Fibonacci Sequence in Nature

Date: 12/08/97 at 17:52:22
From: Naseema Patel
Subject: College Algebra

I need examples of how the Fibonacci sequence is used in nature.
I used the Internet to find examples, but I couldn't find any examples 
except the one I know: the Rabbit example. I need three more examples. 
Can you please help me?

Thank you.

Date: 12/11/97 at 20:25:33
From: Doctor Steven
Subject: Re: College Algebra

Here's one. On a tree when a twig comes out from a branch, the next 
twig to come out from the branch will be slightly farther out on the 
branch and will be rotated about the branch at some angle. The next 
twig will come out farther along the branch, rotated at the same 
angle. Here's the clincher: the number of twigs needed for the last 
twig to come out of the branch at the same angle as the first twig is 
always a number that appears in the Fibonacci sequence, no matter what 
kind of tree it is. An apple tree might need 5 and a pear tree might 
need 8, but the number needed for any tree will always occur in the 
Fibonacci sequence.

Another example of the Fibonacci sequence can be seen in a Nautilus 
shell. Those are those shells that curl around on themselves. Inside 
are little chambers and you guessed it - let's say the first chamber 
has a volume of 1; then the second chamber has a volume of 2, the 
third chamber has a volume of 3, the fourth has a volume of 5, and so 
on: the Fibonacci sequence.

Two more examples: pineapples and turtles. One might ask what turtles 
and pineapples have in common, but the answer is of course the 
Fibonacci sequence (and their shells/rinds).  

A pineapple has shapes on its rind, bigger shapes on the bottom of it 
getting smaller as you go up. If you take one of the shapes on the 
bottom and go diagonally upward (to your right or left) you see that 
these shapes are basically the same only scaled to a smaller size as 
you go up. If we say the bottom has a size of 13, then we'll notice 
that the shape directly to the upper left (or right) has size 8. The 
shape to the upper left (or right) of this next shape has size 5, and 
so on down to one. The size you need to start with depends on the size 
of the pineapple and how many little shapes are in a sequence.

For some turtles basically the same thing happens. Turtles have shells 
that have what I would call tiles on them (although I know that is 
not the correct term). These tiles are the same shape but appear in 
different sizes. The sizes correspond to entries in the Fibonacci 
sequence.  

If you want to read more about these examples, I found them in 
_Applied Combinatorics_ by Fred Roberts. (This book was written to be 
a graduate level textbook in mathematics.)

-Doctor Steven,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/