Tetrachromats could plausibly see more nonspectral colors by activating their extra color receptor and one of the other ones. There would be no exact equivalent in trichromatic vision.

It depends on what their brains decide to do with it- maybe they would just land on an existing color in between. But there's no reason it has to.

So these colors are something that tetrachromats could maybe see while awake, but usually with lots of other colors mixed in. Asleep, maybe the brain starts to play with them and generates pure versions of them. Trichromats wouldn't be very likely to have these dreams because their brains don't have experience of tetrachromat nonspectral colors at all.

I imagine in school the teacher tells the poor girl: Mix the blue and yellow colors to get green, and the girl doesn't understand because she has a lot more complex color model than the teacher.

We see magenta as a separate color because we can distinguish it from green- it's not green, because Red and Blue don't activate our green cones. But we can't distinguish between spectral yellow and R+G because there's no yellow cone: the activations of all our cones are more or less identical in both cases.

If you have a yellow cone, you'd conceivably be able to distinguish between both kinds of yellow: redgreen and yellow could look as different as green and redblue do. I'm not sure how that would look as a "wheel", perhaps a figure-eight with the crossing at yellow.

Whereas a hypothetical tetrachromat with an additional cone type that activates to the yellow wavelength, will have obviously different activations of their cones for the "wavelength of yellow" and "wavelengths of red and green mixed". And hence experience them as different colors.

I'm not familiar enough with the different definitions of color spaces to define how to model this. You could easily define it as a four-dimensional RGBY color space, but to my understanding the other color spaces are defined because they have better properties for combining different colors, or playing better with interpolation between colors.

It would be a challenge to come up with a good four-dimensional color space that is a useful artistic and visual tool, given that there are very few people around to evaluate it! And also because all computer monitors are trichromatic, so it'd be a job in itself to set up a test system.

I wish someone in the interview had asked some of these people how they experience photos on computers!

Tetrachromatic colors could be placed in a tetrahedron, defining a space and being approximated as a sphere. A perceptually uniform representation would probably be curved on the three edges that agree with the spectrum and flat on the three that don't. Green is still probably the biggest. Then you'd add luminance as a fourth dimension because it's still rods.

With tetrachromats we get a four-dimensional space of color. So we need something like RGBX or a two-dimensional hue value like for example H1,H2,S,V. That's what I meant with a second hue circle orthogonal to the one we already know.

What we see as yellow is for the tetrachromat a whole second hue circle varying in the value of second hue.

An example: Yellow is a point on the first hue circle. For the tetrachromat these colors are different, but boring yellow for us:

- pure yellow

- mixture of red and green

- mixture of pure yellow, red and green

In other words, what we see as yellow is a whole dimension of different colors for tetrachromats.

Also, if you mix R+B, that's magenta, and if you start mixing in blue it will turn white or gray. But if you can see yellow, RYB would look different from R+a little g+B, which is what RYB would look for trichromats, I think.

This is honestly one of the saddest parts of human society. Some people, including those in positions of authority, either don't understand or outright reject the notion that different people perceive the world differently.

In secondary school I argued along the lines you describe, that you can mix three colours in a circle, and then a fourth around the circumference mixes only with at most two of the others - and in terms of the FCT is then a 'barrier' allowing re-use of the first three.

That's quite off-topic, but you suddenly jolted my memory of it. More on-topic, being colour-blind (never told which type and I can't work it out either) I find conversations about colours frustrating, and very strange to wonder if we see anything alike at all, or just learn to call the same physical thing alike despite perceiving it perhaps very differently.

Proving some maps can't be colored with three colors is easy, you only need to find one and enumerate all the three-colorings and find none of them work.

It's not really a theorem about colors though, they're just labels.

Re proving any, I replied to a sibling comment. What I was imagining was the inverse - proving that you can't construct one that requires a fifth.

I recognize that this comment includes jargon, but I assume you'll Google it. I'd include diagrams but HN doesn't support them.

But, I understand and trust the mapping of the problem to a graph, so perhaps if I see why the problem is hard on a graph that should be enough for me.

But my reasoning was along the lines (but now in my newfound terminology) of:

K1 - trivially 1 colour needed. K2 - 2.

If we add another region on the map, we either have K3 or K1,2. In the latter case the third region/vertex need only be different to the one it touches, but K3 clearly needs 3 colours.

Adding the fourth region is similar, and again K4 clearly needs 4.

The fifth region touches at most 3 others, since K5 is non-planar. Thus, it can take the colour of the untouched region.

That's not very formal, but why isn't proof by induction easy from there? i.e. if we can't build a map that ever requires a fifth colour, then it can't be that there exists a map which requires more than four?

* It's easy starting from the end, just put one colour in the middle, then alternate second/third colours around the edge until you get back to the start and have to add a fourth. From construction, the worst case is from K1,2 then adding another for K1,3 at which point we need four colours already, but when we add two more on what will be the C5# around they each only connect to one each side and the center. In adding each one you can simply use the colour one hop over on the circumference.

# Since the center of K1,3 must be the center of the final C5-plus-thing, as it already had 3 spokes.

It's just that the C₅-plus-thing requires four colors, even though it doesn't contain any K₄ induced subgraphs. So we could imagine that there might be some larger planar graph that requires 5 colors without containing K₅. As it turns out, there isn't, but that's the thing that's hard to prove.

You could put them in front of light source that generated pure wavelengths designed to selectively activate their extra cone and ask what they see when you start mixing pure blue or green into it. Maybe they'd recognize those colors from their dreams, maybe not. I don't know. But it's a mechanism that would explain how tetrachromats might dream colors that trichromats don't.

We don't see bright, "pure" magenta very much, and when we do it's usually because we've engineered it. Nobody is engineering pigments for tetrachromats to enjoy.

If I'm right, people who are nearly (but not entirely) colorblind (anomalous trichromats) due to a shifted cone would likewise be able to dream in colors they are capable of seeing only dimly while awake, whereas dichromats wouldn't.

The trick to impossible colors is recognizing that our existing receptor spectra overlap. It's not possible to get "100% red" in the sense of only the "red" cone cells firing and none of the "blue" or "green" ones, because those other cells are going to fire at some small rate for the same frequencies.

But our brain can certainly receive such a color even if our eyes don't produce it, so in some sense we "understand" it as a color even if we'll never see it outside a laboratory or hallucination.

There are separately colors which we are wired to hypothetically receive, as you point out (and I find interesting) - IE, if you replace the eye with some sort of artificial replacement, then such a signal could get received by the brain, but with the current sensors it is impossible (also, I suppose a colorblind person could receive such a signal as 100% red without the other colors).

There's also the possibility during a dream of creating objects or sensations that are not physically possible. For instance, I've had dreams where I had a form of telekinesis, and obviously there's no sort of brain wiring to manipulate physical objects outside of the normal motor functions. Still, it made sense in the dream - and it's not like I could describe how I did it, and it's not because the hardware exists but the sensor doesn't.

Although - you could argue that there's a layer of abstraction in the brain of such things in the brain - for instance, when I'm typing these words I don't actually ever have any conscious awareness of my finger motions. I don't even think about the act of typing at all; only the words. So, thinking about it, the brain does have an idea of manipulating objects outside of a need for muscles to do it.

I would think for a tetrachromat the abstract sensation of seeing colors you can't describe to others is itself a familiar one, and that feeling itself is something you might encounter in dreams. My dreams tend to start with a feeling, and then build a context for it.

Tl;dr - dreams are weird.