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

Green Technology and Yellow Afterimages

By Michael H. Brill, Datacolor
A logo shown at the last ISCC meeting evoked a memory from graduate school….

At the recent Baltimore ISCC meeting, David Oakey gave a talk on “Respect for the future through the use of color.” One of his visual aids was the new British Petroleum logo (see below, or search “BP logo” and click on “Image Results”), which spoke of solar power and green energy through its sun-like white center with yellow-bordered rays, surrounded by green leaf-like structures. Staring at the pattern on a large screen, and then at a piece of white paper, I saw a quite distinctive afterimage: bright yellow in the center of the pattern, surrounded by nothing very distinctive. I was surprised that the afterimage was brighter than the paper (the white center should have evoked a dark afterimage), and also by the yellow color (as opposed to blue, induced by the yellow border in the logo). For a smaller image of the logo, I saw something more like what I had expected: a faintly yellowish center with a diffuse purple surround.

This was reminiscent of two effects I found [1] in exploratory efforts as a graduate student under the direction of Jerome Y. Lettvin (MIT).

(1) Extending Abney’s finding [2] that all colors seem to shift toward yellow when mixed with white light, Lettvin [3] proposed that even yellows should get yellower: i.e., a yellow light should become more saturated when mixed with white. Accordingly, I projected sharply focused white spot on the diffuse yellow background produced by shining a white light through a Wratten 15 filter. The apparatus consisted of two quarter-inch light pipes, two American Optical fiber-optic illuminators, two rotary neutral-density filter wedge assemblies, and a focusing lens and diaphragm for the white spot. The white spot indeed seemed a more saturated yellow than the surround when it was not too bright.

(2) When a diffuse, barely discernible blue light (e.g., through a Wratten 98 filter) is shone (e.g., by a projector with no lens) on a white screen in a generally lit room, the shadow cast by an interposed object appears startlingly yellow, and the edge of the shadow appears diffuse no matter how sharp it looked using another light. The shadow can look brighter than the rest of the wall (despite reflecting less light). Furthermore, if the object casting the shadow is a pendulum in motion, the shadow lags the pendulum at the ends of its trajectory (where the acceleration is greatest), in a manner reminiscent of the Pulfrich effect (whereby a pendulum seen binocularly with one eye filter-covered appears to move in 3 dimensions due to the receptor-response lag in the filtered eye). I called the yellow-shadow version a “monocular Pulfrich effect.”

How can all this be explained? One clue is to realize that blue contributes very little to the luminance channel in vision, hence bright yellow has almost the same luminance as white (which matches yellow + blue). Since the luminance channel has much higher resolution both in space and time, it is clear that a border between yellow and white will look blurrier than a border between colors of appreciably different luminance, and will also evoke a time-lagged visual response. That explains the blurriness and time lag of the yellow shadow edge in the “monocular Pulfrich effect.”

Another clue is that the blue receptors also operate in low resolution both in space and time. That is another clue, which together with the first can help explain the BP-logo afterimage and yellow-spot effect. One must also remember that, when looking at the primary pattern, the eye is always moving in a jittering motion to refresh the image.

Anyone care to offer an explanation
for the BP-logo afterimage based
on these clues?

[1] M. H. Brill, Color Vision: An Evolutionary Approach, Ph.D. Dissertation, Syracuse University, 1976, pp. 57-58.
[2] Abney, W. de W. Researches in Normal and Defective Color vision and the Trichromatic Theory, London: Longman, Green and Co., 1913.
[3] J. Y. Lettvin, The Colors of Colored Things, Quarterly Progress Reports of the MIT Research Laboratory of Electronics 87 (1967), 193-225.