Spirals in Time: The Secret Life and Curious Afterlife of Seashells (7 page)

Before we go any further, I should point out that molluscs are not mathematicians. They aren’t aware of the arithmetical elegance of their homes. These patterns simply emerge from the way they grow. You could do the same thing with a tube of toothpaste, albeit a modified version in which the opening can be made wider as you go, so that it produces an expanding cone of minty freshness. Squeeze a coil of toothpaste onto a flat surface and you should see a seashell-like twist appear (our toothpaste coils are solid, unlike the molluscs’ hollow shells). Various different types of toothpaste shell will be made depending on how hard you squeeze and how quickly you move your hand away from the centre of coiling. You can make tightly coiled shapes or expansive ones that veer off and swiftly become enormous. However, keeping the toothpaste on a flat surface limits the types of shell you can make. As you’ve no doubt noticed, mollusc shells are not generally flat: most of them explore a third dimension.

‘The problem is one not of plane but of solid geometry,’ wrote Sir
D’Arcy Wentworth Thompson
in his classic
On Growth and Form
, as he began to grapple with the idea of three-dimensional seashells. While the First World War raged, the professor from St Andrew’s University in Scotland wrote more than a thousand pages packed with his ideas of how mathematics could explain shapes in nature, from horns and honeycombs, beaks and claws, to dolphins’ teeth and the shape of a splash. Thompson brought together many of his predecessors’ theories about the geometry of shells, including those of Sir Christopher Wren, who mused on their architectural beauty. Some earlier shell thinkers rejected the idea of logarithmic spirals, saying they were too simplistic, but Thompson underlined their importance and brought out lots of new examples of shells that were a good fit to this expanding curve. He then set out to find a way of drawing
accurate three-dimensional model shells. His central idea was that coiled shells follow a set of rigid mathematical laws, which all stem from the fact that infant shells are simply smaller versions of their future, grown-up selves.

Molluscs only ever make a single shell, but it’s one they’ll never grow out of. Other creatures with hard exoskeletons tend to do things differently. Crabs, lobsters and all their crustacean relatives break out of their shells every now and then, cast them aside and grow a new version, one size bigger and sometimes in a wildly different shape from the one that came before. Turtles make their shells on the inside by modifying bones in their ribs and pelvis. By contrast, molluscs make their shells on the outside, and they hold on to them. They are among the few animals on the planet that wander around carrying with them the same body armour they had as babies; the pointy tip or innermost whorl is the mollusc’s juvenile shell. Day by day, the mollusc shell slowly expands, making room for the soft animal growing inside.

Thompson visualised the growth of a spiral shell as a two-dimensional shape spinning through three-dimensional space around a central axis; if you imagine poking a needle through a coiling seashell from the tip towards the open end, so that it rotates like a spinning top, then the needle takes the position of the axis. Our toothpaste shells stayed in the same flat plane and didn’t make much use of that axis. Now, imagine what happens if the toothpaste sets solid straight from the tube. You can drop the spiral downwards, along that vertical axis, and create a three-dimensional coiling shell.

Taking this idea (although using paper and pen rather than quick-setting toothpaste), Thompson devised a shell-making model based on four rules: first, the cross-section of the coil must stay the same shape, but grow bigger over time (in other words, slice across our expanding toothpaste cone at any point and you will see the same shape, in this case probably a circle); second, the shell’s curve expands from the centre at a fixed rate (making it logarithmic); third, the
amount of overlap between successive whorls stays the same; finally, and most difficult to visualise (so don’t worry too much), the angle between the spinning whorls and the central axis also remains the same.

These four rules were all Thompson thought were needed to draw any version of a coiling seashell. But what would all those shells look like? That was the question asked by another scientist who, 40 years later, was inspired to customise Thompson’s model and use it to create a shell collection like no other.

The imaginary museum of all possible shells

Standing in the corner of a huge room, you see white walls stretching out in front of you and towering upwards, disappearing as if into the clouds. Suspended in the room’s cavernous space are what at first glance seem to be thousands of glass light bulbs. They dangle in neat rows and columns, beginning just above the floor and reaching up way above your head. Take a closer look and you’ll notice that they aren’t in fact light bulbs but intricate models of seashells.

Despite their glassy appearance, these shells are quite tough and you can push through them without breaking them. As you do you see that the shells differ subtly from one to the next. As you look upwards, the shells gradually become squatter and fatter. Walk forwards and the shells at eye-height flatten out until they are no longer coiled but more flattened, like clams. Stroll on through the museum of all possible shells and you’ll spot both familiar shells and some less familiar shapes.

The architect behind this imaginary museum is palaeontologist David Raup, from Johns Hopkins University in Maryland. In the 1960s he took Thompson’s shell-making model and made a series of adjustments, replacing the original rules with four of his own.

First, Raup defined the rate at which a shell flares outwards. This is the whorl expansion rate, or ‘W’: tightly coiled shells have lower W values compared to more flared,
open shells. Clams and other bivalves have such high W values that they flare right open before having a chance to do much coiling. They may not look it but, in essence, bivalve shells are still spirals.

W: Whorl expansion rate (becoming more clam-like).

Next comes ‘T’, a factor that determines how tall the shell will be (the T in fact stands for Translation, meaning how much the growing spiral travels along its central axis, but it could just as easily mean Tall). In shells with a tall spire, the coil creeps downwards along the axis as it spins round and round. The further it creeps, the taller the shell spire and the greater the value of T.

Raup kept one of Thompson’s rules. He admitted that in the real world, the cross-section of a shell’s growing cone can change, but to keep things simple Raup fixed his shape and made it a circle. He allowed the circle to get bigger as a shell grows but it always stayed the same shape.

T: Translation rate (getting taller).

D: Distance from axis (becoming more wormy).

The final part of Raup’s model is the distance ‘D’ from the whorls to the axis. Adjusting the value of D can produce thin, wormy shells with big gaps between whorls, or chubbier shapes in which the whorls touch or even squash into each other.

Armed with his new idea for plotting shell shapes, Raup did something unusual for a palaeontologist at the time: he used a computer. He bought himself time at the helm of the fastest one available. The enormous mainframe IBM 7090 was a state-of-the-art scientific computer that was intended for the design of missiles, nuclear reactors and supersonic aircraft, but for a short time Raup channelled its power into making shells. He plugged in a few combinations of values for T, W and D and programmed the computer to draw the corresponding shells on a Calcomp x-y plotter. The computer’s output was a series of dots outlining several shells in cross-section, which were then interpreted into three dimensions by an artist. These drawings appeared in
Raup
’s 1962 paper in the journal
Science
. To make more of these scatter plots would have taken way too long and been too expensive on computer time. Raup knew that the available technology was limiting his work.

His next step was to team up with an electrical engineer, Arnold Michelson, and together they tried a more affordable set-up, the PACE TR-10 analogue computer (which despite its gargantuan size was one of the earliest desktop computers,
although presumably only for a rather large desk). They plugged in a wide range of values for T, D and W from Raup’s shell model, hooked the computer up to an oscilloscope that traced the shapes of shells as fast-moving circles across the screen, then stood back and watched.

Unfolding before their eyes was a stunning collection of shells that looked like thousands of tiny X-rays. Among the PACE TR-10 output were examples of just about every type of shell, from the nautilus through to all sorts of coiled snails and even flattened clams and scallops. Their findings were so stunning that one of their virtual shells made it to the front cover of
Science
. Raup and Michelson had shown that from a simple set of rules emerges the great complexity of real shells. And their imaginary shell collection contained plenty more besides. Once they had traced out the shapes of all these shells, Raup turned his attention to the next big question: which of these shells can be seen in the real world?

Back inside the virtual museum of glass seashells, we can now understand how things are arranged: along one wall the thin, wormy shells become chubbier, as values for D shift from zero to one; in another direction the shells become gradually taller, as the values for T start at zero and run along to four; and from ceiling to floor, shells have progressively higher W values, from one to a million, and snails morph into clams. The glass shells are versions of the output from Raup and Michelson’s PACE TR-10 computer program of all possible shells.

Now something changes in our museum. The main lights are dimmed and individual glass shells, here and there, begin to glow (funnily enough, like light bulbs). These are the models that closely resemble real shells, living or extinct. And as parts of the room light up, something becomes obvious: large regions of the museum remain in darkness.

I illuminated the real species among our glass shell models and Raup did a similar thing on paper. He plotted a graph
with three axes (for D, T and W) and shaded in areas where real mollusc shells can be found, as well as the brachiopods, which are only distantly related to molluscs but even so make similar shells. Drawing in the boundaries of reality onto his imaginary shell museum, Raup immediately saw that only a small fraction of his theoretical shells have ever actually evolved. Substantial regions seem to be out of bounds. He theorised that some of the empty space in his museum was filled with ‘bad’ shells that, in reality, don’t work. Maybe they would be too heavy or too weak, or would leave their inhabitants in some way vulnerable to attack? There is an empty region filled with shells that suffer from what Raup referred to as the ‘problem of bivalveness’. Clams, mussels and scallops would be permanently clamped shut if their gentle whorls overlapped (the only option for opening up would be to build a new hinge on the outside and keep moving it as the mollusc grows bigger, and bivalves in the real world don’t do this).