The Mathematical Study of Mollusk Shells
A snail inhabits its shell but can only build onto the shell where it is in contact with the shell exterior, in the neighborhood of the opening. These two images of a small (5mm) unidentified gastropod from Monterey Bay are due to Steve Lonhart and are used with permission. |
To start understanding the mathematical problem a snail faces, imagine that you are a citizen of Molluskville, a Flatland community with very strict zoning laws.
______. | | | | |____________| The Model House
_. |__| house before renovation |
__ | . | |_.| |__| house after renovation |
Of course if you keep on growing you will have to renovate again and again ...
_____________ | | | . | | |_.| | .___|__| house after second renovation |
_____________ | | | . | | |_.| | .___|__| | | | | | | | | | . | | | | | | |_____________ house after third renovation |
and again.
__________________________. | | | | . | | | |_.| | | .___|__| | | | | | | | | | | | . | | | | | | | | | | | | | | | |___________________________________________________| house after fourth renovation
The red dots at the inside edge of each doorway lie on a logarithmic spiral, as can be straightforwardly calculated.
Our rectangular model is simpler than any actual mollusk shell, but it faithfully illustrates the kind of constraints that shape mollusk shell morphology and the way that these constraints force the appearance of the logarithmic spiral.