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. Chris
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 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
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