Pehr Sällström

COLOUR AND PHYSICS

ON THE COMPATIBILITY OF GOETHE’S THEORY OF COLOUR
WITH THAT OF NEWTON

Version date 2021-04-20

Video at https://youtu.be/HtXtFbu-t6M

 

INGRESS
Meeting the Phenomenon of Prismatic Colour Spectra

Once upon a time – back in the nineteensixties – I came across a book by the Norwegian author André Bjerke: “New Contributions to Goethe's Colour Theory”. I did the prismatic experiments described therein, and saw with excitement this variety of colour spectra and their systematic relationships, which was all completely new for me.

If you look through a prism onto this simple picture in black and white – holding the prism horisontally – you will see coloured spectra at the interface between white and black areas, as shown to the right. Each spectrum is a series of colours. One goes from white, via bright yellow, through nuances of orange up to ruby-red, ending in black. The other one, from white, via cyan and nuances of blue, to deep violet, fleeting over into black. These are the so-called "boundary spectra".

Note. In reality there will be boundary colours also at the uppe and lower edge of the picture, but these are hidden by the grey mask. Note also that in reality the colour areas run more smoothly over into each other than in this digital reproduction.

Next, the boundary spectra can be made to merge into secondary spectra, either at a narrow bright opening in darkness, alternatively a dark strip on white surround. The first case renders the well know rainbow spectrum – dominated by blue-violet, green and red – and the second case gives us the inverse spectrum, characterized by yellow, magenta and cyan.

Observe that the spectra can be turned into each other by interchanging black and white in the picture you look at. A perfect symmetry!  It suggests that light and darkness play equivalent roles in the world of prismatic colours.

It is evident that these four spectra together contain all hues needed for a full colour circle: running from yellow via orange and red to magenta, further on to deep violet, blue, cyan and from there, via nuances of green, back to pure yellow. But it is not just a circular series of hues. Some colours seem, by their very nature, closer related to light, others to darkness.

Inspired by Bjerke's critical discussion on current colour theories, I spontaneously thought that I had to find out how this beautiful miniature colour system might be analysed and described in terms of physics. That turned out to be not altogether easy. I knew of course that colour has something to do with the wavelength of light, but how to apply this knowledge to the phenomenon I now had before my eyes, was not immediately given.

I wanted to find out if Goethe had a point with his critical attitude against the Newtonian treatment of colour. And why his own contribution – presented in the pamphlet "Beiträge zur Optik" (1790) – was not regarded as relevant. Was there something essential Newton and his followers had overlooked? And are the two approaches after all compatible?

Searching for a Physical Correlate of Colour

In order to arrive at an appropriate physical description of the prismatic colour spectra and their systematic relationships, I preferred considering what Goethe called “Die objektiven Versuche”. In these you send a flux of light through a prism, instead of looking through it.

If you place a prism on a white cardboard and let the sun shine onto one side of it, the set-up looks like this. (Seen from above. I have shaded off the direct sun-light falling diagonally from the left onto the white board, to be able to observe undisturbed the flux of sun-light passing through the prism and being reflected from the white cardboard. )

Immediately behind the prism there is a region of colourless light, which is however “dispersed”. Which means: light of various wavelengths leave the prism in slightly different directions. This is seen at the edges, where the light beam borders to shadow, and where you get the boundary spectra, corresponding to the spectra Goethe had observed, when looking directly through the prism onto a picture with contrasting black and white areas.

The inherent regularity of the light, falling through the prism, is disclosed as soon as you place a shadow-casting object in the flux of light. The tiny shadow behind the needle turns into a spectrum with yellow, purple and cyan.

Alternatively, if you place a screen with a narrow opening in the flux of colourless light, an ordinary Newtonian spectrum (red, yellow, green, blue, violet) is obtained.

Now I wanted to analyse the spectral compositions of the light at different positions in each of these various kinds of spectra. Starting with the boundary spectra.

Instead of doing it by theoretical analysis, in terms of rays, I found a way to do it experimentally, in a setup like this. (The figure is from an internal report 1966-11-01.)

By sliding screen S1 to various positions, you get on S2 a spectral analysis of the light at that position. Let us see how it looks in practice!

Here we go successively into the yellow boundary spectrum. You see how the full spectrum of light at first looses its violet end, then the green middle, until you are left with red and finally end up in darkness.

You must permit script or ActiveX-controller in order to see the film-sequence

 

Doing the same with the blue boundary spectrum, we cut away components from the long-wave-side. First the red, then green and we are left with the short-wave violet, and finally darkness.

 

Summing up with diagrams, it looks as follows. To the left you have the analysis of the yellow boundary spectrum. To the right, the blue one. Observe the symmetry: We seemingly cut up the full spectral distribution, representing white, in two parts. The corresponding colours are complementary in the sense that if you add them (i.e. superimpose the corresponding light-fluxes) you get back colourless light.

So, by cutting away more and more light from the short-wave side of the distribution, we generate a series of colours continuously running from yellow, via orange to red.
Cutting away light, instead, from the long-wave side, we generate the series cyan, blue, violet.

I have, a bit arbitrarily, marked wavelengths over the visible range 400-700 nm. Actually, in the experiment, it was the varying angle of direction. See the set-up scheme above.

The secondary spectra, wich include green and purple respectively, can be analysed in the same way. They are the result of combining the procedures: Either cutting away from both sides, leaving a region in-between (illustrated to the right in the figure below). Or cutting away light in the middle, leaving a short-wave and a long-wave region (illustrated to the left).

Analysing this inverted spectrum you see a dark region running through the spectrum. The dark region is thinner the farther away from the shadow-casting object you choose the point where you make your measurement. (Upper row, to the right, in the figure below.)

Obviously, when I speak of the ordinary and the inverted spectrum, this does not refer to any specific spectral image, but to two types of spectra. Which concrete spectrum you actually get depends on the relative width of the strip in the middle.

It is worthwhile, in passing by, to remember that Newton had concentrated his study on a limiting case, namely the one seen at the bottom right. His intention was to produce a series of lights of almost pure wavelengths (refrangibility, in his terms), out of which he could extract single ones, for further study. Initially to convince himself (and the reader) that the specific refrangibility is an immutable property of a flux of light, prepared in that way. This, in turn, to get support for the bold idea that rays of specific refrangibility were already at hand in the flux of light arriving from the sun (or, as we nowadays think, from even more distant stellar object).

For the illustration I have chosen an optimal width, making both the ordinary and the inverted spectrum come forth in full colour with optimal brightness.

For any given width, the two types of spectra are complementary, in the sense that if the two images are projected onto each other the result is white. And, of course, even the individual colours of the respective spectra are pairwise complementary, as you see from the diagrams illustrating the compositions. I will return to this in a while.

 

I have now, by help of an analysis in terms of spectral composition, turned Goethe's colourful experimental demonstration into a set of diagrams. The formal relationships among these diagrams correspond to the qualitative relations we saw among the individual colours and colour transitions of the spectral images.

The illustration above represents the logic inherent in the geometry of the experimental set up. Each diagram answers yes or no to the question concerning which wavelengths (= "sorts of rays", in Newton's vocabulary) are to be found in the illumination at the spot where the spectral composition was measured (defined by screen S₁). This is the reason why I call them "compositions". They are strictly binary, unlike the "spectral distributions" we shall meet in a while.

It was pointed out to Goethe by his contemporary physicists that all these various spectra, he showed them, were explainable by help of the ingenious Newtonian concept of "rays of different refrangibility".
To be sure. They are all consistent with geometrical optics. But that is not the point! The point is that a systematic study of the spectral transformations, in the way Goethe did it, gives us information about the physical reality we are confronted with, when we look at coloured objects, illuminated by daylight. It gives us an idea about the basic structure of "the world of colour".

Taken together these simple binary (bright/dark) modifications of the full spectrum constitute a closed system of physically defined colours with interesting symmetry properties - such as polarity and complementarity - and with significant positions for the elementary hues, between the poles white and black.

What Goethe wanted to know was the objective physical grounds for the various colour phenomena we observe in nature. And here he had a first clue to it.

 

WHAT ABOUT THE COLOURS MET WITH IN PRACTICAL LIFE?

Of course these binary spectral compositions are artefacts and one may ask whether they are representative of colours, met with in practical life.

When we compare the idealized spectra with the spectral reflectances of physical objects, it turns out that, yes, on the whole there are similarities. For instance an ordinary yellow surface may have a reflectance curve like this (red curve).

Compare with the corresponding idealized reflectance! (blue) The natural distribution is smoothly varying. But the tendency is the same: strong absorption of light at the short wavelength side.

I have here permitted myself a slight deviation from the perfect ideal distribution, namely by adding a small amount of neutral background and also accepting that the reflectance in practice never reaches the theoretical maximum of 100%. Fact is that the colours of real materials are more or less greyish, as compared to the corresponding ideal colour. There is no place for a grey scale in the system of ideal colours. This is what make them "ideal". Learn more about this in the next lecture "Ideal colours - what is that?".

Here is a series of yellow to orange colour samples, obtained from various concentrations of a yellow colorant. The cut-off wavelength is successively shifted towards the long-wavelength side.

Concurrently with the reddening, the sample by necessity gets darker, because more and more light is absorbed. It is in the nature of yellow to be bright and red to be dark.

To arrive at the very red end of the tint series you have to look through a thick layer of a concentrated solution, as here.

A similar reddening with increasing degree of aborption is found on the blue side; running from cyan, via blue, to dark violet. Even green liquids may switch to red at sufficiently high concentration.

Goethe found this intensification -- "Steigerung" as he called it -- significant. In §519 in his Farbenlehre he says:

This is one of the most important appearances connected with the doctrine of colours, for we here manifestly find that a difference of quantity produces a qualitative impression on our senses.

So far about the boundary spectra. What about other regions of the colour world? The following figure shows an example of reflectance spectra for strongly saturated colour samples, arranged into a colour wheel.

Observe that yellow is placed at the top, due to it being the brightest colour. This is an often used alternative to Goethe’s arrangement with purple at the top. This is Hering's opponent colours circle. The strongly saturated colour samples were prepared by Karl Miescher, in Basle.

This physical colour wheel is certainly not as perfectly symmetric as was the case with the ideal spectra. This is how things look in practice! We are dealing with what Goethe called "chemical colours". Two relatively broad absorption bands in various positions take us around the hue circle.

What happens when we continue from red to purple, deep violet and pure blue? Well, a second absorption band enters on the short-wave side. It keeps the blue and green peaks down, with the consequence that colours in this part of the circle are relatively dark, compared to their opposites, on the warm side. (An asymmetry Goethe keenly observed. He designated yellow as the colour closest to light and blue closest to darkness.) The two sides of the visual "window" are fundamentally different. Short wavelengths face onto energetic radiation, provoking chemical reactions at the molecular level. Longer wavelengths face onto heat radiation and molecular vibrations.

Goethe was well aware of the fact that the idealized colours are highly artificial and searched for colour phenomena in nature, that could serve as concrete examples of the general scheme he had found. He decided on turbid media, displayed in the combined phenomenon of the blue sky and the reddening of the setting sun.

These colours are not based on absorption but on scattering, which is stronger at lower wavelengths, giving distributions like this:

Softly changing curves, but still compatible with the general scheme of ideal colours.

 

SO WHAT HAVE WE LEARNT CONCERNING THE PHYSICAL CORRELATE OF COLOUR?

A popular misunderstanding has been to relate hue to the series of wavelengths, displayed in the ordinary newtonian spectrum.

This goes back to a proposition in Newton's Opticks, where he suggested that:

Every Body reflects the Rays of its own Colour more copiously than the rest, and from their excess and predominance in the reflected Light has its Colour.

This might sound plausible, but as we have seen above, the reflectance of surface colours does not necessarily have a pronounced maximum at a position in the wavelength spectrum corresponding to its hue.

As a rule it is rather the absorption of radiation in a certain wavelength region that determines the colour. Which means that the interplay of light and darkness is decisive, in the formation of colour.

Evidently, it is the distribution over the whole visible range of wavelengths that should be taken as the physical correlate of perceived colour.

Furthermore, since the whole spectral range counts the presence or absence of any particular wavelength is of no decisive importance.

We do not see selected simple rays of light, but we see the reflectance properties of object surfaces.

Of course Newton must have been aware of this .. consider the passage "more copiously than the rest" in the proposition above. Without this qualification the statement would be meaningless.

In practice, the reflectance curve of individual samples can show diverse smaller variations within the total range. How to handle this fact?

My idea is that what we perceive as colour of an object has to do with whether the reflectance function as a whole is articulated towards the long-wave side or the short-wave side, respectively towards the center or the edges of the visible range of electromagnetic radiation. As suggested in the following set of diagrams.

The eye does not perform a detailed spectral analysis. But it is capable of detecting a kind of second order modulation of the light flux. The first order modulation being the overall change in reflected light intensity, leading to more or less darkness of the sample.

Since the number of possible spectral compositions is innumerable, of course many of them will be identical, from the point of view of the second order modulation we perceive as colour. (They impose higher order modulations, but these are invisible, without special tools or procedures. Such as looking through colour filters or looking at the sample in various kinds of illumination; or ultimately doing a spectral analysis. For instance when one wants to identify particular pigments used in a painting.)

Mathematically it can be formalised as follows. Define three functions of wavelength: w, yb and rg, which take care of lightness, yellow/blue-balance and red/green balance. Multiply each one with the reflectance function and integrate over the range 400-700 nm. You get three numbers, characterizing the "colour valency" of the reflectance. All reflectancies having the same valency belong to the same equivalence class.

 

CONCLUSION

By applying a Newtonian analysis to Goethe´s experimental observations I have shown that there is no essential contradition between the two pioneer's contributions to colour theory. They rather complement each other.

The series of elementary rays, introduced by Newton, can be used as a basis for defining and measuring spectral distributions. The way the initial distribution is modified, when incident light is reflected from a surface, or passes through a translucent medium, is what determines the perceived colour of the surface.

To the left a translucent object - a blue colour filter - two the right an opaque pice of blue paper. It reflects the illuminating light. The filter looks blue because the illuminated white wall behind shines through it towards the oberver.

André Bjerke, in his above mentioned book, suggested that the reason why Newton avoided going into the nature of colours we experience in practical life, had to do with the fact that he performed his experiments with rays of light in a dark chamber.

Whereas for Goethe, experimenting with illuminated black/white pictures, it was natural to regard colours as the result of modifications of colourless daylight.

Newton primary interest was to find out the nature of light, the radiation from the sun, and its seeming fundamental constitution in terms of fundamentally different sorts of rays. He also had a practical goal, namely to find a way to get rid of the colored borders that distorted the images in optical devices. But for Goethe the intent was to understand the colours we meet in nature and practical life.

I am sure there lies something in Bjerkes observations. But still, I think there was a deeper reason behind Newton's avoidance of digging into the physics of colour.

After all, optics is the science of vision. How are we capable of perceiving something such as colours? That's an enigma. Like his forerunners - for instance Galilei - Newton held that colours must be understood as subjective sensations, caused by the action of the rays entering the eye. So he delegated the issue of colour to the physiologists.

That the quality of hue is something subjective cannot be denied. But to see the world in colour does not mean that what one sees is subjective. On the contrary: it conveys objective information about the world.

Characteristic of a sense organ is that it is capable of showing something else than itself or products of its own activity. It has the power of making itself transparent for what there is to be perceived. I am inclined to regard the light, which under normal conditions enters the eye, as belonging to the sense organ (or to the visual process, if you like). Hence it is invisible. I see by help of my eyes and by help of the ambient light.

What I see and how I see it are two different questions, and for me as a physicist that first one had priority. But, to be sure, the second one is also intriguing -- and I have later on spent a lot of research efforts in perception psychology -- but that is another story.

Thanks for your attention.

 

EPILOGUE

Listener:
You end by saying that ”how we see colour” is another story. Isn’t that to make it a bit easy for yourself ? In some way or other it must be the light entering through the pupil of your eye that determines what you see. Gives you the image of the outer world, so to speak.

PSm:
You are thinking along the lines of Alhazen, his ”camera obscura”-model of the eye, and a ”point-to-point” relationship between object and image. A way of thinking that where taken up by Descartes, Kepler, Galilei and others, to become a standard model of vision. I would call it a ”causal model”: Our vision of the environment as being caused by the action of light rays entering the eye.

Nowadays we know that this underlying flow of causal events is so many orders of magnitude of scale below the world we consciously live in. It belongs to a micro-world, or even sub-micro-world. So the relationship cannot be strictly deterministic, in a classical mechanical sense, but necessarily only statistical.

Listener 2:
But still, your idea that it is the reflectance of objects that should properly be related to their colours is logically ambiguous. The light, reflected from the object surface, has a spectral energy distribution (SPD) that is the product of the objects spectral reflectance with the SPD of the illumination. How can the vi sual mechanism separate these two factors?

PSm:
This is what make the problem so intriguing, as I said in my lecture. There is no general solution to it. One has to accept that the visul perceptual system makes a qualified guess (like what we try to make machines capable of through artificial intelligence). Mostly it is to the point, sometimes wrong. I suppose the reason why my view has been called a ”computational theory of colour vision” is that it borrows from information theory and technology.

Perception psychologist James J Gibson argued for this way of approach in: Ecological Optics (1961), The senses considered as perceptual systems (1966) and The Ecological Approach to Visual Perception (1979).

***

Let me close this Epilogue with Goethes declaration, in his opposition against Newton:
Hier wird nicht von Ursachen gefragt, sondern nach Bedingungen unter welchen die Phänomene erscheinen. /Das reine Phänomen, 15 Januar 1798/
(I am not asking for causal relations but for conditions under which the various colour phenomena appear.)
In short. What are the physical conditions behind the colours of the objects I see in the world when looking around?

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© Pehr Sällström. October 2017 - September 2018 - last revision April 2021

 

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