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.wingsdailynews.com/wp-content/uploads/2015/04/fib... No doubt it was cherry picked and the spiral placed in the best position, yet the match is abysmal (look at how far the center of the spiral is from the center of the flower). The seeds are just packed tightly together and this produces some patterns due to the seeds on the outside being more developed than those on the inside.

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.

I only meant it as a possible explanation for why it wouldn't be unreasonable to think phi appears often in nature. I don't really have enough knowledge of nature to say if it's the case.

Distribute the seeds of a sunflower radially, placing one seed every 360*x degrees, gradually increasing radius. If x=a/b, after placing b seeds, you will be back to the initial position and the next b seeds will be (radially) shaded from the first b seeds (which I admit, might not be how shading works in practice). If x is irrational, but closely approximated by a/b, the seeds won't line up perfectly, but still enough to shade quite a bit. If x is phi, then they will shade as little as theoretically possible (I think.. This is by no means a proof, just some thoughts). IIRC, the "sunflower seed pattern" is actually achieved only if you simulate such a seed placing with x close to phi.

This is trying to find meaning where there is none. I don't think shading of seeds has any impact on the evolutionary fitness, even if we assume that light is coming in radially which is of course not true at all. Furthermore, even if we make the two (clearly incorrect) assumptions that sunflowers do care about shading of the seeds and that light does come in radially, that does not even constitute a convincing argument that the seeds grow in that pattern. Here is a far more convincing argument along the same lines. Plants care about getting energy. They get energy by absorbing sunlight. Black absorbs the most light. Ergo, plants are black.

Good point. The reason why few to no plants are not black is a very interesting problem, which I haven't heard an answer to.

Still, the number phi has very unique properties, considering its SCF. And the sequence of fibonacci ratios is not an arbitrary sequence converging to it. Whether anything has evolved to utilize this or not, I can not say.

Thing is, Phi's not really all that interesting. It's just a simple root of a quadratic, no more 'mystical' than the square root of five. It doesn't 'emerge' from arithmetic the way e or pi do. It's a fixed point of the sequence of operations: 'take a number; invert it; add one; repeat'... that's, sort of interesting, but 'add one' isn't a very special operation - why not add 12? or add pi? or add phi?

Take a number, invert it, add two, repeat... eventually you get root 2 + 1. And the inverse of that is root 2 - 1! That's pretty magical! Kind of more magical than 'half plus root 5 over 2', anyway. Maybe root 2 + 1 is the platinum ratio!