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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 90o counterclockwise about the corner; the new corner comes from rotating the red dot 90o 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 nth red dot, and (u,v) be the coordinates of the nth 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.