The Mathematical Study of Mollusk Shells
The logarithmic spiral in Molluskville architecture
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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
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),
- These points are clearly on the logarithmic
spiral r= 2^(2theta/pi), with theta = 0, pi/2, pi, 3pi/2, 2pi.
@ Copyright 2000, American Mathematical Society.