The bio-geometry of mollusc shells

The shells of gastropods and cephalopods grow by production of new material, by the mantle, at the lip of the shell. I derive a model of shell form that allows us to describe the morphology of a shell in terms of independently definable biological parameters. These are (1) the relative rates of shell production at different points around the aperture, (2) the total amount of shell produced per time interval, (3) the growth rate of the aperture, (4) aperture shape, and (5) the orientation of the animal within the shell. Describing shell form in these terms allows us to see what biological changes must occur in development in order to change one shell morphology into another and what constraints are associated with particular morphological transformations. The model shows that it is developmentally easy to derive a slightly coiled limpet shell from that of a high-spired ancestor, but difficult to take the next step to a fully conical limpet. Many, if not most, real gastropod shells are not conical but rather have a convex or concave profile. I show that these forms result from a decoupling of shell production rates from the growth rate of the animal within the shell. The model also shows how truly different forms, such as vermetid snails and heteromorph ammonites, escaped the confines of spiral growth - sometimes by rotating the body within the shell, and sometimes by taking up a growth strategy that does not constrain them to coil. This model is compatible with shell morphometric models that have been widely discussed in the literature but strives toward a different goal: understanding the relationships between the various biological processes involved in shell development.