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Fibonacci Sequence in Nature

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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.

Thank you.
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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/
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Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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