B. Lukács

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

Colour is one of those areas of knowledge where everybody has experience but there are lots of interpretations. In the same time colour TV works.

Without claiming completeness, the various paradigms existing amongst recent population can be called as Aristotelian, Newtonian, artistic and female. We can ignore here the last two: they cannot be formulated in ways usual in natural sciences, and the female one cannot be understood by anybody having only one Chromosome X (important genes for red & green cones being on X) [1].

The Aristotelian colour paradigm is scientific; only Aristotle of Stageira had no time to finish it [2]. In some sense Goethe finished it; but it does not work in technical sense.

The most detailed Aristotelian text about colours is in On Sense and Sensible [3], but you can find notes also in Meteorologica [4]. Finally, there is the pseudo-Aristotelian On Colours [5]. In all of them colours are mixtures of black & white (or darkness & light), or they are between them. Books 1 & 2 of On Colours give definite suggestions, how to get crimson and violet by mixing black & white. Still, we cannot get colours from black and white, except one way: by rotating a Fechner-Benham disk, a disk with tricky black&white pattern. It will seem coloured if rotating rapidly; the probable explanation is physiological. I guess that at Bekker number 792a, where the text speaks about the purples of birds' wings, Aristotle observed this effect.

Our present paradigm is based on works of Newton, Young & Maxwell. There are some fundamental colour impressions. Maybe they are seen when exciting one kind of photoreceptors. Other colours are mixtures; they are seen when more than one groups are excited. It seems that anatomy has found the primar photoreceptors (the cones); and it seems that apparatuses built according to these principles work smoothly.

However technical success may suggest oversimplifications. The triality of fundamental impressions is not at all sure: rods have a fourth sensitivity function, so tetrachromacy would be natural. In recent years data are accumulating about female cone tetrachromacy [6]. Sauropsia (Reptilia & Aves) seem to be routinely tetrachromats. And so on.

Now, the recent paradigm can handle such modifications. But in such times I think my earlier works should be saved from oblivion. Enjoy them.

Colour theory uses Riemannian geometry. In the subsequent texts I used it minimally; but the generalisation would need it. For an operational incorporation see most definitely [7], and also [8] for a slight improvement. A tetrachromat woman physicist with General Relativity background could make big breakthrough.

[1] It seems that the genes of red and green paints have different variants. So a woman can be tetrachromat; a man cannot.

[2] In 323 BC Athens revoltad agains Macedon globalization (Alexander III being already dead). In the national upheaval an Eurymedon, priest of Demeter, accused Aristotle with asebeia (impiety, atheism or so) because of declaring Hermeias, Lord of Atarneus, martyr of Greek Cause (with a XXth century Balkan term the Great Cause) hero, i.e. a kind of demigod (see Diogenes Laertius). My guess is that here we see again the old Aristotle-Speusippus feud, but we can pass here the motivations behind. Aristotle, 61 old, leaves the territory of Athens. I think his absence had been believed temporary, but within some months he died in Chalcis. This means that he was not able to finish some works. Later his assistants collected lecture notes and compiled "pseudo-Aristotelian" works. Then the final polish is not of Aristotle; but there may be serious problems if Aristotle was not yet ready. See e.g. Mechanica, and most specially the problem of flying arrow. The problem was not reliably handled until Galileo.

[3] Aristotle, Bk 439-442

[4] Aristotle Bk 372-375

[5] Aristotle Bk 791-799

[6] For the genetics: J. Mollon, Nature 356, 278 (1992). For the discovery of Madam Tetrachromat: G. Zorpette, Red Herring Magazine Nov. 1, 2000. For an evolutionary interpretation: J. Neitz & al., Optics & Photonic News, Jan. 2001, p. 26

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

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

Now Hear This:

Riemannian Geometry is a strong leader in completely unfamiliar situations. We are not lost or confused even near to black hole horizons, in ergospheres, not even in the other half-world of Kerr black hole; and, although confused, not utterly lost even in an acausal world, if we follow the rules of Riemannian Geometry. The same is true for colours. A bichromat can understand (more or less) trichromat situations and a trichromat tetrachromat behaviours if he (rarely: she) applies the correct Riemannian algorithm. While its knowledge is not necessary to understand a lot written here (as you will see), I definitely refuse to deteriorate the equations. But some symbols seem to be improper in some Netscape versions while they seem to be correct in Internet Explorer. Text and Figures are good in both. I cannot help it (maybe my HTML technique should be better); I can only repeat: in some versions the equations are good.

And in the future somebody shows you a non-covariant equation and tells that it is in Riemann Geometry, know that something is wrong. E. g. B_i=C^k is impossible. But B₁=x^5/3 may be correct; or may not. (Here 5/3 is not index but power!)

And now the forgotten texts.

B. Lukács

Central Research Institute for Physics RMKI, H-1525 B. 114. Pf. 49., Budapest, Hungary

Old Egyptian texts clearly mention "greening" stars. According to astrophysics stars radiate approximate Planck spectra, and according to modern colorimetry and Quantum Mechanics a Planck spectrum cannot be green. So there is a controversy between philology and astrophysics. We analyse this controversy to see if we can be calm about our astronomy, which is connected to atomic and nuclear physics.

It is often believed that science is a monolythic unity, answering everything simultaneously and harmonically. It should be true; and it is true in principle. However perfect answers will be reached at infinite time (in the best case), and in finite time some answers are wrong. Then the answers of two different disciplines may contradict each other. Then what is the "scientific opinion"?

This is a question difficult to answer. However even in such cases some answer is needed, at least such one which has the highest possibility to be true. Somehow the statements, methods and reliabilities of the two disciplines must be compared, which is not easy. Such a comparison would need experts familiar with both disciplines; and if such experts existed in large number, then there would not be two disciplines at all, only one.

Any of us can be confronted with such a problenm in any time. So it is useful to see a characteristic example for such contradiction.

Perhaps the two most distant and separate disciplines are astronomy and Egyptian philology; so much that, although both of them belong to science in general sense, the first is done by scientists, the second by scholars. Now, there is a point where a statement of Egyptology seems to contradict to astronomical knowledge. L. Kákosy, a dedicated Egyptologist, from the investigation of old Egyptian texts, concluded that some Egyptians 4000 years ago saw some stars green [1]. Now, astrophysics tells us that stars radiate approximate Planck spectra as usual for heated gases, and according to colorimetry [2] Planck spectra go through the colours red, orange, pale yellow, white, pale blue, pale violet, but not green. Then what?

In this paper we list possible solutions. Among them there is one which is compatible with the present status of both sciences. So, if nobody shows a better solution, then we must accept this one. The problem is not vital, so one can argument for or against without emotional burdens. But, as will be seen, ancient records of stellar colours are not irrelevant at all for astrophysics or even for nuclear physics.

In Old Egypt different ideas existed about cosmology, cosmogony, origin of Sun, Earth, &c. We know only such ideas which survived 4000 years. However papyrus is quite persistent in the Egyptian sand.

The Myth of Celestial Cow survived more than 3000 years. In it the Sun God is creating the stars while leaving Earth. This is a theory, quite reasonable in that time. The only problem is that the text mentions some "greening stars" [3].

There are older texts of similar sense. A pyramid text [4] mentions a goddess scattering stones, and these stones later become stars. This is a viable theory since some precious and semi-precious stones scintillate (as we know, however, only if illuminated). But the text mentions turquiose too.

There was a compilation called Book of Dead used as an itinerary for the deceased in the Underworld. In Chap. 17 the deceased is scattering malachite. According to Kákosy [5] here the deceased wants to identify herself with the goddess mentioned above, in order to take her role in the afterlife, which is indeed an ambitious and worthy effort.

Now, malachite is deep green, and turquoise is green or blue-green. Then it seems as if old Egyptians saw some stars green [1].

The problem is that modern astronomy does not see the stars green. To be sure, in the age of old, visual astronomy some astronomers reported strange hues for stars, but then colorimetry did not exist and colours were artistic terms left to painters, poets and J. W. Goethe. In modern times dedicated astronomers do not look at the stars but photographe them through different filters. Still, green stars must be as rare as white crows. Namely, stellar light comes from the photosphere; there the matter is a thin gas of several thousands of degrees, and the spectrum, according to theoretical astrophysics and many observations, is roughly a Planck spectrum [6]

dI ~ 8πhc^-3x²(exp(hx/T - 1)^-1dx (3.1)

where x stands for the frequency.

Now, this is a two-parameter spectrum, one governing overall brightness, the other, the temperature T, hue. Then one can overimpose the path of Planck spectra of unit intensity on the colour "triangle" of colorimetry [2] and the result is that no Planck spectrum lies in the green sector; see Fig. 1, where Point W in any practical sense lies on line S. One end of S is on the orange-red direction from white, the another in the blue direction. Straight lines starting from W and crossing S in a second point intersect the boundary in red-orange (M), orange (K), yellow (G), blue (B and O); points between (F and A) are generally uncharacteristic white. (To be sure, atmospheric Rayleigh scattering extracts some "sky-blue". That colour is more or less opposite to red. So all colours will slightly redden, this is the reason that M stars are seen red, not red-orange, but in such a way green cannot be obtained.)

Then, in first approach, a contradiction is observed. According to Egyptology, some old Egyptians saw some stars green, but according to natural sciences a star should not be green. An explanation would be needed.

In the following Sections we list some explanations. We can disprove all of them but one. Therefore i) either truth is not a definite thing; ii) or that one explanation is true; iii) or there is a true explanation not yet suggested by anybody.

The first possibility is that the Old Egyptian texts for star colours are unreliable. Indeed, some statements of them are untrue or incorrect. For example, some texts tell that the stars are located on the belly of a huge cow. Modern astronomy tells that they are distant gas spheres, and we accept this latter, in spite of old texts. However this is a case of wrong hypotheses. Records for star colours state observed facts, not hypotheses.

Another possibility is that the texts simply lie. Is any other case when Old Egyptian texts lie?

Yes, there are such cases. One nice example is the Karnak and Abydos texts of Ramesses II (the Great) about the Battle of Qadesh (cca. 1297 BC) with the Hittites. The official inscriptions state that, after some transient Hittite successes, the Pharaoh completely annihilated the Hatti army, Hatti's miserable, defeated chief supplicated for mercy and peace, which he finally got from the gracious Ramesses, going southward.

The text was deciphered sometime in the first half of the last century, was taken as a document of a border war with some tribe, and was taken literally. However in this century Hittite writing had also been deciphered, the peace agreement was found in the Hattusas royal archives, and then it turned out that King Muwatallish had completely defeated the Egyptian army and herded back southward from the border, which is quite a difference.

Another doubtful text, now about astronomy, is a text of Thotmes III, in which a meteorite is reported to have annihilated the Syrian enemy's camp [7]. Now, this is either the only record of an extraordinary meteorite (no other recorded meteorite fall killed anybody), or a propaganda mentioning heavenly help for the just Egyptian case.

However in our case with the colours of stars no political interests seem to be involved. It is probable that if Egyptian texts state that some stars are of malachite and turquoise, then they were similar to malachite and turquoise in something.

There is a possibility that turquoise and malachite were compared to stars in something else than colour. Now, stars are observed optically, by which four properties can be observed: colour, average brightness, brightness fluctuation and apparent size.

Now, average brightness of stars is various, while average brightness of malachite is next to nothing, malachite being a dark green, rather dull material. Stars scintillate, malachite does not scintillate at all. Stars are point-like, and malachite and turquoise chips are generally not.

The last possibility of this Chapter is that turquoise and malachite were similar to stars in some intrinsic property, not directly observed. Possibilities are of infinite variety, and we cannot exhaust all this variety. However the two most probable candidate properties are exiticity and value.

Stars are far and unavailable. Now, malachite is not. Stars may be of great intrinsic value (in any way). Turquoise was also valuable, indeed. However malachite was not. It was an everyday toilet article of higher-class women. They painted their eyelids green with it, and in this habit malachite powder was superior to our ladies' blue paint in two points: i) green eyelids do not indicate overtiredness as blue ones do, and ii) malachite has a benign antibacterial effect preventing trachoma. So it was not distant, unavailable, priceless, or exotic.

Maybe they did not mean stars but planets. Before greeks stars and planets were not strictly distinguished. However the five bright planets are not green at all, and no asteroid exceeds the +6 magnitudo apparent brightness from Earth, which is the usual border of visibility by naked eye. Uranus is slightly green and its apparent brightness changes between +5.7 m and +6.0 m. So in principle it is visible; but at this brightness colour cannot be discriminated, and Uranus was found in 1783 AD by Herschel with a rather substantial telescope. Neptune is quite greenish, but below +7 magnitudo.

Therefore it is better to believe that Old Egyptians saw the colour of some stars similar in some sense to the colour of malachite and turquoise.

Anomalous stars can be of any colour or other property. However, to make a Planck spectrum green in the sense of Fig. 1 one needs a temperature about 6000 K, and absorptions at both ends of the visible spectrum. Starlights are often reddened by dust around the star, but it would need a discontinuous distribution of grain sizes to absorb at 400 mµ and at 700 mµ, but not at 550 mµ.

Another possibility is a double star, lights mixed. Verbally argumenting one would tell that a blue main branch star, say B0, + a yellow supergiant companion, say G0, could give blue + yellow = green. However Fig. 1 shows that by mixing stellar blue with stellar yellow one would get light purple, not light green (to see this, construct the connecting line).

By any chance, not on historical timescales. The Planck spectrum contains the universal constants h and c. These constants can be measured in spectra of distant galaxies. A redshift is seen in them which is some 10% at cca. 1 Gly, so even if redshift were not explained by cosmologic expansion, observable colour changes could not be expected before several hundred million years, not in 4000.

Of course, we may be mistaken in astrophysics, but it is highly improbable that in this point. Anyway, Sun is a star, and Sun was seen yellow during the whole antiquity.

This idea was several times repeatedly popular among historians. The most often detected anomaly in ancient texts is about blue colour. Let us see some examples.

In the Iliad of Homeros some heroes are mentioned as having hairs of violet.

At a place of Iliad the sea (which can be green, blue, or between) is told to be wine-coloured.

The English words black and blue seem to be of common origin (and relatives as blanc, bianco mean white in some languages).

All these observations would get a common explanation if cca. 500 BC the human population had been blue-blind or tritanope, developing blue-sight later. However there are serious counterarguments too, historical, biological and physical. See first historical counterarguments.

In ancient Latin there are words for blue, namely lividus ("plum-colour", maybe dark blue) and caerulus ("sky-colour", maybe light blue).

In Turkish languages blue and black are always discriminated. In Common Turkish blue = kök, black = kara. "Kök" (in Hungarian kék) is meant for the eternal Blue Sky (Kök Tängri), home of the Supreme God, the forefather of the ruler and the Togrul Bird, while "kara" is meant for the earth, the soil, which is the home of, if anybody at all, the inferior chtonic gods of farmers. On the other hand, sometimes yellow and white are confused. "Sarig" or "sharig" is "white" in one Turkish language, "yellow" in another. "Sárga" is "yellow" in Hungarian, but the original Russian name of Volgograd or Stalingrad, Tsaritsin, comes from the Tatar Sarig Sin, meaning "White Wall". So Old Steppe Peoples were not tritanopes, and no colour blindness is known for white.

For biology we can observe that colour vision of apes is practically the same as ours [8], and the human-chimpanzee split is at least 5 million year old.

For physics note that for photons

E = hc/L (9.1)

where L is the wavelength, i.e. in the visible blue photons are the most energetic. Therefore it is highly improbable that they would be hardest to detect.

In general, according to Maxwell's 3-colour theorem [9] 3 different colour-sensitive receptors (cones) are assumed in the human eye, and they are more or less identified (together with the neutral rods) [10]. Then from the incident light V(x) the eye forms 3 weighted integrals

c_i = òA_i(x)V(x)dx (9.2)

where A_i's are the spectral sensitivities, and the 3-dimensional colour coordinates are the raw material for the brain to feel the colour.

If so, then A_i(x) belongs to the hardware of the body, changes indeed very little in 4000 years, and therefore ancients' brains must have got the same signals as today. However then it is just possible that Egyptians (old or new) have different A_i's than Europeans and Americans, just as some Africans are quite differently pigmented in the skin than Caucasians. Now, there exists a global colour sensitivity measurement for Egyptians [11], with so minor difference to White Americans (and with no enhancement in green at all) that it can be ignored henceforth.

Still, c_i is not the colour. The same object more or less have the same colour in morning and noon, while colour photography gives different pictures and multichannel detectors objectively detect different reflected v(x)'s. This is so, because

V(x) = R(x)I(x) (9.3)

where R(x) is the reflectivity of the surface and I(x) is the illumination, say, of Sun through atmosphere. Obviously I(x) depends on time, and so V(x) does too. Colour, the property of the object, should belong to R(x), not to I(x).

According to Weinberg [12], the brain commands the eye to observe both the 3 coordinates of the reflected light and those of the illumination, and from these three data tries to find out the coordinates of R(x). This is mathematically impossible; however it can be done on a 3-dimensional restricted manifold of spectra. He assumed the same forms for V, R and I:

(V, R, I) = exp{p^rP_r(x)} (9.4)

(there is summation for indices occurring twice, above and below). Here P_i(x) is the functional basis and pi are 3 coefficients, characterizing the colours. As far as the forms (9.4) hold, the c_i's of V and I determine the two functions, the ratio is R, and therefore the brain can calculte 3 coordinates to R(x); that is colour.

Natural illuminations and reflectants are not of form (9.4), but it can still be a decent approximation for important ones with the best P_i's. Weinberg got that colour constancy is the best if

A_i(x) = C(x)P_i(x) (9.5)

where C(x) is a single function, roughly the global sensitivity.Then the cone sensitivities A_i(x) are fixed by natural selection. His result is that the P_i's are not too far from the functions

{P_i(x)} ~ {1,x,x²} (9.6)

up to linear combinations of constant coefficients; indeed, checking this on the colour sensitivities, one gets some agreement at middle wavelengths [13]. If so, then eq. (9.4) gives approximate Gaussian spectra, good for one band in reflectance, and inverse Gaussians, good for two bands at extremes. This is indeed good in many environments, and then colour sensitivity is adapted to natural laws via natural selection, i.e. normal colour vision must be unique. As for neglection of blues in old texts, we may accept the professional opinion of C. Lévi-Strauss, the etnographist [14], who tells that when selecting glass beads for Brasilian Indians, one must follow Indian preference, i.e. many white and black, then less red, much less yellow, and then a few blue and green although "it is probable that they will purse their lips on these last ones". So they see blue, but more or less ignore it. His explanation is that i) blue is "cold", and ii) in their environment only a few withering plants are blue. So blue is irrelevant; of course the sky is blue but one has nothing to do with the sky.

However the Weinberg construction cannot be applied on stars. Stars have no R(x), only V(x). True, Old Egyptians may not have known this. But anyways, they could not observed c_i's of I together with those of V. They may have believed that stars reflected solar light (as turquoise and malachite did), but they never saw together stars and Sun on the sky.

Luckily, the form (9.4) cannot be correct. Namely, by introducing a new variable x' instead of x (remember, they did not know anything about wavelengths) the functions V(x), R(x) and I(x) do not transform at all. But V and I should, because they are intensity distributions; indeed R(x) is invariant. So one must change the form (9.4) to

V(x) = W(x)exp{p^rP_r(x)}

R(x) = exp{p^rP_r(x)} (9.7)

I(x) = W(x)exp{p^rP_r(x)}

where W(x) is a further function, transforming with x as a distribution [13]. Natural selection can change W(x) to, e.g., the solar Planck spectrum.

But if so, then there is a default function in the brain for I(x) if not seen. If we do not see any illumination, the brain takes it hidden, assumes the last one, or W(x) or something in close relationship, and determines the colour in such a way. If stellar colours are seen constant (which is more or less true), then this happens. But then again a malachite star has its colour in the same way as a malachite chip does.

But then the problem is still unsolved. Old Egyptians must have had the same Planck spectra, the same Sun, the same cone sensitivities as we; then their brains must have got the same colour coordinates as ours. Then how could they have seen stars green?

Green is not an objective property, but a name. Different peoples indeed call the same colours differently, and translation is needed but not always possible. Japanese "green" (i.e. "go") street lamps are blue-green or turquoise in English. Russian has two words for English "blue": "sinii" (cca. dark blue) and "goluboi" (cca. light blue). Therefore one can translate automatically from Russian to English writing "blue" for "sinii" as well as for "goluboi", but cannot translate English "blue" to Russian without more information in the text.

Present Egyptian Arab language uses a colour naming system quite different in structure from Western and Central European ones. This will be a subject of a forthcoming paper [15], but the green-blue region is definitely different.

There was an Uralic language (so a far relative to Finnish and Hungarian), the Kamassian Samoyed (died out a few years ago), near to the Sayan mountain, under heavy Tatarian (Turkish) influence. On 7 Aug., 1914 some Avdakeya Andzhigatova sang a "ballad" to a Swedish etnographist, K. Donner. In it a line mentions some "kük note" (in rough phonetic writing), which is left behind. Now, there is no doubt that the second word stands for "of my grass" or "of my grassland", and the first is a colour. But the colour name is clearly the Tatarian form of "kök", i.e. "blue", impossible for grass. Interestingly enough, the experts give "of my golden grass" (for details see [16]), and tell that Kamassian "kük" may mean "green", "blue" and "golden" together. This is quite interesting; and, if by any chance, had held for Old Egyptian, could have not be responsible to confuse nonexistent green stars with existent blue and golden (yellow) ones?

No, because the Old Egyptian terms are not technical terms but similes. The goddess and her subject scatter not blue-green and green stones but directly turquoises and malachites. So the texts tell that some stars cause visual impressions similar to turquoises or malachites reflecting sunlight. So it seems as if we were stuck after so many effort. But we are not.

Colours are quadratically diverse. One can distinguish n>>1 "pure" colours (as seen in spectral lines). Now, mixing a pure colour with more and more white, we get more or more diluted (less and less saturated) colours of the same hue. Of them we can distinguish m>>1. That is n*m distinguishable colours, too many to name them individually. Therefore a general practice is to name first the dominant hue (say bluish green or greenish turquoise) and then to tell how much is diluted. Then "very pale malachite" is a colour deviating from the "white" or "neutral" (the reference point of the curvilinear colour triangle, Point W on Fig. 1) towards the malachite point on the curved border of pure colours. Indeed, when we call B stars blue, or K ones orange, we do not claim them to emit monochromatic radiation somewhere in blue or orange; they are diluted blue and orange, but this fact is so obvious that not told at all.

And this observation serves the last possibility to reconcile truths of recent Egyptology and astrophysics. The details are given in Ref. 17.

But first let us determine the white point. Fig. 1 seems to suggest that W is the point of the spectrum of a body of 5704 K temperature (i.e. of Sun; it seems yellow only because atmospheric Rayleigh scattering removes some bluish component from it). However in Chap. 9 we saw that colour belongs not to V(x) but to R(x). Lights cannot define white; the solar spectrum is seen white only because it seems as solar spectrum reflected by a white surface.

Therefore "white" of colorimetry is the colour of a surface reflecting everything with the same efficiency, R(x)=const.

But this cannot have been the definition of pre-industrial societies which could not measure intensity spectra. Their reference points or whites must have belonged to some paints, sheets or such, abundant, reproducible and generally featureless. In northern countries it may have been snow; in other countries e.g. lime, or similar. Now, we can put Old Egyptian White into such a position that for them some stars were at the very edge of malachite and turquoise regions. The result is Fig. 2. Malachite and turquoise are known for modern colorimetry. If Old Egyptian White were in the region of dashed borders, then some stars (yellow or white) were very pale malachite and turqoise for them. This white would be very pale purple in our nomenclature. More "purplish" whites are improbable, because behind the border there are definite modern colour names in the nomenclature, so these reflectants are not "neutral" enough.

So the only possibility, not contradicting anything published so far, is a special Old Egyptian white, according to Fig. 2. But then their white paint should show a depression of reflectivity in the "green" (around cca. 520 mµ). Maybe this can be checked on existing wall paintings, if ageing did not change the paint too much.

Sirius (Sirius A or aCMa A) is a nearby, bright star of A1V spectral type. Its full lifetime is estimated as 400 My. It has a quite separated companion, Sirius B, which is a white dwarf just below 1 solar mass. So its age must be higher than 10 Gy [6]. How is it possible that Sirius A is still in the main branch?

We do not know. But in addition Dogon myths state that Sirius has a companion of 50 year revolution time [18], [19], [20], [21]. Sirius B is very faint for seeing it with naked eye. One explanation was information from alien astronauts. Then it was discovered that some ancient Roman and Greek authors mentioned strange colours for Sirius, as if it had been "reddish" 2000 years ago. If it is true (and they are written sources), then Sirius B finished its red subgiant phase just then, and for a while Dogons could see both revolving around each other with 50 year period.

Astrophysicists do not like this; their equations disprefer the final contraction to be finished in such a short time. But their equations also disprefer Sirius B to be burnt out earlier than Sirius A. If that is so, something is very seriously wrong in our knowledge about nuclear reactions, which is dangerous.

We approach the problem from two directions. The ancients' reference about red Sirius is supported by Dogon knowledge about Sirius B, so let us see first the Dogons. It seems that they informed the French etnographists, M. Griaule and Germaine Dieterlon, during the Second World War in Mali about Sirius B and not backwards; the knowledge about Sirius B is not common among etnographers, especially about its orbital period. Now, telescopic astronomy suspected the existence of Sirius B from the middle of the last century, and it was discovered in 1862, the revolution was measured in the next decades. In 1893 a French expedition led by H. Deslandres, was observing total eclipse not too far from the Dogon lands, and a member of the expedition, called Coculescu, reported in a letter about frequent communication with natives about astronomy [22]. Therefore Dogons might have obtained the basic information about Sirius B from European astronomers. No need to assume a red subgiant Sirius B.

Still, there remain the ancient reports about red Sirius. For references see Ref. 20, further citations therein and Ref. 21. We note first that Dogons are reported to regard Sirius red. According to modern observations Sirius A is A1V, white (or very pale blue), its apparent brightness is -1.6 m; Sirius B is A white dwarf, with apparent brightness +8.3 m. There is nothing in the Sirius system which would be red.

However Neo-Babilonian sources frm 7th century BC mention Sirius as of "copper colour". Horatius in "Hoc Quoque Tiresia" cites a sentence telling: "The red dog star's heat split the speechless statues". The dog star seems to be Sirius, first star of Canis Maior i.e. Big Dog. Seneca states that Sirius is redder than Mars, which latter is red without doubt. Klaudios Ptolemaios classifies it red, together with Aldebaran (K5), Betelgeuse (M0) and Arcturus (K0). We would call K stars orange, M's red. St. Gregory of Tours in the second half of the 6th century AD, in his work "De cursu stellarum ratio" mentions again the red colour of Sirius. However, Al Sufi in the 10th century AD call all of Ptolemaios' red stars red, except for Sirius. A pattern emerges in which Sirius B was a red subgiant, determining the colour of the Sirius system, until the early Middle ages, and it became a white dwarf sometime between the 6th and 10th centuries. As told earlier, astrophysicists do not like this scenario. If our knowledge about nuclear reaction rates, gravitation and heat transfer is not overly wrong, then such a rapid transition is impossible; if they were so wrong, half of the nuclear power plants would have blown up.

Now, as seen in earlier Chapters, ancient colour names need some caution; and, definitely, Seneca could not write "red". In the "red" sector of Figs. 1 and 2 at least 4 different names appear: ruber, rufus, rutilus and rubidus. From vocabularies it seems that "rufus" was used mainly for hairs, and the other three words apply to different parts of the red sector. Similarly, there are at least 5 words in the yellow sector, and 2 in blue. Fig. 3 is a first and premature attempt to locate the Latin colours in the colour "triangle" of colorimetry; but there are two problems.

First, our knowledge is limited. Latin became almost extinct, left everyday use, sometime between St. Gregory of Tours and Al Sufi. In Gallia the critical time is 813 AD when the Synod in Tours orders the priests to preach "in rusticam Romanam linguam" instead of earlier Latin. But even in Italy uneducated people could not use Latin at least from 960 AD. Our present vocabularies, therefore, may not reflect fine nuances felt in the original speech community, and in the best case reflect the usage of poets and orators, not of astronomers. As it will be seen, Neo-Latin languages generally do not inherit the Latin colour names.

Second, there is a possibility that Latin used a system distinguishing not only dominant hue but albedo as well. As seen in Chap. 9, with Gaussian basis reflectances are characterised by 3 data which can be described, for example, as middle wavelength and width of the band, and average (or global) albedo. Modern "scientific" colour names do not specify the third and denote the second by giving the saturation; modern naming systems of clothes industry &c., used mainly by women, use individual names for pastell colours, but do not specify the albedo. But Latin has two words for "white": "albus" and "candidus", of which the second is "bright white". Similarly, "black" can be "ater" = dull black or "niger" = shining black. If this was consequently applied on colours, then some colour names on Fig. 3 differ not in hue or saturation, but in albedo.

Now, in Italy the Latin population survived and remained dominant; however of the 6 fundamental colours of heraldics only 2 continues a Latin word of similar meaning: viridis -> verde and niger -> nero. "Rosso" comes from "roseus" (rosy or pink) not from "ruber". It seems as if in the Middle Ages new colour names had been introduced as international technical terms.

So it is a nontrivial task to translate Latin colour names to modern languages. In addition, heat is red, and in Italy midsummer, or canicula, when Canes, the two Dog constellations, are high on the sky, is hot indeed. We would tell that the red heat of July melts the asphalt on the streets of Rome. Maybe Horatius tried simply to formulate the same observation with the cracking of statute's stones.

So it seems that we do not have to correct our astrophysics and nuclear physics on the basis of ancient reports about star colours; there are alternative explanations concerning ancient colour names. However, remember that for the Egyptian green stars the explanation needs an Egyptian white paint with reduced reflectance about cca. 520 mµ. What happens if archaeologists measure the paints and rule out this possibility? Shall we have to revise our nuclear physics, astrophysics or quantum mechanics?

It would be interesting, but improbable; they are conform with very many physical measurements. For any case, there are sporadic but continuous reports about green stars in the last three centuries. Struve, who was a reliable astronomer, reported a few green stars with spectral class A (which class is white in all astronomy textbooks). And Admiral Smyth in 1864 listed 37 "inexact epiteths" which he had to apply to colours of double stars. Of these 37 names 3 seems to mean a green shade [23]: "apple green" may be "brownish green", "emerald" may be "lucid green" and "sea green" may be "faint cold green". True, double stars' mixed radiation is far from Planck spectrum; however Fig. 2 clearly shows that the mixture should not lie in our green sector.

However this is a problem first for astronomers. If the detailed spectrum turns out to be enhanced in "green", i.e. around 520 mµ, then astrophysics may look for an explanation, say, with specific absorption in the chromosphere. If that is unsuccessful, then, but only then, can be nuclear physicists nervous.

The chances are extremely small for some deep problem of physics. However the analysis of Egyptian malachite stars may serve as an example how complicated a contradiction between far areas of science can be.

As for the problem of having a small white dwarf and a massive young star together in a double system, the most popular explanation is a black hole Sirius C, having stolen mass from the red giant Sirius B. We do not yet see it, but the real problem is that we do not see the gravitational effect of the stolen mass either. If this problem remains open, then we seriously do not understand the Sirius system, no matter if it was seen earlier green, red, pink or violet by anybody.

[1] L. Kákosy, Egyiptomi és antik csillaghit. Akadémiai Kiadó, Budapest, 1978.

[2] Landolt-Börnstein, Zahlenwerte und Funktionen, Technik 3. Teil, Elektrotechnik, Lichttechnik, Röntgentechnik. Springer, Berlin, 1957.

[3] Ch. Maystre, Bull. Inst. Franc. d'Arch. Orient. 40, 81

(1940)

[4] K. Sethe, Die altägyptischen Pyramidentexte. Leipzig, 1908, Text 567

[5] L. Kákosy, Bibl. Orient. 25, 323 (1968)

[6] Eva Novotny, Introduction to Stellar Atmospheres and Interiors. Oxford University Press, New York, 1973

[7] W. Helck, Urkunden der 18. Dinastie. Übersetzungen zu den Heften 17-22. Berlin, 1961. 10.

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

[9] J. C. Maxwell, Trans. Roy. Scottish Soc. Arts 4, 394 (1856)

[10] G. S. Wasserman, Color Vision. J. Wiley & Sons, New York, 1978

[11] I. G. H. Ishak, J. Opt. Soc. Amer. 42, 529 (1952)

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

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

[14] C. Lévi-Strauss, Tristes tropiques. Librairie Plon, Paris, 1955.

[15] M. A. Abdel-Raouf, C. Liegener & B. Lukács, in preparation

[16] J. Lotz, J. Amer. Folk. 67, 369 (1954)

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

[18] Germaine Dieterlon, Le renard pâle. Paris, 1965.

[19] R. K. G. Temple, Das Sirius-Rätsel. München, 1979.

[20] D. B. Herrmann, Rätsel um Sirius. Buchverlag der Morgen, Berlin, 1985.

[21] C. Sagan, Broca's Brain. Ballantine Books, New York, 1980

[22] I. Guman, Die Astronomie in der Mythologie der Dogon. Archenhold-Sternwarte, Berlin-Treptow, 1987.

[23] D. Malin & P. Murdin, Colours of Stars. Cambridge University Press, Cambridge, 1984

Ágnes Holba & B. Lukács

Central Research Institute for Physics RMKI, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

The triality (trichromatism, trivariance) of human colour vision is discussed in the Weinberg scheme: namely that what is this triality, how it works and why it is just trial. At the end some problems, solved and unsolved, are mentioned and the evolutionary path, leading to this triality, is guessed.

In this years the trivariance of colours is a commonplace, because colour TV's work so and they work quite well. However it is important to see, what is this trivariance, how this trivariance works, and why just trivariance exists among colours. We shall see, in addition, that trivariance is not a physical law or a consequence of Maxwell equations, but something originating from the biochemistry of human eye and from the codes in the human brain. There are no colours in Nature, only light waves and intensity distributions; colour is an idea of our brain. However such a well defined idea that one can measure the curvature of its Riemannian geometry.

Sometimes animals are investigated for their colour vision. They cannot be asked about their impressions (except perhaps for some chimpanzees and gorillas familiar with the American sign language of deafs. Therefore conditional reflexes are established and colour discrimination is checked. Since most human daltonians cannot distinguish, say, green and red (and therefore have problems with driving licenses), the idea is that an animal is a trichromat, if it can distinguish colours more or less everywhere in the "visible". However this does not prove either trichromatism or trivariance, as we shall see.

However let us see the results of some investigations. Hess (1973) found some colour vision in fishes, reptiles and birds (Hess, 1973). Cats can distinguish colours, but they rather use brightness differences in recognising the environment (Mello, 1968). Some colour vision is observed in squirrels, horse, pig and in Tupaia (Walls, 1973; DeValois & Jacobs, 1971; Hess, 1973). However most Prosimii are colour blind (Wolin & Massopust, 1970). In spite of this, all observed Simia, except Aotus trivirgatus, are able for colour discrimination (DeValois & Jacobs, 1971), but this ability is suppressed in some New World monkeys. Finally chimpanzees have the same colour vision as humans, but the latter can discriminate slightly better (Passingham, 1982). Fig. 1 comes from Passingham (1982) (originally DeValois & Jacobs (1968)) and Wasserman (1978), showing that lower monkeys may be similar to colour deficient humans (a minority of 1-10%), as Prosimii may be similar to human daltonians.

This suggests that the present human colour vision is the final product of a long evolution from colour blindness to trivariant colour vision, starting from the zero point at Prosimii. However generally Prosimii are put higher on the evolutionary tree than Tupaia; squirrels, cats and horses are not in our branch. In addition, how can some mammals (even lower Primates) be colour blind if some reptiles have colour vision?

Walls (1967) assumes that the colour vision vanished in early mammals. This is indeed possible if early mammals were nocturnal (often assumed) as many Prosimii are. But Tupaia is not nocturnal, and indeed it has some colour vision. In addition, the recent reptiles are only very far relatives of the ancestors of mammals: that branch partly died out partly continued in mammals. So it is hard to reconstruct the colour vision of Permian therapsids at the stem of the mammals.

This means that at the present stage of art it is quite possible that on separate branches of mammals the colour vision evolved separately, without regard to the colour vision of far ancestors. For a moment let us accept for argumentations' sake an extremal but popular idea that during the Mesosoic, when reptiles were large and dominant, proto-mammals were retreated into nocturnal niches. Then they were totally colour blind (we shall see, but anyway, we do know that we are totally colour blind at moonlight). How many different branches of mammal evolution existed at the Cretaceous/Tertiary boundary when on a morning, paralelly to the story of 2Kings 19, 35 the mammals, awakening, found all dinosaurs lifeless?

The exact number is a matter of argumentation. But it is common opinion that then already the Primates Ordo had been separate from the others, e.g. from the Carnivora Ordo containing cats. (The Tupaioidea-Primates connection is doubtful at that time.) So human and felid colour visions, if they started to evolve at the beginning of Tertiary, may have evolved independently. In this case felid colour vision is not necessarily an earlier evolutionary stage compared to the human one, and may even be structurally different.

Here we return to trivariance and tricromatism. We repeat that good distinction of vivid greenish yellow from yellowish green does not prove trivariance.

The notion of trivariance comes from Palmer (1777), Young (1802) and Maxwell (1856). Remember that Newton (1730) postulated seven primary colours. Palmer reduced the number to 3, and postulated that "The superficies of the retina is compounded of particles of three different kinds, analagous to the three rays of light; and each of these particles is moved by his own ray.". Here "is moved" means the process of detection: "is agitated" or so. These 3 primaries Young, after some hesitation, identified as red, green and violet. Finally Maxwell formulated the trivariance in such a way that there are 3 different pure colous sensations; "Every ray of the spectrum gives rise to all three sensations though in different proportions...", and by mixing two rays, the colour of the mixed light lies on the straight line connecting the colours of the two rays, with some weights proportional to the intensities. So according to Maxwell, colour mixing is linear in intensities, the space of colours is of 3 dimensions, and some boundaries of the colour space are convex curves (from experience). The primaries of pure sensations do not exist physically. If the first two statements are true for an animal, then its colour vision is trivariant.

Now let us think backwards. Assume that linearity and 3-dimensionality are established facts from long observation: what can be the biology behind? Obviously colour vision is a harmonic action of eye and brain. It is easy to see that Palmer's assumption explains 3-dimensionality. A "ray" is a light beam, say, with an intensity distribution V(x) where x stands either for wavelength or for frequency or for any simple and convenient function of them. Assume that there are 3 and only 3 different receptors in the eye, each with its own spectral sensitivity A_i(x). Then from V(x) the eye forms 3 and just 3 signals c_i

c_i = òV(x)A_i(x)dx (2.1)

and sends them to the brain. Then the infinite variety of possible V(x)'s is projected into 3 degrees of freedom by the eye, therefore the dimensionality of the colour space cannot be higher than 3 regardless to the abilities of the brain. The boundaries are convex because there are no physical lights (V(x)·0 at all x's) exciting only one receptor. A concave boundary would mean that the linear combination of two physical lights could not be seen at all, and this is contrary to linear mixing.

Eq. (2.1) gives linearity of mixing automatically, and mathematically this form of the functionals c_i(V(x)) is highly specialised. However biologically and physically there is no much choice. The receptors could hardly feel the wave properties of light, and intensities are additive. One must perform an integration that the dummy variable x vanish, and the excitation-signal relation is linear for moderate changes. The logarithmic Weber-Fechner Law may hold, but it can hold in such a way that the logarithm of the integral is sent to the brain; the brain can form any function from the logarithms as well.

Trichromatism means that the eye contains 3 and just 3 pigments in the receptors. This is not exactly the three-dimensionality: the brain can throw away informations or by complicated tricks one may form 3 signals by means of 2 paints too, but obviously if 3-dimensionality and linearity are proven, the simplest and most probable explanation is just 3 receptors of different sensitivity.

Since colour photography and colour TV work, and work with 3 primary light sources + linearity in intensity, let us accept human trivariance in the above sense. But how does it work?

Eq. (2.1) started to show how colour vision works. If the receptors are identified, we are ready with half of the answer. But only with half of it. A method simply determining 3 data of lights reflected from surfaces would not be too worthy in evolution. Namely, the illuminating light, later reflected by the surface, is not unique. Let us write

V(x) = I(x)R(x) (3.1)

and

c_i = c_i(V(x)); c_i(qV)=qc_i(V) (3.2)

Then under unlucky circumstances (special illuminations) the surface of the wolf apple (Aristolochia clematitis) would give the same colour sensation (c_i) as that of the wild apple (Malus silvestris) does under general circumstances. But Aristolochia clematitis is poisonous, so then Homo erectus would have died without inheritors and his genes would have vanished. And rightly so. Probably genes of everybody and everything, using a limited set of receptors and in the same time taking c_i's in face value, long ago vanished. One should identify parameters of R(x) not of V(x). But R(x)=V(x)/I(x). So I must be observed too (no problem); but linear operations on I and V cannot produce functionals of R=V/I. Still the identification of R is needed.

There are two ways, depending on brain capacity: either using many receptors, when practically the I(x) and V(x) functions themselves are measured and then to divide; or to form a limited number of functionals from both and then to perform complicated fitting and root-finding mechanisms. The first is the way of Pecten, a shell-fish (Horváth, 1994) with 40-60 eyes of different spectral sensitivities, the second is the human (and pongid, together anthropoid) way. The difference can be understood, because anthropoids are brainier than most animals; on the other hand during evolution shellfishes lost their head; among recent forms only the most conservative Nucula nucleus keeps some rudiments of head, and one cannot expects able brains without a head. Now the human way is briefly described following Weinberg (1976). One can check at the end if the suggested model satisfies the condition of (moderate) colour constancy, i.e. that at least under "natural" illuminations the colour of the surface must be (almost) constant.

Assume that natural illuminations and natural reflectances can be approximated by an exponential form of 3 basis functions P_i(x), i.e. that

F(x) ~ exp[p^rP_r(x)], i=0,1,2 (3.3)

where F=(I,R), and henceforth there is automatic summation for indices occurring twice, above and below (Einstein convention). Then V's automatically have this form too. Now, the eye can measure 3 parameters of I(x) and V(x) via the c_i's. If the brain can find out p(V) and p(I) from c(V) and c(I), then R is represented by the 3 dimensional p(V)-p(I), and then the reflective surfaces are points in a 3 dimensional space. If this is enough to discriminate benign berries from malign ones, then the method works.There is, of course, the question, why we think that a form (3.3) can be a satisfactory approximation, but this belongs to the next Section.

Then, via a variation principle, Weinberg has shown that the colour constancy is best if

A_i(x) = C(x)P_i(x) (3.4)

where the A_i(x)'s are the sensitivity functions forming c_i's in eq. (2.1). The function C(x) is still arbitrary, but it is some "average" sensitivity, and gives a possibility to evolution to concentrate on "important" wavelenths.

Then Weinberg derived a scalar of luminance H(p)

H(p) = òC(x)exp[p^rP_r(x)]dx (3.5)

and then

c_i = ¶H/¶pⁱ (3.6)

Now, eq. (3.4) has come from a variational problem for colour constancy. Obviously colours should not be discriminated within the errors of colour constancy, so a natural tolerancy region is obtained around any p as

ds² = q²H^-1(¶²H/¶pⁱ¶p^k)dpⁱdp^k (3.7)

where q is a constant characterizing individual overall ability & interest to discriminate. But then we have got a Riemannian geometry in the space of the 3 dimensional parameters p (unit distance is 1 s distance), and therefore we have a Riemannian geometry of the space of colour coordinates c as well. Now this latter geometry is well known (see e.g. McAdam, 1942), and not Euclidean.

Weinberg's scheme needs a minor generalisation. Namely, one can transform the dummy variable x. One possibility is from wavelength to frequency or back, but prehuman brains handling successfully trivariant colour vision did not know too much about wavelength either. So the final results must be invariant under a transformation x'=x'(x). But eq. (3.3) is not invariant. I(x) and V(x) are spectral distributions, so they transform as e.g.

V'(x') = V(x)(dx'/dx)^-1 (3.8)

while

R'(x'(x)) = R(x) (3.9)

so eq. (3.3) must be substituted with

F(x) ~ W(x)exp[p^rP_r(x)], i=0,1,2 (3.10a)

R(x) ~ exp[p^rP_r(x)], i=0,1,2 (3.10b)

F=(I,V) (3.10c)

where W(x) is an intensity distribution, transformed properly (Lukács, 1988). Then W(x) multiplies C(x) in H and everything else remains unchanged.

Now we have 5 functions, C(x), W(x), P_i(x) in the theory. What are they for meaning and form? Obviously C(x) stresses some "important frequencies", and W(x) is the average natural illumination (being W(x) a possible I(x), and there is a freedom in defining W(x) to p=0). So probably W(x) is near to the noon solar irradiation seen on Fig. 2, and W(x)C(x) must be in close connection with either eye sensitivity or with the brightness, luminance or something such of a spectral light of unit intensity. Such "colourfree sensitivity" curves have been measured, and Fig. 3 gives 3 of them, for a frog, a snake and for man; the fist two curves come from Granit (1955) and the third from Ishak (1952). They are rather similar, this part of the visual system did not evolve too much (or if evolved, did it rather paralelly) in the last 250 Mys. One may say that this "luminance" is independent of the colour vision and then its conservativism is not surprising; but it is not necessarily so. As told, in Primates the ancestral vision may have been nocturnal; and in man the really colourless night vision has another sensitivity function peaked quite far from the peak of Fig. 3. We return to this point soon. The curves of Fig. 3 have a common maximum at the maximum of the solar irradiation, 555 mµ, which is quite natural, why C(x) woult try to suppress the best visible range?

As for the P_i's, they are they are the basis for mimicking natural illuminances, reflectances, &c. Their particular forms belong to the next Section, here we only note that if a simple amplification of intensity does not alter the colour (which is advantageous), then the constant function must be within the basis. Since bases which are linear combinations of each other with constant coefficients cannot be distinguished, for a basis, we simply choose

P₀(x) ~ 1 (3.11)

So far so good; such a system can work with enough brain capacity to perform fitting procedures to find p's from c's or vice versa. We cannot dissect a brain to find the fitting codes but we can see if the three receptors with A_i(x) are in the eye or not.

First let us note that human eyes contain two anatomically different types of light receptors (Wasserman, 1978). One kind is called rods, it is connected with night and peripheral vision. That vision is colour-free and more sensitive from the other. Fig. 4 shows the relative spectral sensitivity of human rods; data are from Landolt-Börnstein (1957) and from Rushton (1962); there is a slight difference between the two sources which will not be discussed here. It is centered at 507 nm, far in the blue. But in the center of the vision field, in the fovea or yellow spot, the human eye contains solely another receptors, the cones, and there are different cones with different sensitivities. Rushton (1962) identified pigments of cones most sensitive in red, green or blue, although the last kind of cones were rare. Then let us accept that the 3 different cones are the 3 "particles" of Palmer (1777) and Young was moderately right when guessing the sensitivity regions. There remains a problem, but let us leave it to the last Section.

Maybe the pigments mutated from one ancestral pigment. Then mild biochemical problems lead to one erroneous pigment sensitivity. Then eq. (3.4) does not hold for one A_i, the colour discrimination is worse than normal and the individual is an anomalous trichromat. More serious problems lead to more anomalous trichromats, or to the loss of one pigment. If there are only two pigments, the person is called colour blind or daltonian, and his colour space reduces to 2-dimensional (bichromat). But still such a person sees some colours. the frequency of this defect may be about 1 %. When two cone pigments are absent, then the person cannot discriminate between wavelengths, and in the absence of cones the situation is similar but only peripherial vision remains. These last two defects have a frequency cca. 3*10-6 together.

There is some argumentation about the frequency of (partial) colour blindness and serious anomaly. Canonical numbers are 10% for males and 1% for females, and then the explanation is that some important genes of pigments are on chromosome X for which non-Klinefelter males are hemizygotes. However Wasserman (1978) lists good arguments that females are more reluctant to report their problems with colours and have less possibility to be checked for such.

One may think that trichromatism evolved in 2 steps: first the ancestral pigment splitted into 2 inheritors; then at lower evolutionary steps bichromatism would be expected. But, as we told and shall see, bi- and trichromats cannot be distinguished from simple colour discrimination experiments. So, although Saimiri seems to be protanomal, not protanope, it is not proven.

For the basis P_i(x) in eq. (3.3) Weinberg (1976) suggested

P₀ ~ 1; P₁ ~ x; P₂ ~ x² (4.1)

The reason is that then

R(x) ~ Nexp{±(x-X)²/2s²} (4.2)

For the lower sign this is the general Gaussian distribution, approximating every reflectance having a central peak anywhere with any width and albedo. (If the curve is not symmetric, the approximation is worse.) For the upper sign it mimics two reflection bands at the two ends of the "visible" range (purples). Also, reddening of morning and evening illuminations can be mimicked by multiplying W(x) (the noon one) by Gaussians peaked at long waves, while if Sun's yellow is blocked by clouds, the bluish of the sky can be mimicked Gaussians peaked at short waves. So a well adapted trichromatic vision is not bad to recognise where the reflection is centered at wavelength, how wide is the band, and how bright is the surface. Of course, there are reflectances confused in this system: e.g. two narrow bands in red and yellow can be identified to one wider in orange, but on the whole the ability of the trichromatic system seems quite high.

Then one should check Weinberg's suggestion. He made one check: (4.1) leads to an ellipsis, parabole or hyperbole in the åc_i=const. section of the c coordinates for the locations of the sharp monochromatic spectral lights, and it is really a parabole, except for the violet end. We give here the colour "triangle" as Fig. 5. Another check is given in Lukács (1988) for the "red" and "green" colour coordinates of monochromatic lights: the agreement is quite good between bluish green and almost pure red.

In addition let us refer the simplest possible analytically calculated model (Lukács, 1988) in which the P_i's are as in (4.1) and both W(x) and C(x) are Gaussians, centered anywhere (it is not a very good approximation for the solar spectrum). Then the Riemannian structure of the p space is such that the 3 dimensional space has an SO(1,1)*U(1) symmetry. The U(1) symmetry direction is the total brightness, and the sections of constant brightness are of constant curvature, as the Bolyai plane, so there is no center of the colour space anywhere. Each colour is equally emphasized. In a rich environment this is not bad.

So it is not impossible that evolution prefers the forms (4.1) among trichromats and prefers trichromats to bichromats. Maybe it prefers tetrachromats to trichromats (we will return to tetrachromatism later), but it seems that the advantage of 3 to 2 is larger than that of 4 to 3.

Now Lukács (1988) gives an example for a bichromatic colour vision which is able to discriminate all dominant hues. Let us note, that trichromatic colours can be named by 3 names: brightness, saturation and dominant hue (van der Akker & al., 1947). Roughly the third is the peak of R(x) (X in (4.2)), and the second is its width (s in (4.2)). Now assume that it is not so important to recognise s, only X. Consider now the reduced basis

P₀(x) = 1; P₁(x) = x (or x-x₀) (4.3)

Then, again in the simplest analytic model, the brain assumes Gaussians of fixed width (the same as of the sunlight). Such a person can distinguish red from orange, orange from yellow, yellow from green, green from blue and blue from violet in a sequence of reflectants whose colours we recognise as quite saturated ones.

Now, there seems to be 3 kinds of (partial) colour blindnesses in conformity with the 3 pigments to be lost. Protanopes lack the long wave receptor (they are weak in red), deuteranopes lack the medium wave ones (they are weak in green). As told, the frequency is somewhere about 1%. Then one would expect that tritanopes are weak in blue, confusing green and blue. Tritanopes are rare, below 10^-4, but one unilateral tritanope was investigated, who was able to name the sensations with his defected eye in therms of the trichromatic system learnt with the normal eye (Ohba & Tanino, 1976). He saw approximately normal hues throughout the whole spectrum, but "with a profound desaturation of the portion of the spectrum that normally appear yellow" (Wasserman, 1978). This seems to be the case expected with approximately the basis (4.3).

So an animal with hue discrimination similar throughout the whole "visible" range may be a normal trichromat or a special bichromat, although one guesses that such a system has its limits in discriminating very close wavelengths. For distinguishing special bichromats from trichromats one should apply also colors with the same hue but different saturations.

Since the single tritanope example suggests that there is really a cone with A₀ corresponding to P₀=1 as in (3.4) and there is another with A1 corresponding to a P₁~a+bx, it is possible that human trichromatism evolved through a bichromatic step discriminating hues but not widths. It would be worthwhile to look for such visions among lower monkeys.

Before drawing any more conclusion about the evolution of the human colour vision, we must admit that not all facts and beliefs fit into the emerging pattern. So we list some problems and try to find very briefly possible answers, without pursuing either completeness or final certainty.

According to anatomy one would expect rather tetrachromacity. Namely, besides the three different cones there are the rods with their own sensitivity function. So the eye seems to send not three but four integrals of V(x) to the brain. The rods seem to form a fourth kind of receptor in the midshort range.

To define this statement consider the canonical CIE distribution coefficients. Obviously these coefficients must be linear in the sensitivities A_i(x); only the constant coefficients of possible linear combinations would need discussions, but let it be postponed. One function is peaked just below 450 nm (far blue), one near to the peak of the solar spectrum at 555 nm and one has two peaks, the higher above 600 nm in the red, and the lower in the blue. Now, as far back as 1964 some primate cone sensitivities were known from direct measurements (Marks & al., 1964), and their shortwave ones are peaked at 445 nm. So indeed the rod peak is doubtless between the "blue" and "green" cone peaks. The rod signals would not disturb the 3 dimensionality only if

A_rod(x) = q^rA_r(x) (5.1)

with constant coefficients, and such a relation clearly does not hold. Where is the fourth dimension of the colour space hidden?

True, there are no rods in the human fovea, but we do see colours outside of the centrum of the vision field too. Indeed there are some opinions about tetravariance of the peripheral vision (Trezona, 1976). Wasserman (1978) formulates a statement that "rod activity influences color". But the existence of a fourth dimension is not simply "influence", rather a new degree of freedom. It means discrimination between two reflectants with bands of the same location, width and albedo. Do we see such a phenomenon? It could be easily described because a 4-dimensional Riemannian geometry is not much more complicated than a 3-dimensional one.

Colour printers driven by colour drawing PC codes as e.g. PaintBrush or CorelDraw use subroutines formulated in trichromatic language and the colour printings are not obviously deficient. We, personally, cannot yet recognise the fourth dimension. One possibility will be mentioned here which, however, has been checked and disproven.

It is an old commonplace among painters that it is a really hard task to paint metals on pictures without using metal-containing paints. This problem occurred many times when painting ladies with jewels. Now it would be just marginally possible that metallic shine would need an extra fourth component to be reproduced in colour mixing. The check was performed by a DeskJet printer (which practically meant subtractive mixing). Now, the result is that quite convincing gold shine can be produced by using two different yellow hues in mosaic pattern.

So we do not know where the extra dimension of the colour space is hidden. There is a possibility but its evolutionary gain is doubtful. The brain may neglect the fourth signal in such an extent that for all reasonable reflectants differing only in the fourth coordinate the coordinates are within unit tolerance, so they are not clearly distinguishable for the brain. But what is gained by throwing away information? Maybe the gain is in colour constancy, since at very strong lights rods stop to work and then colour would change if crod were not suppressed.

There is one point where the Weinberg scheme does not work automatically. As seen above, in that scheme the colour belongs not to the reflected light but to the reflection spectrum R(x)=V(x)/I(x). But what happens with a source of its own light? Flames are seen orange-red, some stars and planets are red as Mars or light blue as Vega, and red light bulbs are seen red.

One possible explanation is that this problem disproves the Weinberg scheme; but there is an alternative explanation too. We have seen that the brain assumes an intensity distribution in the (extended) scheme, namely W(x). If illumination is hidden, the brain may (or may not) assume W(x) instead. In this case a star would be seen red if its spectral distribution is similar to a light reflected from a red cloth in full noontime. For light bulbs, if there is only one light bulb in a closed room, then I(x)=V(x), and therefore it seems as if R(x) were constant, i.e. white. Indeed, normal bulbs seem to give more or less white light when they have no concurrents, although the temperature of the wolfram wire is cca. 3000 K, quite in red-orange. Maybe a red bulb (and definitely a red neon light) has a spectral distribution and c coordinates "too far" from "natural" and then the brain stops to fit.

There are some disturbing historical texts. In the Iliad some heroes have violet hairs, and the sea is wine-coloured somewhere. Ptolemy the astronomer classified Sirius red. Some Egyptian texts compare some stars to malachite and turquoise, green anf blue-green for us. The Old Testament does not mention "blue". In the Old German "blue" and "black" both were "bla". And so on. Now we have seen that in the Weinberg scheme the sensitivities A_i(x) i) belong to the hardware of the body and ii) are subjects of evolution but an evolution towards eq. (3.4) where the P_i's are determined more or less by terrestrial illuminations and reflectants. The brain may be somehow reprogrammed but then the colour vision deteriorates, and A_i(x) cannot seriously change in 4000 years if it is quite human for chimpanzees diverging from us for at least 5,000,000 years.

The Egyptian hardware, in fact, was checked by Ishak (1952) (of course the recent one), at least the wavelength-dependence of colourfree brightness. The result was that the sensitivity function differs from that of American whites, but the difference is small and can be explained by the higher pigmentation of the ocular media. The problem was investigated in Lukács (1992) and Lukács (1995a), with the result that the problem belongs to colour naming.

The four "pure" or "simple" colours are of course Red, Yellow, Green and Blue. We did not mention this as a support for tetravariance via rods. We had good reasons not to use such an argument. Again this question, as will be seen, belongs to colour naming.

Colour naming systems differ in different cultures. Let us see some examples.

There was an Uralic language (so a far relative to Finnish and Hungarian, died out a few years ago), Kamassian Samoyed, near to the Sayan Mountain, under heavy Tatarian (Turkish) influence. In a ballad of them a line mentions some "kük" grass or grassland. But the colour name is clearly the Tatarian form of "kök", i.e. "blue", impossible for grass. Interestingly enough, the experts translate it as "golden grass" (Lotz, 1954), and tell that Kamassian "kük" may mean "green", "blue" and "golden" together. Were the Kamassian receptors so different?

Japanese traffic lights (for us) are not red and green, but deep cherry red and blue-green. Are Japanese receptors different?

Probably not. Let us compare recent Italian and Latin; there is no doubt that in Italy the Latin population massively survived the Roman Empire, with some, but smaller, German admixture.

Now the present Italian terminology is similar to English or German or Hungarian: 4 simple "pure" colours plus white, black and purple; most other names are similes or compounds. But in Latin in the "red" sector of Fig. 5 at least 4 different names appear: ruber, rufus, rutilus and rubidus. From vocabularies it seems that "rufus" was used mainly for hairs, and the other three words apply to different parts of the red sector. Similarly, there are at least 5 words in the yellow sector, and 2 in blue. Fig. 6 is a premature attempt to locate the Latin colours in the colour "triangle" of colorimetry; it has been published in Lukács (1995a) but with such a bad printing quality that some names were almost unidentifiable.

The Latin system may have been even more foreign. Latin has two words for "white": "albus" and "candidus", of which the second is "bright white". Similarly, "black" can be "ater" = dull black or "niger" = shining black. If this was consequently applied on colours, then some colour names on Fig. 6 differ not in hue or saturation, but in albedo.

Only two of the present Italian names of "pure" colours are the same as in Latin: viridis -> verde and niger -> nero; and "rosso" (red) comes from "roseus" (rosy or pink) not from "ruber". It seems as if in the Middle Ages new colour names had been introduced as international technical terms. A possibility is that the influence (which unified the systems but not the words everywhere in Western and Central Europe but not outside was heraldics, whose rules were accepted just in Western and Central Europe: in classical heraldics 6 and only 6 paints could be used on shields, arms &c., without mixing: white (or silver), black, red, yellow (or gold), green and blue; purple was also permitted, but only on covers. An example for regions outside of the geographic area of classical heraldics is Russian with 2 pure "blues": "sinii" and "goluboi".

So colour naming is culture, not hardware. To close we mention a preliminary result from a forthcoming paper (Abdoul-Raouf, Liegener & Lukács, in preparation) is mentioned. Present Egyptian Arabic uses a naming system completely different from those in Europe. We were able to locate some coloured objects in the "colour triangle abstracted from Fig. 5. Now the regions of English names are indicated on Fig. 5, while the corresponding Arabic regions seem to be (the top corner is our Green, the right one Red and the left one Blue) those on Fig. 7. The authors of the above mentioned paper have not yet identified the centers of the colour regions in wavelengths, so only a schematic colour triangle is given.

 

Problem 6: What Sets the Ends of the Visible Range?

It is a commonplace that human vision starts at 380 nm and ends at 780 nm. What puts these endpoints? Retinal sensitivity decreases continuously, the exponential forms (3.3) can be continued, and there is still ample illumination beyond 780 nm, as shown by Fig. 2.

However thence I(x) starts to oscillate strongly via H₂O and CO₂ absorption, and the amplitude of this oscillation highly depends on the atmospheric width between us and Sun, so it is much higher at morning than at noon. Such a function could not be easily approximated even with forms (3.10a). So just before 800 nm the colour reconstructing process of the brain breaks down.

This is a difficult question. Rhodopsine is a popular ancestral pigment, but Rushton (1962) showed that rod sensitivities (peaked about 507 nm) are similar to the absorption of rhodopsine, and cone sensitivities should be derived from their "colourless" sensitivity, peaked at 555 nm. For this see Fig. 4, although the explanation of the slight difference between the published dark-adapted sensitivity curves of Landolt-Börnstein and Rushton would need some explanation.

If not, there is trouble, because all modern science is based on it. Aristotelian logic is bivalued, "tertium non datur", human colours are tripolar (if we ignore the rods).

But there is no problem with bivalued logic. It tells that something is "either A, or non-A, tertium non datur". And a colour is either red, or non-red; this is true, only not too informative. Therefore colour language is excellent for tripolar problems as e.g. the physical theory of Quantum Chromodynamics (Prisznyák, 1995) or McLean's "trial brain" (Lukács, 1995b).

Now we have investigated some disturbing problems. Some we have been able to answer, some to ignore and some are in between. Now we can return to the evolution of colour vision. From the above facts, theories and guesses a pattern emerges, maybe good, maybe wrong.

There is no colour vision without cones. But the original rod/cone dichotomy may be connected with nocturnal/diurnal lifestyle. Rods are in trouble in full daytime and cones are blind in night. It is better if cone sensitivities are high at the peak of the solar spectrum (our greenish yellow), but night light is shifted to shorter waves, so the peak of rod sensitivity must be there. If an animal is not purely nocturnal or diurnal, it should have two receptors, rods and neutral cones. With this a limited bichromatic vision is possible, but only in morning or early evening. In noontime rods cannot work and world becomes black-and-white.

To get a permanent diurnal colour vision two different cones are needed. If their spectral sensitivities are fortunate, the animal can reconstruct the characteristic wavelengths of dominant reflection bands. Then it is not handicapped at any part of the visible range. Some mammals, where colour discrimination experiments give doubtful results, may be such special bichromats.

Trichromatic vision would be superior, but the new pigment should be very special, namely from eq. (4.1)

P₂(x)P₀(x) ~ P₁(x)² (5.2)

This needs many steps of mutation; at the beginning of this process either P_i's of the brain are "non-Gaussian", but eq. (3.4) holds, i.e. colour constancy would be good but only for non-natural illuminations and reflectants, or P_i's are Gaussian but eq. (3.4) does not hold, when colour constancy is always wrong. Then this stage corresponds to our "anomalous trichromats". Simultaneous validity of Conds. (3.4) and (4.1) can be more and more approached in the evolutionary processes.

If mammals were indeed nocturnal until the end of Cretaceous to avoid dominant reptiles (for which we do not want to argue), then all non-Primates mammals learnt colour vision independently from and parallelly with us, and reptile colour visions may be much older. Arboreal animals could avoid dinosaurs even daytime, but many Prosimii are nocturnal even now, which may be a hint for nocturnality of Primates in upper Cretaceous (or may be secondary as well). Tupaia is perhaps close to ancestral Primates and sees colours, but Tupaia (and squirrels, unrelated genetically but similar to lifestile) are not nocturnal. All observed Simia sees colours except for Aoutus trivirgatus, but the popular name of that species is "night monkey", so it is nocturnal. We are at the peak of Primates colour vision, together with the Pongids, but it is possible that a future evolutionary stage would call us anomalous trichromats because Relation (5.2) does not yet hold outside of the range (510 - 600) nm. The reported similarity between human and chimpanzee colour visions, whose evolutions are separate for at least the past 5 million years, indicate how slow is now the change of A_i's. So maybe the present stage is already quite satisfactory.

Finally we note that from pure physical viewpoint the triangular scheme would not hold, since for the wavelengths of peaks in sensitivity the 3 receptors are linearly arranged (see Fig. 8). In this, but only in this, sense Green is between Blue and Red. However the brain forms a tripolar scheme from the 3 data.

Extended discussions with Drs. N. Balázs, C. Liegener and M. Abdel-Raouf are warmly acknowledged by B. L.

Abdel-Raouf M. A., Liegener C. & Lukács B (in preparation)

van der Akker & al. (1947): J. Opt. Soc. Amer. 37, 363

DeValois R. L. & Jacobs G. H. (1968): Science 162, 533

DeValois R. L. & Jacobs G. H. (1971): in Behavior of Nonhuman Primates, eds. A. M. Schrier & F. Stollnitz, Vol. 3, p. 107, Academic Press, New York

Granit R. (1955): Receptors and Sensory Perception. Yale University Press, New Haven

Hess E. H. (1973): in Comparative Psychology: A Modern Survey. Eds. D. A. Dewsbury & D. A. Rethlingshafer, McGraw-Hill, New York, p. 349

Horváth G. (1994): Candidate Theses, Hungarian Academy of Sciences, Budapest

Ishak I. G. H. (1952): J. Opt. Soc. Amer. 42, 529

Landolt-Börnstein (1952): Zahlenwerte und Funktionen III. Electrotechnik, Lichttechnik, Röntgentechnik. Springer, Berlin.

Lotz J. (1954): J. Amer. Folk. 67, 369

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

Lukács B. (1992): Acta Antiqua 33, 399

Lukács B. (1995a): Proc. ERÖFI II, ed. A. Rácz, KFKI-1995-11, p. 187

Lukács B. (1995b): This Volume, p. 54

Marks W. B. & al. (1964): Science 143, 1181

Maxwell J. C. (1856): Trans. Roy. Scottish Soc. Arts 4, 394

McAdam D. (1942): J. Opt. Soc. Amer. 32, 247

Mello N. (1968): Neuropsychologia 6, 341

Newton I. (1730): Optics. Innys, London

Ohba N. & Tanino T. (1976): in Colour Vision Deficiencies III ed. G. Verriest, Karger, Basel, p. 331

Palmer G. (1777): Theory of Colours and Vision. Leacroft, London

Passingham R. (1982): The Human Primate. W. H. Freeman & Co. ltd.

Oxford

Prisznyák M. (1995): This Volume, p. 15

Rushton W. A. H. (1962): Visual Pigments of Man. Liverpool University Press, Liverpool

Trezona P. W. (1976): Colour Vision Deficiencies III ed. G. Verriest, Karger, Basel

Walls G. L. (1967): The Vertebrate Eye and its Adaptive Radiation. Hafner, New York

Wasserman G. S. (1978): Color Vision. Wiley Interscience, New York

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

Wolin L. R. & Massopust L. C. (1970): in The Primate Brain, eds. C. R. Noback & W. Montagna, Appleton-Century-Crofts, New York, p. 1

Young T. (1802): Phil. Trans. Roy. Soc. London 92, 387

Really, I think you should read the Trezona paper, and definitely so if you are a male so automatically deficient in your fovea.

And now the paper about an Egyptian goddess scattering malachites & turquoises:

B. Lukács

Central Research Institute for Physics, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

An observation of L. Kákosy, namely that some stars seemed green for ancient Egyptians, is tried to be quantified here. The result is that the only possible explanation for such an effect is the location of ancient Egypt's conventional white colour in our dilute purple region. Semiquantitative attempts are made to reconstruct the structure of the ancient colour space.

Some years ago L. Kákosy arrived at a very remarkable hypothesis about the colour vision of ancient Egyptians [1], namely that some myths and pyramid texts suggest as if they had seen some stars green (Ref. 1, p. 55), even if this colour is not general in the texts, and, of course, mithological references and possibly metaphoric expressions are not to be confused with astronomical statements. The hypothesis was based on the following observations:

1) In a myth of "Celestial Cow" the Sun God, leaving Earth, is creating "greening" stars [2].

2) In a pyramid text a goddess is mentioned to fill the sky with stars by scattering stones, including turquoises and malachites [3].

3) In a standard text the deceased scatters malachite in the Otherworld; according to Kákosy's observation [4] she does so in order to take the rôle of the goddess mentioned in Point 2) for a respectable afterlife.

Since we do not see green stars (details below), these observations are very interesting and important, but their meaning is not obvious. However, according to the above evidences one can accept that there was not impossible for early Egyptians to compare a star to a piece of malachite. Now, a star is observed purely visually, so the similarity is expected in light, which is, in general sense, colour. In this paper we try to specify this statement further.

For the average present observer the overwhelming majority of stars is simply white or colourless, because of faintness. However, ancients were more familiar with naked eye observations, night observations were less disturbed by artificial illumination, and the atmosphere was clearer (and Egypt has even now clear sky and air). So we may accept that they saw some stars coloured.

Telescopic observations show the colours of stars very clearly. Using first qualitative terms, some 99% of stars can be classified into 7 spectral classes, with the approximate colours:

O,B: light blue

A,F: white

G: yellow

K: orange

M: red

and the "exceptional" classes S, R and N are also orange or red [5,6]. So indeed there is no green star. (As we shall see later, this is a consequence of the shape of the Planck spectrum, characteristic for gases in thermal equilibrium. Serious and special deviations from this energy distribution, namely serious suppressions on both ends of the frequency range, may lead to greenish tint, but such stars must be exceptional and rare.)

A planet may be green. However, Mercury and Venus are white, Jupiter and Saturn are yellowish, while Mars is red. Uranus and Neptune possess some greenish tint [6], but Neptune is never visible for unaided eye; Uranus is on the verge of visibility, having an average brightness 5m,8 (the conventional border of visibility is 6m), but it was discovered by Herschel using a rather substantial telescope. So planets are ruled out.

Still, in qualitative terms, it might seem possible seeing a double star (e.g. O+G) green, since some mixture of blue and yellow is green. Therefore we have to turn to quantitative description.

Maxwell's three colour theorem states that colours form a three dimensional space, in which constant luminosity lights can be mapped on a surface. In canonical coordinates (additive in mixing) the map resembles a "curvilinear triangle" [7] displayed on Fig. 1. The boundary consists of a horseshoe and a straight line; the former contains lights from pure spectrum lines, the latter is the purple border composed from mixtures of extreme red and violet lines. The interior points correspond to generic light spectra and can be produced by mixing the "canonical white" with a light from the boundary, i.e. with a spectrum line or with a pure purple [8]. There is no physical light spectrum which could produce coordinates outside the "triangle". For colour, a light is fully defined by its three colour coordinates, normalized to 1 when the intensity is of unity.

Fig. 1: Stars and reflectants in our colour system. Coordinates are normalized to unity. The horseshoe boundary is the location of pure spectral colours (representatives of familiar colours as Red, Orange, Yellow, Yellowish Green, Green, Blue and Violet are indicated by initials), the closing straight line is the purple end. Curve S is the location of blackbody (star) radiation without atmospheric Rayleigh scattering subtracting blue. This effect is not calculated but its general direction is shown by the arrow. Individual points are:

W: our conventional white

W': lamplight white in experiment [8]

M: approximate malachite colour

T: turquoise colour [16].

In this diagram the conventional white practically coincides with solar illumination, so lies roughly on line S. This is the reason that the green sector is not represented by starlight. Note that this Figure contains two quite different sets of points: those of reflectants (W, T, M) and of lights (S, W'). The details of this problem are in the text.

Stellar spectra are well known and, apart from fine details irrelevant here, simple. They are very similar to blackbody or Planck spectra characterized by a single parameter, the temperature. Therefore on the colour map of unit intensity the ensemble of stars occupies a single narrow line S [7], displayed on Fig. 1. This line passes very near the "conventional" white point, where all the colour coordinates are equal and 1/3. It is easy to see that indeed, in order to be in accordance with the general impression about star colours, line S must practically contain the point considered white.

In order to see this let us speak qualitatively about hue and saturation. A colour can be imagined, as already told, as a mixture of a pure one and white. Such a mixture lies on the straight line between the white point and the boundary; then, for orientation, we may tell that the endpoint shows the dominant hue, while the length of the line between white and the colour measures the saturation. Now, assume that W is somewhere above S. Then some points of S lie on lines starting in W and ending on the purple line, i.e. then some stars are (light) purple. We do not see this. In contrast, if W is below S then some stars are light green, and we do not see this either. Our experience can only be understood if W is a part of S (at least in good approximation). Then, going from left to right on S the colours are blue, white, yellow, orange and red, just as in observation.

The "conventional" white is also near to the Planck spectrum at 5700 K temperature, which is the surface temperature of Sun. Still, Sun seems yellow. But this is a consequence of atmospheric Rayleigh scattering, taking out some blue and dispersing it on the whole sky. The full solar illumination is quite white. So at the bottom of the atmosphere all points of S are seen slightly shifted to the right. This is the reason to see really red stars, absent on Fig. 1; the approximate direction of this effect is indicated by an arrow on Fig. 1, the extent of the shift would need detailed calculations. These calculations will not be done here because the arrow does not show into the green sector.

For a double star the mixed light lies on a chord of this line; again, this cannot be in green.

With this we have finished the first part of demonstrating the problem. We now cannot see stars green. If it was possible for ancient Egyptians, something must have been different. Since stellar spectra are and were determined by fundamental physical laws, the only possibilities are

a) different colour vision; or

ß) different colour naming system

in ancient Egypt.

Since colour coordinates vi are additive in mixing, the coordinates of any spectral distribution V(l) can be formed by means of three functions A_i(l) as

v_i(V) = òA_i(l)V(l)dl (4.1)

where l is the wavelength. The functions A_i(l) are well known from colour measurements for average observers, being the colour coordinates of sharp spectral lines of unit intensity [7]. For the biology behind A_i the most probable hypothesis is the existence of three different types of cones in the retina, and then A_i(l) are their sensitivity functions [9]. Indeed, at least two of the functions are experimentally identified [10].

Then some stars might lie in the green region for observers having unorthodox pigments in their cones. However, the pigments belong to the hardware of the human body and one cannot expect major changes in them on 4000 year time scale. Namely:

1) Contemporary human populations (except for ~5% with colour vision defects) exhibit very similar A_i functions. To be sure, American whites have been compared to contemporary Egyptians [11], and minor differences exist in the spectral sensitivity. However, the differences are small indeed, and are, perhaps, connected with the degree of general pigmentation of the body.

2) Mutation rates of various human genes are in the order of ~10^-4/generation [12]. So random mutations cannot result in major changes in 100 generations.

3) Evolutionary trends might lead to major changes, but the measured sensitivity functions are generally conform with the general features of reflected lights (for the idea see Ref. 13, for a quantitative check, Ref. 14). Therefore natural selection is not expected to drive evolutionary changes now or in the near past. (For time scales compare the 4000 years to the ~4 million years of human evolution or to the ~100000 years of Homo sapiens.)

Hence one arrives at the conclusion that Fig. 1 must have been relevant for ancient Egyptians as well, except for the minor differences mentioned in Ref. 11. It would be worthwhile to see the consequences of these differences, but neither their degree nor their nature is enough to transfer any part of line S into the green sector.

If not vision, then maybe naming has changed. Indeed, colour naming systems have substantially changed in historical times. Most contemporary European languages use four "fundamental" colour names (red, yellow, green, blue), although further individual names do exist for "mixed" colours. However, contemporary Russian distinguishes two parts of blue without a generic term; Latin has a great variety of colour names, with a generic term for green, but none for red, yellow or blue. On the other hand, Kamass Samoied has a common word covering blue, green and golden (!) [15]. Such changes may occur; there is nothing in physics or biology to put division lines between sectors of the colour "triangle" (Fig. 1) [13].

However, the system is not involved into the problem if the names are similes. The colour coordinates of real turquoise and malachite can be measured. Indeed, Ref. 16 gives correspondences between colour coordinates and names derived from real materials. Turquoise is listed there; malachite is not, but must be somewhere in the triangle stretched by scarab green, tarragon and mintleaf, which precision is quite enough in this moment for our purposes. So the two materials mentioned in the texts are displayed on Fig. 1.

Of course, a colour "malachite" is not necessarily pure; it is possible that only the dominant hue is malachite. So a colour can be called "malachite" if it lies along (or near) the straight line connecting points M and W, and extended until the boundary on the side of M. The same is true for "turquoise". But these lines meet Curve S again only in the white point. So the problem has still survived.

Still there remains a possibility to be in accordance with everything known. Namely, what is identified by colour vision is not the reflected light but the reflectivity of the surface, loosely speaking the reflected light at average illumination, as emphasized by Ref. 13 and demonstrated by the relative constancy of colours at changing lights. The reflectivity is given by 3 colour coordinates, which are restored in the brain from the coordinates of form (4.1) of the illuminating and reflected lights [13]. To be sure, coloured lights can be seen on dark background (as e.g. stars), but it is reasonable to assume that in such a situation the brain assumes an illumination [14].

But then, strictly speaking, a light does not have a definite colour (although formally it can be put on Fig. 1). The conventionality of light colour is demonstrated by comparing sunlight and electric bulb illumination. In itself the electric bulb seems to shine white, while in daylight illumination the bulb light is reddish yellow. (See Point W' on Fig. 1; it is a well defined lamplight used as white in the experiment described in Ref. 8.) Then it is improper to define white via sunlight; white is the colour of a special reflecting surface, e.g. of a surface reflecting all wavelengths with the same efficiency. The present white, coming from an intenational convention [9], seems to be just such a surface, however a whole region is called loosely white, although one can easily distinguish the included colours in experiments [9].

The wavelength dependence of reflectivity cannot be measured without specific instruments, so our conventional white is an idea which could not be thus formulated by any pre-industrial society. In the colour space there is nothing to single out Point W [13,14], it can be put anywhere in the interior [14].

This fact offers the last possibility to interpret Kákosy's observation. The white, or neutral, or gray, is simply a conventional colour (sufficiently characterless, of course) chosen by the population, to which all other colours are referred. Now, it is possible to find such a new location for W (denoted by W*) that some blackbody radiations (of cca. 8000 K temperature) be in the "green" sector. By requiring that both the TW* and the MW* straight lines have crossing points with Curve S, the actual white point must be somewhere in the region W* on Fig. 2.

Fig. 2: As Fig. 1 but with a different white convention W*. This is the hypothetical colour system of ancient Egypt explaining Kákosy's observation [1]. The exact location of W* cannot yet be reconstructed; it is somewhere below S but restricted by the dashed boundary. The latter is explained in details in the text. For the new points:

C: crocus colour

F: fuchsia colour

P: melon pink colour [16].

The region between the dashed line and S contains colours which for us are dilute purple but nameless [16], so characterless. Lines TW* and MW* do cross Curve S, so the stars at the respective crossing points are in this hypothetical colour system (dilute) turquoise and malachite.

Fig. 2 is only an illustration and first attempt, so here the correction from atmospheric Rayleigh dispersion has not been calculated. The possible region for W* has been restricted by the following way. The left boundary is the line connecting the end of S with "turquoise"; from the left of it even very hot stars could not be seen turquoise. The bottom and right hand side is defined by plant colour names used by us [16] in the light purple region. While ancient Egyptians might have considered other dilute colours important, it is a fact that within the region so obtained Ref. 16 did not found characteristic names for the dilute purples, so those colours are rather neutral indeed. If needs be, more quantitative results could be obtained from detailed calculations.

So our suggestion is that Kákosy's statement about ancient Egyptians having seen some stars greenish, or malachite or turquoise means position W* of the "conventional white" of that c_ivilization. In our naming system that region is nameless dilute purple, nevertheless it may be defined white without self-contradiction. This is the only possibility that the hue of some stars may be named turquoise or malachite. We do not have any reasonable idea why and how that region was chosen white or neutral, only note that such an effect may have caused by a widely accepted white paint having some absorption in the green (around 550 mµ).

Since malachite does not sparkle, the alternative that the similes were chosen not for hue but for brightness (relative intensity) is ruled out, and the possibility that the term was used purely randomly is inconstructive.

Illuminating discussions with Dr. I. Borbély are acknowledged.

[1] L. Kákosy: Egyiptomi és antik csillaghit. Akadémiai, Budapest, 1978 (in Hungarian)

[2] Ch. Maystre: Bulletin de l'Institut Français d'Archéologie Orientale 40, 81 (1941)

[3] K. Sethe: Die altägyptischen Pyramidentexte, Leipzig, 1908, Pyr. 567

[4] L. Kákosy: Bibliotheca Orientalis 25, 323 (1968)

[5] E. Novotny: Introduction to Stellar Atmospheres and Interiors. Oxford University Press, 1973

[6] D. H. Menzel: Astronomy. Random House, N.Y. 1975

[7] Landolt-Börnstein: Zahlenwerte und Funktionen, Technik, 3. Teil, Elektrotechnik, Lichttechnik, Röntgentechnik. Springer, Berlin, 1957

[8] J. A. van den Akker & al.: J. Opt. Soc. Am. 37, 363 (1947)

[9] G. S. Wasserman: Color Vision: An Historical Introduction. J. Wiley & Sons, N.Y. 1978

[10] W. A. H. Rushton: Visual Pigments in Man. Liverpool University Press, Liverpool, 1962

[11] J. A. Fraser Roberts: An Introduction to Medical Genetics. Oxford University Press, 1967

[12] I. G. H. Ishak: J. Opt. Soc. Am. 42, 529 (1952)

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

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

[15] J. Lotz: J. Amer. Folkl. LXVII, 369 (1954)

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

My HomePage, with some other studies, if you are curious.