**The Mathematical Study of Mollusk Shells**

__________________________. | | | | . | | | |_.| | | .___|__| | | | | | | | | | | | . | | | | | | | | | | | | | | | |___________________________________________________| house after fourth renovation

If we take coordinates based at the lower right hand corner of the original house, and let `a = 1`, then the wall holding the original door goes from the corner at `(-2,1)` to the dot at `(-1,1)`. After the first renovation, the new wall goes from `(-2,0)` to `(-2,2)`. The new red dot comes from rotating the old one 90^{o} counterclockwise about the corner; the new corner comes from rotating the red dot 90^{o} clockwise. Applying this construction to the new red dot and the new corner leads to the red dot and the corner in the second renovation, etc.

The red dots are on a logarithmic spiral:

- Let
`(x,y)`be the coordinates of the`n`th red dot, and`(u,v)`be the coordinates of the`n`th corner. Then the coordinates of the`(n+1)`st red dot will be`(-y+u+v,x-u+v)`, and the coordinates of the`(n+1)`st corner will be`(y+u-v,-x+u+v)`, so the two points together transform by the linear map`A(x,y,u,v) = (-y+u+v,x-u+v,y+u-v,-x+u+v).` - Starting with the points in the "original house" and iterating this map backwards converges to
`(-1.6, .8, -1.6, .8)`, with`(-1.6, .8)`therefore the center of the spiral. - Moving the center of the coordinate system to this point gives, for the successive red dots:
`(.6, .2)`,`(-.4, 1.2)`,`(-2.4, -.8)`,`(1.6, -4.8)`,`(9.6, 3.2)`. - Rotating the coordinates counterclockwise by
`arctan(.2/.6)`and scaling them by`2/sqrt(10)`makes them`(1, 0)`,`(0,2)`,`(-4,0)`,`(0,-8)`,`(16,0)`. - These points are clearly on the logarithmic spiral
`r= 2^(2theta/pi)`, with`theta = 0, pi/2, pi, 3pi/2, 2pi`.

- 1. Terminal growth and unchanging form
- 2. Zoning laws in molluskville
- 3.
*Nautilus*and the Ammonites - 4. Things get more complicated in 3 dimensions, but Calculus comes to the rescue.
- 5. Mathematical models of helical shells
- 6. Ray Gildners Java applet for shell sketching
- 7. For further thought