Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why?
I read that one way this is so is that it models the population of honey bees under ideal conditions, which is a quite straight forward argument to understand.
I also read that in many places you see a spiral shape (snail's shell, pinecones, seed heads, leaf arrangements and so on) the Fibonacci sequence is appearing. From what I understand this spiral is constructed by starting with two $1\times 1$ squares, and then putting a $2\times 2$ square on top and then enclosing in a $3\times 3$ square and then introducing a $5\times 5$ square and so on. Then a spiral is drawn by drawing a quarter circle in each square. I know this is not clear, but I guess I don't fully understand it.
Anyway I am wondering if the Fibonacci sequence is unique to this pattern. I mean, the way the spiral is constructed seems arbitrary. Is there another sequence that can be used in this argument? Is there another way of arranging geometric shapes to get these spirals, or some combination of both? Or is the spiral something unique to the Fibonacci sequence?
Lastly, what are some other instances of the Fibonacci sequence appearing in nature? Why is it so special?
The Fibonacci numbers can appear in the length between nodes on Fern leaves, on the rings of pineapples, etc. This can be because because each leaf grows out into an unoccupied space but can't move into the space occupied by the leaves already there.