The mathematics of gnomonic seashells

Parametric Cartesian vector-valued functions are constructed for the purpose of systematically describing various features of spiral shell geometry. The underlying geometrical hypothesis is that molluscan shell shapes can usually, to at least a good first approximation, be developed by rotating a generating curve about a fixed axis whilst simultaneously diminishing it by an “equiangular spiral” scale factor. A first-order symmetry equation is derived; then variational calculus is used to construct energy functionals which establish that Hooke's law is inherent in the formalism and that naturally occurring shell geometries are analogous to those of elastic spiral “clock springs”. The biological requirement that shelly structures must exist in a three-dimensional space is shown to be a sufficiently powerful mathematical constraint to ensure the existence of geometrical artifacts which can, perhaps, be likened to the conservation laws, pseudoforces, and fields of classical physics.