the problem of forming
an image that is both
sharp and bright: the
‘computed lens’. {153}
result of the magic window is that a perfect image of the dolphin appears on the retina. But it is not dark like the image from a tiny pin-hole, because lots of rays (which means torrents of photons) converge from the nose of the dolphin, lots of rays converge from the tail of the dolphin, lots of rays converge from every point on the dolphin to their own particular point on the retina. The magic window has the advantages of a pinhole, without its great disadvantage.
It's all very well to conjure up a so-called ‘magic window’ out of imaginative thin air. But isn't it easier said than done? Think what a complicated calculation the computer attached to the magic window is doing. It is accepting millions of light rays, coming from millions of points out in the world. Every point on the dolphin is sending millions of rays at millions of angles to different points on the surface of the magic window. The rays are criss-crossing one another in a bewildering spaghetti junction of straight lines. The magic window with its associated computer has to deal with each of these millions of rays in turn and calculate its own particular angle, through which it must be precisely turned. Where is this wonderful computer to come from, if not from a complicated miracle? Is this where we meet our Waterloo: an inevitable precipice in our ascent up Mount Improbable?
Remarkably, the answer is no. The computer in the diagram is just an imaginary creation to emphasize the apparent complexity of the task if you look at it in one way. But if you approach the problem in another way the solution turns out to be ludicrously easy. There is a device of preposterous simplicity which happens to have exactly the properties of our magic window, but with no computer, no electronic wizardry, no complication at all. That device is the lens. You don't need a computer because the calculations need never be done explicitly at all. The apparently complicated calculations of millions of ray angles are taken care of, automatically and without fuss, by a curved blob of transparent material. I'll take a little time to explain how lenses work, as a prelude to showing that the evolution of the lens wouldn't have been very difficult.
It is a fact of physics that light rays are bent when they pass from one transparent material into another transparent material (Figure {154}
Figure 5.10 How light is bent. The principle of refraction in a block of glass.
5.10). The angle of bending depends upon which two materials they happen to be, because some substances have a greater refractive index — a measure of the capacity to bend light — than others. If we are talking about glass and water, the angle of bending is slight because the refractive index of water is nearly the same as that of glass. If the junction is between glass and air, the light is bent through a bigger angle because air has a relatively low refractive index. At the junction between water and air, the angle of bending is substantial enough to make an oar look bent.
Figure 5.10 represents a block of glass in air. The thick line is a light ray entering the block, being bent within the glass, then bending back to the original angle as it goes out the other side. But of course there is no reason why a blob of transparent material should have neatly parallel sides. Depending upon the angle of the surface of the blob, a ray can be sent off in any direction you choose. And if the blob is covered with facets at lots of different angles, a set of rays can be sent off in lots of different directions (Figure 5.11). If the blob is curved convexly on one or both of its sides, it will be a lens: the working equivalent of our magic window. Transparent materials are not particularly rare in nature. Air and water, two of the commonest substances on our planet, are both transparent. So are many other liquids. So are some crystals if their surface is polished, for instance by wave action in the sea, to remove surface roughness. Imagine a pebble of some crystalline material, worn into a random shape by the waves. Light rays from a single source are bent in all sorts of directions by the pebble, depending upon the angles of the pebbles surfaces. {155}
Figure 5.11 Random pebbles refract rays in unhelpful directions.
Pebbles come in all sorts of shapes. Quite commonly they are convex on both sides. What will this do to light rays from a particular source like a light bulb?
When the rays emerge from a pebble with vaguely convex sides they wiH tend to converge. Not to a neat, single point such as would reconstruct a perfect image of the light source like our hypothetical ‘magic window’. That would be too much to hope. But there is a definite tendency in the right direction. Any quartz pebble whose weathering happened to make it smoothly curvaceous on both sides would serve as a good ‘magic window’, a true lens capable of forming images which, though far from sharp, are much brighter than a pinhole could produce. Pebbles worn by water usually are, as a matter of fact, convex on both sides. If they happened to be made of transparent material many of them would constitute quite serviceable, though crude, lenses.
A pebble is just one example of an accidental, undesigned object which can happen to work as a crude lens. There are others. A drop of water hanging from a leaf has curved edges. It can't help it. Automatically, without further design from us, it will function as a rudimentary lens. Liquids and gels fall automatically into curved shapes unless there is some force, such as gravity, positively opposing {156} this. This will often mean that they cannot help functioning as lenses. The same is often true of biological materials. A young jellyfish is both lens-shaped and beautifully transparent. It works as a tolerably good lens, even though its lens properties are never actually used in life and there is no suggestion that natural selection has favoured its lens-like properties. The transparency probably is an advantage because it makes it hard for enemies to see, and the curved shape is an advantage for some structural reason having nothing to do with lenses.
Here are some images I projected on to a screen using various crude and undesigned image-forming devices. Figure 5.12a shows a large letter A, as projected on a sheet of paper at the back of a pin-hole camera (a closed cardboard box with a hole in one side). You probably could scarcely read it if you weren't told what to expect, even though I used a very bright light to make die image. In order to get enough light to read it at all, I had to make the ‘pin’ hole quite large, about a centimetre across. I might have sharpened the image by narrowing the pinhole, but then the film would not have registered it — the familiar trade-off we have already discussed.
Now see what a difference even a crude and undesigned lens’ makes. For Figure 5.12b the same letter A was again projected through the same hole on to the back wall of the same cardboard box. But this time I hung a polythene bag filled with water in front of
Figure 5.12 Images seen through the various makeshift holes and crude, makeshift lenses: (a) a plain pinhole; (b) a sagging polythene bag filled with water; (c) a round wine goblet filled with water. {157}
the hole. The bag was not designed to be particularly lens-shaped. It just naturally hangs in a curvaceous shape when you fill it with water. I suspect that a jellyfish, being smoothly curved instead of rucked up into creases, would have produced an even better image. Figure 5.12c (‘CAN YOU READ THIS?’) was made with the same cardboard box and hole, but this time a round wine goblet filled with water was placed in front of the hole instead of a sagging bag. Admittedly the wineglass is a man-made object, but its designers never intended it to be a lens and they gave it its globular shape for other reasons. Once again, an object that was not designed for the purpose turns out to be an adequate lens.
Of course polythene bags and wineglasses were not available to ancestral animals. I am not suggesting that the evolution of the eye went through a polythene-bag stage, any more than it went through a cardboard-box stage. The point about the polythene bag is that, like a raindrop or a jellyfish or a rounded quartz crystal, it was not designed as a lens. It takes on a lens-like shape for some other reason which happens to be influential in nature.
It is not difficult, then, for rudi
mentary lens-like objects to come into existence spontaneously. Any old lump of half-way transparent jelly need only assume a curved shape (there are all sorts of reasons why it might) and it will immediately confer at least a slight improvement over a simple cup or pinhole. Slight improvement is all that is required to inch up the lower slopes of Mount Improbable. What might the intermediates have looked like? Look back at Figure 5.8, and once again I must stress that these animals are modern and must not be thought of as an actual ancestral series. Notice that the cup in Figure 5.8b (marine snail) has a lining of transparent jelly, the ‘vitreous mass’ (vm) which perhaps serves to protect the sensitive photocells from the raw sea water which flows freely through the aperture into the cup. That purely protective vitreous mass has one of the necessary qualities of a lens — transparency — but it lacks the correct curvature and it needs thickening up. Now look at Figures 5.8c, d and e, eyes from a bivalve mollusc, an abalone and a ragworm. In addition to providing yet more examples of {158} cups and intermediates between cups and pinholes, all these eyes show greatly thickened vitreous masses. Vitreous masses, of varying degrees of shapelessness, are ubiquitous in the animal kingdom. As a lens, none of those splodges of jelly would move Mr Zeiss or Mr Nikon to write home. Nevertheless, any lump of jelly that has a little convex curvature would mark significant improvements over an open pinhole.
The biggest difference between a good lens and something like the abalone's vitreous mass is this: for best results the lens should be detached from the retina and separated from it by some distance. The gap need not be empty. It could be filled by more vitreous mass. What is needed is that the lens should have a higher refractive index than the substance that separates the lens from the retina. There are various ways in which this might be achieved, none of them difficult. I'll deal with just one way, in which the lens is condensed from a local region within the front portion of a vitreous mass like that in Figure 5.8e.
First, remember that a refractive index is something that every transparent substance has. It is a measure of its power to bend rays of light. Human lens-makers normally assume that the refractive index of a lump of glass is uniform through the glass. Once a ray of light has entered a particular glass lens and changed direction appropriately, it goes in a straight line until it hits the other side of the lens. The lens-makers art lies in grinding and polishing the surface of the glass into precision shapes, and in joining different lenses together in compound cascades.
You can glue different kinds of glass together in complicated ways to make compound lenses with lots of different refractive indexes in various parts of them. The lens in Figure 5.13a, for instance, has a central core made of a different kind of glass with a higher refractive index. But there are still discrete changes from one refractive index to another. In principle, however, there is no reason why a lens should not have a continuously varying refractive index throughout its interior. This is illustrated in Figure 5.13b. This ‘graded index lens’ is hard for human lens-makers to achieve because of the way they make {159}
Figure 5.13 Two kinds of complex lens.
their lenses out of glass.* But it is easy for living lenses to be built like this because they are not made all at one time: they grow from small beginnings as the young animal develops. And, as a matter of fact, lenses with continuously varying refractive indexes are found in fish, octopuses and many other animals. If you look carefully at Figure 5.8e, you see what might conceivably be a region of differing refractive index in the zone behind the aperture of the eye.
But I was starting to tell the story of how lenses might have evolved in die first place, from a vitreous mass that filled the whole eye. The {160} principle of how it might have happened, and the speed with which it might have been accomplished, has been beautifully demonstrated in a computer model by a pair of Swedish biologists called Dan Nilsson and Susanne Pelger. I shall lead up to explaining their elegant computer model in a slightly oblique way. Instead of going straight to what they actually did, I shall return to our progression from Biomorph to NetSpinner computer models and ask how one could ideally set about making a similar computer model of the evolution of an eye. I shall then explain that this is essentially equivalent to what Nilsson and Pelger did, although they didn't put it in quite the same way.
Recall that the biomorphs evolved by artificial selection: the selecting agent was human taste. We couldn't think of a realistic way of incorporating natural selection into the model so we switched to model spider webs instead. The advantage of spider webs was that, since they do their work in a two-dimensional plane, their efficiency in catching flies could be calculated by the computer automatically. So could their cost in silk, and model webs could therefore be automatically ‘chosen by the computer in a form of natural selection. We agreed that spider webs were exceptional in this respect: we could not easily hope to do the same for the backbone of a hunting cheetah or the fluke of a swimming whale, because the physical details involved in assessing a three-dimensional organ's efficiency are too complicated. But an eye is like a spider web in this respect. The efficiency of a model eye drawn in two dimensions can be assessed automatically by the computer. I am not implying that an eye is a two-dimensional structure, because it isn't. It is just that, if you assume that the eye is circular when seen head on, its efficiency in three dimensions can be assessed from a computer picture of a single vertical slice through the middle. The computer can do a simple ray-tracing analysis and work out the sharpness of image that an eye would be capable of forming. This quality scoring is equivalent to NetSpinner's calculation of the efficiency of a computer spider web at catching computer flies.
Just as NetSpinner webs procreated mutant daughter webs, so we could let model eyes generate mutant daughter eyes. Each daughter eye would have basically the same shape as the parent, but with a small {161} random change to some minor aspect of its shape. Of course some of these computer ‘eyes’ would be so unlike real eyes as not to deserve the title, but no matter. They could still be bred, and their optical quality could still be given a numerical score — presumably it would be very low. We could therefore, in the same way as NetSpinner, evolve improved eyes by natural selection in the computer. We could either start with a fairly good eye and evolve a very good eye. Or we could start with a very poor eye or even with no eye at all.
It is instructive to run a program like NetSpinner as an actual simulation of evolution, setting it off from a rudimentary starting point and waiting to see where it will end up. You could even end up at different culmination points on different evolutionary runs, because there could be alternative accessible peaks of Mount Improbable. We could run our eye model in evolution mode too, and it would make a vivid demonstration. But actually you don't learn much more by letting the model evolve than you would learn by exploring, more systematically, where the upward path(s) on Mount Improbable lead(s). From a given starting point, a path which goes ever upward, never downward, is the path that natural selection would follow. If you ran the model in evolutionary mode, natural selection would follow that path. So it saves computer time if we search systematically for upward paths and for peaks that can be reached from postulated starting points. The important thing is that the rules of the game forbid going downhill. This more systematic search for upward paths is what Nilsson and Pelger did, but you can see why I chose to introduce their work as if we were planning, with them, a NetSpinner-style enactment of evolution.
However we choose to run our model, whether in ‘natural-selection mode’ or in ‘systematic exploration of the mountain mode’, we have to decide upon some rules of embryology: that is, some rules governing how genes control the development of bodies. What aspects of shape do the mutations actually operate upon? And how big, or how small, are the mutations themselves? In the case of NetSpinner, the mutations act upon known aspects of the behaviour of spiders. In the case of biomorphs, mutations act upon the lengths and angles of branches in growing trees. In the case of eyes, Nilsson and Pelge
r began {162} by acknowledging that there are three main types of tissue in a typical ‘camera’ eye. There is an outer casing to the camera, usually opaque to light. There is a layer of light-sensitive ‘photocells’. And there is some kind of transparent material, which may serve as a protective window or which may fill the cavity inside the cup — if, indeed, there is a cup, for we are not taking anything for granted in our simulation. Nilsson and Pelger's starting point — the foot of the mountain — is a flat layer of photocells (grey in Figure 5.14), sitting on a flat backing screen (black) and topped by a flat layer of transparent tissue (off-white). They assumed that mutation works by causing a small percentage change in the size of something, for example a small percentage decrease in the thickness of the transparent layer, or a small percentage increase in the refractive index of a local region of the transparent layer. Their question really is, where can you get to on the mountain if you start from a given base camp and go steadily upwards? Going upwards means mutating, one small step at a time, and only accepting mutations that improve optical performance.
So, where do we get to? Pleasingly, through a smooth upward pathway, starting from no proper eye at all, we reach a familiar fish eye, complete with lens. The lens is not uniform like an ordinary man-made lens. It is a graded index lens such as we met in Figure 5.13b. Its continuously varying refractive index is represented in the diagram by varying shades of grey. The lens has ‘condensed’ out of the vitreous mass by gradual, point by point changes in the refractive index. There is no sleight of hand here. Nilsson and Pelger didn't pre-pro-gram their simulated vitreous mass with a primordial lens just waiting to burst forth. They simply allowed the refractive index of each small bit of transparent material to vary under genetic control. Every smidgen of transparent material was free to vary its refractive index in any direction at random. An infinite number of patterns of varying refractive index could have emerged within the vitreous mass. What made the lens come out lens-shaped’ was unbroken upward mobility, the equivalent of selectively breeding from the best seeing eye in each generation.