As Chronos notes, circles are an abstract concept, unaffected by mass. In the real world we find at best only rough approximations to them.
ETA: What G.A. said.
Dumb indeed. It seems to be a tale of an awkward attempt to designate a rational substitute for an irrational number.
Which is curious, because useful rational approximations have been around since antiquity. Everyone knows of 22/7, accurate to about 1 part in 2500. Much better is 355/113, accurate to better than 1 part in 11 million.
One certainly can, if one desires, speak of circles in spaces with curvature, such as one in the vicinity of a mass and affected according to Einstein’s equation. And if one speaks of such circles, then the ratio of that circle’s circumference to its diameter will not be equal to pi. But one seldom has need to speak of such.
I got curious and went off looking around for more on the subject, and I turned up the following absolute gem:
“I fundamentally believe that it’s related to the curvature of space-time itself. If space-time had a different curvature, I believe that would have a different value. For our 3-dimensional space/time pairing, I suspect that 3.00000000000. . . would be the value if the cosmological constant were exactly one.”
[Pi and Space-time | Physics Forums]
This is such a breathtaking stake in the sand of physics forums that I don’t think I can contribute a single thing to make it glisten any brighter.
Pi is the mathematical definition of the ratio. It is related to the physical result of measuring a specific circle, but measurements belong in the realm of physics, not math. Pi is always exactly pi on a theoretical two-dimensional plane. The measured ratio of a diameter of a circle to its perimeter in our three-dimensional world is never exactly pi. Neither finding contradicts or nitpicks the other.
What is it good for?
Absolutely nothin!
Pi ain’t nothin but a Dedekind partition,
Friend to no one but the mathematician.
Oh, pi has shattered many a Rationalist’s dreams,
Made him confused, bitter and mean.
Life is much too short and precious
To spend calculating digits these days.
Pi can’t give a belief that all numbers have a finite representation,
It can only take it away.
Note that quote is from fringe theories, as stated by the author:
[
](Heretical Physics of John Knouse)
Circles are not rotationally invariant in 4D spacetime so that whole question really has to go back to your choices of projection etc…
In Riemannian geometry, the diameter of a circle is pi while it would be 2 in the Euclidean space.
It also would depend on how you define a circle. If you define it as a line of constant curvature with no endpoints, as several mathematical fields do, and have infinite radius you have a line. If you choose to Lie transformations can turn circles to lines or points, and points are merely zero radius circles.
Note how the answer the OP can change depending on if you choose the Projective, Mobius, Riemannian, high school, or … geometry?
Even if you move to the extremes of hyperbolic geometry circumference is still “2pi sinh(r)” and even pi is greater than 3.14 it doesn’t matter. If spacetime is so curved as to limit your ability to simplify to the euclidean case two observers will simply not agree on anything but the **spacetime interval between any two events **. As spacetime is the ONLY invariant one events are not local, and because it is a scalar yet still a vector field there is another fun place you can find pi hiding in with a “infinitely small circle with an area greater than zero” even if that fact is masked through the tensor calculus na diff geometry conventions.
The question is if it is convenience or convention that cause SO(1,n), SO(n), and SU(2) to be used in relativity and quantum mechanics. I personally don’t know if that is answered or answerable but the value of pi itself is far less important than the properties of Lie groups when working with infinitesimals and the convenience of using Euclid’s third postulate and Euler’s Identity to solve very hard problems.
Anyway I can’t show the math on this here but the more mainstream view is that under strong curvature it is most likely that pi > 3.14 in higher dimensions.
If you use extended reals with infinitesimals, then the OP’s question makes sense, but the ratio is still pi.
Are circles and straight lines the only curves that have constant curvature? That seems true for Euclidean geometry (although I would have no idea how to prove it) but wonder if there might be some ‘novel’ constructions that could also satisfy the condition.
Don’t helices have constant curvature?
Circles and lines are the only smooth curves of constant curvature in a plane in Euclidean geometry.
I am not sure what happens with higher dimensions or with unit circles etc…so someone else will have to answer.
Obviously a sphere would be smooth curves of constant curvature in 3 dimensions as would a plane
Indeed. I meant in the plane.
Sure, but you can specify an arbitrary torsion and get different curves. Anyway, this is in three dimensions, not the Euclidean plane. To make things interesting, if you wanted you could certainly consider curves in the three-dimensional sphere and other Riemannian manifolds; that makes perfect sense.
ETA to define a circle, AFAIK it is the set of all points in the plane at a fixed distance from some specified point. That is, you have the centre and the radius.
See, I was right. Illinois didn’t. 
And yes, I knew it was Indiana. That was deliberate. I possess little shame, just enough to turn me from posting snide, fringe-y jokes about circles. [bad taste omitted]
Would it be threadshitting to say that the question is not well-formed?
Though you should specify that you’re using Euclidean distance - with a different metric you’d get a different shape and possibly a different number of points. With a taxicab metric the set of all points at a fixed distance from a specific point would look square, and if the space was discrete, you would have only a finite number of points on that square (possibly zero).
What the heck are you talking about here? I can’t figure out any way to make this sentence make sense.
If you allow non-Euclidean geometry, then, sure, e.g., following the Bertrand–Diguet–Puiseux theorem, consider a point on a smooth surface and a small geodesic disc centered on it; the Gaussian curvature at that point is equal to the limit, as the radius decreases to 0, of (3/πr[sup]3[/sup])(2πr - the circumference). Note that this does not yield a “different value of π” for small circles in the limit.
ETA I agree circles may be natural geometric objects of interest not only in Euclidean space
Agreed.