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Fibonacci sequence in nature, Golden Mean, Golden Ratio

Date: Tue, 6 Dec 1994 
From: Irene A. Mathews
Subject: Fibonacci sequence in nature

Dear Dr. Math:

My name is Erica Anderson  and I need to know examples of where the 
Fibonacci sequence is found in nature and how that relates to the Golden 
Mean. This is for my Pre Calc. class.

Thank you for your help. Please respond at this address.

Erica Anderson

Date: Tue, 06 Dec 1994
From: Dr. Math
Organization: Swarthmore College Math Doctors
Subject: Re: Fibonacci sequence in nature

Hey Erica,

I love this problem.  In fact I love it so much that one of my
wonderful kind fellow doctors left it for me to answer because he knew it
would make me happy.  And it does.  The Fibonacci sequence happens all the
time in nature.  In fact they occur so much it is amazing.  I am having
trouble not just sitting here and listing all the occurences that I can
think of.  But I will try to resist sending you a seven page e-mail message.

A few stunning examples.

Think about a pine cone.  Have you ever noticed that the petals
kind of spiral up in two directions?  Well, the number of petals it takes to
get once around is almost always a Fibonacci number.

Why are four leaf clovers so rare? (Four isn't a Fibonacci number.)

A great number of flowers also demonstrate Fibonacci sequences.  Check
a few and count their petals.

A few more things to explore are the rotations of the pods of
pussywillows, sunflower blooms, pineapples etc.

Now about the Golden Mean.

The Golden Mean is cool and one of the things that is cool about it
is that the ratio of successive Fibonacci numbers is the Golden Mean.

So the series

   1/0  1/1  2/1  3/1  5/3  8/5  13/8   21/13  34/21  55/34  89/55  144/89
has as its limit the Golden Ratio.

Hope that helps,
Ethan Doctor On Call

Date: Wed, 7 Dec 1994
From: Jane Stavis
Subject: Re: Fibonacci sequence in nature

Dear Erica,

I'll add on, including three of my favorite examples of where the
Fibonacci sequence is found:

As Ethan said, pine cones, pineapples, sunflowers are a few examples of
where the number of spirals are Fibonacci numbers  - and you can actually
count the spirals in two directions, one tighter than the other, (three
for a pineapple). Now is a great time to get pinecones - give it a try! 
(I like to mark my starting spiral with a marker or white out so I can keep
track of what I have counted.) Pine cones are usually 5 and 8 or 8 and 13,
pineapples 13, 21 and 34 (I think), and sunflowers 34 and 55 or 55 and 89
(also, an I think) - give it a try.

You can also draw a logarithmic spiral (like the one in a nautilus
shell) where all the angles are equal based on the Fibonacci sequence.
Using graph paper, draw two 1x1 squares next to each other.  Draw a 2x2 sq
next to them and then spiral round so that the 3x3 is touching a 1x1 and
the 2x2.  The 5x5 shares a side with the 2x2 and 3x3 and it continues on.
When you have run out of paper, you can draw in the spiral itself,
connecting one corner of each square to the opposite corner with a curve.
It make take a try or two before you figure out how the squares line up
against one another, but it's worth it - the result is wonderful!

A third favorite example of mine has to do with the Golden Ratio, which
Ethan described.  Many artists have actually used the Golden Ratio in
their work.  If you can, check out the Time/Life Science Library book on
_Mathematics_.  It has examples of how the Parthenon fits into a Golden
Rectangle, how DaVinci used these proportions (knowingly or not), and how
more modern artists such as Seurat and Mondrian used them quite knowingly.

Good Luck and have fun!

  --Jane Stavis, Westtown School
Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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