1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

My Pi Discovery(Reason why 22/7 works)

  1. Mar 5, 2012 #1
    So, I was working with the Fibonacci sequence this afternoon, and I stumbled onto something that maybe a major discovery to geometry, I don't know if it is or not. So, I began working with the golden spiral, and began drawing it. After drawing it, I noticed you could draw circles and more circles with this sequence and spiral. The values even made sense to the circle sketches.

    This is what I did. Here is an image of it:

    http://img442.imageshack.us/img442/7633/proof1oa.png [Broken]

    What I did was took three numbers and inputted them into the formula I made, r1 / D + r2 / D = pi/2. The first numbers I did this to, I simply just got 2/2. Now, as I went on, the numbers became closer and closer to pi. I realized maybe if I kept going, I would reach a value, which I did. What I found maybe the reason why 22/7 works as an approximation of pi. So, I soon ended up with the diameter as 21. I did 1+1+2+3+5+8 / 21 + 13 / 21 = 3.14/2. Then, this would be 20/21 + 13 / 21. Add these fractions together, and you get 33/21, which simplified would be 11/7. If you were to multiply this by 2, you would get 22/7.

    I was just wondering if this work is legit or not.
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 5, 2012 #2

    Char. Limit

    User Avatar
    Gold Member

    Unfortunately, the only reason that this seems to work is that phi, or 1.618, approximately, is somewhat close to pi/2. You're right that your equation is converging, but wrong in what it's converging to. It's not converging on pi/2, but on phi. Now, let's take a look at this to see why it's true...

    Let's put your denominator as the nth Fibonacci number, F(n). If this is the case, then b will be F(n-1), and a will be the sum of all Fibonacci numbers up to F(n-2). Now, it is the case that the sum of all Fibonacci numbers up to F(n-2) is F(n)-1, or in summation notation:

    [tex]\sum_{k=1}^{n-2} F(k) = F(n)-1[/tex]

    By substituting these in, we get that the left side is equal to:

    [tex]\frac{F(n)+F(n-1)-1}{F(n)} = 1 + \frac{F(n-1)}{F(n)} - \frac{1}{F(n)}[/tex]


    Now, the first term is obviously 1, and the third term, as n goes large, will tend to zero. The second term is the most interesting, however. It's rather well known that the ratio of successive Fibonacci numbers tends to phi, i.e. F(n)/F(n-1) tends to phi for large n. This means that F(n-1)/F(n) tends to 1/phi for large n, and 1/phi can be written as phi-1 (it's one of the properties of phi). Substituting all this in, we get the following result:

    [tex]\lim_{n\to\infty} \frac{F(n)+F(n-1)-1}{F(n)} = \phi[/tex]

    Close to pi/2? Certainly. Converging to pi/2? Not quite.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook