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Topic: Calculating Pi using polygon sides sum / radius
Replies: 40   Last Post: Aug 2, 2013 5:27 PM

 Messages: [ Previous | Next ]
 JT Posts: 1,448 Registered: 4/7/12
Re: Calculating Pi using polygon sides sum / radius
Posted: Jul 30, 2013 11:43 AM

Den tisdagen den 30:e juli 2013 kl. 17:34:10 UTC+2 skrev jonas.t...@gmail.com:
> Den tisdagen den 30:e juli 2013 kl. 17:11:24 UTC+2 skrev Peter Percival:
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> > jonas.thornvall@gmail.com wrote:
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> > > Let say there was a rounding error due to some flaw in the used
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> > > arithmetic and it turn out that Pi actually was rational at some
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> > > fairly early digit.
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> > > But i guess that is ruled out,
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> > It is, you'll find a proof of pi's irrationality in books on real
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> > analysis. Here's a sketch:
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> > Let I_n(alpha) = integral_{-1}^1 (1 - x^2)^n cos alpha x dx. Show that,
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> > if n > 1, alpha^2 = 2n(2n - 1)I_{n-1} - 4n(n - 1)I_{n-2}. Show that,
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> > if n > 0, alpha^{2n + 1}I_n(alpha) = n!(P sin alpha + Q cos alpha),
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> > where P and Q are polynomials in alpha with integral coefficients of
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> > degree < 2n + 1. Show that, if pi/2 = b/a, a and b integers, then
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> > b^{2n + 1}I_n(pi/2)/n! would be an integer. Now consider large n...
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> > > i will try to calculate Pi with my hexagonal javascript, sum the
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> > > vertices as i double them up and calculate the new hypotenuse.
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> > > It will not be 22/7 but who knows maybe the dividend turn out to be
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> > > less then a million before making it rational?
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> > --
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> > Nam Nguyen in sci.logic in the thread 'Q on incompleteness proof'
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> > on 16/07/2013 at 02:16: "there can be such a group where informally
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> > it's impossible to know the truth value of the abelian expression
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> > Axy[x + y = y + x]".
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> I will try to do the calculation tomorrow with a graphical representation in Javascript. Where you can see the fractional representation as you double up number of vertices, and show greatest common factor for the perimeter(Pi?) of each polygon.
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> I am bad at math what is it called when reducing fraction, is it greatest common divisor GCD or greatest common factor?
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> For Pi to be rational i guess i have to prove that at same distant number of vertices, there is no change of perimeter length?
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> Is that correct?

That was not very clear.
To prove that Pi is rational i would have to show that after a certain number of double up of veritces. The perimeter (circumreference) stay the same, it seem impossible unless the difference in height of hypotenuse become zero.

Or that the number converge to a specific fraction, and then it is more about a representational issue with the decimals?

Date Subject Author
7/30/13 JT
7/30/13 JT
7/30/13 Peter Percival
7/30/13 JT
7/30/13 Richard Tobin
7/30/13 JT
7/30/13 Peter Percival
7/30/13 JT
7/30/13 JT
7/30/13 JT
7/30/13 JT
7/31/13 William Elliot
7/31/13 JT
7/31/13 Peter Percival
7/31/13 William Elliot
8/1/13 JT
8/1/13 JT
8/1/13 Virgil
8/1/13 Virgil
8/1/13 JT
8/1/13 JT
8/1/13 Virgil
8/1/13 JT
8/1/13 Virgil
8/1/13 Virgil
8/1/13 Shmuel (Seymour J.) Metz
8/1/13 Virgil
8/2/13 JT
8/2/13 JT
8/2/13 Virgil
8/2/13 JT
8/2/13 Virgil
8/2/13 Brian Q. Hutchings
8/2/13 Brian Q. Hutchings
8/2/13 Virgil
8/2/13 JT
8/2/13 Shmuel (Seymour J.) Metz
8/1/13 JT
8/1/13 Virgil
8/1/13 JT
8/1/13 Virgil