1) Perhaps yes. 2) Mathemtical contstans are purely abstact - pi only exists in a perfectly euclidian plane. It's an abstraction that works for us in a scertain scale because it matches close enough. In real life, there aren't circles.. they're just close enough. So no - a universe could have completely different maths required to describe it. 3) Math is purely abstract and detached from the universe. It operates on theoretical models that try to model reality. Math has it's own universal truths, but reality does not. The universe came before math - math is a tool invented by man to analyze the universe.

> So no - a universe could have completely different maths required to describe it.

That wasn't the question, though. Pi is the ratio between circumference and diameter of a perfect circle, whether or not you can create a perfect circle in your universe. We can't even create one in ours, and yet pi still "exists" here. Undoubtedly, wiggly-space mathematicians would also have a conception of pi, though it might be as esoteric to them as toroidial universes are to us.

To put that another way: Math has very simple axioms (e.g. set theory), and everything else derives from them. What theories can be proven from those axioms in this universe, can be proven from those axioms in any universe (and a universe that doesn't obey those axioms likely wouldn't "function" as a universe—it wouldn't have any reason to be causal, for example.) The maths describing a particular universe is its physics, which do change from universe to universe—however, those physics are all just different subsets of the set of mathematical physical models—which are, themselves, a subset of the mathematical theories provable from the axioms.

No, it isn't universal, even in our universe. If you go and draw a circle on the surface of the Earth, with a 1km diameter, the ratio of the diameter to the circumference will not be pi, due to the curvature of the Earth's surface. I suppose that you could say that I wasn't measuring the right diameter in this case, and that the true diameter passes underneath the surface of the Earth. But there is an similar deformation introduced by the curvature of spacetime caused by the Earth's mass, so you still won't get pi as a result when measuring the ratio between the circumference and the diameter.

That said, I would very much like to understand why c = 2 x pi x r actually works. pi itself can be derived from pure mathematics, knowing the relationship e^(i x pi) = -1. So why a circle should have such a relationship with this constant is to me a fascinating question.

> Why [should] a circle should have such a relationship with this constant?

Because the "circles" we talk about in mathematics are found in Euclidean space—which is defined to have simple metric properties. Our own space is non-Euclidean.

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

Umm, yes, I understand that c = 2 x pi x r only holds in Euclidean space (indeed, my original post gave two examples of circles in non-Euclidean spaces, to illustrate my point), but my question was more about why it holds at all. That is, if you consider pi as being a constant that you can derive from e (which is itself just a power series), and i (again, just a mathematical construct), it seems quite stunning to me that this constant (pi) should have such strong ties of Euclidean space. It implies a deeper connection between Euclidean Space, e, and i. Why should that be so?

1. Taking e to a complex power (using θi) walks you through the complex plane at a given angle θ. (That is, f(x) "draws" a continuous curve, where the "pen" has a constant rotational torque of θi.)

2. When θ = pi, f(x) is periodic (the angle is such that f(x) intersects itself or "loops.")

3. The set of complex numbers is isomorphic to a 2D Euclidean plane. Therefore, a walk at angle i*pi through the set of complex numbers is isomorphic to a circle.