It seems to be frequently declared that spiral mollusc shells spiral logarithmically outward. In fact, it's often claimed that Nautilus spirals in a perfect Golden Ratio spiral. I'm sort of annoyed by this, because it seems to me to be an oversimplification, and frequently wrong. I started thinking on this because there seem to be people using the mathematical mystique to sell nautilus shells to science teachers, and since I've been reading up on the dangers of overfishing of nautilus for their shells, this is frustrating. However, in further thinking about it, I'm dubious. First, there's no reason I can think of why Nautilus would prefer to have a ratio of phi involved than anything else, and it seems like nautiloids, ammonoids, spirula, and such have all sorts of different values. I suppose it's worth mentioning that a logarithmic spiral (which I'd more properly call an exponential spiral) is one where the radius grows exponentially as the angle sweeps around, which is consistent with the animal growing at a typical rate as it builds its shell. The math of this is r = k exp ( c a ) where exp is the exponential function (e to the x ), r is the radius, a is the angle, which is understood not to stop at 360 degrees but to keep growing as it sweeps around, and k and c are constants. A "golden" spiral chooses k and c so that the golden ratio phi is the change in radius over 90 degrees, to get some kooky Greek-aesthetic rectangle silliness to happen. Anyway, the consequence of this is that if you draw a line out from the center of a nautilus past a few layers, the distance between successive layers (or thickness of the tube) grows significantly as you go out: the inner whorls are close together, but as you get further out, the animal has grown, and so it needs a bigger tube. The thing is, for many fossil cephalopods, I see shells that sure don't look much like that. Many ammonites seem to have started that way, but quite rapidly developed their body size to a fixed width, and then they grew their shells much more "along" but without changing the width much. This makes shells that look more like a spiral of rope on the ground, where the thickness of the rope is the same everywhere, and doesn't get wider as it goes further out, like this picture: http://www.stairropes.com/stonkwebimages/spiralmat2img.jpg Although I'm suspicious that logarithmic spirals are an over-generalization even for modern molluscs, I'm wondering if the non-logarithmic nature of ammonites in particular can tell us anything about how their body plans differed from those of Nautilus or gastropods. It seems like a safe generalization to say that nautiloids more readily kept the logarithmic shape. The cone-shaped shells seem to show linear growth in width, so that their cross sections are triangles, not some sort of weird shape where they get wider faster as they get bigger. Some of this is because things are different when you parameterize by length rather than angle: as you go linearly along the cone, it's parameterized by length, but as you go around the angle of a spiral, the length you travel per degree of angle becomes longer as the radius increases... so I suspect that it's better to avoid parameterizing by angle, and see if saying "the radius of the living chamber increases linearly with the length of the shell the animal has produced," which might be what results in the logarithmic spiral and the triangular cones. Again, though, with a lot of ammonites (spiral and not) after a short initial growth spurt, many seem to keep the radius of their shells close to constant for much of their lives. Looking at Monks & Palmer, and measuring my meager collection, it does look as if I exaggerate, and most spiral ammonites do grow exponentially, just with a much lower exponent, so the growth per whorl is much lower than the "golden ratio" enthusiasts would like. I suppose evolutionary aesthetes might argue that the reason Nautilus survived the mass extinctions was somehow because of the aesthetic superiority of its shell shape, but I am skeptical. In any case, it would be interesting to measure sequential whorls of ammonites and see if my instinct is correct that the successive whorls increase more slowly than one would expect for exponential growth, which would suggest that, at least in width, the adult animals had a growth spurt in their youth, but then grew more slowly, at least compared to the rate they added new shell and septa, once they were older. Most reconstructions also have the body cavity much longer and thinner than modern Nautilus, also, of course... how that relates I'm not sure.

http://www.math.tamu.edu/~dallen/digitalcam/spiral/spiral.htm More equiangular than archimedian However see http://www.geo.uw.edu.pl/agp/table/pdf/55-2/korn.pdf

I was lucky enough to get a summer job driving Gary (Geerat) and his family around NZ a few years back (he had a grant to look at mollusc morphology all round the world!), what an interesting man and taught me so much about using senses other than sight to investigate the world (he's blind but still "sees" so much!) J