Seashells: the plainness and beauty of their mathematical description

Jorge Picado

Departamento de Matemática
Universidade de Coimbra
picado@mat.uc.pt

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?   Mathematics is very much like poetry ... what makes a good poem -- a great poem -- is that there is a large amount of thought expressed in very few words. In this sense formulas like or are poems.   Mathematics possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature ... sublimely pure, and capable of a stern perfection such as only the greatest art can show. Albert Einstein Lipman Bers Bertrand Russell

One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells that one may find classified in a good seashell bookguide.

The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells.

Index:

1. How seashells grow

1.1. The equiangular spiral

1.2. Growth of the horns of animals

1.3. Gnomonic growth of the nautilus

1.4. Gnomonic growth of cones

1.5. Gnomonic growth of bivalves

2. The basic model

3. The complete model

4. Index of examples

5. The origins of limpets

    Bibliography

Next: 1. How seashells grow