I am in love with this comment and want to have its babies:
pi appears as a constant in many formula of physics. General relativity says that it isn’t constant. Is it the origin of the pi particle, aka pion?
A curmudgeonly literalist might, when faced with a question such as this, harrumph a simple “No.” A more loquacious sort might explain that general relativity does not say that π is not a contstant. Pi is not a parameter of physics like the fine-structure constant, which could conceivably be different or even variable from place to place. It’s a universal answer to a fixed question, to wit: what is the ratio of the circumference of a circle to its diameter, as measured in Euclidean geometry? The answer is of course 3.141592653589793…, or any number of representations in terms of infinite series.
But the point of the question is that GR says we don’t live in Euclidean space; we move through a curved spacetime manifold. That’s okay. In a curved space, we could imagine defining the “diameter” of a circle as the maximum geodesic distance connecting two of its points, and taking the ratio of the circumference with that diameter, and indeed it would typically not give us 3.14159… But that doesn’t mean π is changing from place to place; it just means that the ratio of circumference to diameter (defined this way) in a curved space doesn’t equal π. If the circumference/diameter ratio is less than π, you are in a positively curved space, such as a sphere; if it is greater than π, you are in a negatively curved space, such as a saddle. Geometry can also be much more complicated than that, with different ratios depending on how the circle is oriented in space, which is why curvature is properly measured by tensors rather than by a simple number.
(Parenthetically, one of the dumbest mathematical arguments ever given was put forward by the world’s smartest person, Marilyn Vos Savant. The columnist wrote an entire book criticizing Andrew Wiles’s proof of Fermat’s Last Theorem. Her argument: Wyles made use of non-Euclidean geometry, but what if geometry is really Euclidean? Touche!)
However … despite the fact that π doesn’t really change from place to place in general relativity, the geometry does change from place to place, and there is a particle associated with those dynamics — the graviton. Although the formulation of the original question isn’t accurate, the spirit is very much in the right place. And I, for one, will henceforth be perpetually sad that the physics community missed a chance by attaching the word pion to the lightest quark-antiquark bound state, rather than to the particle associated with deviations from Euclidean geometry. That would have been awesome.
In mathematics, choice of definitions is not an absolute, but based on convenience and mathematical aesthetics.
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.
Haven’t we proved that the Universe is flat? I seem to recall a number of studies supporting that conclusion.