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.