How do we know Pi is Constant?

SteamKing

If pi wasn't constant, then potentially every day could be pi day.

bcrowell

Is this in response to my #15? I agree with what you said, and it doesn't contradict #15. #15 wasn't about pi, it was about C/d. What depends on Euclidean geometry is the statement that C/d is the same for all circles (in which case it then also equals pi).

lavinia

You ask a good question. I guess I was making a few points.

- I wanted to mention another way of seeing how pi is constant.

- I wanted to show that pi in an intrinsic property as well as extrinsic. For the circle in the Euclidean plane its length ( intrinsic) is compared to it radius(extrinsic) so the classic ratio is an extrinsic measure of pi. Gauss Bonnet gives it as intrinsic.

- it was said here that the definition of pi is a matter of convenience. I guess this means that pi is defined in the context of the subject. While true, I believe that pi is intrinsic to mathematics and appears in different guises. But it is hardly a matter of convenience. I wanted the example of the Gauss Bonnet theorem to suggest this. One arrives at pi by necessity. I wanted to show that the ratio example is only one way that this happens.

BTW:

- Your example of ratios on a curved surface seemed to suggest that pi is not constant.
It omitted the limit of these ratios as the polar radius goes to zero. This would have given another definition of pi since this limit is the same a every point on every surface, yet another proof that pi is constant.

- Another way to get at this would be though the idea of winding number.

bcrowell

Not true. I clearly stated that I was not defining pi as C/d.

That wouldn't qualify as a proof unless you specified some set of axioms to be used in the proof. Riemannian geometry is locally Euclidean, so defining pi using this limit simply defines pi as C/d in the Euclidean case. It then remains to prove the constancy of C/d in the Euclidean case, presumably based on Euclid's postulates. That would then establish that the definition of pi as C/d was consistent.