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Fibonacci Sequence - An Example

Date: 05/12/99 at 23:59:13
From: Chris Chow
Subject: Fibonacci Sequences

Dear Dr. Math,

I have seen that many other people have asked for examples of the 
Fibonacci Sequence in Nature, but my math teacher wants unique ones.  
I can't use the bunny one, or the pineapple, or the shells, or the 
plants, etc. Examples like the Stock Market following the Fibonacci 
sequence would be greatly appreciated.  

Thank you for your time.

Date: 05/13/99 at 06:16:07
From: Doctor Floor
Subject: Re: Fibonacci Sequences

Hi, Chris,

Thanks for your question.

I might have something for you that your teacher doesn't know.

Consider two glass plates against each other. We send in a ray of 
light from above. This ray of light can be reflected several times 
within the plates before it gets out, like this:

   \  /\          /                  This ray of light has been
 ___\/__\________/_________________  reflected 5 times before
         \  /\  /                    getting out.

We will consider the sequence of numbers of routes r(n) with n 
reflections. Of course r(0) = 1, that is, the ray of light can go 
through with no reflections in one way.

Now let us count the number of possible routes that a ray of light can 
take inside these glass plates with one reflection:

   \  /     \      /
 ___\/_______\____/________________  There are 2 possible routes
              \  /                   with 1 reflection, so r(1) = 2.

Two reflections:

   \  /\     \        \      /\
 ___\/__\_____\________\____/__\___  There are 3 possible routes
         \     \  /\    \  /    \    with 2 reflections, so r(2) = 3.
           \         \            \

Three reflections:

   \  /\  /   \  /\      /    \      /\  /
                    \  /        \  /

   \      /\      /     \          /
 ___\____/__\____/_______\________/_____  There are 5 possible routes
     \  /    \  /         \  /\  /        with 3 reflections,
 _____\/______\/___________\/__\/_______   so r(3) = 5.

The sequence we find is 1, 2, 3, 5,.... How do we show that this is 
indeed the Fibonacci sequence, and goes on 8, 13, 21,... etc.?

Suppose we want to know the number of routes with n reflections, and 
we do know r(n-1) and r(n-2). 

Suppose that for n reflections the ray of light should leave the glass 
plates downwards. 

We know there are r(n-2) routes such that the ray of light leaves 
downwards with n-2 reflections. These routes can be extended with a 
route via the middle line, and then they form all possible routes with 
n reflections and the last reflection in the middle line. 

We also know there are r(n-1) routes, so that the ray of light leaves 
upwards with n-1 relfections. If these are reflected back downwards, 
then they form all possible routes with n reflections with the last 
reflection in the top line.

That gives that the total of possible routes with n reflections  
r(n) = r(n-2)+r(n-1). This is exactly Fibonacci's formula.

I hope this example will help you and surprise your teacher.

Best regards,
- Doctor Floor, The Math Forum   

Date: 05/13/99 at 17:52:36
From: Chris Chow
Subject: Re: Fibonacci Sequences

Thank you so much!  I will give it to her tomorrow.

Associated Topics:
High School Analysis
High School Fibonacci Sequence/Golden Ratio
High School Sequences, Series

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