Modeling Natural Shapes: Seashells

O conchiglia marina, figlia
della pietra e del mare biancheggiante,
tu meravigli la mente dei fanciulli.

La conchiglia di Alceo. (Traduzione di Salvatore Quasimodo, da Lirici greci, 1940) 

Lat. 54.352° N, Long. 13.363° E.

Walking on the beach of this wonderful island, it is easy to remains of defunct mollusks: seashells. These marvelous and elaborate sculptures of calcium carbonate, are the variation of a common mathematics theme: the three-dimensional logarithmic spiral. This structure is very common in nature and some example are shown in the following collage of pictures:


One of the first mathematical biologist to discuss the mathematics behind the shapes of seashell and other natural spiral forms was Sir D’arcy Thompson in his magnificent book On Growth and Form [1]. Several models for the description of the shape of these marvels have been proposed (for an interesting discussion of one of these model see for example Dawkins [2]). One of the most complete is the Cortie’s one [3] that take into account 16 parameters to describe not only the overall shape but also complex details of the surface of some of the species. The model is based on the translations and scaling along an helico-spiral of the shape of the aperture of the shell as schematically indicated in the following slides.


Mathematically this procedure corresponds to the following set of equations with the parameters indicated in the Figure (see [4]).


A simplified version of the model is implemented in a Maple(TM) script that can be download from here: Maple Script.* By varying the parameters in the script, it is possible to obtain most of the model described in the original paper by Cortie. Here some example of outputs.



  1. D’Arcy W. Thompson. On Growth and Form. (2nd edition 1944).
  2. Richards Dawkins. Climbing Mount Impossible (1996)
  3. M. Cortie, Models for mollusk shell shape, South African Journal of Science, vol. 85, pp. 454–460, 1989.
  4. J. Picado,  Seashells: the plainness and beauty of their mathematical description, MAA Mathematical Sciences Digital Library, 2009.


(*)I would like to thank Miss Shannon Woolis for her contribution to developing this script as part of her final project for the Bachelor degree in Mathematics.

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