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### Fibonacci Series

```
From: Jeff Jones
Date: Mon, 7 Nov 94 14:08:53 EST
Subject: Fibonacci series

Hi,

I was helping an Algebra student with a "bonus" problem recently.  It
What is this series?  Does it draw a spiral?  Any help would be
appreciated.

Thanks,
Jeff Jones
```

```
Date: Mon, 7 Nov 1994 14:33:31 -0500
X-Sender: vmotto1@cc.swarthmore.edu

Thanks for writing!

Jeff, Looking back at an old calculus book of my own, I can give you the
definition of the Fibonacci sequence, which is constructed using a function
that is defined as follows:

For each positive integer n, then

/ 1  , if n = 1 or n = 2
f(n) =      /
\  the sum of the two preceding values of f  ,  if n > 2
\
For example,
f(1) = 1
f(2) = 1
f(3) = f(1) + f(2) = 2
f(4) = f(2) + f(3) = 3
f(5) = f(3) + f(4) = 5

and so on.

I am not that familiar with Fibonacci numbers myself, but if one of the
other math doctors knows some more, they will write you soon. In the
meantime, I hope this will give you an idea of what is going on with the
sequence and how it formed.

-Vanessa, M.D.
```

```
From: Dr. Ethan
Date: Mon, 7 Nov 1994 14:41:01 -0500 (EST)

I am so excited--this is so great--I love the Fibonacci sequence.  I am
actually so pumped to be able to answer this question. It sure does draw a
spiral and it is a really neat one as well.  I am just freaking out in the joy
I am receiving from answering this question.

Unfortunately, describing how to get it to draw such a spiral might
be a little tricky via email but we can give it a try.

First think superimposed rectangles.

Okay, let's start with 1,1.  So draw a square of side length 1.  Then
choose one side, extend its length to 2 units, and draw the larger rectangle
so it looks like a rectangle of side lengths 1 and 2 that is divided in
half.  Then we start to get a pattern.  Expand along the direction of the
shortest side to the length of the next Fibonacci number.  This creates a
cycle of ever expanding rectangles that produce a beautiful sprial.  That I
believe is the same spiral that conch shells make as they spiral out.

Hope you can figure this out.  If not, write again and I will try later.

-Ethan, Dr. on call.
```

```
From: Dr. Ken
Date: Mon, 7 Nov 1994 16:50:10 -0500 (EST)

Jeff!

In looking at our responses to your question, I think the most helpful thing
we could do is direct you to another source, one that I think gives a very
interesting overview of Fibonacci Sequences and things that they are related
to (as well as lots and lots of other interesting and beautiful mathematics):

The book is called the Joy of Mathematics, and it's written by Theoni
Pappas.  I'm not sure who publishes it, but I know that its sequel (More Joy
good place to check first.  I'm almost positive that it's the same publisher.

You won't be disappointed if you investigate the Fibonacci numbers further.
There's a wealth of fascinating, accessible, and beautiful mathematics going
on with them.  Good luck!

-Ken "Dr." Math
```

```
From: Jeff Jones
Date: Mon, 7 Nov 94 15:14:04 EST

Hi Ken,

Glad I could bring you such a pleasure today.

The Algebra book from which I took the problem said to draw right
triangles stacked up.  For instance, draw a 45 degree right triangle
with the hypotenuse sloping up and to the right, and the right angle on
the bottom-right.  Imagine that the 45 degree vertex at the left will
be the origin of the spiral.  Now, draw a line beginning at the far end
of the hypotenuse, at a right angle to the hypotenuse, for a distance
equal to the next Fibonacci number.  It should look like below.
Continue using the hypotenuse of one triangle as the base for the next
triangle.

I think you're right, the book said something about conch shells.

Jeff

3
. . . . . . . . . . . . . . .
. r .                        r. .
.       .                       .   .
.           .                     .     .  2
5    .               .                   .       .
.                   .                 .         .
.                       .               .           .
.                           .             .            r.
.           .           . |
.         .         .   |
.       .       .     |
.     .     .       |   1
.   .   .         |
. . .           |
O____________r|

1

(the "r" is the right angle)
```

```
From: Dr. Ken
Date: Tue, 8 Nov 1994 14:18:17 -0500 (EST)

Hello there, Jeff!

I think that if you take a closer look, you'll find that the lengths of the
sides are not 1,1,2,3,5,8,13,21,... (the Fibonacci numbers), but the square
roots of those numbers.  I mean, you may have already seen this, but I
thought I'd point it out anyway.  Enjoy!

-Ken "Dr." Math
```
Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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