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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.

Chris

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

Hi, Chris,

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.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/

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.

Chris

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

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