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
asked something about drawing a spiral using the Fibonacci series.
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 
of Mathematics) is published by Wide World Publishing/Tetra, so that's a 
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.

Thanks for your help,
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