Departamento de Matemática

Universidade de Coimbra

picado@mat.uc.pt

Abstract:

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.

Full article text

This article uses LiveGraphics3D, a Java applet which displays three-dimensional figures as generated by Mathematica. Any figure displayed with LiveGraphics3D can be rotated by clicking and dragging the mouse on the image.

Contents:

• Introduction
• 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 bivalvesv
• 2. The basic model
• 3. The complete model
• 4. Index of examples
• 5. The origins of limpets
• Bibliography

Also available:

PDF version

Portuguese language version (HTML, external link)

Portuguese language version (PDF, external link)