Mapping out Tetrachromat Color Categories

In Describing Non-human Vision, I wrote

tetrachromacy is kind of convenient because you still have only 2 dimensions of hue, so you can actually diagram out what the color regions are, and just tell people "y'all already know how brightness and saturation work, so I don't need to put those on the chart".

    But I didn't actually try to make such a diagram. However, the last two episodes of George Corley's Tongues and Runes stream on Draconic gave me a solid motivation to figure out how to do it.

    Any such diagram will have to use some kind of false-color convention. We could try subdividing the spectrum to treat, e.g., yellow or cyan like a 4th physical primary for producing color combinations, and that might be the most accurate if you're trying to represent the color space of a tetrachromat whose total visual spectrum lies within ours, just divided up more finely--but the resulting diagrams are really hard to interpret. It's even worse if you try to stretch the human visible spectrum into the infrared or ultraviolet, 'cause you end up shifting colors around so that, e.g., what you would actually perceive as magenta ends up represented as green on the chart. The best option I could come up with was to map the "extra" spectral color--the color you can't see if it happens to be ultraviolet or infrared--to black, and use luminance to represent varying contributions of that cone to composite colors. Critically, if you don't want to work out the exact spectral response curves for a theoretical tetrachromatic creature to calculate their neurological opponent channels, you can map out the color space in purely physical terms, like we do with RGB color as opposed to, e.g., YCrCb or HSV color spaces. That doesn't require any ahead-of-time knowledge of which color combinations are psychologically salient.

    My first intuition on how to map out the 2D hue space was to arrange the axes along spectral hue--exactly parallel to the human sense of hue--and non-spectral hue, which essentially measures the distance between two simultaneous spectral stimuli. As the non-spectral hue gets larger, the space that you have to wiggle an interval back and forth before one end runs off the edge of the visible spectrum shrinks, so the space ends up looking like a triangle:

    This particular diagram was intended for describing the vision of RGBU tetrachromats, with black representing UV off the blue end of the spectrum; you could put black representing IR at the other end, but ultimately the perceivable spectrum ends up being cyclic so it doesn't really matter. If you want the extra cone to be yellow or cyan-receptive, though.. eh, that gets complicated, and any false-color representation will be bad. But, that highlights a general deficiency of this representation: it does a really bad job of showing which colors are adjacent at the boundaries. The top-edges spectrum is properly cyclic, but otherwise the edges don't match up, so you can't just roll this into a cone.

    Another possible representation is based on the triangle diagram of trichromat color space:

    Each physical primary goes at the corner of a simplex, and each point within the simplex is colored based on the relative distance from each corner. This shows you both hue, with the spectrum running along the exterior edges, and saturation, with minimal saturation (equal amount of all primaries) in the center. We can easily extend this idea to tetrachromacy, where the 4-point simplex is a tetrahedron:

    The two-dimensional hue space exists on the exterior surface and edges of the tetrahedron, with either saturation or luma mapped to the interior space. Note that one triangular face of the tetrahedron is the trichromat color triangle, but the center of that face no longer represents white.If we call the extra primary Q (so as not to bias the interpretation towards UV, IR, or anything else), the the center of the RGB face represents not white, but anti-Q, which we percieve as white, but which is distinct from white to a tetrachromat. This is precisely analogous to how the center of the dichromat spectrum is "white", but what a dichromat (whose spectral range is identical as hours) sees as white could be any of white, green, or magenta to us. Similarly, what we see as white could be actual 4-white, or anti-Q.

    Since the surface of a tetrahedron is still 2D, we can unfold the tetrahedron into another flat triangle:

    Here, in is unfolded around the RGB face, but that is arbitrary--it could equally well be unfolded around any other face, with a tertiary anti-color in the center, and that would make no difference to a tetrachromat, you as spinning a color wheel makes no difference to you. Note that, after unfolding, the Q vertex is represented three times, and every edge color is represented twice--mirrored along the unfolded edges. This becomes slightly more obvious if we discretize the diagram:

    Primary colors at the vertices, secondary colors along the edges, tertiary colors (which don't exist in trichromat vision) on the faces. This arrangement, despite the duplications, makes it very easy to to put specific labels on distinct regions of the space--although the particular manner in which the color space is divided up is somewhat artificial. And the duplications actually help to show what's going on with the unfolded faces--yes, the Q vertex shows up three times, but note that the total area of the discretized region around the Q vertex is exactly the same size as the area around the R, G, and B vertices.

    If we return to the trichromat triangle, note that you can obtain a color wheel simply be warping it into a circle; the spectrum of fully-saturated hues runs along the outside edge either way. Similarly, we can "inflate" the tetrahedron to get a color ball.

    If we want it flattened out again, any old map projection will do, but we have to keep in mind that the choice of poles is arbitrary; here's the cylindrical projection along the Q-anti-Q axis:

    And here's a polar projection centered on anti-Q:

    This ends up looking quite a lot like a standard color wheel, just extended past full saturation to show darkening as well as lightening; note the fully saturated ring at half the radius. However, the interpretation is quite different; remember, that center color isn't actually white. True tetrachromat white exists at the center of the ball, and doesn't show up on this diagram. And the false-color black around the edge isn't just background, it's the Q pole. If you need extra help to get your brain out of the rut of looking at this as a trichromat wheel, we can look at 7 other equally-valid polar projections that show exactly the same tetrachromatic hue information:

The Q pole.

The B pole

The anti-B pole.

The R pole

The anti-R pole

The G pole

The anti-G pole

(I probably should've done some scaling for equal area on these; the opposite poles end up looking like they take up way more of the color gamut than they actually do, and the false-color-black Q pole ends up getting washed out as a result. But I don't really expect anybody to use these alternate projections for labelling regions of hue--they're just to help you understand that the space really is a sphere, not a wheel!)

    And we could produce alternately-oriented cylindrical projections as well, if we wanted to.

    Of course, the full tetrachromat color space still contains two more whole dimensions--saturation and luminosity. But those work exactly the same way as they do for trichromats. Thus, if you want to create separate named color categories for tetrachromatic equivalents of, say, brown (dark orange) or pink (light red), you can still place them on the map by identifying the relevant range of hues and then just adding a note to say, e.g., "this region is called X when saturated, but Y when desaturated".

    Now, go forth and create language for non-human speakers with appropriate lexical structure in color terms!