Project Manaia

Seashells with Spiral Structures

Collection of seashells

According to Encyclopedia Britannica, a seashell is a mollusk’s multi-layered exoskeleton that guards their body and life. Many are drawn to their varying outward appearances, as they differ in color, shape, texture, and size. Pearls, which are microstructures within the innermost layer of the shell, are particularly fascinating to people, and are used in arts and crafts hobbies like jewelry-making.

Pigment, the biological cause of their color and pattern is determined, regulated and released by the cells within the shells. The spiral structure and form is also a unique quality of a seashell. Mollusks, the life forms which live in the shells, require an increasing amount of space as they grow. This is the reason behind a shell’s cone-like shape. As the mollusk grows larger, the shell’s material is only multiplied at a faster rate near the opening, causing a spiral structure to take form.

Growth rate determines the variance in a shell’s outward appearance, and this growth can be described mathematically, even by using simple geometry.

seashell spiral

Logarithmic Spirals

Their mathematical appearance can be described as a “generating spiral” or “generating curve”. The rotation of the semi-circle shaped opening and the upwards lift causes expansion and a spiral, with only the width increasing, and the angles staying constant.

Mathematical shells are obtained by the displacement of an exponentially growing curve along a constant, or logarithmic spiral. Each time there is an angle rotation, the upwards distance from the starting point increases, along with the width between the innermost center point and outer rim of the shell. Nautilus shells are one example of the equiangular spiral taking place.

halfed seashell photo


To generate a seashell shape using mathematics, computer graphic software like Wolfram Programming Lab can be tested. Additionally, Desmos and GeoGebra allow one to experiment with functions as a way to create “logarithmic” or “equiangular” spirals.

If you would like to see an example of seashell growth using mathematics, the video shown below titled “The Math of the Shells” visually demonstrates how seashells can be generated with equations and parameters as input.

Written by Miette Broussard

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