shell2

$e-MATH$

The Mathematical Study of Mollusk Shells

2. Zoning Laws in Molluskville

To understand 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.

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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. Furthermore the building material used in Molluskville is such that once put up it cannot be torn down. Also a house once built may not be abandoned until its inhabitant dies.

Suppose you have built such a house and are living in it, and suppose that now you have grown then and that your house has become uncomfortably small. You cannot leave, and you cannot tear down the walls you have. The only possibility is to build on, and there is only one way this can be done: if your house dimensions were 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, you will be in compliance with the zoning law. And 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 fairly simply be calculated.

Tony Phillips
Stony Brook