is very little colour down there to hold on to. Hex mode makes you name the code instead, and hue sort asks you to put a scrambled set back into spectral order. There are more of them here, and a daily round if you want just one.
For the neighbouring measurements, see why you see afterimages for the mechanism all three chimerical recipes depend on, the visible spectrum for what a screen does to the reachable edge, magenta for a colour that has no wavelength but is entirely possible, and how many colours you can see for the counting problem underneath all of it.