Averaging Phi Options

[PJS]

x^3 + 2x^2 = 1

2x^2 = 1 - x^3
x^2 = (1 - x^3)/2
x = sqrt (1 - x^3)/2
1/x = averaged phi

For example: x = 0.6
x = sqrt (1 - 0.216)/2
x = sqrt (0.784 / 2) = 0.392
x = sqrt 0.392 = 0.62609903369994111499456862724476
1/x = 1.5971914124998497831494097633795

next enter x = 0.62609903369994111499456862724476

Continue to enter each subsequent x into sqrt (1 - x^3)/2 then invert the answer 1/x for averaged phi. After a few entries the number phi will start to be produced.

P.j.S

[Flex]

Sum (n=0 to infinity) tr^n =t+tr+tr^2...tr^n

Ok so this is the series expansion whose sum converges to t/1-r

Let now look at the solution we get to this series expansion when we look at the sum where n=i and the function itself is e^-ui/KbT and ui=ixKbT

Reduces to sum n=i of e^-i and when t=1 and r=1/e we get the ratio of e/(e-1) = 1.5819..... Pretty damn close to phi eh?

Now if we take this and square it, we get (e/e-1)^2 = 3.16395

Notice that the first number is less than phi, but close, and the squared term is more than pi, but close. If we carry this on forever, I believe we get close and closer to true values of both numbers. The reason we never get a real solution is that the answer is relating two irrational numbers, resulting in an irrational solution. Looks like a fractal to me

*I apologize for the chemistry variables I used, I came the this conclusion in class the other day lol. Basically the idea is that if you take something like (1/e^0)+(1/e^1)+(1/e^n) you near phi, and when we square phi, we near pi.

I do not think it any coincidence the random ass solution I came up with in class happens to be very similar to the time constant. In fact, to me it almost seems like two sides of the same coin.

http://en.wikipedia.org/wiki/Time_constant

[P JayS]

Sum (n=0 to infinity) tr^n =t+tr+tr^2...tr^n

Ok so this is the series expansion whose sum converges to t/1-r

Let now look at the solution we get to this series expansion when we look at the sum where n=i and the function itself is e^-ui/KbT and ui=ixKbT

Reduces to sum n=i of e^-i and when t=1 and r=1/e we get the ratio of e/(e-1) = 1.5819..... Pretty damn close to phi eh?

Now if we take this and square it, we get (e/e-1)^2 = 3.16395

Notice that the first number is less than phi, but close, and the squared term is more than pi, but close. If we carry this on forever, I believe we get close and closer to true values of both numbers. The reason we never get a real solution is that the answer is relating two irrational numbers, resulting in an irrational solution. Looks like a fractal to me

*I apologize for the chemistry variables I used, I came the this conclusion in class the other day lol. Basically the idea is that if you take something like (1/e^0)+(1/e^1)+(1/e^n) you near phi, and when we square phi, we near pi.

I do not think it any coincidence the random ass solution I came up with in class happens to be very similar to the time constant. In fact, to me it almost seems like two sides of the same coin.

http://en.wikipedia.org/wiki/Time_constant

When you square phi you get 1 + phi. But when you multiply phi by 2 you get a figure near pi.

2(phi) - 1 - sqrt 5 = 0
2(phi) = 1 + sqrt 5

How does multiplying your answer by 2 instead of squaring affect your answer 2(e/e-1) approximately pi?