Feynman’s Paint-Mixing Problem

Physics Nobel Laureate Richard Feynman not only played bongo drums in nightclubs, but also wrote two chapters on color and vision in his Lectures on Physics. And that’s not all: There’s also…

Feynman’s Paint-Mixing Problem

Richard Feynman tells an interesting story [1] about revealing a painter's trick in mixing red and white paint to get yellow. Here's how it goes:

Feynman: "I don't know how you get yellow without using yellow."
Painter: "Well, if you mix red and white, you'll get yellow."
Feynman: "Are you sure you don't mean pink?"
Painter: "No, you'll get yellow."
Feynman: "It must be some kind of chemical change. Were you using some special pigments that make a chemical change?"
Painter: "No. Any old pigments will work."

So Feynman got a can of red and a can of white paint, and the painter began to mix them. It kept looking pink to Feynman. But then:

Painter: "I used to have a little tube of yellow here, to sharpen it up a bit---then this'll be yellow."
Feynman: "Oh! Of course! You add yellow, and you can get yellow, but you couldn't do it without the yellow."

Touché. Feynman wins.

But did he really? I remember looking at a white wall through a vial of yellow food-coloring liquid, and seeing it as red. That’s because the transmission spectrum goes from very low at the short-wavelength (blue) end of the spectrum to nearly 1 at the long-wavelength (red) end of the spectrum. As one piles on more layers of the same fluid, the transmission spectrum multiplies by itself wavelength-by-wavelength (an action known as Beer’s law, which by coincidence also happens when you look through beer). Therefore, at the wavelength where one ply of the liquid transmits half the incident energy, two ply of the liquid transmits only 1/4 of the energy. On the other hand, at wavelengths where one ply transmits all the energy, two ply will transmit all the energy as well. For a transmission coefficient that increases monotonically in wavelength (such as most yellows), the transmitted-light spectrum becomes biased toward longer wavelengths (i.e., is redder) when the layer is thicker.

So there’s at least one way red and white can will mix to give yellow: a clear diluting vehicle for the white and a red Beer's-law ink that transmits enough light at medium wavelengths so it yellows up when you see through less of it. Of course, you must have a reflecting background---let's make it matte white. As a numerical example, suppose a unit optical thickness of the red ink has transmittance zero for wavelengths below 540 nm, t for wavelengths between 540 and 640 nm, and 1 for wavelengths above 640 nm. The light reflected from the background through a unit thickness of ink can then be represented as the triplet (0, t², 1). That triplet will change to (0, t^2x, 1) when the optical thickness is changed to x. A deep red ink will have, say, t² = 0.1, whereupon ten-fold dilution of the ink (x = 0.1) will produce t^2x = 0.7943. The layer will therefore be substantially yellow.

You can also do this exercise (at least theoretically) with opaque red and white paints that obey Kubelka-Munk mixture algebra. [2]. I’ll elaborate about that in a future publication.

It seems, then, that the painter could have made a yellow by mixing particular red and white paints, contrary to Feynman’s intuition. But it certainly couldn’t be expected for all reds and whites as asserted by the painter. For example, drinkers of red wine (instead of beer) won’t see the yellowing effect---diluted red wine looks pink, not yellowish. Why should wine obey Feynman’s intuition where beer does not? The subject is worth much experimentation. Care to join me?

[1] R. P. Feynman and R. Leighton, Surely You're Joking, Mr. Feynman (Norton, New York, 1997), pp. 82-83.
[2] G. Wyszecki and W. S. Stiles, Color Science, (2nd Ed., Wiley, New York, 1982), p. 785.

Michael H. Brill, Datacolor