Three-Color Vision
I have heard somewhere that three colors are not enough to reproduce ALL colors which the eye can see. Is that true?
(Adapted from question asked on Physhare, a list server of physics teachers)
Reply
It is true.
I have looked up this question in the writings of the true master, Richard Feynman, who dealt with color vision in chapters 35 and 36 of "The Feynman Lectures", volume 1. It is remarkable how far his interests have ranged, how few were the topics he did NOT look into. My own field of magnetospheric physics is unfortunately one of those few, and that only because his sister Joan (who is still with us) worked in it. When her brother showed interest, she told him to butt out, leave something for her to do. He did, and now we are stuck with all sorts of unresolved problems.
You should read Feynman's exposition, in particular chapter 35, which I tried to flesh out a bit here. Feynman is so much more articulate than I can ever be, and what I wrote below is anyway my own interpretation, not necessarily the same thing.
The important thing to do is to pose the proper question: what exactly do we want to claim? There is no question that the normal eye has 3 kinds of receptors, which we may designate (R,G.B) for (red,green,blue), although each is sensitive to a wide spectral range beyond the "pure" rainbow colors of red, green and blue.
When we look at some colored object, with some standard brightness (I choose here a standard brightness for all colors, so that only relative intensity matters), if the (R,G,B) sensors detect intensities (a,b,c), we will see SOME color, and that color will be characterized by the 3 numbers (a,b,c). The range of values for (a,b,c)--say, from 0 to 1 for each, after we have standardized intensities--covers ALL colors the eye can see. There exists nothing more.
The real question to ask is--is it always possible to "fake out" the color (a,b,c) by presenting the eye with a superposition of 3 colored sources, of three standard colors (U,V,W). These can be pure spectral colors, or not. (Red, Green, Blue) are a good choice, though, because each stimulates primarily one kind of receptor. Therefore, by twiddling with their intensities, we can make each type of receptor receive an EQUAL intensity, and the result registers in our mind as white. Let this be an assumed restriction on the choice of "primary" colors, requiring that we CAN get white; three different hues of green may never be able to do so.
Let the color U create responses (a1, b1, c1) in the 3 receptors of the eye. Similarly:
--the color V creates responses (a2, b2, c2), and
--the color W creates responses (a3, b3, c3)
These numbers (a1 . . . c3) are all COMPLETELY determined by our choice of reference colors (U,V,W).
Suppose we want to illuminate a white surface with a combination (a', b', c') of the 3 sources, in such a way that the eye will be made believe that it sees a GIVEN color (a,b,c). Then
a = a1 a' + a2 b' + a3 c'
b = b1 a' + b2 b' + b3 c'
c = c1 a' + c2 b' + c3 c'
It is now necessary to invert the equations: (a,b,c) are given, (a', b', c') are to be determined. (You can write it in matrix notation if you wish, and you then need invert some matrix). There always exists a solution (unless the equations are dependent--e.g. the 3rd is the sum of the first two), so it SHOULD be possible.
Having reached that point, Feynman raises a warning. A solution exists, but do we have a guarantee that (a', b', c') are all positive? They have to be if we add light from three sources--we cannot subtract (except maybe by using a colored screen, something he briefly discusses).
If we choose (U,V,W) wisely, the coefficients will be positive for a wide range of color combinations (a,b,c), all of which can be represented by combinations of our reference colors. But there may exist some oddball combinations (a,b,c) where the solution gives some negative coefficient. NO combination of (U,V,B) can then fool the eye to believe that it sees that color. Feynman claims that there will always exist such colors, and at this point I tend to believe him.
Unless there is some flaw here, you may pass all this to your class. Let them see Feynman's book, too!