Fibonacci Ratio, Golden Ratio, and 8:5

Date: 06/01/97 at 01:20:39
From: Kevin Brown
Subject: Fibonacci Sequences\Numbers and Golden Ratio

I was asked to measure and record the body lengths in a diagram. I was 
then asked to put them into a ratio and these are the ratios I got:

8:5
2:1
79:50
40:89
8:5

I was then asked to comment on the ratios. So far three of them are 
close to the 8:5 ratio (which is the Golden Ratio, isn't it?). Do you 
think they are all supposed to be the 8:5 ratio? If so, I'll have to 
measure it again.

Also, I am supposed to investigate whether this statement is 
mathematically true and why: 

The total number of keys in an octave on the piano, the number of 
white keys, and the number of black keys are all Fibonacci numbers.

I found out that there are 8 white keys per octave and 5 black keys 
per octave. Are these Fibonacci numbers? Why? I know that 8:5 is the 
Golden Ratio, but is it a Fibonacci number? 

Thank you for your time,
Kevin

Date: 06/01/97 at 06:02:16
From: Doctor Mitteldorf
Subject: Re: Fibonacci Sequences\Numbers and Golden Ratio

Dear Kevin,

You're thinking very well about this, but there's some information 
they're not giving you.  Also, sometimes questions just aren't asked 
very clearly - the teachers and also the people who write the 
textbooks are only human, and often they don't realize what it's like 
to be on the other side of the question.
   
The ratios of Fibonacci numbers are:
 
1/1     = 1
2/1     = 2
3/2     = 1.5
5/3     = 1.6666...
8/5     = 1.6
13/8    = 1.625
21/13   = 1.6153...
34/21   = 1.6190...
    
If you keep on going for a long time, the ratios settle down.  They 
all get closer and closer to a number without ever quite reaching it.  
There's a name for this phenomenon when it happens in math - it's 
called a "limit". You can say that the limiting ratio of Fibonacci 
numbers as the numbers get higher and higher is the definition of the 
Golden Ratio.  It's not exactly 8/5, but it's close.  The exact value 
of the Golden Ratio is (1+sqrt(5))/2, which is about 1.6180339...
    
I don't know if you've studied algebra yet, or if you know about 
quadratic equations and the tricks people use for solving them.  In 
case these things are familiar and interesting to you, here's a fact 
you might investigate: 
    
The ratios of Fibonacci numbers get closer and closer to the Golden
Ratio, but they never quite get there.  You can prove that if they DID 
ever get there, then they wouldn't change any more, but would stay 
equal to the Golden Ratio forever after that.  
     
Suppose you get to a Fibonacci number called a.  The next one is b, 
and the ratio b/a = (1+sqrt(5))/2.  Then the next Fibonacci number 
would be a+b, so the ratio of (a+b)/b would be what?  In other words, 
if b/a = (1+sqrt(5))/2, what is (a+b)/b?  Can you solve that?  Can you 
simplify your answer?  It comes out in a pretty cool way.

8/5 and 89/55 and 43/21 are all Fibonacci ratios, but they're not THE 
golden ratio.  I don't know what your teacher was thinking with those
examples s/he gave you.  Maybe s/he just wanted you to think about all 
this and then write about it.

An octave on the piano is really 5 black notes and 7 white ones.  You 
can make it 5 and 8 if you include the white notes at both ends.  But 
you could equally well make it 6 and 7 by including the black notes at 
both ends.  And in any case, by the time you got to 2 octaves or 3, 
you wouldn't have a true Fibonacci ratio any more.  I think this goes 
to show that some of the things people say about Fibonacci numbers are 
just superstition.  Same way some great astronomers of 500 years ago 
used to try to relate the ratios of the orbital diameters of different 
planets in our solar system to mathematical ratios from geometry.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/