In a non-Euclidean plane, the ratio of circumference to diameter may be more or less than 3.14159...--but that still may not mean that Pi has changed.
No, it’s only equal to that value in Euclidean space. In elliptical space, it tends toward that value locally, and gets smaller at larger scales, while in hyperbolic space, it tends toward the same value locally, and gets bigger at larger scales. In Minkowsy space, it doesn’t exist in any meaningful way, except in the Euclidean subspaces. In both L1 and L∞ it is 4.
At least as far as I know.
@Melvar - I believe Pi is defined as being in Euclidean space. The ratio of a circle's diameter and radius will change in non-Euclidean spaces, but they'll need their own name, because 'Pi' is already taken, so I think bmonk is right.
You can avoid the need for a Euclidean space by defining Pi by 4 times the sum from i=0 to infinity of (-1)^i / (2*i+1).
Then, Pi is always 3.14159... in base ten, anyway.
Pi is different in base 2 or base 3 or...
Well, pi is the same value, but it has different notation or "decimal" expansion in non-ten bases.
Yup and different notation is my argument.
Well, everything is arbitrary...
The English language is arbitrary, the way we write it down is arbitrary, and even mathematical notation is arbitrary... no matter how we denote mathematics, the ideas behind the notation remain true. All possible notations are "isomorphic".
Shall I break out the Xi and Xi-bar's to make it clearer? :-P
There IS a difference that can occur in this notation - it is the position of the "...". So you can say for engineers it is this, for physicists it is that, for school children it is 3.14...
I'm just going to post in order to see what the avatar looks like for my e-mail.
The girls suck at math! As usual :)
(this is supposed to be a joke)
It can be proven that in arbitary norm (of R^2), pi can be any number in [3,4].
I thought pi was defined as the ratio between the circumference and the diameter. Doesn't that mean pi is the same across all spaces? (Just not necessarily the same value across all of them.)