Ágnes Holba & B. Lukács

CRIP RMKI, H-1525 Bp. 114. Pf. 49, Budapest, Hungary

holba@rmki.kfki.hu

lukacs@rmki.kfki.hu

Human tetrachromacy is extensively discussed in the literature recently, and it seems that we already understand the genetic background of at least some kinds of it. However its mathematics has not yet been analysed in details. Here we are going to discuss the dimensionality and topology of the colour space. As for its Riemannian geometry (so distances) the data are not yet sufficient.

Human tetrachromacy is practically the surprise of New Age or Third Millenium. After 150 years of Maxwelian trichromacy, empirically verified by industries as colour photography, TV, printing & clothes the genetic possibility was suggested in the 90's (see e.g. [1]), and the story of the first well-checked subject (the famous but anonymous "Mrs. Tetrachromat") started to be well-known in 2000.

Human tetrachromacy was completely unexpected, and while experiences must have existed earlier, for one reason or another they were never formulated not only in science but also in widespread common speech. So technical terms are missing and words may mean quite different impressions for researcher and subject. But then 1) a researcher (say, trichromat) cannot discuss the extra dimension of tetrachromats with another researcher without the danger of misunderstanding; and 2) misunderstanding is almost inevitable between a trichromat researcher and a tetrachromat subject. Indeed, it seems that tetrachromat subjects generally use either trichromat language, or no semi-scientific one at all.

In addition, it seems that different kinds of tetrachromacies exist, in which case even two tetrachromats may regard each other as a trichromat. To help future discussions, here we enumerate the "unusual" consequences of four (4) different sensors ion the retina. The simplest case seems to be the occurrence of 4 different opsins in the cones, which is genetically understood quite well now, and alleles of the L- and M-opsins are known. Still problems remain, but we return to this point later. Here we only note that 4 opsins are the necessary, but not the sufficient, condition. Namely assume, say, 2 different L-cones in the retina. If they are innerved into the vision center of the brain separately, then tetrachromacy is possible, but if it happens in a mixed way, the result will be a slightly anomalous trichromacy.

However since the 70's there are repeated comments about the existence of rods. (See e.g. [2], [3], [4], and, for a new review, [5].) Namely any "normal" retina does possess four sensors: L-, M- and S-cones and rods, and these four kinds of sensors are surely separately innerved. True, rods are saturated at strong illuminations, but still in mesopic situations one would expect tetrachromacy; but this kind of tetrachromacy is seldom reported if at all. Maybe the brain neglects rod signals when evaluating colours, and maybe this happens for colour constancy, against Purkyne effect. Still it seems that nobody yet has any clear idea how this happens and/or how general is this neglection in the population.

We personally do know person(s) with proven or very probable tetrachromacy albeit in this moment we still lack access to quantitative measuring apparatuses. One of these tetrachromats permitted to report her all observations while nothing personal except that she is female and well past her teens. We will refer her as Ms. Y, where Y is in no connection with her name; it is only an exotic letter, good as reference.

We give only the information in consensus in vision community. We have not performed any genetic research on this topics, simply collected the data necessary here.

Nobody knows the time of the first divergence leading to separate rod (scotopic) and cone (photopic) opsins. Maybe it is almost as old as vision itself. Note that terrestrial vision has 3 different illumination levels:

1) With natural point sources. Starry sky with Venus at her brightest is not brighter than -5 magnitudo, and Venus is visible only just before or after sunset. Without Venus it is rather -3 m at peak. Without planets, the brightest is Sirius, -1.6 m.

2) With Moon. On most nights Moon is visible, but only in part of time. This means cca. -11 m luminance at peak, but then it may be anything between -11 m and the -5 (or -3) m of the previous level, depending on the actual lunar phase & clouds.

3) With Sun. The peak luminosity of Sun is cca. -26 m; then this may decrease to almost anything depending on the degree of overcast and time of day. After sunset an originally slightly lower luminance gradually fades into Level 2 or 1 in cca. 1.5 hours.

So this is 25 magnitudo range, or 10 decimal orders of magnitude. Obviously the range is too big to use one kind of sensors. Human vision uses two: rods are appropriate for night vision (Levels 1 & 2) and cones for day vision. However note that even night vision must cover 10 magnitudos or 4 orders of magnitude, so different mechanisms for Levels 1 and 2 might be advantageous for some animals.

Human night vision seems completely colourless, so human rods are uniform. As far as we know, this seems to be true also for other mammals, birds and reptiles. For other vertebrates and for invertebrates we do not dare to generalise.

The human rod opsine, rhodopsine, is coded on Chromosome 3. Errors lead to bad night vision; but deficiency in Vitamine A, plus probably a lot of deficiencies & innate troubles of eye may lead to similar night vision problems, so it is more modest to refrain from quantitative statements here. The rhodopsine peak absorption is at 507 mm, and maximum sensitivity of night vision is reported in the blue-green in general and seems to have almost the same wavelength [6].

Now it seems that cone opsins have a very old dichotomy; it may go back as far as 700 My. This was a blue-yellow dichotomy. Then, again, it seems that both opsins were doubled (in ancient amphibians?), and now birds, direct ancestors of diapsid reptiles, are tetrachromats, with very roughly a "red", a "green", a "blue" and a "violet", or "near UV" cone. We wonder, what opsin may be homologous, or even paralogous to human ones.

Now, mammals evolved on another clade, independent for at least 310 My [7]; and it is doubtful if at the stem of mammal tree (in the Triassic, cca. 180 My ago) this branch even reached trichromacy. If it did, later, in the Jurassic and Cretaceous, mammals lived "in the shadows of" much larger dinosaurs, and one modest tactics of avoiding them was night activity. (Another was arborealism, and this was the way of our ancestors at least since Uppermost Cretaceous). Night activity may lead to atrophied day vision (albeit not its total absence), so maybe the clock of evolution of colour vision was reset at the C/T boundary. Now primates are di- and trichromats, but most other mammals are dichromats at most.

So we may use the working hypothesis that at the C/T boundary mammal cones exhibited a Blue-Yellow dichotomy. Now the Short opsin, very possibly the inheritor of the old Blue, is coded on Chromosome 7, an autosome.

However, the other two opsins seem to have originated very recently from the old Yellow (details come immediately) and both genes are on Chromosome X, a sexual chromosome. We think that originally the position of the Yellow gene had nothing to do with sexes; this position on the precursor(s) of human X may be much older than the present mammal sex determination mechanism (note that mammal and avian determinations fundamentally differ while alligator sex determination might even not be chromosomal). However the present X position leads to much higher dichromacy rate in males than in females.

Human Red and Green opsins are rather similar both for structure and for amino acid sequence, in accordance with a gene doubling, maybe not older than 20 My ago. The Red and Green genes are still adjacent, and some erroneous crossovers leading to hybrid "RG" opsins are recently reported.

Indeed, generally Prosimii seem to be dichromats, and also the New World Anthropoidea, with the exception coming in next Chapter. So then Red-Green opposition is a new ability, and primate Red and Green opsins cannot be homologous with reptile or avian ones.

Genetic and colour vision measurements seem to be not fully conform. In vitro measurements lead to peak absorptions cca.

In the same time "greenest green" of colours seem to appear rather at 520 mm, and colour measurements would suggest an L peak much behind 560 mm. Also, the present knowledge is not enough to settle the "violet mystery". Most people (but seemingly not all!) see Violet at the extreme short wavelengths. Now this Violet is generally reported as a mixture of deep Blue + moderate Red. (This is the reason that it is not utterly meaningless to close the spectral horseshoe into a "colour circle", albeit more reddish purples obviously do not exist among monochromatic distributions. We shall return later to the spectral horseshoe, but with neglecting anything beyond its parabolic approximation.) The reappearance of Red impression may have a simple explanation with a secondary maximum on the Red opsin at cca. 400 mm ; the problem is that no in vitro measurement shows this secondary maximum, and also some in vivo colour matching experiments lack it. For this study we will not discuss the violet mystery, only note that alleles of opsins may be behind.

So then: canonical human vision seems to use four pigments. "Blue-green" in the rods, "Red", "Green" and "Blue" in the cones. Canonical colour recognition is 3-dimensional, which gets its easy explanation if only the signals of the 3 cone types are used there.

As told, it is well known that men are much more frequently "Daltonians", so red- or green-deficient than women. The reason is clear. Normal men have only 1 X chromosomes, and gene defect there leads to abnormal or defunct Red or Green gene. With 2 X's (normal women) one of the X's will be eliminated later, but in a multicellular phase, so there will be mosaicism in the retina and cca. half of the endangered cones will still be normal. Klinefeldter men (genotype XXY) then would exhibit the lower, female, incidence, but this syndrome is rare, and leading to multiple anomalies colour vision is not the first to check. Turner women (Xø) should also exhibit male incidence, but again the Turner syndrome leads to multiple anomalies. Triple-X "superwomen" are known, and obviously colour vision problems should be rarer for them than for XX women, but we do not know any measurement.

Now, alleles for both L and M opsins have been found. Generally the differences in peak absorptions are not more than a few mm, so they can be classified either as L opsins or as M ones, and may call them L' and M'. However "hybrid" opsins are sometimes reported or/and discussed, and then they must have peaks between M and L. We denote here this opsin as H.

Obviously alleles of S opsin are possible too, but research do not seem to concentrate on them.

Now let us stop here, with solely cone opsins. Then men may have the following geno- and phenotypes:

where "..." stands for very similar phenotypes.

Blue mosaicism would mean four cone opsins altogether, but this phenotype seems not to have been reported. 0 stands for a very defected gene coding a defunct (or very bad) opsin. Alleles L' or M' will cause minor individual differences, but it would need very much self-knowledge or great empathy to distinguish SSML from SSM'L. But SSHL clearly differs from SSML, and the first is detected surely as a deuteranomale with "anomalous green".

However there are much more combinatorial possibilities in women. Again not to give explicitly minor differences, the classes are at least

Phenotypes SS00LL and such, with defect on both homologous chromosomes are of course possible but too rare to discuss them here. But it seems that red or green mosaicisms must be frequent. Also, if we estimate the occurrence of gene H from male colour anomalies (up to 5-10 %), "fundamental" female tetrachromacy should be quite common.

The term "fundamental" means that the, say, L and H opsins differ much more than L and L'. In the second case in vitro experiments suggest 4-6 mm difference, near to the edge of colour distinction [8], while H may differ both from L and from M as much as 15 mm. It may happen that nobody observes a difference between retinae MM and MM' "for first sight", but MM and MH phenotypes should differ perceptibly.

And while there have been found a few female tetrachromats, their occurrence seems well below the expected 10 % level. While lots of diverse causes may cause this, obviously one possibility is the problem with communicating her own colour experience with trichromats. In a good communication the tetrachromacy of the subject would stand out; but look at the standard terminology.

Boys learn a simple and logical systems. There are a few words for "pure" or "saturated" colours as "red", "yellow", "green" and such; almost all other colours are either mixtures of them (e.g. "bluish green"), or pastels of them (pale green, dirty green &c.), or both (pale bluish green). The system is clearly three-dimensional.

Girls generally learn colour names from mothers, but the female system is seldom so logical. Lots of individual colour names may occur (as e.g. periwinkle, crocus & mauve), and it is far from being clear if two girls communicating use exactly the same system or not (just under investigation). In addition, SSMHLL tetrachromats have SSHL father and SSMMLL mother or SSML father and SSMHLL mother, neglecting the rarer possibilities. Now, in the first situation both parents are trichromats, so the genetically tetrachromat girl cannot learn her extra degree of freedom from the parents.

In such cases the present study may help.

We accept that tetrachromacy does exist. We also accept the common opinion about the 3+1 fundamental opsin types. However we do not want to use any more hypotheses about vision mechanisms; as e.g. opponent theories, saturation of rods &c. For example, it is not a fundamental question in recognising the subject's own tetrachromacy for the subject, how exactly it happens. However clearly at least 3 major different types may appear. The 3 "canonical" receptors used by colour vision will be denoted by L (Red), M (Green) and S (Blue).

Case 1: Close alleles. In a female SSMMLL' there are two alleles of L, but both are "fundamentally red". It is possible that the simultaneous occurrence of L and L' result in better colour discrimination at the red end, it is even possible that the subject recognises the better than average discrimination, but the result is probably "not dramatic". Mutatis mutandis, similar resultscome from MM' or SS' alleles. Clearly in females this results in 27 slightly different colour visions. In males the number of classes are 12.

Case 2: The occurrence of a hybrid kind of "yellow" gene. Erroneous crossovers are reported resulting in a gene seriously differing both from the original L and from the original M. While surely there are several such variants, here we deliberately restrict ourselves to a single H variant, exactly between, for general oversight. Then the maximum of the absorption of the H opsin is cca. at 548 mm. Then the subject has an extra receptor at Yellow or Orange. (There is some contradiction here between wavelengths and colour names. We guess that this problem is the same as the contradiction between in vitro peaks and in vivo impressions; and this latter contradiction may get its final explanation in some details of the vision mechanisms.) This Yellow cone is then not the original primate Yellow 30 My ago; but its action is similar.

This tetrachromacy can occur only in females. It has two subclasses, as SSMHLL and SSMMHL. In the first case the extra degree of freedom is most prominent in the "Green-Yellow" range, in the second case in the "Yellow-Red" one. The quotation marks indicate that in both cases the subject's Yellow impression is not homologous with ours, but her Yellow, coming from maximal excitation of her H cone occurs roughly when we observe Yellow (cca. equal excitation of our L and M). We will return to this point later.

Besides of better colour distinction, SSMMHL tetrachromats can easily be detected via usual matching apparatuses, used in traffic tests. Because of higher dimensionality of their colour space they generally are unable to match a prescribed orange shade from the Red and Yellow lights of the apparatus. The reason will be clear in the next Chapter, but we tell already here that this inability comes from better colour vision: their colour vision is better than should be for the measuring apparatus. Daltonians always can make a match which they accept.

We personally know such a tetrachromat, not the same as Ms. Y. She, however, up to now did not permit us to publish the details of her vision.

Case 3: An extra receptor between M and S. While this is a logical possibility, and mutations can cause almost anything, for first sight one would not expect the extra cone between M and S; e.g. crossover hardly could produce such a gene. Still, Ms. Y's self-reports seem to show this combination. In addition, she was measured at the Technical University of Budapest by Dr. Klára Wenzel & collaborators, with the result of anomalously good discrimination in green, with the maximum intensity of the "other green" (explanation will come) at 513 mm. Since this wavelength is rather short for H genes, this must have been classified into the present Class 3.

Of course, it is possible (as far as we know now) that there is here some influence of rods on the cone system (anyhow); we are rather near to the rod (rhodopsine) peak at 507 mm, but it is better to remain at facts. A very short resume of Ms. Y's reports can be found in Chap. 9.

This Chapter is about "normal trichromats". They compose the majority of the population, colour industry is based on trichromatic theories and we all learn these theories in school (if at all). So all the content of this Chapter should be known. However remember Moliere's hero who was surprised when a rhetoric expert told him that he spoke in prose. Normal trichromats generally evaluate the impressions automatically, not observing themselves. So:

For a trichromat:

the colour space is 3 dimensional;

for constant luminance there are surfaces;

their border is a horseshoe & the purples;

the horseshoe

is called spectral loop;

is concave;

the colours on the horseshoe are maximally saturated;

they are monochromatic;

they can be one-dimensionally arranged uniquely;

while purples are not monochromatic and do not fit into the sequence.

Now come the explanations:

The colour space is 3 dimensional, so a colour can be denoted & reproduced by means of 3 independent numbers; and generally 2 is not enough. For simplicity's sake remain well in the middle of the space. There a prefixed Red, a Green and a Blue lamp are enough. By slots you can use more or less lights from any of them, and by an appropriate set of weights you can form any colours there. Such a statement is not true globally, but only because of the concavity of the border. This 3 dimensionality means the inherent triality of stimuli going into the brain, then the infinite variance of the frequency distributions result between eye and brain in the manifold of a number triad. This is automatically so if only the 3 kinds of cones contribute to Colour. Maybe rods are neglected.

Luminance is somehow easily separable. We can easily recognise that two lights are "essentially the same, except for neutral amplification". In planar geometry we can easily recognise that two triangles are not identical but similar, meaning that all sizes are different but proportional, and angles are pairwise equal. The similarity in hearing similar sounds transposed an octave away is somewhat misleading. In Riemann geometrical colour theories (e.g. [9]) this means the existence of a conformal Killing vector, but it is enough to be simply mentioned. Indeed simple enough properties of colour decoding in the brain can lead to such structure [9], [10], which seems to be true, at least in good approximation. A saturated turquoise remains "similar" if brighter or dimmer. "Really different" colours occupy surfaces, on which, say, the sum of the three intensities are 1.

Now, these surfaces are bounded. Observations unequivocally show that the boundary is composed of two quite different segments. There are, first, 2 limiting colours: extreme red and extreme blue (or extreme violet). Extreme red impression is made via monochromatic lights of longer and longer wavelengths. The impression is purer and purer red, with less and less luminance. If we think of an L receptor with a long-wave peak, extreme long wave lights already do not excite M and S, but of course, beyond the peak also the L excitation decreases. Amplifying, however, the light appropriately, the limiting impression is a "pure red" of unit intensity. This is the purest excitation: only L responds. Redder impressions in the brain are simply impossible. The short-wave end may be similar (Blue as fundamental colour), or the L receptor may have a second maximum at short wavelengths (this is again the Violet Mystery). Anyways again we get a limiting impression beyond which one cannot go. Between the two extremes there must be always a boundary line, because the sum of the three coordinate is 1 and none can be negative (negative excitation being impossible). However we can mix Extreme Red and Extreme Blue too. The resulting excitation from such a mixture, however, can hardly be anything other than the sum of the 2 extremal excitations, so on the 2 dimensional section of the 3 dimensional coordinate space the loci of the mixture of extremes could hardly be anything else than a straight line. Purples are, by definition, extreme red + extreme blue (or violet).

However there is another segment of the border between the extremes. Note that we cannot subtract lights, only add to. So some weights must be positive semidefinite. Now, clearly monochromatic lights are physically possible, and their sum with nonnegative coefficients too. If we mix monochromatic lights, they will make different colour impressions. If we reached the point where one weight is 0, we cannot go further into that direction because that would make one weight negative. Now the spectral loop consist of monochromatic lights. Therefore spectral lights must be at the boundaries, although not all the boundaries are spectral lights.

The loci of the monochromatic or spectral lights are generally on a convex curve (although limiting linear parts are possible). Consider the 3 receptors, L, M and S. Until S is not excited, a stimulus of shorter and shorter waves result in less excitation in L but more in M. But the sum is fixed. The result is a straight line. However with shorter and shorter wavelengths S will be excited also. Now consider an M with a peak between L and S. Par excellence longwave stimuli will result in much L excitation, less M and even less S. Par excellence short-wave ones will result in much S, less M and even less L; and both kinds of points are on the boundary. Now, it is easy to see that the arc of two boundary points are inside: the mixture of two boundary lights are physically possible, so their result must be also on the surface, generally not at the boundary. An exactly triangular boundary (so with a straight boundary between Extreme Red, and an Ideal Green and another between Ideal Green and Extreme Blue) would need ideal and practically impossible sensitivity curves. For further discussions see [9].

If you add a neutral light to a chosen light, you get a less saturated one. There is a light called white, defined purely traditionally, with 3 coefficients whose ratios are constant (and all 3 are constant on a surface of constant luminance). So on this surface White is a fixed point somewhere far from both the horseshoe and the purple line [11]. Now take a point on the horseshoe and connect it to the White point by a straight line. When going inward, the light is whiter and whiter, so, by definition, more or more neutral. So it is the least neutral at the boundary. This is just the way saturation is defined.

The monochromaticity of the boundary colours was already discussed. But be careful: colours are not monochromatic. However monochromatic lights made impressions of boundary colours. This is so because wavelength distributions with locally negative weights are impossible to be produced.

Now, from the above points it follows that the boundaries are the spectral horseshoe and the purple line. On the horseshoe there are the points which can be produced with monochromatic lights. (Maybe not only with monochromatic lights, but they can be made with them anyways.) For the manifold of monochromatic lights there are only two degrees of freedom: the total intensity and the wavelength. But by scaling out the luminance absolute intensity is a unique function of wavelength, and not independent anyway. So monochromatics of unit luminosity can be arranged according to wavelength only, which is one-dimensional. A popular way is the warm/cold sequence. Reds are longwave and warm, blues (or violets) are shortwave and cold, others are in between. The usual names are red>orange>yellow>green>turquoise>blue(>violet).

Purples are neither warm nor cold; but reddish purples are rather warm and bluish purples rather cold.

So with N receptors the colour space is N-dimensional, the hypersurface of unit luminances has the dimensionality N-1, and the boundary of undiluted colours of maximal saturation is of N-2 dimensions. On the other hand the spectral loop is always of 1 dimension, from pure physical reasons.

The maximally emotionless and reproducible colour naming system for trichromats is used by men of industrial societies. Since the colour space is 3 dimensional, you must give 3 independent data. One is total intensity for lights or, say, albedo for reflecting surfaces. A light can be intense, medium or dim, or anything between. An albedo can be almost 1, can be medium, or can be almost 0.

Now, defining the intensity, we can act further as if the intensity were unit. Then we use the White point defined by the community (e.g. white is limestone, or chalk or the paint to whitewash walls, or pure snow, depending on environment). Then virtually, in our brains, we subtract as much whiteness from the actual colour as possible.

Then we arrive at the boundary, either at the spectral horseshoe or at the purple line. On the horseshoe some "pure" colours are fixed in the community. Red is the extreme longwave impression, Orange is maybe an orange or a carrot, Yellow is either similar to gold, or if we are unfamiliar with gold, then brass, or a lemon. Pure green is perhaps grass or the foliage of a leafy tree (but not a pine), under good watering. Blue is either a unique colour defined by some fruit, or two (Russian sinii and goluboi, Latin lividus, "plum-colour" and coerulus/coelurus "sky-colour"). Now colours in between can be named by a compound of two neighbouring colours, as reddish orange and yellowish orange; or greenish yellow vs. yellowish green. On the purple line there are first very reddish purples, then reddish purples, purples, bluish purples and maybe very bluish purples are already violets.

By means of this system all possible colours can be well approximated. Among males there are two substantial deviations from the system. "Pink" is sometimes "pale rose", so even paler red; but more frequently a pale (reddish?) purple. But this colour name always carries some emotion. The more fundamental deviation is "brown". Brown is cca. Yellow (or Orange) plus Black, so a low intensity/albedo pastel Yellow. Still it is handled as a new colour.

The Maxwellian trichromatic theory is the scientific form of the system of the previous Chapter. Take 3 lights, as saturated as possible. It is best to take a pure Red, a pure Green and a pure Blue, pure here means that "from the boundary", and for Blue as shortwave and for Red as longwave as technically appropriate. Then all the lights you can compose are in a topsy-turvy pyramid, infinite upwards. Again you can take a section of unit luminance, neutral amplification being trivial. Then your original lights are 3 points and you can produce all the mixed lights on and within the triangle spanned by the three points/lights. Of course still the horseshoe is convex and you cannot produce the lights between the curved horseshoe and the borders of the triangle, but you can make that area minimal by choosing a best Green, and cannot help about the remaining minimal inability. (This is the reason for multicolour printing of expensive catalogs & such, and this does not disturb the 3-dimensionality of the space at all.)

Now you can start with Red. Adding more and more Green, you get Flame Red, Reddish Orange, Orange, Yellowish Orange and Yellow. Yellow is qualitatively "half of Red, half of Green", the exact numbers depending on the choice of Green. According to colour matching experiences the boundary is almost straight until Yellow. So no normal trichromat can distinguish even very pure Oranges from proper mixtures of a very saturated Red and of a very saturated Yellow.

But you cannot mix a very pure Yellow from Red and Green, because the boundary starts to curve from Yellow to Coldward. This happens because the S receptors start to be excited too. Still Yellow is Red + Green. Now if you add more Greens and less Reds, you are going into Yellowish Green, then into Greenish Yellow. When you completely switch off Red, you are in Green, of course.

Now you start to add Blue to Green. The midpoint is Turquoise, or Cyan. Before you had Bluish Green, after Greenish Blue. Then you arrive finally to Blue; or if your extreme shortwave light was Violet, then Blue is Violet + a small amount of Green; but with a Violet light more area is irreproducible between the boundary and the triangle (and we are not interested now in the Violet Mystery).

After the Blue (or Violet) corner you may add Blue and Red, and then you move on the Purple Line, if your Red and Blue were indeed the good representations of the results of extreme long- and shortwave stimuli. Namely, as told, the purple boundary is necessarily straight line.

The inside of the triangle is Red + Green + Blue. Let us define now White as 1/3*(R+G+B). (This is not necessary but simple.) Take an inside point, say

P = x*R + y*G + (1-x-y)*B

none of the coefficients negative. Now take the smallest coefficient. There are 3 cases but all them go likewise, so let us assume that x is the smallest. If so, the interior light can be composed from a limiting colour (which is now Bluish Green or Greenish Blue) and from White, as

P = 3x*W + (y - x)*G + (1 - 2x -y)*B

all the new coefficients nonnegative by construction. So P is a mixture of some White, + a limiting colour, namely, normalised to unit luminance,

[(y-x)/(1-3x)]*G+[(1-2x-y)/(1-3x)]*B

Colours are warm on the boundary from R to G, cold from G to B, and purple from B to R. They are saturated if the weight of White is almost 0 and diluted or pastel if its weight is almost 1.

Now we turned the colour visions of a trichromat into mere numbers (except two narrow lunules between the triangle and the horseshoe), and from this moment everything is reproducible either without humans. This is colour TV, colour photography, colour printing & such.

The two lunules appear because you cannot produce a stimulus exciting only the M receptor (the M peak being between the other two). The irreproducible area (even with optimal lights) appears for monochromes exciting all the three receptors. If there are substantial trial overlaps, the borders are not even similar to straight lines, and then very saturated greenish lights cannot be produced with our 3 lights although the space is 3-dimensional. Alas, mixing is a linear operation.

But what happens if a tetrachromat is present?

For simplicity it is better to assume that the mutant opsins form a finite set, and in this Chapter I assume that the only possible mutant opsin is H, halfway between L and M. As told above, some tetrachromats seem to have such genes.

As you start to change the wavelength from extreme long downward, first only L is excited, then starts also H. M is still far away.

Now for her Orange is Red and Yellow. This is just as for us, but her Yellow is not the trichromat's Yellow. In first approximation, and this is necessary to emphasize now (wait a moment, why) Red is the colour of a monochrome light say with 630 mm, orange is another with 600 mm, and yellow is a third with, say, 580 mm. Then what is the difference between her and us at all?

Look: where we have 2 receptors, R and G, she has three, R, H and G. Our Orange is (almost) a mix of Red and Yellow, and Red is the colour of the monochrome of 630 mm, while Yellow is (say) that of 580 mm. For us a proper sum of a sharp peak at 630 mm and one at 580 mm (almost) makes the same impression as one peak at 600 mm. But for her NOT!

Namely for us a 600 mm monochrome excites strongly L, somewhat less M and practically not S. The 630 mm signal practically excites only L, and a 580 mm one equally L and M, and still not S. So it is the same for us if we get a signal of one peak at 600 mm or a mix of two peaks at 630 mm and 580 mm.

But the two signals are not equivalent for her, because she has 3 receptors in the range. So her Yellow is not the monochromatic lght between Red and Green, different from the equal mix of Red and Green! (Or who knows what she call Yellow; it depends on childhood language lessons.) She can distinguish between two peaks and one peak, and Yellow for her is not the mix of equal Red and Green. We got a lunule between Red and Green where (saturated enough) Yellows could not be mixed from Green and Red lights; she, even if tetrachromat TV technique were at reach, would feel TV colours flat enough. Another such lunule would appear from Yellow to Green. Below 510 mm the situation is roughly as for us, because there already L is not excited and even H is hardly.

But the biggest difference is not simply 3 lunules instead of 2. The dimensionality of the trichromat colour space is 3, that of the (hyper)surface of equal luminances is 2, and so that of the border horseshoe is 1. So all the fully saturated non-purple colours are also monochromatic colours. All non-purple bright colours have an equivalent wavelength, or, what is the same, a colour temperature. (OK, warm colours have low colour temperatures and vice versa, but this simply means that psychology is not physics.)

However for our Hybrid Tetrachromat, having L, H, M and S there are fully saturated, maximally chromatic colours which cannot be produced via a monochromatic stimulus. And it is easy to understand, why.

Consider L, H and G; S is far down. First let us give a monochromatic stimulus. Much above we told that the peak sensitivity of L is at 560 mm, H's is perhaps at 548 mm and M's is at 535 mm. Consequently (this mere word stands for a lot of calculations and thumb's rule guesses), using the very forms of the peaks, for signal wavelengths above 600 mm L is strongestly excited, H is second and M is third, between 600 mm and 580 mm H is strongestly excited, L is second and M is third, then between 580 mm and maybe 560 mm H is strongestly excited, M is second and L is third. Below 560 mm for a while (until 500 mm?) M is first, H is second and L is third.

If somebody believes that all the logical possibilities were narrated, let us see them in formulae:

But for three receptors there is six schemes. Our formulae do not contain L > M > H and M > L > H!

Wait a moment until saying that would be unnatural! With two ingeniously placed narrow peaks you can excite stronger the M and L receptors than H, and this is the reason that a Hybride Tetrachromat (somebody also with the hybrid of L and M as well) can feel more colour impressions than trichromats.

This is so simple; even we, without a H receptor (and one of us a mere male although at least not a Daltonian) can discuss what a Hybrid Tetrachromat can distinguish. We do know one who is certainly a tetrachromat, and very probably one with a H. One of us asked her and indeed her Flame Red was something rather Rosy for us, and she reported that she was not able to make matches admissible for herself of Orange to Red + Yellow on a standard optical machine originally manufactured to detect protanomales and deuteranomales for traffic licence checks. Of course, the machine produced Standard Trichromat Reds, Yellows and Oranges. She can distinguish some Oranges even from best sums of Reds and Yellows.

Of course it is not easy for tetrachromats. They see differences between colours for which they did not learn different names. They cannot arrange uniquely buttons of paints or colour fibre pens into a sequence, and then they maybe believe that their colour vision is wrong. But simple mathematics shows that their bordering, purest & strongest colours are on a 2 dimensional ribbon, while the wanted arranging would need a poorer colour vision. There is a colour temperature for H>L>M>S, maybe cca. 3000 K (stars of spectral class K, maybe; more research is needed). But from elementary physical reasons no blackbody radiation can produce L>M>H>S while they may see such colours.

Although chance hybridisation of 3 opsin genes might have resulted in 6 types of tetrachromacy, really only the hybridisation discussed in the previous Chapter is possible. Namely for crossover the genes must be on the same chromosome (and very near, in addition). The rod opsin is coded on Chromosome 3, the S gene is on Chromosome 7 and the M and L ones on Chromosome X. So H can result, but no other hybridisation is possible.

And still there is Ms. Y. In her teens she observed various strange vision impressions and gradually discovered that, in spite of the statements of her relatives, "Green is not a boring colour". Sooner or later she observed that the green leaves of bush can easily be distinguished from the green wire-fencing.

To separate Hungarian cultural exotica from biology & physics, let us write down that it is usual in Hungary to paint the garden fences green. That green is not always the same green, but it is always a chemical product of organic industry. It does not mimic clorophylle green even for trichromats; still the difference is not too great to trichromats between fence green (in this Chapter simply Fency) and some plant green (in this Chapter simply Grassy). We note that in Hungarian this terminology of Ms. Y (mirror-translated) would not be forced at all. Magyar is an Uralic language, but "green" is "zöld" from Osetian "zelde" = "(a kind of) grass".

Mere trichromats can see distinctly any plant even on the fence background, but from bigger distance they tend to merge. Not for Ms. Y who tells that she easily sees the borderline "from any distance"; if any merging occurs, then, at great distances, the fence merges with blue objects. Also she lacks Fency (and her favoured Turquoises) in TV's and of course cannot mix it in computers. She tells that European colour photography (or colour prints) rather use Grassy while Japanese ones rather Fency. On a national holyday she detected difference between the national Red-White-Green flags on neighbouring houses: one was Red-White-Grassy, the other Red-White-Fency. (The law defining the national flag needs 2/3 majority so it will be difficult to upgrade it). She was rather hilarious when discovering that on the signboard of a bio shop the desirable plants & herbs were painted with Fency. She reports that all the Plant Kingdom, with one sure and one possible exception, uses Grassy. This is rather natural, being the plant chlorophylle a definite spectrum. The sure exception (according to her) is Blue-Green Algae (or maybe some moss?). And look: blue-green algae aka Cyanophytae, of course contain clorophylle, but also a special chromatine, phycocyan, and for trichromats the result is a blue-green chromatoplasm. Hence the very name; and the Classical Greek "cyanos" when does not mean simply undefined "blue", means blue-green. (True, you must read and reread Aristotle's De Anima until understanding this; De Coloribus is not sufficiently detailed.)

The probable exception is lemon fruit, but first we must discuss again mixing. The scheme is rather similar to that of the previous Chapter, but now there seems to be two receptors in the middle region. Let us use now the characteristic colours for the receptor, so there is

We do not really know the peak position of F, but Ms. Y reports a measurement where she had the maximal Fency impression at 513 mm. Because of overlaps we could guess slightly longer value, but hard to tell, what. This value is good enough to tell the story.

This scheme suggests that the trichromat representation of Fency is slightly bluish green, similar to the Scarab Green reported at 510 mm average and 70 mm transmission width in Godlove's classic article [12]. Indeed, Ms. Y tells that some beetle greens seem more or less Fency for her. If so, Fency seems a simple distribution with one band, but this is not yet sure.

Indeed, in many cases her Fency is more bluish for trichromats than her Grassy; but not always. And of course as with an independent H receptor in the previous Chapter, it is not necessary that Fency have higher colour temperature than Grassy at all. For a while on the two sides of trichromatic M or Green the border of her colour space widens from a line to a ribbon, not very wide but of 2 dimensions. The spectral loop is a line on this ribbon, and the colours of monochromatic lights do have colour temperatures. But not all behind trichromatic Greens. Maybe Fency can be monochromatic and Grassy not (suggested only by the Fency nature of Scarab Green, but that was identified by trichromat Godlove), but we seem to feel the opposite more probable; detailed comparison of European and Japanese colour printing paints might help, but it is not easy to start a new research with a single object of measurements.

Now let us see mixing. Obviously in the red, orange, violet and deep blue sectors the G and F receptors are already far, so very handmade stimuli would be needed to feel anything from the doubling of the Middle receptor. There the ribbon of the fully saturated colours has practically narrowed into the usual one-dimensional line. However from Yellow to Turquoise the situation is not hopeless.

For trichromats Yellow = Red + Green. However for Ms. Y two kinds of Yellows should be possible (if not, even the reality of the feeling could be doubted) and Red + Grassy should give a "Yellow" definitely different from Red + Fency (although of course with smaller difference than between Grassy and Fency). Now, the subject does report differences between a "cold" Yellow and a "warm" one. In the best case the terminology is arbitrary. However the subject reports that all yellow fruits are "warm" yellow, with the possible exception of lemon fruit.

Now, if it is true then, this would mean that plant green, Grassy, + Red is the "warm" Yellow, but then the very young "green" lemon fruit should be Fency. The situation is tempting and we are waiting for young lemons on sheltered lemon trees in Budapest, North latitude 47°, but truly lemon yellow is somewhat "colder" yellow even for trichromats (although this means surely that lemon yellow is greener for trichromats). We are translating private terminology of a single individual, not the optimal situation for linguistic evolution.

The shortwave side of Grassy and Fency is rather obscure up to now. The subject should be able to distinguish Grassy + Blue from Fency + Blue; the second would be more frequent and so more familiar than the first, because a single spectral line between say 500 m and 445 m always would excite F and B stronger than G, and only special 2-band reflection spectra would result in G + B. But so far the subject's reports are equivocal. Sometimes she reports that she does see "only one kind of Turquoise", but sometimes she tells that Grassy + Blue "is simply not Turquoise, but a boring Green + Blue", and sometimes she reports tiredness of eye. However her colour distinction is abnormally good in the green range, not deteriorating where it generally should be [8], [13], so it would be hard to doubt an extra receptor; but we must admit that we do not have so far viable ideas about the mechanism.

[1] J. Mollon: Nature 356, 378 (1992)

[2] P. W. Trezona: in Colour Vision Deficiencies III, ed. G. Verriest, Karger, Basel, 1976, p. 52

[3] Ágnes Holba & B. Lukács: in Proc. 6st Symp. On Matter Evolution, eds. B. Lukács & al., KFKI-1995-21, p. 60

[4] J. Winderickx & al: Nature 356, 431 (1992)

[5] L. T. Sharpe & al., in K. R. Gegenfurtner & L. T. Sharpe (eds.): Color Vision: from Genes to Perception. Cambridge University Press, NY, 1999

[6] Landolt-Börnstein: Zahlenwerte und Funktionen. Elektrotechnik, Lichttechnik, Röntgentechnik. Springer, Berlin, 1952

[7] S. Kumar & S. B. Hedges: Nature 392, 917 (1998)

[8] R. E. Passingham: The Human Primate. W. H. Freeman & al., Oxford, 1973

[9] J. Weinberg: Gen. Rel. Grav. 7, 135 (1976)

[10] B. Lukács: Acta Phys. Pol. B19, 243 (1988)

[11] B. Lukács: Acta Antiqua 33, 399 (1990-92)

[12] I. H. Godlove: J. Opt. Soc. Amer. 37, 778 (1947)

[13] G. S. Wasserman: Color Vision. Wiley Interscience, NY, 1978

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