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 November 6th, 2010, 01:16 AM #1 Member   Joined: Jun 2010 Posts: 71 Thanks: 0 Pi Approximation I discovered a nice approximation of $\: \pi \:$ involving the golden ratio while working on something. $\pi \Phi^3 \: \approx \: \frac {3327} {250}$ $\pi \: \approx \: \frac {3327} {250 \Phi^3} \: \approx \: 3.14159264456 \:$( good to 7 decimal digits ) Written in a curious form: $\pi \: \approx \: \frac { 3.3 \: 10^3 \: + \: 3^3 } { 2 \: 5^3 \: \Phi^3 }$
 November 7th, 2010, 06:27 PM #2 Senior Member   Joined: Apr 2008 Posts: 435 Thanks: 0 Re: Pi Approximation That's interesting. How did you come across this?
 November 7th, 2010, 09:03 PM #3 Member   Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0 Re: Pi Approximation Here is another interesting one phi =1.6180339887 and 7*pi() / 5*e = 1.6180182897 3.1415926536*7 = 21.9911485751 2.7182818285 *5 = 13.5914091423 Seems to me that you are trying to find phi in Natural Constants - am I right ?
 November 7th, 2010, 10:18 PM #4 Member   Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0 Re: Pi Approximation There is another interesting property when you split the numbers into two digit groups The prime factorization of 3327 is 3*1109, split into 33 and 27, 3 * 11=33 and 3*9=27
 November 8th, 2010, 12:04 AM #5 Member   Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0 Re: Pi Approximation You can replace phi^3 = phi^2+phi^1 You can move phi^3 to the top because phi^n = 1/PHI^n Your 321 is the integer component of phi^12, for all odd powers you can get the integer component by phi^n-PHI^n (PHI=1/phi) You can replace 321 with phi^13-PHI^13-200=521.0019193787-0.0019193787-200=321 And this is the distance between two very important primes in Quantum Physics 137 and 337. Why are they so important ? They have a common property. You can swap the numbers and get another prime. 337, 373, 733 and each of these numbers can be written as the sum of two squares, which contain all the important quantum numbers. 137 is even more powerful. You get 3 primes that can be expressed as the sum of 2 squares and 3 composites that have only two prime factors. So you have 6 Quantum number triplets. You can also drop a number - drop 1 get 37 = prime, swap 73=prime Drop 3 get 17 = prime, swap 71=prime, drop 7 get 13, swap 31= prime The digit sum of 137 is 11 the prime index of 137 = 33 I have another question - are you decoding numbers found in the Pyramids of Giza ?
November 8th, 2010, 03:39 AM   #6
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Re: Pi Approximation

Quote:
 That's interesting. How did you come across this?
Quote:
 Seems to me that you are trying to find phi in Natural Constants - am I right ?
Actually I was investigating the natural appearance of the golden ratio in circles, and using that fact to approximate $\pi$.

Quote:
 ... two very important primes in Quantum Physics 137 and 337
Their sum is a palindrome

$337 \: \equiv \: 3.3.7 \: ( mod \: 137 )$

$137 \: \equiv \: 1+3+7 \: ( mod \: 3.3.7 )$

$\frac {337+1} 2= ( 3+3+7 )^2$

$200 + 137= 337$

$200 - 137= 3.3.7$

Quote:
 I have another question - are you decoding numbers found in the Pyramids of Giza ?

November 8th, 2010, 09:09 PM   #7
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Re: Pi Approximation

Quote:
Originally Posted by Wissam
Quote:
Because the Golden Ratio turns up in these monuments, especially in the Kings Chamber.

Are you especially looking for an integer approximation ? If not you can use

phi = 2*cos(pi()/5) = 2*cos(36 degrees). You get a very interesting function with x = 2*cos(n*pi()/5)

All primes have either the value phi or -PHI and their products have the same values. You can actually use this property to factorize semi primes.

 November 9th, 2010, 02:18 AM #8 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Pi Approximation A fraction of integers for $\varphi$, the golden ratio, can be approached by using Fibonacci's numbers, so that $\varphi \approx \frac{F_{n+1}}{F_n}$. As n, which is integer, increases, the approximation gets more and more accurate. Fibonacci's numbers can be found by Binet's formula. You could also use Egyption fractions to approach $\varphi$. Hoempa
 November 9th, 2010, 03:12 AM #9 Member   Joined: Jun 2010 Posts: 71 Thanks: 0 Re: Pi Approximation Another approximation of $\pi$ $\pi \: \approx \: \Phi^2 \: \sqrt[17]{2 \Phi^5} \: \approx \: 3.141592708 \:$ ( good to 6 decimal places ) I also found that $\frac 1 {\Phi^3} \:= \: \sum_{n=1}^{\infty} \: {\frac {(-1)^{n+1} \: n } {\Phi^n} }$
November 9th, 2010, 06:35 PM   #10
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Re: Pi Approximation

Quote:
Originally Posted by Wissam
Quote:
 Actually I was investigating the natural appearance of the golden ratio in circles, and using that fact to approximate $\pi$.
What where the circles that lead to this approximation? I am working on a quantum model based on Platonic solids in an fcc lattice. The golden ratio
appears when you construct an Icosahedron inside an Octahedron (see Avatar). When you do that the Icosahedron points divide the Octahedron side
by the Golden ratio. I want to find out to which geometries your circles belong.

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