To get some idea of what we are supposed to be thinking of when we think of the Museum of All Possible Animals, this chapter will deal with a particular case which can, more or less, be confined to three dimensions. In the next chapter I shall return to the controversy with which this chapter began and try to make a constructive overture towards the other side (for I am a known partisan). This chapter's three-dimensional special case is that of snail shells and other coiled shells. The reason the galleries of shells can be confined to three dimensions is that most of the important variation among shells can be expressed as change in only three numbers. In what follows, I shall be following in the footsteps of David Raup, a distinguished palaeontologist from the University of Chicago. Raup, in turn, was inspired by the celebrated D'Arcy Wentworth Thompson, of the ancient and distinguished Scottish University of St Andrews, whose book, On Growth and Form (first published in 1919), has been a persistent, if not quite mainstream, influence on zoologists for most of the twentieth century. It is one of the minor tragedies of biology that D'Arcy Thompson died just before the computer age, for almost every page of his great book cries out for a computer. Raup wrote a program to generate shell form, and I have written a similar program to illustrate this chapter although — as might be expected — I incorporated it in a Blind Watchmaker-style artificial selection program.
The shells of snails and other molluscs, and also the shells of creatures called brachiopods which have no connection with molluscs but superficially resemble them, all grow in the same kind of way, which is different from the way we grow. We start small and grow all over (with some bits growing faster than others). You can't take a man and dissect out the bit of him that was him as a baby. With a mollusc shell you can {201}
Figure 6.1 A section through the shell of a Nautilus. The animal itself lives in the largest, most recent, chamber.
do just that. A mollusc shell starts small and grows at the margins, so the innermost part of the adult coil is the baby shell. Each animal carries its own infant form around with it, as the narrowest part of its shell. The shell of Nautilus (already mentioned for its pinhole eye) is divided into air-filled flotation compartments, all except for the largest and most recently built compartment at the growing margin, in which the animal itself lives at any one time (Figure 6.1).
Figure 6.2 Kinds of spirals: (a) Archimedean spiral; (b) logarithmic spiral with slow rate of opening; (c) logarithmic spiral with rapid rate of opening. {202}
Because of their method of expanding at the margin, shells all have the same general form. It is a solid version of the so-called logarithmic or equiangular spiral. The logarithmic spiral is different from the Archimedean spiral which is what a sailor produces when he coils a rope on the deck. No matter how many turns the rope takes, each successive turn is still the same width — one thickness of the rope. In a logarithmic spiral, by contrast, the spiral opens out as it propels itself away from the centre. Different spirals open out at different rates, but it is always a particular rate for any particular spiral. Figure 6.2 shows, in addition to an Archimedean coiled-rope spiral, two logarithmic spirals with different rates of opening out.
A shell grows, not as a line, but as a tube. The tube doesnt have to be circular in cross-section like a French horn but, just for the moment, we'll assume that it is. We'll also assume that the spiral drawn represents the outer margin of the tube. The diameter of the tube could happen to expand at just the right rate to keep the inner margin fitting snugly against the previous whorl of the spiral, as in Figure 6.3a. But it doesn't have to. If the tube's diameter expands more slowly than the outer margin of the spiral, a gap of increasing size will be left between successive whorls, as in Figure 6.3b. The more ‘gappy’ the shell, the more it seems suitable for a worm rather than for a snail.
Raup described the spirals of shells using three numbers, which he called W, D and T. I hope it will not be thought too quaint if I
Figure 6.3 Two tubes with same spiral but different tube size: (a) tube big enough to fill the gap between successive whorls of the spiral; (b) tube narrow enough to leave thin air (or open water) between successive whorls of the spiral. {203}
rename them flare, verm and spire. It is easier to remember which is which than in the case of the mathematical letters. Flare is a measure of the expansion rate of the spiral. If the flare is 2, this means that, for every complete turn around the spiral, the spiral opens out to twice its previous size. This is true of Figure 6.2b. For every turn around Figure 6.2b, the width of the spiral doubles. Figure 6.2c, being a much more open shell, has a flare of 10. For each complete circuit around this spiral, the width would increase tenfold (although in practice the spiral comes to an end before it has time to complete a whole circuit). Something like a cockle, which opens out so rapidly that you don't even think of it as coiling, has a flare value up in the thousands.
When describing flare I was careful not to say that it measures the rate of increase of the diameter of the tube. This is where the second number, verm, comes in. We need verm because the tube does not have to fill, snugly, the space made available by the expanding spiral. The shell can be ‘gappy’, like the one in Figure 6.3b. Verm gets its name from ‘vermiform’ which means ‘worm-shaped’. Figure 6.3a and Figure 6.3b have identical flare values (2) but Figure 6.3b has a higher verm score (0.7) than Figure 6.3a's 0.5. A verm of 0.7 means that the distance from the centre of the spiral to the inner margin of the tube is 70 per cent of the distance from the centre of the spiral to the outer margin of the tube. It doesn't matter which part of the tube you use to make the measurement, the verm score is the same (this doesn't logically have to be true but it seems often to be true of real shells and we shall assume it unless otherwise stated). You can easily see that a very high verm like 0.99 makes for a very thin, threadlike tube, because the inner margin of the tube is 99 per cent of the distance to the outer margin of the tube.
What verm value is needed to ensure snug fitting, as in Figure 6.3a? That depends on the flare. To be precise, the critical verm value for a snug fit is exactly the reciprocal of the flare value (that is, one divided by flare). The flare is 2 in both parts of Figure 6.3, so the critical verm for a snug fit is 0.5, and this is what Figure 6.3a has. Figure 6.3b has a verm which is higher than its ‘snug critical’ value, which is why the shell appears gappy. If we take a shell like Figure 6.2c with a flare value of 10, the snug critical verm score would be 0.1. {204}
What if the verm value is smaller than the snug critical value? Could we imagine a tube so fat that it goes beyond snug fitting and actually encroaches inside the territory of the previous whorl — for example, a spiral like those in Figure 6.3 but with a verm value of, say, 0.4? There are two ways in which the clash can be resolved. One is simply to allow the tube to enclose earlier whorls of itself. Nautilus does this. It means that the shape of the cross-section of the available tube can no longer be a plain circle but has a ‘bite’ taken out of it. But this is no disaster because, as you'll remember, it was only an arbitrary decision to assume that the tube has a circular section in any case. Many molluscs live happily in a tube which is far from circular in section, and we shall come on to them. In some cases the best way to interpret the non-circular shape of the cross-section of the tube is as a means of accommodating previous whorls of the tube.
The other way to resolve the would-be encroachment of previous whorls of the tube is to move out of the plane. This brings us to the third of our shell signature numbers, spire. Think of the expanding spiral as moving sideways as it expands, making a conical shape like a top. The third shell signature number, spire, is the rate at which successive whorls of the spiral creep along the length of the cone. Nautilus happens to have a spire value of 0: all its successive windings are in one plane.
So, we have three shell signature numbers, flare, verm and spire (Figure 6.4). If we ignore one of these, say spire, we can plot a graph of the other two on a flat piece of paper. Every point on the graph has a unique combination
of flare and verm values, and we can program the computer to draw, at that point, the shell that would be produced. Figure 6.5 shows twenty-five regularly spaced points on the graph. As you move from left to right across the graph, the computer shells become progressively more ‘wormy’ as verm increases. As you move from top to bottom and flare increases, the spirals become progressively more open until they don't look obviously spiral at all. In order to get a good spread as we go down, we make flare increase logarithmically. This means that each equal step down the page corresponds to multiplying by some number (in this case ten) rather than, as in a normal graph and as in the progression {205}
Figure 6.4 Shells to illustrate flare, verm and spire: (a) high flare. Liconcha castrensis, a bivalve mollusc; (b) high verm: Spirula; (c) high spire. Turritella terehra.
Figure 6.5 Table of computer-generated shells systematically varying verm and flare. Changes in the third dimension, spire, would not be visible in this view. The flare axis is logarithmic — equal steps down the page represent a tenfold multiplication of the value of flare. On the verm axis, equal steps across the page represent a fixed addition to the ‘worminess’ score. A few named real animals are written in approximately their right places on the chart.
of verm-scores across the page, adding a number with each step. This is necessary in order to accommodate shells like cockles and clams, at the bottom left of the picture (which have flare values up in the thousands where small changes don't make much difference) in the same graph as ammonites and snails (which typically have flare scores in low, single figures where small changes make a big difference). In various parts of the graph you can see shapes that resemble ammonites, Nautilus, clams, rams’ horns and tubeworms, and I've written labels in approximately the right places.
My computer program can draw shells in two views. Figure 6.5 shows one view, emphasizing the shape of the spiral itself. Figure 6.6 shows the other view, ‘X-ray’ cross-sections, giving an impression of the solid shape of the shells. Figure 6.7 is an actual X-ray photograph of real shells to explain the nature of this view. The four shells of Figure 6.6 are computer shells chosen, like the real shells of Figure 6.4, to illustrate different values of flare, verm and spire.
Figure 6.8 is a graph, similar to Figure 6.5 except that the computer shells are shown in X-ray view, and the axes are flare and spire instead of flare and verm.
One could also, of course, plot verm versus spire, but I won't take the space to do this. Instead, I'll go straight to Raup's famous cube (Figure 6.9). Because three numbers suffice to define a shell (leaving aside the question of cross-sectional shape of the tube) you can place each shell in its own unique spot in a three-dimensional box. The Museum of Possible Shells, unlike, say, the Museum of Possible Pelvis Bones, is a simple tower block. One dimension corresponds to each of the three shell signature numbers. Stand in the Museum of Possible Shells and walk, say, north, which we'll designate the verm dimension. As you make your way along the gallery, the shells that you pass steadily become more ‘wormy’, while keeping everything else constant. If at any point you turn left and walk west, the shells that you pass steadily increase their spire value, becoming more cone-shaped, while keeping other things constant. Finally if, at any point, you stop moving either west/east or north/south and climb directly downwards instead — the flare dimension — you encounter shells with a steadily increasing rate of opening out. You can get from any shell {207}
Figure 6.6 Four computer shells in ‘X-ray’ view, to show different flare, verm and spire values. {208}
Figure 6.7 Photograph
of .1 real shell in X-ray
view
Figure 6.8 Graph of computer shells (‘X-ray view’) plotting flare (labelled W, down the page) versus spire (labelled T, across the page). As in Figure 6.5, the flare scale is logarithmic, but here flare is confined to low values — none of the shells opens out very far. {209}
Figure 6.9 Raup's cube. David M. Raup drew a three-dimensional graph of flare (which he called W) down the page, against spire (which he called T) across the page from right to left, against verm (which he called D) backwards into the page. ‘X-ray views’ of computer-drawn shells are sampled at strategic points in the cube. Regions of the cube in which real-life shells can be found are shaded. The unshaded regions house theoretically conceivable shells that do not actually exist.
to any other by burrowing through the cube at the appropriate angle, and you'll pass a continuous series of intermediate shells on the way. Figures 6.5 and 6.8 can be thought of as two outer faces of Raup's cube. On two-dimensional paper we could print any slice, at any particular angle, through the cube.
Raup wrote the original computer program that is the inspiration for mine. In his published diagram, rather than attempt the impractical task of drawing all the shells in the cube, Raup sampled particular points. The pictures round the edge of Figure 6.9 represent the theoretical shells that you would find at the designated {210} points of the space. Some of them look like actual shells that you might find on a beach. Others look like nothing on earth, but they still belong in the space of all computable shells. Raup shaded in on his picture those regions of the space where actual shells are to be found.
Ammonites, those once-ubiquitous Nautilus-relatives who seem to have come to the same sad end (whatever it was) as the dinosaurs, had coded shells but, unlike snails, their coils were nearly always limited to one plane. Their spire value was zero. At least, this is true of typical ammonites. Pleasingly, however, a few of them, such as the Cretaceous genus Turrilites evolved a high spire value, thereby independently inventing the snail form. Such exceptional forms apart, the ammonites are housed along the eastern wall of the Museum of Shells (names like ‘east’ and ‘south’ are, of course, arbitrary labels for the diagram). The cabmets of typical ammonites don't occupy more than the southern half of the eastern wall, and only the top few storeys. Snails and their kind overlap with the Ammonite Corridor but they also spread far to the west (the spire dimension) and they penetrate a little further down towards the lower storeys of the tower block. But most of the lower storeys — where the flare rate is large and shells open out rapidly — belong to the two great groups of double-shelled creatures. Bivalve molluscs stretch a little to the west — they have a slight twist on them like snails but their tube opens out so fast that they don't look like snails. Brachiopods or ‘lamp-shells’, which, as we have seen, are not molluscs at all but superficially resemble bivalve molluscs, share with ammonites a ‘coil’ that is entirely in one plane. As with the molluscan bivalves, brachiopod tubes typically flare completely open before they have time to build up a ‘coil’ worthy of the name.
Any particular evolutionary history is a trajectory through the Museum of All Possible Shells and I have represented this by embedding my shell-drawing computer procedure in the larger, Blind Watchmaker artificial-selection program. I simply removed the tree-growing embryology from the Blind Watchmaker program and dropped a shell-growing embryology into its place instead. The combined program is called Blind Snailmaker. Mutation is equivalent to small {211} movements in the museum — remember that all shells are surrounded by their most similar neighbours. In the program, the three shell signature numbers are each represented by one gene locus whose numerical value can vary. So we have three classes of mutation, small changes in flare, small changes in verm and small changes in spire. These muta-tional changes can be positive or negative, within limits. The flare gene has a minimum value of I (smaller values would indicate a shrinking rather than a growth process) and no fixed maximum value. The verm gene's value is a proportion, varying from 0 to just below I (a verm of I would indicate a tube so thin and wormy as to be non-existent). Spire has no limits: negative values trivially indicate an upside-down shell. Following the original Blind Watchmaker program, Blind Snailmaker presents a parent shell in the middle of the computer screen, surrounded by a litter of asexual offspring — its randomly mutated near neighbours in the Museum of All Shells.
The human selector clicks the mouse to choose one of the shells for breeding. It glides to the parental position in the centre, and the screen fills up with a litter of its offspring. The process recycles as long as the selector has patience. Slowly, you feel yourself creeping through the Museum of All Possible Shells. Sometimes you are walking amidst familiar shells, of the kind that you could pick up on any beach. At other times you stray outside the bounds of reality, into mathematical spaces where no real shells have ever existed.
I earlier explained that, although the set of all possible shells can largely be described with only three numbers, this does include a simplifying assumption which is wrong: the assumption that the cross-sectional shape of the tube is always a circle. It seems to be generally true that, as the tube flares out, it remains the same shape, but it is by no means true that that shape is always a circle. It can be an oval and my computer model incorporates a fourth ‘gene’, called shape, whose value is the height of the oval tube divided by its width. A circle is the special case of a shape of 1. Incorporation of this gene adds surprisingly to the power of the model to represent real shells. But it still is not enough. Many real shells have a variety of more complicated cross-sectional shapes which are neither circular nor oval, and which don't lend themselves to simple mathematical description. Figure 6.10 shows a range {212}
Climbing Mount Improbable Page 20