The Shapes of Sea Shells: Mathematical Beauty in the Natural World

Part of the natural beauty of seashell derives from the visual rhythm of their self replicating whorls.   We give a mathematical model that captures sea shell morphology using space curves, parameterized surfaces, and Frenet frames.  We show how the growth parameters of this model may be measured from x-rays or cross sections of actual shells, and then 3-D computer generated simulations of the intact shell produced.  We conclude with a gallery of ‘fantasy’ shells, that is, shells created solely for their artistic appeal, but which also demonstrate the effects of manipulating the parameters of the model.

The basic shell model may be taught at the undergraduate level (e.g. calculus III, vector calculus, or differential geometry), and provides a beautiful and engaging introduction to mathematical modeling. It is available as a ‘discovery-learning’ module through ILAP/COMAP.