The Mathematical Study of Mollusk Shells
## 2. Zoning Laws in Molluskville
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.
______. | | | | |____________| 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.
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.
Of course if you keep on growing you will have to renovate again and again ...
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. |
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