  Accretion of mineralized thin wall-like structures via localized growth along
their edges is observed in a range of physical and biological systems ranging
from molluscan and brachiopod shells to carbonate-silica composite
precipitates. To understand the shape of these mineralized structures, we
develop a mathematical framework that treats the thin-walled shells as a smooth
surface left in the wake of the growth front that can be described as an
evolving space curve. Our theory then takes an explicit geometric form for the
prescription of the velocity of the growth front curve, along with some
compatibility relations and a closure equation related to the nature of surface
curling. The result is a set of equations for the geometrical dynamics of a
curve that leaves behind a compatible surface. Solutions of these equations
capture a range of geometric precipitate patterns seen in abiotic and biotic
forms across scales. In addition to providing a framework for the growth and
form of these thin-walled morphologies, our theory suggests a new class of
dynamical systems involving moving space curves that are compatible with
non-Euclidean embeddings of surfaces.
