calculation1

**The Mathematical Study of Mollusk Shells**

## The logarithmic spiral in Molluskville architecture

__________________________. | | | | . | | | |_.| | | .___|__| | | | | | | | | | | | . | | | | | | | | | | | | | | | |___________________________________________________| 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`.

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