Footnotes to the Film

The idea behind the film was not to present already established facts and theories in a popular form. It rather presents as series of demonstrations that raise questions which force us to start thinking.

In physics, observation goes hand-in-hand with imagination. How can we avoid getting snared by the magic of the appealing demonstrations of classical optics? What are the underlying postulates that were never explicitly formulated?  What does it mean to understand the world through mathematical concepts? For instance, to understand the nature of Light in terms of "Rays"?

I think a good film should illuminate a number of questions without trying to answer them. However, should you like to get a somewhat more elaborated commentary to what is shown in the film than the very concise statements that accompany the visual aspect of it  – here follows some supplementary notes.  

You are invited to contribute to the notes! Send a mail to and I will put your comment into this page.


An important aspect of the kind of investigation here documented is the careful attention to each phenomenon as a whole. Trying to keep all relevant aspects in sight. We may call it a  holistic approach.  I start my exposition with the statement that each image has its counter image.  This idea is intuitively self-evident as long as we are concerned with simple pictures in black and white. Just interchange the black and white areas! It was the ingenious idea of Goethe to study such complementary designs in black and white through a prism, thus establishing a rule of pairwise correspondence between colours, directly following from the symmetries of the geometrical conditions. He saw therein a ground for the notion of  "complementary colours", since long figuring in discourses on the art of painting.

His own interpretation of the phenomenon was that colour always presents itself as a totality. The given instance before ones eyes implicates its opponent colour, which is  potentially present and ready to appear "the moment a suggesting cause presents itself", as Goethe says in §69 of his "Farbenlehre", where he discusses the phenomenon of coloured shadows.


The complementarity between the two types of spectra is clarified in the diagram below.

Suppose you make a small hole in a plate and use it to produce a Newtonspectrum, by placing it in a flux of dispersed light. The small disk that you had to cut out to make the hole, if put into the light flux, casts a shadow in the shape of  a Goethespectrum. Of course, if you try to produce the two spectra simultaneously at one and the same place by putting back the small disk, the result will be total darkness. Thus we may properly call the two kinds of  spectra complementary, in so far as the conditions for there existence are complementary.  Furthermore, if the plate is a mirror, on the reflecting side you will get a Goethespectrum and the small disk will reflect a Newtonspectrum. Again, if we put the reflecting objects together, the two spectra will inhibit each other, in this case with plain white as a result.

Furthermore, complementarity is not just a question of adding two coloured light fluxes together, by superimposing them, and getting white light as a result. Nor is it a question of putting two transparencies on top of each other and getting opacity as a result.  No, the complementarity we get to know here is more fundamental. It holds between aspects of one and the same entity. Aspects, or moments, that by necessity belong to each other, as do the two sides of one and the same paper. This is what we call "the formal ontology of light and colour".  

In any theory formal ontological clarifications are necessarily prior to the empirical investigation. This is what Goethe alludes at, when saying: "to know that heaven is everywhere blue, you don’t have to travel the world around".   It is also therefore he is entitled to say, in his conversation with Eckermann, on the 18:th of March 1831: "…meine Farbenlehre is so alt wie die Welt."

It seems to me that, in general, Goethes experimental procedure aims at clairfying necessary relations that hold between aspects of one and the same whole, aspects that we are used to describe as being independently existing. And therefore tend to treat as if they were independently existing.


If light does not illuminate some object it is invisible. So the flow of light and the colour phenomena produced in it need to be visualized. One way of doing this is by help of fine dust or mist. Another often used and more convenient method is to project the light flow onto a sloping white board. The rays we see on the board are not coincident with the light rays themselves, but it is in our case a fair representation of these.  Because of the way a prism acts, due to its shape, the geometrical optical phenomena we are going to study are essentially two-dimensional, which permits this simple way of simultaneously visualizing the various stages of the metamorfosis of the colour spectra along the light way.

Height of prism: 85 mm

Width of light path: 55 mm

Slit in mirror:  1 mm

Note, that a "ray" is a concept, not an observable physical entity. The only observable involved is the event of absorption of a photon. And the place where it happens is the endpoint of a ray. An experimental observation involves a statistics of such absorption events at a certain place and time. Then you repeat the observation after moving the detector to another place, assuming a stationary state of the flux of light, so that it does not matter that this observation is done at a later time.  It turns out that the material, cleverly analysed and ordered, can suitably be described as the flux of light following straight lines in space, the so called rays. Next step is to visualize this concept in a proper way. (In my film I produce a kind of "meta-rays", being constituted by a series of end-points of rays.)


"Dispersed light"

One thing that especially caught my imagination when reading Holtsmarks article was the idea of a space illuminated by seemingly ordinary white light, but having the pecular inner quality of being "dispersed" after having passed the border between two media of different optical density.  Not until you start experimenting with various objects in that light its secrets are revealed. That was the reason why I wanted a prism as big as possible and it was also the reason why I choose to fade out the boundary colours of my "light road".

As a matter of fact "white light" is far from a simple concept. The quality of daylight varies appreciably, depending on the time of the day, weather conditions etc., and we humans have invented various kinds of artificial light sources – all giving an illumination that falling onto a white paper looks colourless, neutral, i.e. just "white".  As soon as colour samples are introduced into the illuminated field they may look shockingly different from their ordinary look, revealing the qualitative peculiarity of the light. I had earlier studied effects of this kind (so called "metamerism") quite extensively, in connection with an evaluation of the colour rendering properties of artificial light sources. So I knew that this way of demonstrating the relationship between light and colour is both spectacular and thought provocative. 


As you know, left and right are interchanged when you look into a mirror. So the reflected image, with a blue boundary spectrum, in the figure below,  actually corresponds to the spectrum that would appear at the left side of the original light-path.  (Whereas the yellow-red boundary spectrum appears at the right side, as you see.)  When you see this demonstrated in the film, the first time, it could be a bit confusing.




The diagram shown in the film is taken from Propr. II, Theor. II, Exper. 6,  Fig. 18, in Optics (1704) and the quotation is from Book One, Part I.

Newton further says:

All homogeneal Light has its proper Colour answering to its degree of Refractibility, and that Colour cannot be changed by Reflexions and Refractions. Opticks,  Book One, Part II. (PROP. II THEOR. II)



Concerning the concept of "Rays"

The ray concept itself is old – rays are seen already on old egyptian paintings and reliefs. How did humans come to think about light as rays? Because there is also darkness. The idea of the rectilinear propagation of light wouldn’t have appeard if there had not been objects casting shadows, linearly drawn by the sun on the ground or floor. Rays signify the experience of direction, of light arriving as illumination from a certain direction – the direction of the sun or any other localized light-source.

On the other hand, in an illuminated space light is just present. It is a state of the space, in the same way as total darkness is a possible state of the space. So what gives us the right to speak of light as fragmented into rays?

Most textbooks in optics either avoid defining the concept of "light-rays" or else name it as "the path along which light travels" or "path of shortest optical length". The concept is however more thoroughly discussed by the mathematician Georg Unger, in his book "Von Bilden Physikalische Begriffe", Teil II (1961), and similar ideas are found in the writings of the American perception psychologist James Gibson.

Georg Unger:  Ohne materielle (genauer: undurchsichtige) Körper kann die Geradlinigkeit der Lichtausbreitung nicht festgestellt werden. (13)

Wenn immer wir Lichtstrahlen Zeichnen, sind wir uns bewusst, dass ihre Berechtigung darin besteht, dass sie durch geeignete Blenden als Licht-Schatten-Grenzen verwirklicht werden können. Die Lichtstrahlen sind mögliche Licht-Schatten oder Bedeckungsgrenzen. (14)

Die geometrischen Konstruktionen und phoronomischen Bilder, die mit der Ausbreitung des Lichts verknüpft werden, beziehen sich grundsätzlich auf Erscheinungsort und Verschwindungsstelle unter berücksichtigung, praktisch der am Lichtwege liegenden, theoretisch aber aller im Raum befindlichen, materiellen Körper. (15)

James Gibson: Ecological Optics (1960): In ecological optics a ray should not be conceived as a beam of light, not even as one which vanishes to a geometrical line, but as a transition between one beam, and the next. It is the locus of change in light energy over the array.

The concept of a beam of light which becomes vanishingly thin but still retains a given intensity and spectral character is troublesome. Such a fiction may be useful for geometrical optics, and convenient for the tracing of rays through refracting media, but it cannot stimulate an eye. A transition, however, can stimulate an eye. The rays of ecological optics, being the loci of transitions, are not infinitely dense as they are supposed to be in pure geometry. Ecological optics does not have to be concerned with the problem of waves or particles, nor with the laws of refraction, reflection, and diffraction. It is primarily concerned with margins, borders, contrasts, ratios, differences and textures in the optic array.  (258)

Now listen to how Isaak Newton in his time defined Rays,  in his famous book Opticks:

By the Rays of Light I understand its least Parts, and those as well Successive in the same Lines, as Contemporary in several lines  (...) The least Light or Part of Light, which may be stopped alone without the rest of the Light, or propagated alone, or do or suffer any thing alone, which the rest of the Light doth not or suffers not, I call a Ray of Light.

Note 1: Newton does not speak of the Light of the Sun, but of the Rays of the Sun. This is in plural!

Note 2: Newton's view was not totally unlike the way many a physicist of today still looks upon light. Namely as a flux of an enormous number of quasi-particles, the photons, each photon being solitary and totally "unaware" of the others.

But an important difference is this: there are no different kinds of photons. The photon is one elementary particle among others, perculiar by being massless and having spin one. The frequency (and wavelength) is not a property of the photon. It depends on the quantum of energy that it happens to transmit -- and that is statistically determined by the particular material configuration, including the light source. And that energy influences how the photon behaves on its way between the point of emission and that of absorption. Furthermore: the photon is not a "Ray", nor a  point-wise localized particle, nor a wave-motion in some medium or field. It is rather a coincidence between two events in space-time: the emission and the absorption of a quantum of energy.   

In fact these modern rays, the photons, have a very solitary existence. They do not know of each other, just being momentary coincidences of emission- and absorption-events in an eternity of time and infinity of space. Light fluxes that are superimposed just add without any interaction. A light beam does not "take up space", does not (like matter) make itself impenetrable.


The important feature of Holtsmark's approach in the paper from 1960 is the consequent use of  mirrors instead of absorbing plates to delimit and diverge the light flux.  With mirrors no light is eliminated – the light flux is split in two parts, which can both be followed and studied on equal footing with each other. This helps us see clearly how the material conditions by necessity give rise to complementarity in the optical phenomena.

Two crucial diagrams from Torger Holtsmark's article from 1970.

Figur 11 describes the arrangement implemented in the film. Figur 10 describes an alternative arrangement, in which you can see how a dark absorptionline in an ordinary spectrum corresponds to a white line in the invertred spectrum. In our set-up with parallel slit-mirrors the two cases in principle look like this:




Explanation in terms of supersposition of complementary spectra.

How do you explain the purple ray?  What kind of illusion is this? Is it not typical for illusionists to make trick with mirrors! The picture below can perhaps help you understand what goes on. 


Turkois ray coming from below. Seems to continue on the upper side. Nothing comes from below. G-spektrum on the upper side. White light from below. Totally white field on the upper side.

What propagates through the hole is not a purple ray. But a dark ray that however gets illuminated on the other side, thus turning into a purple ray!  It gets its colour by being an illuminated shadow. Darkness cannot bring its colour with it …

The phenomenon of "coloured shadows" (shadows that are not just black och grey, due to the interplay between at least two light sources of different spectral quality) is important to the understanding of human colour perception and Goethe played a lot of attention to it, in his old days discussing with Eckermann how it might be explained. In many practical situations the phenomenon is not ideally realized (the shadow does not appear on a perfectly white field) due to circumstances, and the visual process then tends to change the visual image in the direction of the ideal case, through chromatic adaptation. Because of this it has usually (as by Goethe) been designated as a physiological colour, or subjective colour. But in the present case it is as objective (or subjective, if you prefer) as the green colour of the monochromatic light ray on the dark side of our apparatus.

How to explain the phenomenon along the lines of Newton's ideas

You can explain the purple line with help of the newtonian concept of  rays of light that superimpose. To simplify: Think of the light falling towards the arrangement of mirrors as being constituted by the superimposed light fluxes from three lamps: a red, a green and a  blue, mounted a little beside each other, thus shining from slightly different directions. And apply the well-known rule that the light fluxes  pairwise add to cyan, magenta, yellow and all three together add up to white.

First put on only the green light.  With suitably arranged mirrors you get a green ray on the dark side and a black ray in green environment, on the bright side of the mirrors. 

Then put on the red and the blue lamp. The black ray turns into a purple ray, since it is illuminated from the sides by red and blue light, adding up to purple. So the purple ray is a kind of coloured shadow!

Look upon this display from increasing distance and you will see how the image of a purple line on grey background emerges. 


Note. The purple ray in the light field could be used as a signal. It could be interpreted as information about something being the case, in connection with that specific direction.  The  direction of propagation of information is however not the same as the direction of propagation of energy in the field, as it would be in the ordinary case, when light rays are used to mediate the transmission. 





Besides that I present Goethe's idea of the principle of complementarity as central in colour theory,  and that I show how the use of mirrors is essential, when you want to demonstrate the principle, my study mainly concerns the concept of "Rays of Light". My conclusion is that Rays (even when you suggestively see them before your eyes and photograph them) are just constructs, artefacts, "Gebilde", in the medium of light.  But never light itself! It remains enigmatic.  



The case of "comb filtering", i.e. when neighbouring N- and G-spectra are made to overlap more and more, ending up in just grey, is shown in the clip below.



PSColour/fotnoter_uk.htm 2010-05-03 revised 2010-09-16