Fibonacci Flim Flam

Post by fvirexi » Tue, Sep 01, 2015, 10:31 PM UTC

fvirexi wrote: post #30

jameshart wrote: post #27

Earlier quoted context omitted.

"it is _the_ number that is hardest to approximate by any rational number" I'm not sure what you mean by this. Any irrational number can be approximated to an infinite number of different levels of precision by an infinite number of different rational numbers. And rational approximations to phi can be trivially generated by any Lucas sequence, starting from an infinite number of different possible seeds (not just the…

I apologize for the vagueness. "Hardest to approximate" does require certain definitions first. The reasoning behind this is simple continued fractions, ie. fractions like a_0 + 1/(a_1 + 1/(a_2 + ...)) = [a_0; a_1, a_2, ...], with a_i\in N. Every irrational number corresponds uniquely to an infinite continued fraction, and the finite "steps" of the SCF are the fractions that best* approximates the irrational numbers.…

EDIT3: 233/114 should be 233/144, and it's off by ~10^{-5}

Post by lawpoop » Wed, Sep 02, 2015, 2:33 AM UTC

jules wrote: post #19

fvirexi wrote: post #17

Earlier quoted context omitted.

I don't know about limbs, but it could explain sunflower seeds and branches of trees. If they are distributed with a period, every n'th time around they will shade for other seeds (or branches). Approximating \phi is then a good way to reduce such shading.

That argument is too hand wavy to be convincing. Can you make it precise? How exactly are the seeds or branches being distributed in 2d/3d space, and how does that minimize shading? Why would sunflower seeds want to minimize shading in the first place? There's also the trouble that sunflower seeds and branches of trees do not actually approximate phi in any meaningful way. Check out this picture: http://www.wingsdail…

They actually do have a relationship to Phi, but not the one in the in picture you link, which is obviously bad.

If you draw a pattern of dots according by rotating points 137.5 degrees (which you see on a lot of seed heads and fruits, such as sunflowers, pineapples, pine cones, romanesco broccoli, various succulents, etc.), you create a pattern were certain spirals 'jump out' at you. If you count the number of 'arms' in each successive set of spirals, the numbers are the Fibonacci sequence. http://momath.org/home/fibonacci-numbers-of-sunflower-seed-s...

Somewhere along the line I read on some website that it might have to do with optimal packing theory -- distributing the maximum number of seeds over the seedhead, but I think it was just a guess. It does seem to show up in a variety of plants, more than just chance would lead you to expect.

Re: Fibonacci Flim Flam

Re: Fibonacci Flim Flam