What's the significance of pi?

[Naty1]

Back in 2008 I posted the same topic [thread now closed, due to age??:

C:\Users\Owner\Documents\PHYSICS\What's the significance of pi.mht

We missed a fascinating answer I just noticed:

In natural units, commonly used in high energy physics, where the Coulomb constant is 1/4π and c = ħ = 1, the value of the fine structure constant is α = ε²/∏

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

and is also a component of the gauge coupling constant...

 

[DiracPool]

Pi is just the universal constant of rotation, of curvature. What does 3.14…. mean? I don’t know, why don’t we just set it to one and meditate on how profound it is. What I want to know is how many decimal places do we need to go before a circle is a perfect circle as far as we can tell 1) psychophysically, and 2) practically as far as engineering applications. I think that would be an interesting figure.

One more thing, Pi may actually be overrated. Have you heard of Tau? Tau is just 2 time Pi, and it seems even a more parsimonious and natural number to use than Pi. See Khan’s cool talk on it.



However, I don't think we're going to see a conversion here soon among contemporary scientists and engineers any more than we saw a swift conversion to the metric system in the USA.

 

[DiracPool]

BTW, is that squiggly E you used in the fine structure constant equation the permitivitty of free space or something else?

 

[Borek]

https://www.physicsforums.com/showthread.php?t=263423

Some time ago micromass showed me a nice infinite sum expressing pi using prime numbers, but I can't remember it right now and quick googling didn't yield it.

 

[Naty1]

BTW, is that squiggly E you used in the fine structure constant equation the permitivity of free space or something else?

who knows...it's from the 'quick symbol' list on the right of my screen when I post...
It doesn't look quite right, but I figured 'close enough' !

 

[Dale]

$\pi$ is highly significant, with p<.0001

 

[micromass]

https://www.physicsforums.com/showthread.php?t=263423

Some time ago micromass showed me a nice infinite sum expressing pi using prime numbers, but I can't remember it right now and quick googling didn't yield it.

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-2}} = \frac{\pi^2}{6}$$

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

 

[Borek]

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-2}} = \frac{\pi^2}{6}$$

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

Thanks. It wasn't sum, it was a product, which is probably why I couldn't find it.

 

[demoncore]

I think this is from Dirac -- (4pi^3 + pi^2 + pi^1)^(-1) = fine-structure constant (to a surprisingly good accuracy!)

 

[DiracPool]

I think this is from Dirac -- (4pi^3 + pi^2 + pi^1)^(-1) = fine-structure constant (to a surprisingly good accuracy!)

Don't get me wrong, I love looking for deeper meaning in constants like Pi and e, and others, but where does healthy interest end and numerology begin? Is there a "line in the sand" that contemporary physicists agree upon, or is it more subjective?

 

[Drakkith]

Don't get me wrong, I love looking for deeper meaning in constants like Pi and e, and others, but where does healthy interest end and numerology begin? Is there a "line in the sand" that contemporary physicists agree upon, or is it more subjective?

Maybe not, but there's definitely a line in the pie. Made by whipped cream. The pie on this side is mine, and you shall not pass or you'll get a radian in your...

 

[jedishrfu]

Hey Borek,

Mathworld had the series equivalent to get pi/4 called the Gregory Liebnitz series:

pi/4 = 1 - 1/3 + 1/5 - 1/7 ...

http://mathworld.wolfram.com/PiFormulas.html

 

[demoncore]

Don't get me wrong, I love looking for deeper meaning in constants like Pi and e, and others, but where does healthy interest end and numerology begin? Is there a "line in the sand" that contemporary physicists agree upon, or is it more subjective?

If there is a line, the Dirac/Jung investigation into the fine-structure constant definitely crossed it. It is clearly numerology, but still often pretty interesting, in a speculative way. It's more than a little surprising that the first 4 natural numbers and a (purely?) mathematical constant can be related to a physical, dimensionless constant in such an elegant way.

 

[rbj]

BTW, is that squiggly E you used in the fine structure constant equation the permitivitty of free space or something else?

i think it was supposed to be the elementary charge.

in any units:

$$\alpha \ = \ \frac{e^2}{(4\pi\epsilon_0) \hbar c}$$

depending on the variant of "natural units" 3 of the four variable symbols on the right can go to 1 (or any other predetermined constant).

 

[Borek]

Hey Borek,

Mathworld had the series equivalent to get pi/4 called the Gregory Liebnitz series:

pi/4 = 1 - 1/3 + 1/5 - 1/7 ...

Yes, but that uses just odd numbers, the formula micromass posted uses primes. That's what caught my attention - you would think primes are just too "random" to be able to produce constant like pi, fact that they do shows there are really deep links between different branches of math.

Or at least that's how I see it, I can be wrong (and happy with it ).

 

[jedishrfu]

Yes, but that uses just odd numbers, the formula micromass posted uses primes. That's what caught my attention - you would think primes are just too "random" to be able to produce constant like pi, fact that they do shows there are really deep links between different branches of math.

Or at least that's how I see it, I can be wrong (and happy with it ).

Yes, true. In Carl Sagan's book Contact (not the movie) the protagonist finds the image of a circle embedded deep within the digits of pi and that as a sign that there is some intelligence behind the creation of the universe.

In contrast, there was the story of how pi was embedded within the measurements of the pyramid as in 100 cubits high vs 100 pi cubits on edge. Whereupon an engineer figured that the designers had rolled a 1 cubit diameter disk a hundred times to measure out the edge. (not sure if its true or a 4500 yr old urban legend)

 

[mfb]

I think this is from Dirac -- (4pi^3 + pi^2 + pi^1)^(-1) = fine-structure constant (to a surprisingly good accuracy!)

Numerology. If you search hard enough, you can find many good approximations. This one cannot be exact:

137.036303775878 = 4pi^3 + pi^2 + pi^1
137.035999074(44) = 1/alpha

Deviates by 0.0003, this corresponds to ~7000 standard deviations of the uncertainty.

 

[DiracPool]

Deviates by 0.0003, this corresponds to ~7000 standard deviations of the uncertainty.

IDK, I think that raising Pi by an exponent is unaesthetic anyway, it doesn't make sense.