The original definition of Pi was indeed the ratio of the circumference to the diameter, in an era when all geometry was assumed to be Euclidean. This ratio was subsequently calculated as a mathematical constant, and turned up in all sorts of other places that had nothing obvious to do with circles or even geometry.

So it became convenient to switch the fundamental definition from circles to one of these mathematical alternatives. It gave the same answer, so what harm was done?

But then mathematicians discovered that Euclidean geometry was not the only alternative, and that you could have curved spaces, and indeed spaces with different metrics (definitions of the distance between points). So all of a sudden, the definitions could conflict. But by now, the earlier value had wormed its way into so many different areas of mathematics that it would have caused chaos to try to extract it and call it something else, so mathematicians just made the switch of definition formal. Pi was now equal to the number, and it just happened to be the ratio of circumference to diameter in the special case of Euclidean geometry.

But there’s nothing “wrong” about going back to the original definition, especially for the sake of engaging people’s interests in mathematics, so long as you’re clear that it’s not the same as the definition of Pi as a number.

However isn’t the problem resolved if you accept that pi is a concept? It’s value as we experience it may be 3.14…, but in other, ah, curvatures?, it could have different specific values?

But I have another question: I had a professor mention in passing that coordinate transformations in General relativity are not path independent. Could someone elaborate on that somewhat?

You cannot construct or observe a perfect sphere (or spherical shell, or a planar slice through these) in nature where one or more of the following exists:

a) non-negligible spacetime curvature;
b) non-negligibly dynamical spacetime;
c) non-negligibly dynamical matter embedded in spacetime;
d) non-continuous matter;
e) non-continuous spacetime.

Furthermore the c^2 term in spacetime intervals and the Heisenberg uncertainty principle impose limits on what one can say about the shape of an object carefully set up to be an instantaneous spherical shell of *any* radius. This observational uncertainty dwarfs the uncertainty in our calculations of the number pi.

So I asked a very clever person who I thought might know the answer. Prof. Ian Stewart, FRS, Emeritus Professor of Mathematics, Warwick University replied:

pi is always the same. The definition in terms of a circle refers specifically to euclidean geometry. In fact pi is usually defined by analytic methods, for example as half the period of the sine function (which is in turn defined as a power series).

In non-Euclidean geometry, the formula for the circumference of a circle differs from the euclidean one. In general the circumference is bigger than it would be for the same radius in Euclidean geometry, if we work in hyperbolic space (negative curvature). It is smaller if we work in elliptic geometry (positive curvature). And how much bigger or smaller depends on the radius.

But this doesn’t change the mathematical definition of pi.

Thanks Ian. I hope you don’t mind me reproducing your reply.

By the way, his Cabinet of Mathematical Curiosities is worth taking a look at…

In light-hearted mood one might ask whether gravitation could save from this trouble and allow to speak about circumference of circle also in p-adic context. By replacing plane with a cone (this requires cosmic string;-)), 2pi defined as ratio of length of circle to its radius becomes k*2pi and could therefore be also rational.

General Relativity does not say such a thing.

Pi is a number, and as such the nature of spacetime does not affect it.

It is true that on surfaces of non-zero curvature, the ratio of circumference to diameter of a circle is not [math]pi[/math]. But …