The Relation of the Golden Ratio and the Fibonacci Series

Date: 1/28/96 at 22:52:8
From: Anonymous
Subject: The connection between the Golden Ratio and the Fibonacci Series

  I am trying to figure out why the Golden Ratio and the 
Fibonacci series are related.  I have figured out that the ratio 
of a number in the Fibonacci series over the previous becomes 
increasingly closer to the golden ratio, but I have no idea how 
to relate that occurrence to some mathmatical reason or logic. I 
would appreciate any help.  

Thanks.

Date: 1/29/96 at 11:23:8
From: Doctor Ethan
Subject: Re: The connection between the Golden Ratio and the Fibonacci Series

Great question,

You have done the sticky part. Now let's look at the beautiful 
part.

You may have noticed that if you have a Golden rectangle and you 
cut off a square with side lengths equal to the length shorter 
rectangle side, then what remains is another Golden rectangle.

This could go on forever. You can just keep cutting off these 
big squares and getting smaller and smaller Golden rectangles.

The idea with the Fibonacci series is to do the same thing in 
reverse.

Here what you do is start with a square (1 by 1), find the 
longer side, and add a square of that size to the whole thing to 
form a new rectangle.

So when we start with a 1 by 1 square the longest side is one, 
so we add another square to it.  Now we have a 2 by 1 rectangle

Then the longest side is 2, so we connect a 2 by 2 square to our 
2 by 1 rectangle to get a 3 by 2 rectangle.  This continues, and 
the sides of the rectangle will always be a successive Fibonacci 
number.  Eventually the rectangle will be very close to a Golden Rectangle.

I will do a few steps to let you see it in action:

###
# #    1 by 1, so we add.   1 by 1 to get...
###

######
# ## #  Now it is 2 by 1, so we add 2 by 2 to get......
######

######
# ## #
######
######
#    #   Now it is 2 by 3, so we add a 3 by 3 to get.......
#    #
#    #
#    #
######

###############
# ## ##       #
#######       #
#    ##       #
#    ##       #  Now it is 3 by 5, so we would add a 5 by 5 square.
#    ##       #
#    ##       #
###############

I hope that you can follow this and it helps.

-Doctor Ethan,  The Math Forum