The compound (insect) eye
Let us now return to biology. The human eye is not the only kind of eye. In the vertebrates, almost all eyes are essentially like the human eye. However, in the lower animals there are many other kinds of eyes: eye spots, various eye cups, and other less sensitive things, which we have no time to discuss. But there is one other highly developed eye among the invertebrates, the compound eye of the insect. (Most insects having large compound eyes also have various additional simpler eyes as well.) A bee is an insect whose vision has been studied very carefully. It is easy to study the properties of the vision of bees because they are attracted to honey, and we can make experiments in which we identify the honey by putting it on blue paper or red paper, and see which one they come to. By this method some very interesting things have been discovered about the vision of the bee. In the first place, in trying to measure how acutely bees could see the color difference between two pieces of “white” paper, some researchers found they were not very good, and others found they were fantastically good. Even if the two pieces of white paper were almost exactly the same, the bees could still tell the difference. The experimenters used zinc white for one piece of paper and lead white for the other, and although these look exactly the same to us, the bee could easily distinguish them, because they reflect a different amount in the ultraviolet. In this way it was discovered that the bee’s eye is sensitive over a wider range of the spectrum than is our own. Our eye works from $7000$ angstroms to $4000$ angstroms, from red to violet, but the bee’s can see down to $3000$ angstroms into the ultraviolet!  This makes for a number of different interesting effects. In the first place, bees can distinguish between many flowers which to us look alike. Of course, we must realize that the colors of flowers are not designed for our eyes, but for the bee; they are signals to attract the bees to a specific flower. We all know that there are many “white” flowers. Apparently white is not very interesting to the bees, because it turns out that all of the white flowers have different proportions of reflection in the ultraviolet; they do not reflect one hundred percent of the ultraviolet as would a true white. All the light is not coming back, the ultraviolet is missing, and that is a color, just as, for us, if the blue is missing, it comes out yellow. So, all the flowers are colored for the bees. However, we also know that red cannot be seen by bees. Thus we might expect that all red flowers should look black to the bee. Not so! A careful study of red flowers shows, first, that even with our own eye we can see that a great majority of red flowers have a bluish tinge because they are mainly reflecting an additional amount in the blue, which is the part that the bee sees. Furthermore, experiments also show that flowers vary in their reflection of the ultraviolet over different parts of the petals, and so on. So if we could see the flowers as bees see them they would be even more beautiful and varied! It has been shown, however, that there are a few red flowers which do not reflect in the blue or in the ultraviolet, and would, therefore, appear black to the bee! This was of quite some concern to the people who worry about this matter, because black does not seem like an interesting color, since it is hard to tell from a dirty old shadow. It actually turned out that these flowers were not visited by bees, these are the flowers that are visited by hummingbirds, and hummingbirds can see the red! Another interesting aspect of the vision of the bee is that bees can apparently tell the direction of the sun by looking at a patch of blue sky, without seeing the sun itself. We cannot easily do this. If we look out the window at the sky and see that it is blue, in which direction is the sun? The bee can tell, because the bee is quite sensitive to the polarization of light, and the scattered light of the sky is polarized.2 There is still some debate about how this sensitivity operates. Whether it is because the reflections of the light are different in different circumstances, or the bee’s eye is directly sensitive, is not yet known.3 It is also said that the bee can notice flicker up to $200$ oscillations per second, while we see it only up to $20$. The motions of bees in the hives are very quick; the feet move and the wings vibrate, but it is very hard for us to see these motions with our eye. However, if we could see more rapidly we would be able to see the motion. It is probably very important to the bee that its eye has such a rapid response. Now let us discuss the visual acuity we could expect from the bee. The eye of a bee is a compound eye, and it is made of a large number of special cells called ommatidia, which are arranged conically on the surface of a sphere (roughly) on the outside of the bee’s head. Figure 36–7 shows a picture of one such ommatidium. At the top there is a transparent area, a kind of “lens,” but actually it is more like a filter or light pipe to make the light come down along the narrow fiber, which is where the absorption presumably occurs. Out of the other end of it comes the nerve fiber. The central fiber is surrounded on its sides by six cells which, in fact, have secreted the fiber. That is enough description for our purposes; the point is that it is a conical thing and many can fit next to each other all over the surface of the eye of the bee. Now let us discuss the resolution of the eye of the bee. If we draw lines (Fig. 36–8) to represent the ommatidia on the surface, which we suppose is a sphere of radius $r$, we may actually calculate how wide each ommatidium is by using our brains, and assuming that evolution is as clever as we are! If we have a very large ommatidium we do not have much resolution. That is, one cell gets a piece of information from one direction, and the adjacent cell gets a piece of information from another direction, and so on, and the bee cannot see things in between very well. So the uncertainty of visual acuity in the eye will surely correspond to an angle, the angle of the end of the ommatidium relative to the center of curvature of the eye. (The eye cells, of course, exist only at the surface of the sphere; inside that is the head of the bee.) This angle, from one ommatidium to the next, is, of course, the diameter of the ommatidia divided by the radius of the eye surface: \begin{equation} \label{Eq:I:36:1} \Delta\theta_g=\delta/r. \end{equation} So, we may say, “The finer we make the $\delta$, the more the visual acuity. So why doesn’t the bee just use very, very fine ommatidia?” Answer: We know enough physics to realize that if we are trying to get light down into a narrow slot, we cannot see accurately in a given direction because of the diffraction effect. The light that comes from several directions can enter and, due to diffraction, we will get light coming in at angle $\Delta\theta_d$ such that \begin{equation} \label{Eq:I:36:2} \Delta\theta_d=\lambda/\delta. \end{equation} Now we see that if we make the $\delta$ too small, then each ommatidium does not look in only one direction, because of diffraction! If we make them too big, each one sees in a definite direction, but there are not enough of them to get a good view of the scene. So we adjust the distance $\delta$ in order to make minimal the total effect of these two. If we add the two together, and find the place where the sum has a minimum (Fig. 36–9), we find that \begin{equation} \label{Eq:I:36:3} \ddt{(\Delta\theta_g+\Delta\theta_d)}{\delta}=0 =\frac{1}{r}-\frac{\lambda}{\delta^2}, \end{equation} which gives us a distance \begin{equation} \label{Eq:I:36:4} \delta=\sqrt{\lambda r}. \end{equation} If we guess that $r$ is about $3$ millimeters, take the light that the bee sees as $4000$ angstroms, and put the two together and take the square root, we find \begin{align} \delta &=(3\times10^{-3}\times4\times10^{-7})^{1/2}\text{ m}\notag\\[1ex] \label{Eq:I:36:5} &=3.5\times10^{-5}\text{ m}=35\text{ $\mu$m}. \end{align} The book says the diameter is $30$ $\mu$m, so that is rather good agreement!  So, apparently, it really works, and we can understand what determines the size of the bee’s eye! It is also easy to put the above number back in and find out how good the bee’s eye actually is in angular resolution; it is very poor relative to our own. We can see things that are thirty times smaller in apparent size than the bee; the bee has a rather fuzzy out-of-focus image relative to what we can see. Nevertheless it is all right, and it is the best they can do. We might ask why the bees do not develop a good eye like our own, with a lens and so on. There are several interesting reasons. In the first place, the bee is too small; if it had an eye like ours, but on his scale, the opening would be about $30$ $\mu$m in size and diffraction would be so important that it would not be able to see very well anyway. The eye is not good if it is too small. Secondly, if it were as big as the bee’s head, then the eye would occupy the whole head of the bee. The beauty of the compound eye is that it takes up no space, it is just a very thin layer on the surface of the bee. So when we argue that they should have done it our way, we must remember that they had their own problems!