Friday, March 23, 2012

phyllotaxis & primes

I was inspired to make some pictures of phyllotaxis with primes colored in by this post at Maxwell's Demon, which has not only nice pictures, but also explanations for why the primes appear to line up on certain curves. Like the spirals there, the ones in these pictures are not numbered in the "naturally" occurring way - in a phyllotaxis spiral that occurs in nature (as in a sunflower or pine-cone) the outer dots/seeds grow first and are pushed out by the younger seeds that emerge from the center.

It is a lot easier to simulate a phyllotaxis spiral as a function of the radius (described here) rather than modeling the pushing-out from the center that happens in plants (a good description of this is found in The Book of Numbers by Conway and Guy).

Drawing number spirals with primes on them is some sort of recreational mathematics meme - the ulam spiral is the most famous, they are also drawn on quadratic spirals and others.