Logarithmic spirals and the nautilus

I just composed the following observations in connection with a MoneyLaw post and thought that the Jurisdynamics audience might appreciate them even more:


The chambers of a nautilus are arranged according to an approximate logarithmic spiral that can be calculated in polar coordinates according to this simple formula:

r = ae^bθ

where r represents the radial coordinate, θ represents the angular coordinate, e is the base of natural logarithms, and a and b are constants that (1) are arbitrary in modeling and (2) are empirically determined in real-world applications of logarithmic spirals.

I post this picture of the nautilus, which graces the banner for BioLaw: Law and the Life Sciences, because it is beautiful. It reminds us that beautiful things are often beautiful because they work. The nautilus and its relatives, after all, have cruised the seas for half a billion years with very few evolutionary adaptations.

For technical details, see Wikipedia's excellent articles on logarithmic spirals and polar coordinates. Extra intellectual credit goes to readers who tackle the article on spherical coordinates as well.