The Mathematical Study of Mollusk Shells

## 2. Zoning Laws in Molluskville

 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.

• Every house must look exactly like the Model House, which is rectangular, twice as long as it is wide, with a door at one end of one of the long walls. The opening is exactly half as wide as the wall is long. There is a red dot at the near end of the doorway.
      ______.
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The Model House

• A house may be bigger than or smaller than the Model House, but the proportions of walls and door must be exactly the same.
• The only way a house may be altered is by extensions on the side of the door.
• A house once built may not be abandoned until its inhabitant dies.
Suppose you have built such a house and are living in it, but suppose that you have been growing and that your house has become uncomfortably small. You cannot leave, but you can no longer fit in the house as it is. The only possibility is to build on, and there is only one way this can be done: if your house dimensions are a feet wide and 2a feet long, you add on a 2a x 3a room to the side with the door. Now your house is 2a feet wide and 4a feet long. If you put in a door a feet wide at the end of the new long wall, and a new red dot in the right place, you will be in compliance with the zoning law. This is the only way you can expand your living quarters.
  _. |__| 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.


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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.

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