Exploring the Relationship Between Pi and Equations

[Lokhtar]

I have a amateur question regarding pi. I know what pi is, in a general sense, but I am looking at equations (e.g, the cosmological constant, Heisenberg's uncertainty principle, etc) where pi appears, but I do not see any relation to circles in those equations. I am sure there is, but I just don't see it. Can anyone explain where that comes in and why pi appears there?

 

[pam]

Pi is not only related to circles.
The relation between C and D for a circle is only one application of pi.
Because it is simplest to visualize, it is the circle that is taught in elementary school.

 

[tiny-tim]

Hi Lokhtar! Welcome to PF!

(btw, if you type alt-p, it prints π)

I think it's mostly because these πs tend to be the result of "integrating" something over all of space, and the something is usually spherically symmetrical, so you integrate over a sphere and make the sphere very large, and the π comes from that.

For example, the integral of the normal (Gaussian) distribution over all of space, [tex]\int e^{-r^2}[/tex], is (I think) π√π.

 

[Andy Resnick]

That's a great question! In fact, Eugene Wigner, in his great essay "The Unreasonable Effectiveness of Mathematics" opens with nearly the exact same point:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

You may be interested to read the full essay.

 

[rbj]

Pi is not only related to circles.
The relation between C and D for a circle is only one application of pi.
Because it is simplest to visualize, it is the circle that is taught in elementary school.

actually pam, i sort of disagree. i think that eventually, in any mathematical equation with [itex]\pi[/itex] in it, you'll find that it eventually will get back to the circle. whether it's Euler's formula (which leads to the expoential representation of sinusoids which plays a role in Heisenberg uncertainty) or the integral of the Gaussian bell curve, it gets back to a circle.

 

[kenewbie]

Pi is not only related to circles.
The relation between C and D for a circle is only one application of pi.
Because it is simplest to visualize, it is the circle that is taught in elementary school.

Huh? One application of pi? We didn't invent PI and then check where it might happen to fit, no? It was the other way around, we tried squaring the circle and found that the ratio between the circumference and the diameter of the circle was always the same. Thus, rather than using r^2*4 when calculating the area of a circle (which gives the area of a square which encompass the circle), we use r^2*3.14 (which approximates the actual area of the circle).

Without having ANY clue as to the math of the uncertainty principle or the cosmological constant, I can go out on a super-thin branch here and venture a guess. If the universe started with a big bang, it should be spherical in form, no? I can see pi there. Same reasoning with the uncertainty principle: If you have a "cloud" of possible positions for something, based around a single point, which is equally probable in all directions, this "cloud" would form a sphere as well.

I may be way way way off the mark here, so keep in mind that I am neither a mathematician or a physicist :p

k

 

[Hootenanny]

Just to weigh in at this point, in agreement with both rbj and kenewbie, we define pi to be the ratio of a circle's circuference to it's diameter.